From 96947556948310b4e977e4ebbc3c271041075c97 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 24 Sep 2020 06:36:42 +0200 Subject: [PATCH 1/2] typo in week39 --- doc/pub/week39/html/._week39-bs000.html | 2 +- doc/pub/week39/html/._week39-bs025.html | 4 +- doc/pub/week39/html/._week39-bs026.html | 6 +- doc/pub/week39/html/._week39-bs028.html | 14 +-- doc/pub/week39/html/._week39-bs037.html | 2 +- doc/pub/week39/html/._week39-bs038.html | 22 ++-- doc/pub/week39/html/._week39-bs046.html | 2 +- doc/pub/week39/html/._week39-bs047.html | 4 +- doc/pub/week39/html/._week39-bs048.html | 14 +-- doc/pub/week39/html/._week39-bs049.html | 4 +- doc/pub/week39/html/._week39-bs050.html | 4 +- doc/pub/week39/html/._week39-bs051.html | 2 +- doc/pub/week39/html/._week39-bs052.html | 6 +- doc/pub/week39/html/._week39-bs053.html | 4 +- doc/pub/week39/html/._week39-bs054.html | 2 +- doc/pub/week39/html/._week39-bs055.html | 4 +- doc/pub/week39/html/._week39-bs065.html | 4 +- doc/pub/week39/html/week39-bs.html | 2 +- doc/pub/week39/html/week39-reveal.html | 100 +++++++++---------- doc/pub/week39/html/week39-solarized.html | 100 +++++++++---------- doc/pub/week39/html/week39.html | 100 +++++++++---------- doc/pub/week39/ipynb/ipynb-week39-src.tar.gz | Bin 197 -> 191 bytes doc/pub/week39/ipynb/week39.ipynb | 6 +- doc/src/week39/week39.do.txt | 2 +- 24 files changed, 205 insertions(+), 205 deletions(-) diff --git a/doc/pub/week39/html/._week39-bs000.html b/doc/pub/week39/html/._week39-bs000.html index ee689d11b..d1107c7b3 100644 --- a/doc/pub/week39/html/._week39-bs000.html +++ b/doc/pub/week39/html/._week39-bs000.html @@ -301,7 +301,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Sep 22, 2020

+

Sep 24, 2020


diff --git a/doc/pub/week39/html/._week39-bs025.html b/doc/pub/week39/html/._week39-bs025.html index 2f5e565c5..8ae4a5c59 100644 --- a/doc/pub/week39/html/._week39-bs025.html +++ b/doc/pub/week39/html/._week39-bs025.html @@ -305,7 +305,7 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) +print(beta_linreg) beta = np.random.randn(2,1) eta = 0.1 @@ -315,7 +315,7 @@ Niterations = 1 gradients = 2.0/m*xb.T.dot(xb.dot(beta)-y) beta -= eta*gradients -print(beta) +print(beta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] ypredict = xbnew.dot(beta) diff --git a/doc/pub/week39/html/._week39-bs026.html b/doc/pub/week39/html/._week39-bs026.html index 47e982f89..73af92e68 100644 --- a/doc/pub/week39/html/._week39-bs026.html +++ b/doc/pub/week39/html/._week39-bs026.html @@ -298,10 +298,10 @@ y = 4+3* xb = np.c_[np.ones((100,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) -sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1) +print(beta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print(sgdreg.intercept_, sgdreg.coef_) +print(sgdreg.intercept_, sgdreg.coef_)

diff --git a/doc/pub/week39/html/._week39-bs028.html b/doc/pub/week39/html/._week39-bs028.html index b9f46b937..9163594df 100644 --- a/doc/pub/week39/html/._week39-bs028.html +++ b/doc/pub/week39/html/._week39-bs028.html @@ -300,14 +300,14 @@ x = 2*np y = 4+3*x+np.random.randn(m,1) xb = np.c_[np.ones((m,1)), x] -XT_X = xb.T @ xb +XT_X = xb.T @ xb #Ridge parameter lambda lmbda = 0.001 Id = lmbda* np.eye(XT_X.shape[0]) -beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y -print(beta_linreg) +beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y +print(beta_linreg) # Start plain gradient descent beta = np.random.randn(2,1) @@ -315,12 +315,12 @@ eta = 0.1= 100 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta + gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta beta -= eta*gradients -print(beta) -ypredict = xb @ beta -ypredict2 = xb @ beta_linreg +print(beta) +ypredict = xb @ beta +ypredict2 = xb @ beta_linreg plt.plot(x, ypredict, "r-") plt.plot(x, ypredict2, "b-") plt.plot(x, y ,'ro') diff --git a/doc/pub/week39/html/._week39-bs037.html b/doc/pub/week39/html/._week39-bs037.html index 032687147..b0a146d89 100644 --- a/doc/pub/week39/html/._week39-bs037.html +++ b/doc/pub/week39/html/._week39-bs037.html @@ -325,7 +325,7 @@ j = 0 gamma_j = step_length(t,t0,t1) j += 1 -print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) +print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))

diff --git a/doc/pub/week39/html/._week39-bs038.html b/doc/pub/week39/html/._week39-bs038.html index 3b004e4a2..1c4d57d52 100644 --- a/doc/pub/week39/html/._week39-bs038.html +++ b/doc/pub/week39/html/._week39-bs038.html @@ -300,12 +300,12 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print("Own inversion") -print(theta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) +print("Own inversion") +print(theta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print("sgdreg from scikit") -print(sgdreg.intercept_, sgdreg.coef_) +print("sgdreg from scikit") +print(sgdreg.intercept_, sgdreg.coef_) theta = np.random.randn(2,1) @@ -314,10 +314,10 @@ Niterations = 1 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ ((xb @ theta)-y) + gradients = 2.0/m*xb.T @ ((xb @ theta)-y) theta -= eta*gradients -print("theta frm own gd") -print(theta) +print("theta frm own gd") +print(theta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] @@ -337,11 +337,11 @@ theta = np.= np.random.randint(m) xi = xb[random_index:random_index+1] yi = y[random_index:random_index+1] - gradients = 2 * xi.T @ ((xi @ theta)-yi) + gradients = 2 * xi.T @ ((xi @ theta)-yi) eta = learning_schedule(epoch*m+i) theta = theta - eta*gradients -print("theta from own sdg") -print(theta) +print("theta from own sdg") +print(theta) plt.plot(xnew, ypredict, "r-") plt.plot(xnew, ypredict2, "b-") diff --git a/doc/pub/week39/html/._week39-bs046.html b/doc/pub/week39/html/._week39-bs046.html index 39104ee0b..5bda4b981 100644 --- a/doc/pub/week39/html/._week39-bs046.html +++ b/doc/pub/week39/html/._week39-bs046.html @@ -362,7 +362,7 @@ plt.legend() plt.show() -print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic)))) +print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))

diff --git a/doc/pub/week39/html/._week39-bs047.html b/doc/pub/week39/html/._week39-bs047.html index cd3d8e82b..ef9d36bc4 100644 --- a/doc/pub/week39/html/._week39-bs047.html +++ b/doc/pub/week39/html/._week39-bs047.html @@ -306,11 +306,11 @@ f1_grad = grad(f1) a = 1.0 # See the evaluated gradient at a using autograd: -print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) +print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) # Compare with the analytical derivative, that is f1'(x) = 3*x**2 grad_analytical = 3*a**2 -print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical)) +print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))

diff --git a/doc/pub/week39/html/._week39-bs048.html b/doc/pub/week39/html/._week39-bs048.html index 557adc05a..02361db35 100644 --- a/doc/pub/week39/html/._week39-bs048.html +++ b/doc/pub/week39/html/._week39-bs048.html @@ -306,8 +306,8 @@ f2_grad_x2 = grad(f2,= 1.0 x2 = 3.0 -print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) -print("-"*30) +print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) +print("-"*30) # Compare with the analytical derivatives: @@ -318,13 +318,13 @@ f2_grad_x1_analytical = = x1 - 5 # See the evaluated derivations: -print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print() +print() -print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) -print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))

Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. diff --git a/doc/pub/week39/html/._week39-bs049.html b/doc/pub/week39/html/._week39-bs049.html index 7e6074737..c16a02e3f 100644 --- a/doc/pub/week39/html/._week39-bs049.html +++ b/doc/pub/week39/html/._week39-bs049.html @@ -297,13 +297,13 @@ f3_grad = grad(f3) x = np.linspace(0,4,5) # Print the computed gradient: -print("The computed gradient of f3 is: ", f3_grad(x)) +print("The computed gradient of f3 is: ", f3_grad(x)) # The analytical gradient is: (2, 3, 5, 7, 22*x[4]) f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]]) # Print the analytical gradient: -print("The analytical gradient of f3 is: ", f3_grad_analytical) +print("The analytical gradient of f3 is: ", f3_grad_analytical)

Note that in this case, when sending an array as input argument, the diff --git a/doc/pub/week39/html/._week39-bs050.html b/doc/pub/week39/html/._week39-bs050.html index 2729af8b2..b464865be 100644 --- a/doc/pub/week39/html/._week39-bs050.html +++ b/doc/pub/week39/html/._week39-bs050.html @@ -297,13 +297,13 @@ f4_grad = grad(f4) x = 2.7 # Print the computed derivative: -print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) +print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) # The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi # Print the analytical gradient: -print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical)) +print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))

diff --git a/doc/pub/week39/html/._week39-bs051.html b/doc/pub/week39/html/._week39-bs051.html index 8ae82d470..9ce938862 100644 --- a/doc/pub/week39/html/._week39-bs051.html +++ b/doc/pub/week39/html/._week39-bs051.html @@ -300,7 +300,7 @@ f5_grad = grad(f5) x = 2.7 # Print the computed derivative: -print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) +print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))

diff --git a/doc/pub/week39/html/._week39-bs052.html b/doc/pub/week39/html/._week39-bs052.html index 7115363fe..a2953fc82 100644 --- a/doc/pub/week39/html/._week39-bs052.html +++ b/doc/pub/week39/html/._week39-bs052.html @@ -309,8 +309,8 @@ f6_while_grad = grad(f6_while) x = 0.5 # Print the computed derivaties of f6_for and f6_while -print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) -print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x))) +print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) +print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))

@@ -323,7 +323,7 @@ f6_grad_analytical = for i in range(10): f6_grad_analytical += i*x**(i-1) -print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) +print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))

diff --git a/doc/pub/week39/html/._week39-bs053.html b/doc/pub/week39/html/._week39-bs053.html index 13559746a..f899539ed 100644 --- a/doc/pub/week39/html/._week39-bs053.html +++ b/doc/pub/week39/html/._week39-bs053.html @@ -299,7 +299,7 @@ f7_grad = grad(f7) n = 2.0 -print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) +print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) # The function f7 is an implementation of the factorial of n. # By using the product rule, one can find that the derivative is: @@ -312,7 +312,7 @@ f7_grad_analytical = *= (n - k) f7_grad_analytical += tmp -print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical)) +print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))

Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. diff --git a/doc/pub/week39/html/._week39-bs054.html b/doc/pub/week39/html/._week39-bs054.html index ce003f63f..aea05d9ec 100644 --- a/doc/pub/week39/html/._week39-bs054.html +++ b/doc/pub/week39/html/._week39-bs054.html @@ -300,7 +300,7 @@ f8_grad = grad(f8) x = 8.4 -print("The derivative of f8 is:",f8_grad(x)) +print("The derivative of f8 is:",f8_grad(x))

Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. diff --git a/doc/pub/week39/html/._week39-bs055.html b/doc/pub/week39/html/._week39-bs055.html index 40f96bf8b..247d23d72 100644 --- a/doc/pub/week39/html/._week39-bs055.html +++ b/doc/pub/week39/html/._week39-bs055.html @@ -296,7 +296,7 @@ f9_grad = grad(f9) x = np.array([1.0,0.0]) -print("The derivative of f9 is:",f9_grad(x)) +print("The derivative of f9 is:",f9_grad(x))

Here we are told that the 'dot' function does not belong to Autograd's @@ -316,7 +316,7 @@ f9_alternative_grad = grad(f9_alternative) x = np.array([3.0,0.0]) -print("The gradient of f9 is:",f9_alternative_grad(x)) +print("The gradient of f9 is:",f9_alternative_grad(x)) # The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively # w.r.t x is (b_1, b_2). diff --git a/doc/pub/week39/html/._week39-bs065.html b/doc/pub/week39/html/._week39-bs065.html index 26c2773f9..ca4bb41cd 100644 --- a/doc/pub/week39/html/._week39-bs065.html +++ b/doc/pub/week39/html/._week39-bs065.html @@ -304,7 +304,7 @@ MathJax.Hub.Config({ fig = pt.figure() ax = fig.gca(projection="3d") -xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] +xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] fmesh = f(np.array([xmesh, ymesh])) ax.plot_surface(xmesh, ymesh, fmesh) @@ -336,7 +336,7 @@ Run it! alpha_opt = sopt.golden(f1d) next_guess = x + alpha_opt * s guesses.append(next_guess) -print(next_guess) +print(next_guess)

What happened? diff --git a/doc/pub/week39/html/week39-bs.html b/doc/pub/week39/html/week39-bs.html index ee689d11b..d1107c7b3 100644 --- a/doc/pub/week39/html/week39-bs.html +++ b/doc/pub/week39/html/week39-bs.html @@ -301,7 +301,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Sep 22, 2020

+

Sep 24, 2020


diff --git a/doc/pub/week39/html/week39-reveal.html b/doc/pub/week39/html/week39-reveal.html index e89e0b683..382f963eb 100644 --- a/doc/pub/week39/html/week39-reveal.html +++ b/doc/pub/week39/html/week39-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

 
-

Sep 22, 2020

+

Sep 24, 2020


@@ -790,7 +790,7 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) +print(beta_linreg) beta = np.random.randn(2,1) eta = 0.1 @@ -800,7 +800,7 @@ Niterations = 1000 gradients = 2.0/m*xb.T.dot(xb.dot(beta)-y) beta -= eta*gradients -print(beta) +print(beta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] ypredict = xbnew.dot(beta) @@ -834,10 +834,10 @@ y = 4+3* xb = np.c_[np.ones((100,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) -sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1) +print(beta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print(sgdreg.intercept_, sgdreg.coef_) +print(sgdreg.intercept_, sgdreg.coef_) @@ -892,14 +892,14 @@ x = 2*np.random.rand(m,4+3*x+np.random.randn(m,1) xb = np.c_[np.ones((m,1)), x] -XT_X = xb.T @ xb +XT_X = xb.T @ xb #Ridge parameter lambda lmbda = 0.001 Id = lmbda* np.eye(XT_X.shape[0]) -beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y -print(beta_linreg) +beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y +print(beta_linreg) # Start plain gradient descent beta = np.random.randn(2,1) @@ -907,12 +907,12 @@ eta = 0.1 Niterations = 100 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta + gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta beta -= eta*gradients -print(beta) -ypredict = xb @ beta -ypredict2 = xb @ beta_linreg +print(beta) +ypredict = xb @ beta +ypredict2 = xb @ beta_linreg plt.plot(x, ypredict, "r-") plt.plot(x, ypredict2, "b-") plt.plot(x, y ,'ro') @@ -1128,7 +1128,7 @@ j = 0 gamma_j = step_length(t,t0,t1) j += 1 -print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) +print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) @@ -1152,12 +1152,12 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print("Own inversion") -print(theta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) +print("Own inversion") +print(theta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print("sgdreg from scikit") -print(sgdreg.intercept_, sgdreg.coef_) +print("sgdreg from scikit") +print(sgdreg.intercept_, sgdreg.coef_) theta = np.random.randn(2,1) @@ -1166,10 +1166,10 @@ Niterations = 1000 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ ((xb @ theta)-y) + gradients = 2.0/m*xb.T @ ((xb @ theta)-y) theta -= eta*gradients -print("theta frm own gd") -print(theta) +print("theta frm own gd") +print(theta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] @@ -1189,11 +1189,11 @@ theta = np.random.randn(2,1] yi = y[random_index:random_index+1] - gradients = 2 * xi.T @ ((xi @ theta)-yi) + gradients = 2 * xi.T @ ((xi @ theta)-yi) eta = learning_schedule(epoch*m+i) theta = theta - eta*gradients -print("theta from own sdg") -print(theta) +print("theta from own sdg") +print(theta) plt.plot(xnew, ypredict, "r-") plt.plot(xnew, ypredict2, "b-") @@ -1561,7 +1561,7 @@ plt.legend() plt.show() -print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic)))) +print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic)))) @@ -1591,11 +1591,11 @@ f1_grad = grad(f1) a = 1.0 # See the evaluated gradient at a using autograd: -print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) +print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) # Compare with the analytical derivative, that is f1'(x) = 3*x**2 grad_analytical = 3*a**2 -print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical)) +print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical)) @@ -1625,8 +1625,8 @@ f2_grad_x2 = grad(f2,1) x1 = 1.0 x2 = 3.0 -print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) -print("-"*30) +print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) +print("-"*30) # Compare with the analytical derivatives: @@ -1637,13 +1637,13 @@ f2_grad_x1_analytical = 9*x1**5 # See the evaluated derivations: -print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print() +print() -print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) -print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))

Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. @@ -1666,13 +1666,13 @@ f3_grad = grad(f3) x = np.linspace(0,4,5) # Print the computed gradient: -print("The computed gradient of f3 is: ", f3_grad(x)) +print("The computed gradient of f3 is: ", f3_grad(x)) # The analytical gradient is: (2, 3, 5, 7, 22*x[4]) f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]]) # Print the analytical gradient: -print("The analytical gradient of f3 is: ", f3_grad_analytical) +print("The analytical gradient of f3 is: ", f3_grad_analytical)

Note that in this case, when sending an array as input argument, the @@ -1700,13 +1700,13 @@ f4_grad = grad(f4) x = 2.7 # Print the computed derivative: -print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) +print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) # The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi # Print the analytical gradient: -print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical)) +print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical)) @@ -1730,7 +1730,7 @@ f5_grad = grad(f5) x = 2.7 # Print the computed derivative: -print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) +print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) @@ -1763,8 +1763,8 @@ f6_while_grad = grad(f6_while) x = 0.5 # Print the computed derivaties of f6_for and f6_while -print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) -print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x))) +print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) +print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))

@@ -1777,7 +1777,7 @@ f6_grad_analytical = 0 for i in range(10): f6_grad_analytical += i*x**(i-1) -print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) +print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) @@ -1800,7 +1800,7 @@ f7_grad = grad(f7) n = 2.0 -print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) +print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) # The function f7 is an implementation of the factorial of n. # By using the product rule, one can find that the derivative is: @@ -1813,7 +1813,7 @@ f7_grad_analytical = 0 tmp *= (n - k) f7_grad_analytical += tmp -print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical)) +print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))

Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. @@ -1839,7 +1839,7 @@ f8_grad = grad(f8) x = 8.4 -print("The derivative of f8 is:",f8_grad(x)) +print("The derivative of f8 is:",f8_grad(x))

Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. @@ -1861,7 +1861,7 @@ f9_grad = grad(f9) x = np.array([1.0,0.0]) -print("The derivative of f9 is:",f9_grad(x)) +print("The derivative of f9 is:",f9_grad(x))

Here we are told that the 'dot' function does not belong to Autograd's @@ -1881,7 +1881,7 @@ f9_alternative_grad = grad(f9_alternative) x = np.array([3.0,0.0]) -print("The gradient of f9 is:",f9_alternative_grad(x)) +print("The gradient of f9 is:",f9_alternative_grad(x)) # The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively # w.r.t x is (b_1, b_2). @@ -2174,7 +2174,7 @@ $$ fig = pt.figure() ax = fig.gca(projection="3d") -xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] +xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] fmesh = f(np.array([xmesh, ymesh])) ax.plot_surface(xmesh, ymesh, fmesh) @@ -2206,7 +2206,7 @@ Run it! alpha_opt = sopt.golden(f1d) next_guess = x + alpha_opt * s guesses.append(next_guess) -print(next_guess) +print(next_guess)

What happened? diff --git a/doc/pub/week39/html/week39-solarized.html b/doc/pub/week39/html/week39-solarized.html index f229262d6..b6409b568 100644 --- a/doc/pub/week39/html/week39-solarized.html +++ b/doc/pub/week39/html/week39-solarized.html @@ -216,7 +216,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Sep 22, 2020

+

Sep 24, 2020












@@ -801,7 +801,7 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) +print(beta_linreg) beta = np.random.randn(2,1) eta = 0.1 @@ -811,7 +811,7 @@ Niterations = 1000 gradients = 2.0/m*xb.T.dot(xb.dot(beta)-y) beta -= eta*gradients -print(beta) +print(beta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] ypredict = xbnew.dot(beta) @@ -844,10 +844,10 @@ y = 4+3* xb = np.c_[np.ones((100,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) -sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1) +print(beta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print(sgdreg.intercept_, sgdreg.coef_) +print(sgdreg.intercept_, sgdreg.coef_)

@@ -895,14 +895,14 @@ x = 2*np.random.rand(m,4+3*x+np.random.randn(m,1) xb = np.c_[np.ones((m,1)), x] -XT_X = xb.T @ xb +XT_X = xb.T @ xb #Ridge parameter lambda lmbda = 0.001 Id = lmbda* np.eye(XT_X.shape[0]) -beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y -print(beta_linreg) +beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y +print(beta_linreg) # Start plain gradient descent beta = np.random.randn(2,1) @@ -910,12 +910,12 @@ eta = 0.1 Niterations = 100 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta + gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta beta -= eta*gradients -print(beta) -ypredict = xb @ beta -ypredict2 = xb @ beta_linreg +print(beta) +ypredict = xb @ beta +ypredict2 = xb @ beta_linreg plt.plot(x, ypredict, "r-") plt.plot(x, ypredict2, "b-") plt.plot(x, y ,'ro') @@ -1119,7 +1119,7 @@ j = 0 gamma_j = step_length(t,t0,t1) j += 1 -print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) +print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))











@@ -1142,12 +1142,12 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print("Own inversion") -print(theta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) +print("Own inversion") +print(theta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print("sgdreg from scikit") -print(sgdreg.intercept_, sgdreg.coef_) +print("sgdreg from scikit") +print(sgdreg.intercept_, sgdreg.coef_) theta = np.random.randn(2,1) @@ -1156,10 +1156,10 @@ Niterations = 1000 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ ((xb @ theta)-y) + gradients = 2.0/m*xb.T @ ((xb @ theta)-y) theta -= eta*gradients -print("theta frm own gd") -print(theta) +print("theta frm own gd") +print(theta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] @@ -1179,11 +1179,11 @@ theta = np.random.randn(2,1] yi = y[random_index:random_index+1] - gradients = 2 * xi.T @ ((xi @ theta)-yi) + gradients = 2 * xi.T @ ((xi @ theta)-yi) eta = learning_schedule(epoch*m+i) theta = theta - eta*gradients -print("theta from own sdg") -print(theta) +print("theta from own sdg") +print(theta) plt.plot(xnew, ypredict, "r-") plt.plot(xnew, ypredict2, "b-") @@ -1525,7 +1525,7 @@ plt.legend() plt.show() -print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic)))) +print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))

@@ -1554,11 +1554,11 @@ f1_grad = grad(f1) a = 1.0 # See the evaluated gradient at a using autograd: -print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) +print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) # Compare with the analytical derivative, that is f1'(x) = 3*x**2 grad_analytical = 3*a**2 -print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical)) +print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))











@@ -1587,8 +1587,8 @@ f2_grad_x2 = grad(f2,1) x1 = 1.0 x2 = 3.0 -print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) -print("-"*30) +print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) +print("-"*30) # Compare with the analytical derivatives: @@ -1599,13 +1599,13 @@ f2_grad_x1_analytical = 9*x1**5 # See the evaluated derivations: -print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print() +print() -print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) -print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))

Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. @@ -1628,13 +1628,13 @@ f3_grad = grad(f3) x = np.linspace(0,4,5) # Print the computed gradient: -print("The computed gradient of f3 is: ", f3_grad(x)) +print("The computed gradient of f3 is: ", f3_grad(x)) # The analytical gradient is: (2, 3, 5, 7, 22*x[4]) f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]]) # Print the analytical gradient: -print("The analytical gradient of f3 is: ", f3_grad_analytical) +print("The analytical gradient of f3 is: ", f3_grad_analytical)

Note that in this case, when sending an array as input argument, the @@ -1662,13 +1662,13 @@ f4_grad = grad(f4) x = 2.7 # Print the computed derivative: -print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) +print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) # The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi # Print the analytical gradient: -print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical)) +print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))











@@ -1691,7 +1691,7 @@ f5_grad = grad(f5) x = 2.7 # Print the computed derivative: -print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) +print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))











@@ -1723,8 +1723,8 @@ f6_while_grad = grad(f6_while) x = 0.5 # Print the computed derivaties of f6_for and f6_while -print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) -print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x))) +print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) +print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))

@@ -1737,7 +1737,7 @@ f6_grad_analytical = 0 for i in range(10): f6_grad_analytical += i*x**(i-1) -print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) +print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))











@@ -1759,7 +1759,7 @@ f7_grad = grad(f7) n = 2.0 -print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) +print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) # The function f7 is an implementation of the factorial of n. # By using the product rule, one can find that the derivative is: @@ -1772,7 +1772,7 @@ f7_grad_analytical = 0 tmp *= (n - k) f7_grad_analytical += tmp -print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical)) +print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))

Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. @@ -1798,7 +1798,7 @@ f8_grad = grad(f8) x = 8.4 -print("The derivative of f8 is:",f8_grad(x)) +print("The derivative of f8 is:",f8_grad(x))

Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. @@ -1820,7 +1820,7 @@ f9_grad = grad(f9) x = np.array([1.0,0.0]) -print("The derivative of f9 is:",f9_grad(x)) +print("The derivative of f9 is:",f9_grad(x))

Here we are told that the 'dot' function does not belong to Autograd's @@ -1840,7 +1840,7 @@ f9_alternative_grad = grad(f9_alternative) x = np.array([3.0,0.0]) -print("The gradient of f9 is:",f9_alternative_grad(x)) +print("The gradient of f9 is:",f9_alternative_grad(x)) # The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively # w.r.t x is (b_1, b_2). @@ -2113,7 +2113,7 @@ $$ fig = pt.figure() ax = fig.gca(projection="3d") -xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] +xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] fmesh = f(np.array([xmesh, ymesh])) ax.plot_surface(xmesh, ymesh, fmesh) @@ -2145,7 +2145,7 @@ Run it! alpha_opt = sopt.golden(f1d) next_guess = x + alpha_opt * s guesses.append(next_guess) -print(next_guess) +print(next_guess)

What happened? diff --git a/doc/pub/week39/html/week39.html b/doc/pub/week39/html/week39.html index 3310762ca..23b999f95 100644 --- a/doc/pub/week39/html/week39.html +++ b/doc/pub/week39/html/week39.html @@ -221,7 +221,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Sep 22, 2020

+

Sep 24, 2020












@@ -806,7 +806,7 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) +print(beta_linreg) beta = np.random.randn(2,1) eta = 0.1 @@ -816,7 +816,7 @@ Niterations = 1 gradients = 2.0/m*xb.T.dot(xb.dot(beta)-y) beta -= eta*gradients -print(beta) +print(beta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] ypredict = xbnew.dot(beta) @@ -849,10 +849,10 @@ y = 4+3* xb = np.c_[np.ones((100,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) -sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1) +print(beta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print(sgdreg.intercept_, sgdreg.coef_) +print(sgdreg.intercept_, sgdreg.coef_)

@@ -900,14 +900,14 @@ x = 2*np y = 4+3*x+np.random.randn(m,1) xb = np.c_[np.ones((m,1)), x] -XT_X = xb.T @ xb +XT_X = xb.T @ xb #Ridge parameter lambda lmbda = 0.001 Id = lmbda* np.eye(XT_X.shape[0]) -beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y -print(beta_linreg) +beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y +print(beta_linreg) # Start plain gradient descent beta = np.random.randn(2,1) @@ -915,12 +915,12 @@ eta = 0.1= 100 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta + gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta beta -= eta*gradients -print(beta) -ypredict = xb @ beta -ypredict2 = xb @ beta_linreg +print(beta) +ypredict = xb @ beta +ypredict2 = xb @ beta_linreg plt.plot(x, ypredict, "r-") plt.plot(x, ypredict2, "b-") plt.plot(x, y ,'ro') @@ -1124,7 +1124,7 @@ j = 0 gamma_j = step_length(t,t0,t1) j += 1 -print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) +print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))











@@ -1147,12 +1147,12 @@ y = 4+3* xb = np.c_[np.ones((m,1)), x] theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print("Own inversion") -print(theta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) +print("Own inversion") +print(theta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) -print("sgdreg from scikit") -print(sgdreg.intercept_, sgdreg.coef_) +print("sgdreg from scikit") +print(sgdreg.intercept_, sgdreg.coef_) theta = np.random.randn(2,1) @@ -1161,10 +1161,10 @@ Niterations = 1 for iter in range(Niterations): - gradients = 2.0/m*xb.T @ ((xb @ theta)-y) + gradients = 2.0/m*xb.T @ ((xb @ theta)-y) theta -= eta*gradients -print("theta frm own gd") -print(theta) +print("theta frm own gd") +print(theta) xnew = np.array([[0],[2]]) xbnew = np.c_[np.ones((2,1)), xnew] @@ -1184,11 +1184,11 @@ theta = np.= np.random.randint(m) xi = xb[random_index:random_index+1] yi = y[random_index:random_index+1] - gradients = 2 * xi.T @ ((xi @ theta)-yi) + gradients = 2 * xi.T @ ((xi @ theta)-yi) eta = learning_schedule(epoch*m+i) theta = theta - eta*gradients -print("theta from own sdg") -print(theta) +print("theta from own sdg") +print(theta) plt.plot(xnew, ypredict, "r-") plt.plot(xnew, ypredict2, "b-") @@ -1530,7 +1530,7 @@ plt.legend() plt.show() -print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic)))) +print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))

@@ -1559,11 +1559,11 @@ f1_grad = grad(f1) a = 1.0 # See the evaluated gradient at a using autograd: -print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) +print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) # Compare with the analytical derivative, that is f1'(x) = 3*x**2 grad_analytical = 3*a**2 -print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical)) +print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))











@@ -1592,8 +1592,8 @@ f2_grad_x2 = grad(f2,= 1.0 x2 = 3.0 -print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) -print("-"*30) +print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) +print("-"*30) # Compare with the analytical derivatives: @@ -1604,13 +1604,13 @@ f2_grad_x1_analytical = = x1 - 5 # See the evaluated derivations: -print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print() +print() -print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) -print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))

Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. @@ -1633,13 +1633,13 @@ f3_grad = grad(f3) x = np.linspace(0,4,5) # Print the computed gradient: -print("The computed gradient of f3 is: ", f3_grad(x)) +print("The computed gradient of f3 is: ", f3_grad(x)) # The analytical gradient is: (2, 3, 5, 7, 22*x[4]) f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]]) # Print the analytical gradient: -print("The analytical gradient of f3 is: ", f3_grad_analytical) +print("The analytical gradient of f3 is: ", f3_grad_analytical)

Note that in this case, when sending an array as input argument, the @@ -1667,13 +1667,13 @@ f4_grad = grad(f4) x = 2.7 # Print the computed derivative: -print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) +print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) # The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi # Print the analytical gradient: -print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical)) +print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))











@@ -1696,7 +1696,7 @@ f5_grad = grad(f5) x = 2.7 # Print the computed derivative: -print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) +print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))











@@ -1728,8 +1728,8 @@ f6_while_grad = grad(f6_while) x = 0.5 # Print the computed derivaties of f6_for and f6_while -print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) -print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x))) +print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) +print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))

@@ -1742,7 +1742,7 @@ f6_grad_analytical = for i in range(10): f6_grad_analytical += i*x**(i-1) -print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) +print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))











@@ -1764,7 +1764,7 @@ f7_grad = grad(f7) n = 2.0 -print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) +print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) # The function f7 is an implementation of the factorial of n. # By using the product rule, one can find that the derivative is: @@ -1777,7 +1777,7 @@ f7_grad_analytical = *= (n - k) f7_grad_analytical += tmp -print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical)) +print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))

Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. @@ -1803,7 +1803,7 @@ f8_grad = grad(f8) x = 8.4 -print("The derivative of f8 is:",f8_grad(x)) +print("The derivative of f8 is:",f8_grad(x))

Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. @@ -1825,7 +1825,7 @@ f9_grad = grad(f9) x = np.array([1.0,0.0]) -print("The derivative of f9 is:",f9_grad(x)) +print("The derivative of f9 is:",f9_grad(x))

Here we are told that the 'dot' function does not belong to Autograd's @@ -1845,7 +1845,7 @@ f9_alternative_grad = grad(f9_alternative) x = np.array([3.0,0.0]) -print("The gradient of f9 is:",f9_alternative_grad(x)) +print("The gradient of f9 is:",f9_alternative_grad(x)) # The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively # w.r.t x is (b_1, b_2). @@ -2118,7 +2118,7 @@ $$ fig = pt.figure() ax = fig.gca(projection="3d") -xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] +xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] fmesh = f(np.array([xmesh, ymesh])) ax.plot_surface(xmesh, ymesh, fmesh) @@ -2150,7 +2150,7 @@ Run it! alpha_opt = sopt.golden(f1d) next_guess = x + alpha_opt * s guesses.append(next_guess) -print(next_guess) +print(next_guess)

What happened? diff --git a/doc/pub/week39/ipynb/ipynb-week39-src.tar.gz b/doc/pub/week39/ipynb/ipynb-week39-src.tar.gz index 7b4b0bd7d04a6caee96f35e4ff25cd32f91eaca3..e1e5227affc6fc1eee30898c3706695c6fa4bd50 100644 GIT binary patch literal 191 zcmV;w06_mAiwFQ1B5Yp(1MSaC3c@fD2H>uHia9}{g};8r^E}V{+5`F=LG1tt007*_SVRB- literal 197 zcmb2|=3r14&5CDWetUjn-XQ~#*2MERM{Aoj{A85eR3xlIoJ92#L{>*M@b(Fwy&b&k zK+_BXr@#B3@;vAHUUzo+=JS;k>o--+`gFY_M{nirWqZ!rN11zF&yAIgmAt$vD14Wp z*Qw`oOG@8t_EpLf4=#K7?_J>kWz(|npFW;+; \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Sep 22, 2020**\n", + "Date: **Sep 24, 2020**\n", "\n", "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -812,7 +812,7 @@ "xb = np.c_[np.ones((100,1)), x]\n", "beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)\n", "print(beta_linreg)\n", - "sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1)\n", + "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", "sgdreg.fit(x,y.ravel())\n", "print(sgdreg.intercept_, sgdreg.coef_)" ] @@ -3043,5 +3043,5 @@ ], "metadata": {}, "nbformat": 4, - "nbformat_minor": 2 + "nbformat_minor": 4 } diff --git a/doc/src/week39/week39.do.txt b/doc/src/week39/week39.do.txt index 67959c123..9ed997293 100644 --- a/doc/src/week39/week39.do.txt +++ b/doc/src/week39/week39.do.txt @@ -553,7 +553,7 @@ y = 4+3*x+np.random.randn(100,1) xb = np.c_[np.ones((100,1)), x] beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) print(beta_linreg) -sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) sgdreg.fit(x,y.ravel()) print(sgdreg.intercept_, sgdreg.coef_) From 6df59f6e6d187c38539aaa6410da06443f3dcfb0 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 24 Sep 2020 06:54:17 +0200 Subject: [PATCH 2/2] updating week39 --- doc/pub/week39/html/._week39-bs000.html | 229 ++- doc/pub/week39/html/._week39-bs001.html | 229 ++- doc/pub/week39/html/._week39-bs002.html | 229 ++- doc/pub/week39/html/._week39-bs003.html | 229 ++- doc/pub/week39/html/._week39-bs004.html | 229 ++- doc/pub/week39/html/._week39-bs005.html | 229 ++- doc/pub/week39/html/._week39-bs006.html | 229 ++- doc/pub/week39/html/._week39-bs007.html | 229 ++- doc/pub/week39/html/._week39-bs008.html | 229 ++- doc/pub/week39/html/._week39-bs009.html | 229 ++- doc/pub/week39/html/._week39-bs010.html | 229 ++- doc/pub/week39/html/._week39-bs011.html | 229 ++- doc/pub/week39/html/._week39-bs012.html | 229 ++- doc/pub/week39/html/._week39-bs013.html | 229 ++- doc/pub/week39/html/._week39-bs014.html | 229 ++- 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doc/pub/week39/html/week39-solarized.html | 1275 ++++++-------- doc/pub/week39/html/week39.html | 1275 ++++++-------- doc/pub/week39/ipynb/ipynb-week39-src.tar.gz | Bin 191 -> 192 bytes doc/pub/week39/ipynb/week39.ipynb | 1594 ++++++++---------- doc/src/week39/week39.do.txt | 871 ++++------ 78 files changed, 11412 insertions(+), 13885 deletions(-) diff --git a/doc/pub/week39/html/._week39-bs000.html b/doc/pub/week39/html/._week39-bs000.html index d1107c7b3..2e9c5fff5 100644 --- a/doc/pub/week39/html/._week39-bs000.html +++ b/doc/pub/week39/html/._week39-bs000.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian 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'___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({

  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -325,7 +302,7 @@ MathJax.Hub.Config({
  • 9
  • 10
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs001.html b/doc/pub/week39/html/._week39-bs001.html index 8c4eb24d6..09906fc71 100644 --- a/doc/pub/week39/html/._week39-bs001.html +++ b/doc/pub/week39/html/._week39-bs001.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -308,7 +285,7 @@ Reading suggestions for both days: 10
  • 11
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs002.html b/doc/pub/week39/html/._week39-bs002.html index e367f6cfd..92e3800db 100644 --- a/doc/pub/week39/html/._week39-bs002.html +++ b/doc/pub/week39/html/._week39-bs002.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
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  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • More complicated functions using the elements of their arguments directly
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  • Conjugate gradient method
    • -
  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
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  • Friday September 25
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  • Computation of gradients
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    • +
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    • +
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  • Functions using mathematical functions from Numpy
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    • +
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    • +
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    • @@ -305,7 +282,7 @@ MathJax.Hub.Config({
  • 11
  • 12
  • ...
    • -
  • 76
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  • More on convex functions
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  • Revisiting our first homework
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
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  • Simple program
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  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Friday September 25
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  • Simple example code
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
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  • The routine for the steepest descent method
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Friday September 25
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    • @@ -313,7 +290,7 @@ some approximative/numerical method to compute the minimum.
  • 12
  • 13
  • ...
    • -
  • 76
    • +
  • 71
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    • diff --git a/doc/pub/week39/html/._week39-bs004.html b/doc/pub/week39/html/._week39-bs004.html index 2c623810e..8d735bf5b 100644 --- a/doc/pub/week39/html/._week39-bs004.html +++ b/doc/pub/week39/html/._week39-bs004.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - 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  • When do we stop?
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  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -321,7 +298,7 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas
  • 13
  • 14
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs005.html b/doc/pub/week39/html/._week39-bs005.html index 2f2e88eea..e3dcc869f 100644 --- a/doc/pub/week39/html/._week39-bs005.html +++ b/doc/pub/week39/html/._week39-bs005.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
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  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • The gradient step
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Second moment of the gradient
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  • The syntax a.dot(b) when finding the dot product
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  • Recommended to avoid
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Code examples for steepest descent
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  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
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  • The routine for the steepest descent method
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method and iterations
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  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
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  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method and iterations
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • Simple program
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  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
    • +
  • SGD example
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  • The gradient step
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  • Simple example code
    • +
  • When do we stop?
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  • Slightly different approach
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  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -326,7 +303,7 @@ This defines what is called the Hessian matrix.
  • 14
  • 15
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs006.html b/doc/pub/week39/html/._week39-bs006.html index fb2334de4..a6b5d20ea 100644 --- a/doc/pub/week39/html/._week39-bs006.html +++ b/doc/pub/week39/html/._week39-bs006.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
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    • -
  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
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  • Simple program
    • -
  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • Simple example code
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Gradient method
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  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
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  • Gradient descent example
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • Simple program
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  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
    • +
  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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    • @@ -327,7 +304,7 @@ If we can compute these matrices, in particular the Hessian, the above is often
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/week39/html/._week39-bs007.html b/doc/pub/week39/html/._week39-bs007.html index b63c0673f..9c8ad410d 100644 --- a/doc/pub/week39/html/._week39-bs007.html +++ b/doc/pub/week39/html/._week39-bs007.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - 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  • More on convex functions
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  • The Hessian matrix
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  • Simple program
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Momentum parameter
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    • +
  • Recommended to avoid
    • @@ -319,7 +296,7 @@ normally discourage the use of this method.
  • 16
  • 17
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs008.html b/doc/pub/week39/html/._week39-bs008.html index b62b324a6..f801e22fe 100644 --- a/doc/pub/week39/html/._week39-bs008.html +++ b/doc/pub/week39/html/._week39-bs008.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
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    • -
  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
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    • -
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Simple example code
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method and iterations
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  • Conjugate gradient method
    • -
  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
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  • Broyden–Fletcher–Goldfarb–Shanno algorithm
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  • Standard steepest descent
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
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  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
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  • Slightly different approach
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  • RMS prop
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    • +
  • Recommended to avoid
    • @@ -339,7 +316,7 @@ $$
  • 17
  • 18
  • ...
    • -
  • 76
    • +
  • 71
  • »
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  • More on convex functions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • The derivative of the cost/loss function
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  • Using gradient descent methods, limitations
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  • Friday September 25
    • +
  • Stochastic Gradient Descent
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  • Computation of gradients
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  • Recommended to avoid
    • @@ -322,7 +299,7 @@ vanishes, then Newton-Raphson may fail totally
  • 18
  • 19
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs010.html b/doc/pub/week39/html/._week39-bs010.html index 27260c959..5adfecdc0 100644 --- a/doc/pub/week39/html/._week39-bs010.html +++ b/doc/pub/week39/html/._week39-bs010.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • Simple example code
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
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  • Gradient descent example
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  • The derivative of the cost/loss function
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  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Computation of gradients
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  • SGD example
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  • The gradient step
    • +
  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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    • @@ -360,7 +337,7 @@ more than two non-linear equations. In our case, the Jacobian matrix is given by
  • 19
  • 20
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs011.html b/doc/pub/week39/html/._week39-bs011.html index e0ae207fe..53191b59b 100644 --- a/doc/pub/week39/html/._week39-bs011.html +++ b/doc/pub/week39/html/._week39-bs011.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - 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  • More on convex functions
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  • Revisiting our first homework
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • Simple program
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  • Gradient descent and Ridge
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  • Simple example code
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
    • +
  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • Simple example code
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  • Automatic differentiation
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  • Using autograd
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  • Functions using mathematical functions from Numpy
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  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -329,7 +306,7 @@ we are always moving towards smaller function values, i.e a minimum.
  • 20
  • 21
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs012.html b/doc/pub/week39/html/._week39-bs012.html index faec60080..65643fd8e 100644 --- a/doc/pub/week39/html/._week39-bs012.html +++ b/doc/pub/week39/html/._week39-bs012.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
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  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Computation of gradients
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  • SGD example
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • Second moment of the gradient
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  • RMS prop
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method and iterations
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
    • +
  • Gradient descent example
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  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
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    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Momentum based GD
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  • Momentum parameter
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  • RMS prop
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  • Practical tips
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    • +
  • Recommended to avoid
    • @@ -324,7 +301,7 @@ the learning rate within the context of Machine Learning.
  • 21
  • 22
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs013.html b/doc/pub/week39/html/._week39-bs013.html index 2069d79ae..c840bc480 100644 --- a/doc/pub/week39/html/._week39-bs013.html +++ b/doc/pub/week39/html/._week39-bs013.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - 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  • More on convex functions
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  • Revisiting our first homework
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • Simple program
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  • Gradient descent and Ridge
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  • Friday September 25
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  • Simple example code
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  • Slightly different approach
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  • Gradient method
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  • Steepest descent method
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  • Final expressions
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  • The routine for the steepest descent method
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
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  • Broyden–Fletcher–Goldfarb–Shanno algorithm
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  • Standard steepest descent
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  • Gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • Simple program
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  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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  • ADAM optimizer
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  • Practical tips
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  • Automatic differentiation
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  • Using autograd
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  • Autograd with more complicated functions
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  • Functions using mathematical functions from Numpy
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  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -331,7 +308,7 @@ Note that the gradient is a function of \( \mathbf{x} =
  • 22
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  • ...
    • -
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    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs014.html b/doc/pub/week39/html/._week39-bs014.html index 3adf21aef..8a6f50e7a 100644 --- a/doc/pub/week39/html/._week39-bs014.html +++ b/doc/pub/week39/html/._week39-bs014.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • More on momentum based approaches
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  • Second moment of the gradient
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  • RMS prop
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
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  • Broyden–Fletcher–Goldfarb–Shanno algorithm
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
    • +
  • Gradient descent example
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  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
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  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
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  • The gradient step
    • +
  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • More on momentum based approaches
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  • Momentum parameter
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  • RMS prop
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    • +
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    • @@ -324,7 +301,7 @@ randomness. One such method is that of Stochastic Gradient Descent
  • 23
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  • ...
    • -
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    • +
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  • Steepest descent example
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  • Conjugate gradient method
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  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Stochastic Gradient Descent
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  • Autograd with more complicated functions
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    • @@ -325,7 +302,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).
  • 24
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  • ...
    • -
  • 76
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'___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
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  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
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  • Simple program
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  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Friday September 25
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  • SGD example
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Second moment of the gradient
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method and iterations
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
    • +
  • Gradient descent example
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Practical tips
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  • Steepest descent example
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  • Recommended to avoid
    • @@ -349,7 +326,7 @@ This condition is particularly useful since it gives us an procedure for determi
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    • -
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    • +
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  • »
    • diff --git a/doc/pub/week39/html/._week39-bs018.html b/doc/pub/week39/html/._week39-bs018.html index c86f2308c..eef7ba5a5 100644 --- a/doc/pub/week39/html/._week39-bs018.html +++ b/doc/pub/week39/html/._week39-bs018.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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  • More on convex functions
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  • The Hessian matrix
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  • Simple program
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  • Friday September 25
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  • Simple example code
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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    • @@ -336,7 +313,7 @@ This result means that if we know that the cost/loss function is convex and we a
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    • -
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    • diff --git a/doc/pub/week39/html/._week39-bs019.html b/doc/pub/week39/html/._week39-bs019.html index 95bc2bbea..73f887c94 100644 --- a/doc/pub/week39/html/._week39-bs019.html +++ b/doc/pub/week39/html/._week39-bs019.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - 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2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
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  • Gradient descent example
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  • The derivative of the cost/loss function
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  • The Hessian matrix
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  • Simple program
    • -
  • Gradient Descent Example
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  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
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  • Using gradient descent methods, limitations
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  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
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  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • More on momentum based approaches
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  • Second moment of the gradient
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  • RMS prop
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  • ADAM optimizer
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  • The syntax a.dot(b) when finding the dot product
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  • Recommended to avoid
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  • Standard steepest descent
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Code examples for steepest descent
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  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method and iterations
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  • Conjugate gradient method
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  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
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  • Standard steepest descent
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
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    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
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  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
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  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
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  • Practical tips
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  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
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  • More complicated functions using the elements of their arguments directly
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  • Functions using mathematical functions from Numpy
    • +
  • More autograd
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  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
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  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -332,7 +309,7 @@ Using the definition of convexity, try to show that a function satisfying the pr
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    • diff --git a/doc/pub/week39/html/._week39-bs020.html b/doc/pub/week39/html/._week39-bs020.html index 314e6521a..394900c6e 100644 --- a/doc/pub/week39/html/._week39-bs020.html +++ b/doc/pub/week39/html/._week39-bs020.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - 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  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
    • -
  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • More on momentum based approaches
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  • Momentum parameter
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  • Second moment of the gradient
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  • RMS prop
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  • ADAM optimizer
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  • Practical tips
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  • Automatic differentiation
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  • Using autograd
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  • Autograd with more complicated functions
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  • More complicated functions using the elements of their arguments directly
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    • -
  • More autograd
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  • Unsupported functions
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  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
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  • Standard steepest descent
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Code examples for steepest descent
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  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
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  • Conjugate gradient method
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  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
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  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
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  • And with loops
    • +
  • Using recursion
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  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -280,37 +257,40 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Revisiting our first homework

    +

    Standard steepest descent

    -We will use linear regression as a case study for the gradient descent -methods. Linear regression is a great test case for the gradient -descent methods discussed in the lectures since it has several -desirable properties such as: +Before we proceed, we would like to discuss the approach called the +standard Steepest descent, which again leads to us having to be able +to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). -

      -
    1. An analytical solution (recall homework set 1).
      2. -
    2. The gradient can be computed analytically.
      3. -
    3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates
      4. -
    - -We revisit the example from homework set 1 where we had +

    +The success of the CG method +for finding solutions of non-linear problems is based on the theory +of conjugate gradients for linear systems of equations. It belongs to +the class of iterative methods for solving problems from linear +algebra of the type $$ -y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100 +\begin{equation*} +\hat{A}\hat{x} = \hat{b}. +\end{equation*} $$ -with \( x_i \in [0,1] \) chosen randomly with a uniform distribution. Additionally \( \xi_i \) represents stochastic noise chosen according to a normal distribution \( \cal {N}(0,1) \). -The linear regression model is given by +

    +In the iterative process we end up with a problem like + $$ -h_\beta(x) = \hat{y} = \beta_0 + \beta_1 x, +\begin{equation*} + \hat{r}= \hat{b}-\hat{A}\hat{x}, +\end{equation*} $$ -such that -$$ -\hat{y}_i = \beta_0 + \beta_1 x_i. -$$ +where \( \hat{r} \) is the so-called residual or error in the iterative process. + +

    +When we have found the exact solution, \( \hat{r}=0 \).

    @@ -338,7 +318,7 @@ $$

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    • diff --git a/doc/pub/week39/html/._week39-bs021.html b/doc/pub/week39/html/._week39-bs021.html index 6ee655550..ed02517e3 100644 --- a/doc/pub/week39/html/._week39-bs021.html +++ b/doc/pub/week39/html/._week39-bs021.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -280,29 +257,21 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Gradient descent example

    +

    Gradient method

    -Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \) +The residual is zero when we reach the minimum of the quadratic equation +$$ +\begin{equation*} + P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b}, +\end{equation*} +$$

    -It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by -$$ -X \equiv \begin{bmatrix} -1 & x_1 \\ -\vdots & \vdots \\ -1 & x_{100} & \\ -\end{bmatrix}. -$$ - -The loss function is given by -$$ -C(\beta) = ||X\beta-\mathbf{y}||^2 = ||X\beta||^2 - 2 \mathbf{y}^T X\beta + ||\mathbf{y}||^2 = \sum_{i=1}^{100} (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2 -$$ - -and we want to find \( \beta \) such that \( C(\beta) \) is minimized. +with the constraint that the matrix \( \hat{A} \) is positive definite and +symmetric. This defines also the Hessian and we want it to be positive definite.

    @@ -330,7 +299,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.

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    • diff --git a/doc/pub/week39/html/._week39-bs022.html b/doc/pub/week39/html/._week39-bs022.html index 9b5dc9569..0e7fda90a 100644 --- a/doc/pub/week39/html/._week39-bs022.html +++ b/doc/pub/week39/html/._week39-bs022.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,17 +259,25 @@ MathJax.Hub.Config({ -

    The derivative of the cost/loss function

    +

    Steepest descent method

    -Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as +We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). +We can assume without loss of generality that $$ -\nabla_{\beta} C(\beta) = (\partial C(\beta) / \partial \beta_0, \partial C(\beta) / \partial \beta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} = 2X^T(X\beta - \mathbf{y}), +\begin{equation*} +\hat{x}_0=0, +\end{equation*} $$ -where \( X \) is the design matrix defined above. +or consider the system +$$ +\begin{equation*} +\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, +\end{equation*} +$$ + +instead.

    @@ -320,7 +305,7 @@ where \( X \) is the design matrix defined above.

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    • diff --git a/doc/pub/week39/html/._week39-bs023.html b/doc/pub/week39/html/._week39-bs023.html index 93fb691e8..21711e289 100644 --- a/doc/pub/week39/html/._week39-bs023.html +++ b/doc/pub/week39/html/._week39-bs023.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
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  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
    • -
  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • More on momentum based approaches
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  • Second moment of the gradient
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  • ADAM optimizer
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  • Practical tips
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  • More complicated functions using the elements of their arguments directly
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  • Functions using mathematical functions from Numpy
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  • More autograd
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  • And with loops
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  • Unsupported functions
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  • The syntax a.dot(b) when finding the dot product
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  • Recommended to avoid
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  • Standard steepest descent
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  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
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  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
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  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,16 +259,33 @@ MathJax.Hub.Config({ -

    The Hessian matrix

    -The Hessian matrix of \( C(\beta) \) is given by +

    Steepest descent method

    +
    +
    +

    +One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form $$ -\hat{H} \equiv \begin{bmatrix} -\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ -\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ -\end{bmatrix} = 2X^T X. +\begin{equation*} + f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. +\end{equation*} $$ -This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite. +This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) +to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), +which equals +$$ +\begin{equation*} +\hat{A}\hat{x}_0-\hat{b}, +\end{equation*} +$$ + +and +\( \hat{x}_0=0 \) it is equal \( -\hat{b} \). + +

    +

    +
    +

    @@ -319,7 +313,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(

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    • diff --git a/doc/pub/week39/html/._week39-bs024.html b/doc/pub/week39/html/._week39-bs024.html index ebc3b52a9..37e4be7cd 100644 --- a/doc/pub/week39/html/._week39-bs024.html +++ b/doc/pub/week39/html/._week39-bs024.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,22 +259,46 @@ MathJax.Hub.Config({ -

    Simple program

    - -

    -We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to +

    Final expressions

    +
    +
    +

    +We can compute the residual iteratively as $$ -\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots +\begin{equation*} +\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, + \end{equation*} $$ -

    -We can use the expression we computed for the gradient and let use a -\( \beta_0 \) be chosen randomly and let \( \gamma = 0.001 \). Stop iterating -when \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \). +which equals +$$ +\begin{equation*} +\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k), + \end{equation*} +$$ + +or +$$ +\begin{equation*} +(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k, + \end{equation*} +$$ + +which gives + +$$ +\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k} +$$ + +leading to the iterative scheme +$$ +\begin{equation*} +\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k}, + \end{equation*} +$$ +

    +
    -

    -And finally we can compare our solution for \( \beta \) with the analytic result given by -\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).

    @@ -325,7 +326,7 @@ And finally we can compare our solution for \( \beta \) with the analytic result

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    • diff --git a/doc/pub/week39/html/._week39-bs025.html b/doc/pub/week39/html/._week39-bs025.html index 8ae4a5c59..61cd7646e 100644 --- a/doc/pub/week39/html/._week39-bs025.html +++ b/doc/pub/week39/html/._week39-bs025.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,52 +259,71 @@ MathJax.Hub.Config({ -

    Gradient Descent Example

    +

    Steepest descent example

    -

    -Here our simple example

    -

    # Importing various packages
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.mplot3d import Axes3D
-from matplotlib import cm
-from matplotlib.ticker import LinearLocator, FormatStrFormatter
-import sys
+
    import numpy as np
+import numpy.linalg as la
 
-# the number of datapoints
-m = 100
-x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
+import scipy.optimize as sopt
 
-xb = np.c_[np.ones((m,1)), x]
-beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
-print(beta_linreg)
-beta = np.random.randn(2,1)
+import matplotlib.pyplot as pt
+from mpl_toolkits.mplot3d import axes3d
 
-eta = 0.1
-Niterations = 1000
+def f(x):
+    return 0.5*x[0]**2 + 2.5*x[1]**2
 
-for iter in range(Niterations):
-    gradients = 2.0/m*xb.T.dot(xb.dot(beta)-y)
-    beta -= eta*gradients
+def df(x):
+    return np.array([x[0], 5*x[1]])
 
-print(beta)
-xnew = np.array([[0],[2]])
-xbnew = np.c_[np.ones((2,1)), xnew]
-ypredict = xbnew.dot(beta)
-ypredict2 = xbnew.dot(beta_linreg)
-plt.plot(xnew, ypredict, "r-")
-plt.plot(xnew, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Gradient descent example')
-plt.show()
+fig = pt.figure()
+ax = fig.gca(projection="3d")
+
+xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
+fmesh = f(np.array([xmesh, ymesh]))
+ax.plot_surface(xmesh, ymesh, fmesh)
+
    
+
    
+And then as countor plot
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh)
+guesses = [np.array([2, 2./5])]
+
    
+
    
+Find guesses
+
    
+
+
+
    x = guesses[-1]
+s = -df(x)
+
    
+
    
+Run it!
+
    
+
+
+
    def f1d(alpha):
+    return f(x + alpha*s)
+
+alpha_opt = sopt.golden(f1d)
+next_guess = x + alpha_opt * s
+guesses.append(next_guess)
+print(next_guess)
+
    
+
    
+What happened?
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh, 50)
+it_array = np.array(guesses)
+pt.plot(it_array.T[0], it_array.T[1], "x-")
 
    
 
    
 
    
@@ -355,7 +351,7 @@ plt.show()
   
  • 34
    • 
   
  • 35
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs026.html b/doc/pub/week39/html/._week39-bs026.html
index 73af92e68..706f78a27 100644
--- a/doc/pub/week39/html/._week39-bs026.html
+++ b/doc/pub/week39/html/._week39-bs026.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,27 +259,34 @@ MathJax.Hub.Config({
 
 
 
-
    And a corresponding example using scikit-learn 
    
+
    Conjugate gradient method 
    
+
    
+
    
+
     
+In the CG method we define so-called conjugate directions and two vectors 
+\( \hat{s} \) and \( \hat{t} \)
+are said to be
+conjugate if
+$$
+\begin{equation*}
+\hat{s}^T\hat{A}\hat{t}= 0.
+\end{equation*}
+$$
 
-
    
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \hat{x}_i \) obeying the above criterion, namely
+$$
+\begin{equation*}
+\hat{x}_i^T\hat{A}\hat{x}_j= 0.
+\end{equation*}
+$$
 
-
-
    # Importing various packages
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
+Two vectors are conjugate if they are orthogonal with respect to 
+this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
+
    
+
    
 
-x = 2*np.random.rand(100,1)
-y = 4+3*x+np.random.randn(100,1)
 
-xb = np.c_[np.ones((100,1)), x]
-beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
-print(beta_linreg)
-sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
-sgdreg.fit(x,y.ravel())
-print(sgdreg.intercept_, sgdreg.coef_)
-

    @@ -329,7 +313,7 @@ sgdreg.fit(x,y.

  • 35
  • 36
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs027.html b/doc/pub/week39/html/._week39-bs027.html index de681a49d..0c99ee4be 100644 --- a/doc/pub/week39/html/._week39-bs027.html +++ b/doc/pub/week39/html/._week39-bs027.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -280,29 +257,23 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Gradient descent and Ridge

    - -

    -We have also discussed Ridge regression where the loss function contains a regularized term given by the \( L_2 \) norm of \( \beta \), +

    Conjugate gradient method

    +
    +
    +

    +An example is given by the eigenvectors of the matrix $$ -C_{\text{ridge}}(\beta) = ||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. +\begin{equation*} +\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j, +\end{equation*} $$ -

    -In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows -$$ -\nabla_\beta C_{\text{ridge}}(\beta) = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta). -$$ +which is zero unless \( i=j \). +

    +
    -

    -We can easily extend our program to minimize \( C_{\text{ridge}}(\beta) \) using gradient descent and compare with the analytical solution given by -$$ -\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}. -$$

    @@ -330,7 +301,7 @@ $$

  • 36
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  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs028.html b/doc/pub/week39/html/._week39-bs028.html index 9163594df..30af301c5 100644 --- a/doc/pub/week39/html/._week39-bs028.html +++ b/doc/pub/week39/html/._week39-bs028.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,54 +259,31 @@ MathJax.Hub.Config({ -

    Program example for gradient descent with Ridge Regression

    -

    +

    Conjugate gradient method

    +
    +
    +

    +Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size +\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector +$$ +\begin{equation*} +\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. +\end{equation*} +$$ - -

    from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.mplot3d import Axes3D
-from matplotlib import cm
-from matplotlib.ticker import LinearLocator, FormatStrFormatter
-import sys
+We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. 
+Then the \( \hat{p}_{i} \)  form a basis of \( R^n \) and we can expand the solution 
+$  \hat{A}\hat{x} = \hat{b}$ in this basis, namely
 
-# the number of datapoints
-m = 100
-x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
+$$
+\begin{equation*}
+  \hat{x}  = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
+\end{equation*}
+$$
+
    +
    -xb = np.c_[np.ones((m,1)), x] -XT_X = xb.T @ xb -#Ridge parameter lambda -lmbda = 0.001 -Id = lmbda* np.eye(XT_X.shape[0]) - -beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y -print(beta_linreg) -# Start plain gradient descent -beta = np.random.randn(2,1) - -eta = 0.1 -Niterations = 100 - -for iter in range(Niterations): - gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta - beta -= eta*gradients - -print(beta) -ypredict = xb @ beta -ypredict2 = xb @ beta_linreg -plt.plot(x, ypredict, "r-") -plt.plot(x, ypredict2, "b-") -plt.plot(x, y ,'ro') -plt.axis([0,2.0,0, 15.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Gradient descent example for Ridge') -plt.show() -

    @@ -356,7 +310,7 @@ plt.show()

  • 37
  • 38
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs029.html b/doc/pub/week39/html/._week39-bs029.html index 32c3b7d04..8ac2ba899 100644 --- a/doc/pub/week39/html/._week39-bs029.html +++ b/doc/pub/week39/html/._week39-bs029.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,17 +259,37 @@ MathJax.Hub.Config({ -

    Using gradient descent methods, limitations

    +

    Conjugate gradient method

    +
    +
    +

    +The coefficients are given by +$$ +\begin{equation*} + \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. +\end{equation*} +$$ -

      -
    • Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our energy function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
      • -
    • GD is sensitive to initial conditions. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.
      • -
    • Gradients are computationally expensive to calculate for large datasets. In many cases in statistics and ML, the energy function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over all \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm.
      • -
    • GD is very sensitive to choices of learning rates. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would adaptively choose the learning rates to match the landscape.
      • -
    • GD treats all directions in parameter space uniformly. Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.
      • -
    • GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.
      • -
    +Multiplying with \( \hat{p}_k^T \) from the left gives +$$ +\begin{equation*} + \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b}, +\end{equation*} +$$ + +and we can define the coefficients \( \alpha_k \) as + +$$ +\begin{equation*} + \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k} +\end{equation*} +$$ +
    +
    + + +

    diff --git a/doc/pub/week39/html/._week39-bs030.html b/doc/pub/week39/html/._week39-bs030.html index 020f2691d..cfcb8b8d4 100644 --- a/doc/pub/week39/html/._week39-bs030.html +++ b/doc/pub/week39/html/._week39-bs030.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,7 +259,39 @@ MathJax.Hub.Config({ -

    Friday September 25

    +

    Conjugate gradient method and iterations

    +
    +
    +

    + +

    +If we choose the conjugate vectors \( \hat{p}_k \) carefully, +then we may not need all of them to obtain a good approximation to the solution +\( \hat{x} \). +We want to regard the conjugate gradient method as an iterative method. +This will us to solve systems where \( n \) is so large that the direct +method would take too much time. + +

    +We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). +We can assume without loss of generality that +$$ +\begin{equation*} +\hat{x}_0=0, +\end{equation*} +$$ + +or consider the system +$$ +\begin{equation*} +\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, +\end{equation*} +$$ + +instead. +

    +
    +

    @@ -310,7 +319,7 @@ MathJax.Hub.Config({

  • 39
  • 40
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs031.html b/doc/pub/week39/html/._week39-bs031.html index 1ffb2de0f..ec79c4982 100644 --- a/doc/pub/week39/html/._week39-bs031.html +++ b/doc/pub/week39/html/._week39-bs031.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
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    • -
  • Revisiting our first homework
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    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
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  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
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  • Simple example code
    • -
  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • More on momentum based approaches
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  • Momentum parameter
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  • Second moment of the gradient
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  • RMS prop
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  • ADAM optimizer
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  • Practical tips
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  • Automatic differentiation
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  • The syntax a.dot(b) when finding the dot product
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  • Recommended to avoid
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Code examples for steepest descent
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  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
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  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
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  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,21 +259,34 @@ MathJax.Hub.Config({ -

    Stochastic Gradient Descent

    - -

    -Stochastic gradient descent (SGD) and variants thereof address some of -the shortcomings of the Gradient descent method discussed above. - -

    -The underlying idea of SGD comes from the observation that the cost -function, which we want to minimize, can almost always be written as a -sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \), +

    Conjugate gradient method

    +
    +
    +

    +One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form $$ -C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, -\mathbf{\beta}). +\begin{equation*} + f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. +\end{equation*} $$ +This suggests taking the first basis vector \( \hat{p}_1 \) +to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), +which equals +$$ +\begin{equation*} +\hat{A}\hat{x}_0-\hat{b}, +\end{equation*} +$$ + +and +\( \hat{x}_0=0 \) it is equal \( -\hat{b} \). +The other vectors in the basis will be conjugate to the gradient, +hence the name conjugate gradient method. +

    +
    + +

    @@ -323,7 +313,7 @@ $$

  • 40
  • 41
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs032.html b/doc/pub/week39/html/._week39-bs032.html index 131c59f90..282ae88ab 100644 --- a/doc/pub/week39/html/._week39-bs032.html +++ b/doc/pub/week39/html/._week39-bs032.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
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  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
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  • Stochastic Gradient Descent
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  • Computation of gradients
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  • SGD example
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  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
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  • Slightly different approach
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  • Program for stochastic gradient
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  • Momentum based GD
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  • More on momentum based approaches
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  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
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  • ADAM optimizer
    • -
  • Practical tips
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  • Automatic differentiation
    • -
  • Using autograd
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  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
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  • Functions using mathematical functions from Numpy
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  • More autograd
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  • And with loops
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  • Using recursion
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  • Unsupported functions
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  • The syntax a.dot(b) when finding the dot product
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  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,22 +259,32 @@ MathJax.Hub.Config({ -

    Computation of gradients

    - -

    -This in turn means that the gradient can be -computed as a sum over \( i \)-gradients +

    Conjugate gradient method

    +
    +
    +

    +Let \( \hat{r}_k \) be the residual at the \( k \)-th step: $$ -\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}). +\begin{equation*} +\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k. +\end{equation*} $$ -

    -Stochasticity/randomness is introduced by only taking the -gradient on a subset of the data called minibatches. If there are \( n \) -data points and the size of each minibatch is \( M \), there will be \( n/M \) -minibatches. We denote these minibatches by \( B_k \) where -\( k=1,\cdots,n/M \). +Note that \( \hat{r}_k \) is the negative gradient of \( f \) at +\( \hat{x}=\hat{x}_k \), +so the gradient descent method would be to move in the direction \( \hat{r}_k \). +Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other, +so we take the direction closest to the gradient \( \hat{r}_k \) +under the conjugacy constraint. +This gives the following expression +$$ +\begin{equation*} +\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k. +\end{equation*} +$$ +

    +
    +

    @@ -325,7 +312,7 @@ minibatches. We denote these minibatches by \( B_k \) where

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    • diff --git a/doc/pub/week39/html/._week39-bs033.html b/doc/pub/week39/html/._week39-bs033.html index 78843ef81..4eb0da943 100644 --- a/doc/pub/week39/html/._week39-bs033.html +++ b/doc/pub/week39/html/._week39-bs033.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
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  • ADAM optimizer
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  • Practical tips
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  • Automatic differentiation
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  • Using autograd
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  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
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  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,27 +259,42 @@ MathJax.Hub.Config({ -

    SGD example

    -As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) -and we choose to have \( M=5 \) minibathces, -then each minibatch contains two data points. In particular we have -\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = -(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you -have only a single batch with all data points and on the other extreme, -you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e -\( B_k = \mathbf{x}_k \). +

    Conjugate gradient method

    +
    +
    +

    +We can also compute the residual iteratively as +$$ +\begin{equation*} +\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, + \end{equation*} +$$ -

    -The idea is now to approximate the gradient by replacing the sum over -all data points with a sum over the data points in one the minibatches -picked at random in each gradient descent step +which equals $$ -\nabla_{\beta} -C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta -c_i(\mathbf{x}_i, \mathbf{\beta}). +\begin{equation*} +\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k), + \end{equation*} $$ +or +$$ +\begin{equation*} +(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k, + \end{equation*} +$$ + +which gives + +$$ +\begin{equation*} +\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k}, + \end{equation*} +$$ +

    +
    + +

    @@ -329,7 +321,7 @@ $$

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    • diff --git a/doc/pub/week39/html/._week39-bs034.html b/doc/pub/week39/html/._week39-bs034.html index 6c021bfb2..7f28fb2cb 100644 --- a/doc/pub/week39/html/._week39-bs034.html +++ b/doc/pub/week39/html/._week39-bs034.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
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  • ADAM optimizer
    • -
  • Practical tips
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  • Automatic differentiation
    • -
  • Using autograd
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  • More complicated functions using the elements of their arguments directly
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  • Functions using mathematical functions from Numpy
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  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -280,23 +257,37 @@ MathJax.Hub.Config({

     

     

     

    - + -

    The gradient step

    +

    Revisiting our first homework

    -Thus a gradient descent step now looks like +We will use linear regression as a case study for the gradient descent +methods. Linear regression is a great test case for the gradient +descent methods discussed in the lectures since it has several +desirable properties such as: + +

      +
    1. An analytical solution (recall homework set 1).
      2. +
    2. The gradient can be computed analytically.
      3. +
    3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates
      4. +
    + +We revisit the example from homework set 1 where we had $$ -\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) +y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100 $$ -

    -where \( k \) is picked at random with equal -probability from \( [1,n/M] \). An iteration over the number of -minibathces (n/M) is commonly referred to as an epoch. Thus it is -typical to choose a number of epochs and for each epoch iterate over -the number of minibatches, as exemplified in the code below. +with \( x_i \in [0,1] \) chosen randomly with a uniform distribution. Additionally \( \xi_i \) represents stochastic noise chosen according to a normal distribution \( \cal {N}(0,1) \). +The linear regression model is given by +$$ +h_\beta(x) = \hat{y} = \beta_0 + \beta_1 x, +$$ + +such that +$$ +\hat{y}_i = \beta_0 + \beta_1 x_i. +$$

    @@ -324,7 +315,7 @@ the number of minibatches, as exemplified in the code below.

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    • +
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  • »
    • diff --git a/doc/pub/week39/html/._week39-bs035.html b/doc/pub/week39/html/._week39-bs035.html index 945b02ea2..fbeabf47a 100644 --- a/doc/pub/week39/html/._week39-bs035.html +++ b/doc/pub/week39/html/._week39-bs035.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -280,36 +257,29 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Simple example code

    +

    Gradient descent example

    +Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \) - -

    import numpy as np 
-
-n = 100 #100 datapoints 
-M = 5   #size of each minibatch
-m = int(n/M) #number of minibatches
-n_epochs = 10 #number of epochs
-
-j = 0
-for epoch in range(1,n_epochs+1):
-    for i in range(m):
-        k = np.random.randint(m) #Pick the k-th minibatch at random
-        #Compute the gradient using the data in minibatch Bk
-        #Compute new suggestion for 
-        j += 1
-

    -Taking the gradient only on a subset of the data has two important -benefits. First, it introduces randomness which decreases the chance -that our opmization scheme gets stuck in a local minima. Second, if -the size of the minibatches are small relative to the number of -datapoints (\( M < n \)), the computation of the gradient is much -cheaper since we sum over the datapoints in the \( k-th \) minibatch and not -all \( n \) datapoints. +It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by +$$ +X \equiv \begin{bmatrix} +1 &; x_1 \\ +\vdots &; \vdots \\ +1 &; x_{100} &; \\ +\end{bmatrix}. +$$ + +The loss function is given by +$$ +C(\beta) = ||X\beta-\mathbf{y}||^2 = ||X\beta||^2 - 2 \mathbf{y}^T X\beta + ||\mathbf{y}||^2 = \sum_{i=1}^{100} (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2 +$$ + +and we want to find \( \beta \) such that \( C(\beta) \) is minimized.

    @@ -337,7 +307,7 @@ all \( n \) datapoints.

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    • -
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    • +
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    • diff --git a/doc/pub/week39/html/._week39-bs036.html b/doc/pub/week39/html/._week39-bs036.html index 420267b2c..57ee1ffe1 100644 --- a/doc/pub/week39/html/._week39-bs036.html +++ b/doc/pub/week39/html/._week39-bs036.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,19 +259,17 @@ MathJax.Hub.Config({ -

    When do we stop?

    +

    The derivative of the cost/loss function

    -A natural question is when do we stop the search for a new minimum? -One possibility is to compute the full gradient after a given number -of epochs and check if the norm of the gradient is smaller than some -threshold and stop if true. However, the condition that the gradient -is zero is valid also for local minima, so this would only tell us -that we are close to a local/global minimum. However, we could also -evaluate the cost function at this point, store the result and -continue the search. If the test kicks in at a later stage we can -compare the values of the cost function and keep the \( \beta \) that -gave the lowest value. +Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as +$$ +\nabla_{\beta} C(\beta) = (\partial C(\beta) / \partial \beta_0, \partial C(\beta) / \partial \beta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ +\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ +\end{bmatrix} = 2X^T(X\beta - \mathbf{y}), +$$ + +where \( X \) is the design matrix defined above.

    @@ -322,7 +297,7 @@ gave the lowest value.

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    • diff --git a/doc/pub/week39/html/._week39-bs037.html b/doc/pub/week39/html/._week39-bs037.html index b0a146d89..f64a6c2dc 100644 --- a/doc/pub/week39/html/._week39-bs037.html +++ b/doc/pub/week39/html/._week39-bs037.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,51 +259,17 @@ MathJax.Hub.Config({ -

    Slightly different approach

    +

    The Hessian matrix

    +The Hessian matrix of \( C(\beta) \) is given by +$$ +\hat{H} \equiv \begin{bmatrix} +\frac{\partial^2 C(\beta)}{\partial \beta_0^2} &; \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ +\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} &; \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &; \\ +\end{bmatrix} = 2X^T X. +$$ -

    -Another approach is to let the step length \( \gamma_j \) depend on the -number of epochs in such a way that it becomes very small after a -reasonable time such that we do not move at all. +This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite. -

    -As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in time \( t \). - -

    -In this way we can fix the number of epochs, compute \( \beta \) and -evaluate the cost function at the end. Repeating the computation will -give a different result since the scheme is random by design. Then we -pick the final \( \beta \) that gives the lowest value of the cost -function. - -

    - - -

    import numpy as np 
-
-def step_length(t,t0,t1):
-    return t0/(t+t1)
-
-n = 100 #100 datapoints 
-M = 5   #size of each minibatch
-m = int(n/M) #number of minibatches
-n_epochs = 500 #number of epochs
-t0 = 1.0
-t1 = 10
-
-gamma_j = t0/t1
-j = 0
-for epoch in range(1,n_epochs+1):
-    for i in range(m):
-        k = np.random.randint(m) #Pick the k-th minibatch at random
-        #Compute the gradient using the data in minibatch Bk
-        #Compute new suggestion for beta
-        t = epoch*m+i
-        gamma_j = step_length(t,t0,t1)
-        j += 1
-
-print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
-

    @@ -353,7 +296,7 @@ j = 0

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    • diff --git a/doc/pub/week39/html/._week39-bs038.html b/doc/pub/week39/html/._week39-bs038.html index 1c4d57d52..fd8a6e3bb 100644 --- a/doc/pub/week39/html/._week39-bs038.html +++ b/doc/pub/week39/html/._week39-bs038.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,78 +259,22 @@ MathJax.Hub.Config({ -

    Program for stochastic gradient

    +

    Simple program

    +We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to +$$ +\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots +$$ - -

    # Importing various packages
-from math import exp, sqrt
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
-
-m = 100
-x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
-
-xb = np.c_[np.ones((m,1)), x]
-theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
-print("Own inversion")
-print(theta_linreg)
-sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
-sgdreg.fit(x,y.ravel())
-print("sgdreg from scikit")
-print(sgdreg.intercept_, sgdreg.coef_)
-
-
-theta = np.random.randn(2,1)
-eta = 0.1
-Niterations = 1000
-
-
-for iter in range(Niterations):
-    gradients = 2.0/m*xb.T @ ((xb @ theta)-y)
-    theta -= eta*gradients
-print("theta frm own gd")
-print(theta)
-
-xnew = np.array([[0],[2]])
-xbnew = np.c_[np.ones((2,1)), xnew]
-ypredict = xbnew.dot(theta)
-ypredict2 = xbnew.dot(theta_linreg)
-
-
-n_epochs = 50
-t0, t1 = 5, 50
-def learning_schedule(t):
-    return t0/(t+t1)
-
-theta = np.random.randn(2,1)
-
-for epoch in range(n_epochs):
-    for i in range(m):
-        random_index = np.random.randint(m)
-        xi = xb[random_index:random_index+1]
-        yi = y[random_index:random_index+1]
-        gradients = 2 * xi.T @ ((xi @ theta)-yi)
-        eta = learning_schedule(epoch*m+i)
-        theta = theta - eta*gradients
-print("theta from own sdg")
-print(theta)
-
-plt.plot(xnew, ypredict, "r-")
-plt.plot(xnew, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Random numbers ')
-plt.show()
-

    -Challenge: try to write a similar code for a Logistic Regression case. +We can use the expression we computed for the gradient and let use a +\( \beta_0 \) be chosen randomly and let \( \gamma = 0.001 \). Stop iterating +when \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \). + +

    +And finally we can compare our solution for \( \beta \) with the analytic result given by +\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).

    @@ -381,7 +302,7 @@ plt.show()

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    • diff --git a/doc/pub/week39/html/._week39-bs039.html b/doc/pub/week39/html/._week39-bs039.html index 68c46b74d..e7497eb6e 100644 --- a/doc/pub/week39/html/._week39-bs039.html +++ b/doc/pub/week39/html/._week39-bs039.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,40 +259,53 @@ MathJax.Hub.Config({ -

    Momentum based GD

    +

    Gradient Descent Example

    -The stochastic gradient descent (SGD) is almost always used with a -momentum or inertia term that serves as a memory of the direction we -are moving in parameter space. This is typically implemented as -follows - -$$ -\begin{align} -\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\ -\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}, -\tag{2} -\end{align} -$$ - +Here our simple example

    -where we have introduced a momentum parameter \( \gamma \), with -\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to -indicate the gradient is to be taken over a different mini-batch at -each step. We call this algorithm gradient descent with momentum -(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a -running average of recently encountered gradients and -\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory -used in the averaging procedure. Consistent with this, when -\( \gamma=0 \), this just reduces down to ordinary SGD as discussed -earlier. An equivalent way of writing the updates is -$$ -\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t), -$$ + +

    # Importing various packages
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+from matplotlib.ticker import LinearLocator, FormatStrFormatter
+import sys
 
-where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).
+# the number of datapoints
+m = 100
+x = 2*np.random.rand(m,1)
+y = 4+3*x+np.random.randn(m,1)
 
+xb = np.c_[np.ones((m,1)), x]
+beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
+print(beta_linreg)
+beta = np.random.randn(2,1)
+
+eta = 0.1
+Niterations = 1000
+
+for iter in range(Niterations):
+    gradients = 2.0/m*xb.T.dot(xb.dot(beta)-y)
+    beta -= eta*gradients
+
+print(beta)
+xnew = np.array([[0],[2]])
+xbnew = np.c_[np.ones((2,1)), xnew]
+ypredict = xbnew.dot(beta)
+ypredict2 = xbnew.dot(beta_linreg)
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Gradient descent example')
+plt.show()
+

    @@ -342,7 +332,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\

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    • diff --git a/doc/pub/week39/html/._week39-bs040.html b/doc/pub/week39/html/._week39-bs040.html index 1a90db0aa..a4d0d4056 100644 --- a/doc/pub/week39/html/._week39-bs040.html +++ b/doc/pub/week39/html/._week39-bs040.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,33 +259,27 @@ MathJax.Hub.Config({ -

    More on momentum based approaches

    +

    And a corresponding example using scikit-learn

    -Let us try to get more intuition from these equations. It is helpful -to consider a simple physical analogy with a particle of mass \( m \) -moving in a viscous medium with drag coefficient \( \mu \) and potential -\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \), -then its motion is described by -$$ -m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}). -$$ + +

    # Importing various packages
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import SGDRegressor
 
-
    
-We can discretize this equation in the usual way to get
-
-$$
-m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
-$$
-
-
    
-Rearranging this equation, we can rewrite this as
-
-$$
-\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
-$$
+x = 2*np.random.rand(100,1)
+y = 4+3*x+np.random.randn(100,1)
 
+xb = np.c_[np.ones((100,1)), x]
+beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
+print(beta_linreg)
+sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
+sgdreg.fit(x,y.ravel())
+print(sgdreg.intercept_, sgdreg.coef_)
+

    @@ -335,7 +306,7 @@ $$

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    • diff --git a/doc/pub/week39/html/._week39-bs041.html b/doc/pub/week39/html/._week39-bs041.html index d4cfb5787..cf169d4f2 100644 --- a/doc/pub/week39/html/._week39-bs041.html +++ b/doc/pub/week39/html/._week39-bs041.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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'___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
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  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
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  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
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  • Conjugate gradient method
    • -
  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
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  • Standard steepest descent
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  • Gradient method
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  • Steepest descent method
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  • Steepest descent method
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  • Final expressions
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  • Steepest descent example
    • +
  • Conjugate gradient method
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  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -280,60 +257,29 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Momentum parameter

    +

    Gradient descent and Ridge

    -Notice that this equation is identical to previous one if we identify -the position of the particle, \( \mathbf{w} \), with the parameters -\( \boldsymbol{\theta} \). This allows us to identify the momentum -parameter and learning rate with the mass of the particle and the -viscous drag as: - +We have also discussed Ridge regression where the loss function contains a regularized term given by the \( L_2 \) norm of \( \beta \), $$ -\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}. +C_{\text{ridge}}(\beta) = ||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. $$

    -Thus, as the name suggests, the momentum parameter is proportional to -the mass of the particle and effectively provides inertia. -Furthermore, in the large viscosity/small learning rate limit, our -memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \). - -

    -Why is momentum useful? SGD momentum helps the gradient descent -algorithm gain speed in directions with persistent but small gradients -even in the presence of stochasticity, while suppressing oscillations -in high-curvature directions. This becomes especially important in -situations where the landscape is shallow and flat in some directions -and narrow and steep in others. It has been argued that first-order -methods (with appropriate initial conditions) can perform comparable -to more expensive second order methods, especially in the context of -complex deep learning models. - -

    -These beneficial properties of momentum can sometimes become even more -pronounced by using a slight modification of the classical momentum -algorithm called Nesterov Accelerated Gradient (NAG). - -

    -In the NAG algorithm, rather than calculating the gradient at the -current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one -calculates the gradient at the expected value of the parameters given -our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma -\mathbf{v}_{t-1}) \). This yields the NAG update rule - +In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows $$ -\begin{align} -\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\ -\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}. -\tag{3} -\end{align} +\nabla_\beta C_{\text{ridge}}(\beta) = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ +\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ +\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta). $$

    -One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \). +We can easily extend our program to minimize \( C_{\text{ridge}}(\beta) \) using gradient descent and compare with the analytical solution given by +$$ +\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}. +$$

    @@ -361,7 +307,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea

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    • diff --git a/doc/pub/week39/html/._week39-bs042.html b/doc/pub/week39/html/._week39-bs042.html index b86a81c79..4c806df95 100644 --- a/doc/pub/week39/html/._week39-bs042.html +++ b/doc/pub/week39/html/._week39-bs042.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,31 +259,54 @@ MathJax.Hub.Config({ -

    Second moment of the gradient

    - +

    Program example for gradient descent with Ridge Regression

    -In stochastic gradient descent, with and without momentum, we still -have to specify a schedule for tuning the learning rates \( \eta_t \) -as a function of time. As discussed in the context of Newton's -method, this presents a number of dilemmas. The learning rate is -limited by the steepest direction which can change depending on the -current position in the landscape. To circumvent this problem, ideally -our algorithm would keep track of curvature and take large steps in -shallow, flat directions and small steps in steep, narrow directions. -Second-order methods accomplish this by calculating or approximating -the Hessian and normalizing the learning rate by the -curvature. However, this is very computationally expensive for -extremely large models. Ideally, we would like to be able to -adaptively change the step size to match the landscape without paying -the steep computational price of calculating or approximating -Hessians. -

    -Recently, a number of methods have been introduced that accomplish -this by tracking not only the gradient, but also the second moment of -the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and -ADAM. + +

    from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+from matplotlib.ticker import LinearLocator, FormatStrFormatter
+import sys
 
+# the number of datapoints
+m = 100
+x = 2*np.random.rand(m,1)
+y = 4+3*x+np.random.randn(m,1)
+
+xb = np.c_[np.ones((m,1)), x]
+XT_X = xb.T @ xb
+
+#Ridge parameter lambda
+lmbda  = 0.001
+Id = lmbda* np.eye(XT_X.shape[0])
+
+beta_linreg = np.linalg.inv(XT_X+Id) @ xb.T @ y
+print(beta_linreg)
+# Start plain gradient descent
+beta = np.random.randn(2,1)
+
+eta = 0.1
+Niterations = 100
+
+for iter in range(Niterations):
+    gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta
+    beta -= eta*gradients
+
+print(beta)
+ypredict = xb @ beta
+ypredict2 = xb @ beta_linreg
+plt.plot(x, ypredict, "r-")
+plt.plot(x, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Gradient descent example for Ridge')
+plt.show()
+

    @@ -333,7 +333,7 @@ ADAM.

  • 51
  • 52
  • ...
    • -
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    • diff --git a/doc/pub/week39/html/._week39-bs043.html b/doc/pub/week39/html/._week39-bs043.html index 3377e6091..8556d99f0 100644 --- a/doc/pub/week39/html/._week39-bs043.html +++ b/doc/pub/week39/html/._week39-bs043.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,35 +259,17 @@ MathJax.Hub.Config({ -

    RMS prop

    +

    Using gradient descent methods, limitations

    -

    -In RMS prop, in addition to keeping a running average of the first -moment of the gradient, we also keep track of the second moment -denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule -for RMS prop is given by +

    -$$ -\begin{align} -\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) -\tag{4}\\ -\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\ -\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber -\end{align} -$$ - -

    -where \( \beta \) controls the averaging time of the second moment and is -typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate -typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8} \) is a -small regularization constant to prevent divergences. Multiplication -and division by vectors is understood as an element-wise operation. It -is clear from this formula that the learning rate is reduced in -directions where the norm of the gradient is consistently large. This -greatly speeds up the convergence by allowing us to use a larger -learning rate for flat directions. - -

    diff --git a/doc/pub/week39/html/._week39-bs044.html b/doc/pub/week39/html/._week39-bs044.html index f7a7c0eda..14ce0290a 100644 --- a/doc/pub/week39/html/._week39-bs044.html +++ b/doc/pub/week39/html/._week39-bs044.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,51 +259,7 @@ MathJax.Hub.Config({ -

    ADAM optimizer

    - -

    -A related algorithm is the ADAM optimizer. In ADAM, we keep a running -average of both the first and second moment of the gradient and use -this information to adaptively change the learning rate for different -parameters. In addition to keeping a running average of the first and -second moments of the gradient -(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and -\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM -performs an additional bias correction to account for the fact that we -are estimating the first two moments of the gradient using a running -average (denoted by the hats in the update rule below). The update -rule for ADAM is given by (where multiplication and division are once -again understood to be element-wise operations below) - -$$ -\begin{align} -\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) -\tag{5}\\ -\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\ -\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\ -\hat{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\ -\hat{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\ -\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \hat{\mathbf{m}}_t \over \sqrt{\hat{\mathbf{s}}_t} +\epsilon}, \nonumber \\ -\tag{6} -\end{align} -$$ - -

    -where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and -second moment and are typically taken to be \( 0.9 \) and \( 0.99 \) -respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop. - -

    -Like in RMSprop, the effective step size of a parameter depends on the -magnitude of its gradient squared. To understand this better, let us -rewrite this expression in terms of the variance -\( \boldsymbol{\sigma}_t^2 = \hat{\mathbf{s}}_t - -(\hat{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The -update rule for this parameter is given by - -$$ -\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}. -$$ +

    Friday September 25

    @@ -354,7 +287,7 @@ $$

  • 53
  • 54
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs045.html b/doc/pub/week39/html/._week39-bs045.html index 7390fc2de..3e96d5202 100644 --- a/doc/pub/week39/html/._week39-bs045.html +++ b/doc/pub/week39/html/._week39-bs045.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - 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'___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
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  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
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  • And with loops
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  • Using recursion
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  • Unsupported functions
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  • The syntax a.dot(b) when finding the dot product
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  • Recommended to avoid
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  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
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  • Final expressions
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  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Conjugate gradient method
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  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
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  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
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  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,16 +259,20 @@ MathJax.Hub.Config({ -

    Practical tips

    +

    Stochastic Gradient Descent

    - +

    +Stochastic gradient descent (SGD) and variants thereof address some of +the shortcomings of the Gradient descent method discussed above. -Geron's text, see chapter 11, has several interesting discussions. +

    +The underlying idea of SGD comes from the observation that the cost +function, which we want to minimize, can almost always be written as a +sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \), +$$ +C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, +\mathbf{\beta}). +$$

    @@ -319,7 +300,7 @@ Geron's text, see chapter 11, has several interesting discussions.

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    • diff --git a/doc/pub/week39/html/._week39-bs046.html b/doc/pub/week39/html/._week39-bs046.html index 5bda4b981..620dd11be 100644 --- a/doc/pub/week39/html/._week39-bs046.html +++ b/doc/pub/week39/html/._week39-bs046.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,88 +259,23 @@ MathJax.Hub.Config({ -

    Automatic differentiation

    +

    Computation of gradients

    -Automatic differentiation (AD), -also called algorithmic -differentiation or computational differentiation,is a set of -techniques to numerically evaluate the derivative of a function -specified by a computer program. AD exploits the fact that every -computer program, no matter how complicated, executes a sequence of -elementary arithmetic operations (addition, subtraction, -multiplication, division, etc.) and elementary functions (exp, log, -sin, cos, etc.). By applying the chain rule repeatedly to these -operations, derivatives of arbitrary order can be computed -automatically, accurately to working precision, and using at most a -small constant factor more arithmetic operations than the original -program. +This in turn means that the gradient can be +computed as a sum over \( i \)-gradients +$$ +\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}). +$$

    -Automatic differentiation is neither: +Stochasticity/randomness is introduced by only taking the +gradient on a subset of the data called minibatches. If there are \( n \) +data points and the size of each minibatch is \( M \), there will be \( n/M \) +minibatches. We denote these minibatches by \( B_k \) where +\( k=1,\cdots,n/M \). -

    - -Symbolic differentiation can lead to inefficient code and faces the -difficulty of converting a computer program into a single expression, -while numerical differentiation can introduce round-off errors in the -discretization process and cancellation - -

    -Python has tools for so-called automatic differentiation. -Consider the following example -$$ -f(x) = \sin\left(2\pi x + x^2\right) -$$ - -which has the following derivative -$$ -f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) -$$ - -Using autograd we have - -

    - - -

    import autograd.numpy as np
-
-# To do elementwise differentiation:
-from autograd import elementwise_grad as egrad 
-
-# To plot:
-import matplotlib.pyplot as plt 
-
-
-def f(x):
-    return np.sin(2*np.pi*x + x**2)
-
-def f_grad_analytic(x):
-    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)
-
-# Do the comparison:
-x = np.linspace(0,1,1000)
-
-f_grad = egrad(f)
-
-computed = f_grad(x)
-analytic = f_grad_analytic(x)
-
-plt.title('Derivative computed from Autograd compared with the analytical derivative')
-plt.plot(x,computed,label='autograd')
-plt.plot(x,analytic,label='analytic')
-
-plt.xlabel('x')
-plt.ylabel('y')
-plt.legend()
-
-plt.show()
-
-print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))
-

    @@ -390,7 +302,7 @@ plt.show()

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    • diff --git a/doc/pub/week39/html/._week39-bs047.html b/doc/pub/week39/html/._week39-bs047.html index ef9d36bc4..c6c4eabce 100644 --- a/doc/pub/week39/html/._week39-bs047.html +++ b/doc/pub/week39/html/._week39-bs047.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -280,38 +257,29 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Using autograd

    +

    SGD example

    +As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) +and we choose to have \( M=5 \) minibathces, +then each minibatch contains two data points. In particular we have +\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = +(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you +have only a single batch with all data points and on the other extreme, +you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e +\( B_k = \mathbf{x}_k \).

    -Here we -experiment with what kind of functions Autograd is capable -of finding the gradient of. The following Python functions are just -meant to illustrate what Autograd can do, but please feel free to -experiment with other, possibly more complicated, functions as well. +The idea is now to approximate the gradient by replacing the sum over +all data points with a sum over the data points in one the minibatches +picked at random in each gradient descent step +$$ +\nabla_{\beta} +C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta +c_i(\mathbf{x}_i, \mathbf{\beta}). +$$ -

    - - -

    import autograd.numpy as np
-from autograd import grad
-
-def f1(x):
-    return x**3 + 1
-
-f1_grad = grad(f1)
-
-# Remember to send in float as argument to the computed gradient from Autograd!
-a = 1.0
-
-# See the evaluated gradient at a using autograd:
-print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
-
-# Compare with the analytical derivative, that is f1'(x) = 3*x**2 
-grad_analytical = 3*a**2
-print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
-

    @@ -338,7 +306,7 @@ grad_analytical = 56

  • 57
  • ...
    • -
  • 76
    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs048.html b/doc/pub/week39/html/._week39-bs048.html index 02361db35..7eb896ff1 100644 --- a/doc/pub/week39/html/._week39-bs048.html +++ b/doc/pub/week39/html/._week39-bs048.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,52 +259,21 @@ MathJax.Hub.Config({ -

    Autograd with more complicated functions

    +

    The gradient step

    -To differentiate with respect to two (or more) arguments of a Python -function, Autograd need to know at which variable the function if -being differentiated with respect to. +Thus a gradient descent step now looks like +$$ +\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) +$$

    - - -

    import autograd.numpy as np
-from autograd import grad
-def f2(x1,x2):
-    return 3*x1**3 + x2*(x1 - 5) + 1
-
-# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
-f2_grad_x1 = grad(f2,0)
-
-# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
-f2_grad_x2 = grad(f2,1)
-
-x1 = 1.0
-x2 = 3.0 
-
-print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
-print("-"*30)
-
-# Compare with the analytical derivatives:
-
-# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
-f2_grad_x1_analytical = 9*x1**2 + x2
-
-# Derivative of f2 w.r.t x2 is: x1 - 5:
-f2_grad_x2_analytical = x1 - 5
-
-# See the evaluated derivations:
-print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-
-print()
-
-print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-
    -

    -Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. +where \( k \) is picked at random with equal +probability from \( [1,n/M] \). An iteration over the number of +minibathces (n/M) is commonly referred to as an epoch. Thus it is +typical to choose a number of epochs and for each epoch iterate over +the number of minibatches, as exemplified in the code below.

    @@ -355,7 +301,7 @@ Note that the grad function will not produce the true gradient of the function.

  • 57
  • 58
  • ...
    • -
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    • +
  • 71
  • »
    • diff --git a/doc/pub/week39/html/._week39-bs049.html b/doc/pub/week39/html/._week39-bs049.html index c16a02e3f..d55e02a73 100644 --- a/doc/pub/week39/html/._week39-bs049.html +++ b/doc/pub/week39/html/._week39-bs049.html @@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source ('Conditions on convex functions', 2, None, '___sec16'), ('More on convex functions', 2, None, '___sec17'), ('Some simple problems', 2, None, '___sec18'), - ('Revisiting our first homework', 2, None, '___sec19'), - ('Gradient descent example', 2, None, '___sec20'), - ('The derivative of the cost/loss function', 2, None, '___sec21'), - ('The Hessian matrix', 2, None, '___sec22'), - ('Simple program', 2, None, '___sec23'), - ('Gradient Descent Example', 2, None, '___sec24'), + ('Standard steepest descent', 2, None, '___sec19'), + ('Gradient method', 2, None, '___sec20'), + ('Steepest descent method', 2, None, '___sec21'), + ('Steepest descent method', 2, None, '___sec22'), + ('Final expressions', 2, None, '___sec23'), + ('Steepest descent example', 2, None, '___sec24'), + ('Conjugate gradient method', 2, None, '___sec25'), + ('Conjugate gradient method', 2, None, '___sec26'), + ('Conjugate gradient method', 2, None, '___sec27'), + ('Conjugate gradient method', 2, None, '___sec28'), + ('Conjugate gradient method and iterations', 2, None, '___sec29'), + ('Conjugate gradient method', 2, None, '___sec30'), + ('Conjugate gradient method', 2, None, '___sec31'), + ('Conjugate gradient method', 2, None, '___sec32'), + ('Revisiting our first homework', 2, None, '___sec33'), + ('Gradient descent example', 2, None, '___sec34'), + ('The derivative of the cost/loss function', 2, None, '___sec35'), + ('The Hessian matrix', 2, None, '___sec36'), + ('Simple program', 2, None, '___sec37'), + ('Gradient Descent Example', 2, None, '___sec38'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec25'), - ('Gradient descent and Ridge', 2, None, '___sec26'), + '___sec39'), + ('Gradient descent and Ridge', 2, None, '___sec40'), ('Program example for gradient descent with Ridge Regression', 2, None, - '___sec27'), + '___sec41'), ('Using gradient descent methods, limitations', 2, None, - '___sec28'), - ('Friday September 25', 2, None, '___sec29'), - ('Stochastic Gradient Descent', 2, None, '___sec30'), - ('Computation of gradients', 2, None, '___sec31'), - ('SGD example', 2, None, '___sec32'), - ('The gradient step', 2, None, '___sec33'), - ('Simple example code', 2, None, '___sec34'), - ('When do we stop?', 2, None, '___sec35'), - ('Slightly different approach', 2, None, '___sec36'), - ('Program for stochastic gradient', 2, None, '___sec37'), - ('Momentum based GD', 2, None, '___sec38'), - ('More on momentum based approaches', 2, None, '___sec39'), - ('Momentum parameter', 2, None, '___sec40'), - ('Second moment of the gradient', 2, None, '___sec41'), - ('RMS prop', 2, None, '___sec42'), - ('ADAM optimizer', 2, None, '___sec43'), - ('Practical tips', 2, None, '___sec44'), - ('Automatic differentiation', 2, None, '___sec45'), - ('Using autograd', 2, None, '___sec46'), - ('Autograd with more complicated functions', 2, None, '___sec47'), + '___sec42'), + ('Friday September 25', 2, None, '___sec43'), + ('Stochastic Gradient Descent', 2, None, '___sec44'), + ('Computation of gradients', 2, None, '___sec45'), + ('SGD example', 2, None, '___sec46'), + ('The gradient step', 2, None, '___sec47'), + ('Simple example code', 2, None, '___sec48'), + ('When do we stop?', 2, None, '___sec49'), + ('Slightly different approach', 2, None, '___sec50'), + ('Program for stochastic gradient', 2, None, '___sec51'), + ('Momentum based GD', 2, None, '___sec52'), + ('More on momentum based approaches', 2, None, '___sec53'), + ('Momentum parameter', 2, None, '___sec54'), + ('Second moment of the gradient', 2, None, '___sec55'), + ('RMS prop', 2, None, '___sec56'), + ('ADAM optimizer', 2, None, '___sec57'), + ('Practical tips', 2, None, '___sec58'), + ('Automatic differentiation', 2, None, '___sec59'), + ('Using autograd', 2, None, '___sec60'), + ('Autograd with more complicated functions', 2, None, '___sec61'), ('More complicated functions using the elements of their ' 'arguments directly', 2, None, - '___sec48'), + '___sec62'), ('Functions using mathematical functions from Numpy', 2, None, - '___sec49'), - ('More autograd', 2, None, '___sec50'), - ('And with loops', 2, None, '___sec51'), - ('Using recursion', 2, None, '___sec52'), - ('Unsupported functions', 2, None, '___sec53'), + '___sec63'), + ('More autograd', 2, None, '___sec64'), + ('And with loops', 2, None, '___sec65'), + ('Using recursion', 2, None, '___sec66'), + ('Unsupported functions', 2, None, '___sec67'), ('The syntax a.dot(b) when finding the dot product', 2, None, - '___sec54'), - ('Recommended to avoid', 2, None, '___sec55'), - ('Standard steepest descent', 2, None, '___sec56'), - ('Gradient method', 2, None, '___sec57'), - ('Steepest descent method', 2, None, '___sec58'), - ('Steepest descent method', 2, None, '___sec59'), - ('Final expressions', 2, None, '___sec60'), - ('Code examples for steepest descent', 2, None, '___sec61'), - ('Simple codes for steepest descent and conjugate gradient ' - 'using a $2\\times 2$ matrix, in c++, Python code to come', - 2, - None, - '___sec62'), - ('The routine for the steepest descent method', - 2, - None, - '___sec63'), - ('Steepest descent example', 2, None, '___sec64'), - ('Conjugate gradient method', 2, None, '___sec65'), - ('Conjugate gradient method', 2, None, '___sec66'), - ('Conjugate gradient method', 2, None, '___sec67'), - ('Conjugate gradient method', 2, None, '___sec68'), - ('Conjugate gradient method and iterations', 2, None, '___sec69'), - ('Conjugate gradient method', 2, None, '___sec70'), - ('Conjugate gradient method', 2, None, '___sec71'), - ('Conjugate gradient method', 2, None, '___sec72'), - ('Simple implementation of the Conjugate gradient algorithm', - 2, - None, - '___sec73'), - ('Broyden–Fletcher–Goldfarb–Shanno algorithm', - 2, - None, - '___sec74')]} + '___sec68'), + ('Recommended to avoid', 2, None, '___sec69')]} end of tocinfo --> @@ -211,62 +193,57 @@ MathJax.Hub.Config({
  • Conditions on convex functions
  • More on convex functions
  • Some simple problems
    • -
  • Revisiting our first homework
    • -
  • Gradient descent example
    • -
  • The derivative of the cost/loss function
    • -
  • The Hessian matrix
    • -
  • Simple program
    • -
  • Gradient Descent Example
    • -
  • And a corresponding example using scikit-learn
    • -
  • Gradient descent and Ridge
    • -
  • Program example for gradient descent with Ridge Regression
    • -
  • Using gradient descent methods, limitations
    • -
  • Friday September 25
    • -
  • Stochastic Gradient Descent
    • -
  • Computation of gradients
    • -
  • SGD example
    • -
  • The gradient step
    • -
  • Simple example code
    • -
  • When do we stop?
    • -
  • Slightly different approach
    • -
  • Program for stochastic gradient
    • -
  • Momentum based GD
    • -
  • More on momentum based approaches
    • -
  • Momentum parameter
    • -
  • Second moment of the gradient
    • -
  • RMS prop
    • -
  • ADAM optimizer
    • -
  • Practical tips
    • -
  • Automatic differentiation
    • -
  • Using autograd
    • -
  • Autograd with more complicated functions
    • -
  • More complicated functions using the elements of their arguments directly
    • -
  • Functions using mathematical functions from Numpy
    • -
  • More autograd
    • -
  • And with loops
    • -
  • Using recursion
    • -
  • Unsupported functions
    • -
  • The syntax a.dot(b) when finding the dot product
    • -
  • Recommended to avoid
    • -
  • Standard steepest descent
    • -
  • Gradient method
    • -
  • Steepest descent method
    • -
  • Steepest descent method
    • -
  • Final expressions
    • -
  • Code examples for steepest descent
    • -
  • Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • -
  • The routine for the steepest descent method
    • -
  • Steepest descent example
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method and iterations
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Conjugate gradient method
    • -
  • Simple implementation of the Conjugate gradient algorithm
    • -
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • +
  • Standard steepest descent
    • +
  • Gradient method
    • +
  • Steepest descent method
    • +
  • Steepest descent method
    • +
  • Final expressions
    • +
  • Steepest descent example
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method and iterations
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Conjugate gradient method
    • +
  • Revisiting our first homework
    • +
  • Gradient descent example
    • +
  • The derivative of the cost/loss function
    • +
  • The Hessian matrix
    • +
  • Simple program
    • +
  • Gradient Descent Example
    • +
  • And a corresponding example using scikit-learn
    • +
  • Gradient descent and Ridge
    • +
  • Program example for gradient descent with Ridge Regression
    • +
  • Using gradient descent methods, limitations
    • +
  • Friday September 25
    • +
  • Stochastic Gradient Descent
    • +
  • Computation of gradients
    • +
  • SGD example
    • +
  • The gradient step
    • +
  • Simple example code
    • +
  • When do we stop?
    • +
  • Slightly different approach
    • +
  • Program for stochastic gradient
    • +
  • Momentum based GD
    • +
  • More on momentum based approaches
    • +
  • Momentum parameter
    • +
  • Second moment of the gradient
    • +
  • RMS prop
    • +
  • ADAM optimizer
    • +
  • Practical tips
    • +
  • Automatic differentiation
    • +
  • Using autograd
    • +
  • Autograd with more complicated functions
    • +
  • More complicated functions using the elements of their arguments directly
    • +
  • Functions using mathematical functions from Numpy
    • +
  • More autograd
    • +
  • And with loops
    • +
  • Using recursion
    • +
  • Unsupported functions
    • +
  • The syntax a.dot(b) when finding the dot product
    • +
  • Recommended to avoid
    • @@ -282,36 +259,34 @@ MathJax.Hub.Config({ -

    More complicated functions using the elements of their arguments directly

    +

    Simple example code

    -

    import autograd.numpy as np
-from autograd import grad
-def f3(x): # Assumes x is an array of length 5 or higher
-    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
+
    import numpy as np 
 
-f3_grad = grad(f3)
+n = 100 #100 datapoints 
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+n_epochs = 10 #number of epochs
 
-x = np.linspace(0,4,5)
-
-# Print the computed gradient:
-print("The computed gradient of f3 is: ", f3_grad(x))
-
-# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
-f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
-
-# Print the analytical gradient:
-print("The analytical gradient of f3 is: ", f3_grad_analytical)
+j = 0
+for epoch in range(1,n_epochs+1):
+    for i in range(m):
+        k = np.random.randint(m) #Pick the k-th minibatch at random
+        #Compute the gradient using the data in minibatch Bk
+        #Compute new suggestion for 
+        j += 1
 
    
 
    
-Note that in this case, when sending an array as input argument, the
-output from Autograd is another array. This is the true gradient of
-the function, as opposed to the function in the previous example. By
-using arrays to represent the variables, the output from Autograd
-might be easier to work with, as the output is closer to what one
-could expect form a gradient-evaluting function.
+Taking the gradient only on a subset of the data has two important
+benefits. First, it introduces randomness which decreases the chance
+that our opmization scheme gets stuck in a local minima. Second, if
+the size of the minibatches are small relative to the number of
+datapoints (\( M <  n \)), the computation of the gradient is much
+cheaper since we sum over the datapoints in the \( k-th \) minibatch and not
+all \( n \) datapoints.
 
 
    
 
    
@@ -339,7 +314,7 @@ could expect form a gradient-evaluting function.
   
  • 58
    • 
   
  • 59
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs050.html b/doc/pub/week39/html/._week39-bs050.html
index b464865be..c034781f0 100644
--- a/doc/pub/week39/html/._week39-bs050.html
+++ b/doc/pub/week39/html/._week39-bs050.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -280,31 +257,22 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Functions using mathematical functions from Numpy 
    
+
    When do we stop? 
    
 
 
    
+A natural question is when do we stop the search for a new minimum?
+One possibility is to compute the full gradient after a given number
+of epochs and check if the norm of the gradient is smaller than some
+threshold and stop if true. However, the condition that the gradient
+is zero is valid also for local minima, so this would only tell us
+that we are close to a local/global minimum. However, we could also
+evaluate the cost function at this point, store the result and
+continue the search. If the test kicks in at a later stage we can
+compare the values of the cost function and keep the \( \beta \) that
+gave the lowest value.
 
-
-
    import autograd.numpy as np
-from autograd import grad
-def f4(x):
-    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
-
-f4_grad = grad(f4)
-
-x = 2.7
-
-# Print the computed derivative:
-print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
-
-# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
-f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
-
-# Print the analytical gradient:
-print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
-
    
 
    
 
    
 
@@ -331,7 +299,7 @@ f4_grad_analytical = x59
   
  • 60
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs051.html b/doc/pub/week39/html/._week39-bs051.html
index 9ce938862..525d2670e 100644
--- a/doc/pub/week39/html/._week39-bs051.html
+++ b/doc/pub/week39/html/._week39-bs051.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,25 +259,50 @@ MathJax.Hub.Config({
 
 
 
-
    More autograd 
    
+
    Slightly different approach 
    
+
+
    
+Another approach is to let the step length \( \gamma_j \) depend on the
+number of epochs in such a way that it becomes very small after a
+reasonable time such that we do not move at all.
+
+
    
+As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in time \( t \).
+
+
    
+In this way we can fix the number of epochs, compute \( \beta \) and
+evaluate the cost function at the end. Repeating the computation will
+give a different result since the scheme is random by design. Then we
+pick the final \( \beta \) that gives the lowest value of the cost
+function.
 
 
    
 
 
-
    import autograd.numpy as np
-from autograd import grad
-def f5(x):
-    if x >= 0:
-        return x**2
-    else:
-        return -3*x + 1
+
    import numpy as np 
 
-f5_grad = grad(f5)
+def step_length(t,t0,t1):
+    return t0/(t+t1)
 
-x = 2.7
+n = 100 #100 datapoints 
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+n_epochs = 500 #number of epochs
+t0 = 1.0
+t1 = 10
 
-# Print the computed derivative:
-print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
+gamma_j = t0/t1
+j = 0
+for epoch in range(1,n_epochs+1):
+    for i in range(m):
+        k = np.random.randint(m) #Pick the k-th minibatch at random
+        #Compute the gradient using the data in minibatch Bk
+        #Compute new suggestion for beta
+        t = epoch*m+i
+        gamma_j = step_length(t,t0,t1)
+        j += 1
+
+print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
 
    
 
    
 
    
@@ -328,7 +330,7 @@ x = 2.7
   
  • 60
    • 
   
  • 61
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs052.html b/doc/pub/week39/html/._week39-bs052.html
index a2953fc82..0712f84da 100644
--- a/doc/pub/week39/html/._week39-bs052.html
+++ b/doc/pub/week39/html/._week39-bs052.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,49 +259,79 @@ MathJax.Hub.Config({
 
 
 
-
    And  with loops 
    
+
    Program for stochastic gradient 
    
 
 
    
 
 
-
    import autograd.numpy as np
-from autograd import grad
-def f6_for(x):
-    val = 0
-    for i in range(10):
-        val = val + x**i
-    return val
+
    # Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import SGDRegressor
 
-def f6_while(x):
-    val = 0
-    i = 0
-    while i < 10:
-        val = val + x**i
-        i = i + 1
-    return val
+m = 100
+x = 2*np.random.rand(m,1)
+y = 4+3*x+np.random.randn(m,1)
 
-f6_for_grad = grad(f6_for)
-f6_while_grad = grad(f6_while)
+xb = np.c_[np.ones((m,1)), x]
+theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
+print("Own inversion")
+print(theta_linreg)
+sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
+sgdreg.fit(x,y.ravel())
+print("sgdreg from scikit")
+print(sgdreg.intercept_, sgdreg.coef_)
 
-x = 0.5
 
-# Print the computed derivaties of f6_for and f6_while
-print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
-print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+theta = np.random.randn(2,1)
+eta = 0.1
+Niterations = 1000
+
+
+for iter in range(Niterations):
+    gradients = 2.0/m*xb.T @ ((xb @ theta)-y)
+    theta -= eta*gradients
+print("theta frm own gd")
+print(theta)
+
+xnew = np.array([[0],[2]])
+xbnew = np.c_[np.ones((2,1)), xnew]
+ypredict = xbnew.dot(theta)
+ypredict2 = xbnew.dot(theta_linreg)
+
+
+n_epochs = 50
+t0, t1 = 5, 50
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+    for i in range(m):
+        random_index = np.random.randint(m)
+        xi = xb[random_index:random_index+1]
+        yi = y[random_index:random_index+1]
+        gradients = 2 * xi.T @ ((xi @ theta)-yi)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print("theta from own sdg")
+print(theta)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Random numbers ')
+plt.show()
 
    
 
    
+Challenge: try to write a similar code for a Logistic Regression case.
 
-
-
    import autograd.numpy as np
-from autograd import grad
-# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9
-# The analytical derivative is: sum(i*x**(i-1)) 
-f6_grad_analytical = 0
-for i in range(10):
-    f6_grad_analytical += i*x**(i-1)
-
-print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
-
    
 
    
 
    
 
@@ -351,7 +358,7 @@ f6_grad_analytical = 61
   
  • 62
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs053.html b/doc/pub/week39/html/._week39-bs053.html
index f899539ed..d66241573 100644
--- a/doc/pub/week39/html/._week39-bs053.html
+++ b/doc/pub/week39/html/._week39-bs053.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,40 +259,39 @@ MathJax.Hub.Config({
 
 
 
-
    Using recursion 
    
+
    Momentum based GD 
    
+
 
    
+The stochastic gradient descent (SGD) is almost always used with a
+momentum or inertia term that serves as a memory of the direction we
+are moving in parameter space.  This is typically implemented as
+follows
 
-
-
    import autograd.numpy as np
-from autograd import grad
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\tag{2}
+\end{align}
+$$
 
-def f7(n): # Assume that n is an integer
-    if n == 1 or n == 0:
-        return 1
-    else:
-        return n*f7(n-1)
-
-f7_grad = grad(f7)
-
-n = 2.0
-
-print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
-
-# The function f7 is an implementation of the factorial of n.
-# By using the product rule, one can find that the derivative is:
-
-f7_grad_analytical = 0
-for i in range(int(n)-1):
-    tmp = 1
-    for k in range(int(n)-1):
-        if k != i:
-            tmp *= (n - k)
-    f7_grad_analytical += tmp
-
-print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
-
    
 
    
-Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
+where we have introduced a momentum parameter \( \gamma \), with
+\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to
+indicate the gradient is to be taken over a different mini-batch at
+each step. We call this algorithm gradient descent with momentum
+(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a
+running average of recently encountered gradients and
+\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory
+used in the averaging procedure. Consistent with this, when
+\( \gamma=0 \), this just reduces down to ordinary SGD as discussed
+earlier. An equivalent way of writing the updates is
+
+$$
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
+$$
+
+where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).
 
 
    
 
    
@@ -343,7 +319,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
   
  • 62
    • 
   
  • 63
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs054.html b/doc/pub/week39/html/._week39-bs054.html
index aea05d9ec..c5af9ae1a 100644
--- a/doc/pub/week39/html/._week39-bs054.html
+++ b/doc/pub/week39/html/._week39-bs054.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,28 +259,32 @@ MathJax.Hub.Config({
 
 
 
-
    Unsupported functions 
    
-Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
+
    More on momentum based approaches 
    
 
 
    
-Assigning a value to the variable being differentiated with respect to
+Let us try to get more intuition from these equations. It is helpful
+to consider a simple physical analogy with a particle of mass \( m \)
+moving in a viscous medium with drag coefficient \( \mu \) and potential
+\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \),
+then its motion is described by
+
+$$
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+$$
+
 
    
+We can discretize this equation in the usual way to get
 
-
-
    import autograd.numpy as np
-from autograd import grad
-def f8(x): # Assume x is an array
-    x[2] = 3
-    return x*2
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
 
-f8_grad = grad(f8)
-
-x = 8.4
-
-print("The derivative of f8 is:",f8_grad(x))
-
    
 
    
-Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
+Rearranging this equation, we can rewrite this as
+
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
 
 
    
 
    
@@ -331,7 +312,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
   
  • 63
    • 
   
  • 64
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs055.html b/doc/pub/week39/html/._week39-bs055.html
index 247d23d72..86920f8b9 100644
--- a/doc/pub/week39/html/._week39-bs055.html
+++ b/doc/pub/week39/html/._week39-bs055.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,45 +259,59 @@ MathJax.Hub.Config({
 
 
 
-
    The syntax a.dot(b) when finding the dot product 
    
-
    
-
-
-
    import autograd.numpy as np
-from autograd import grad
-def f9(a): # Assume a is an array with 2 elements
-    b = np.array([1.0,2.0])
-    return a.dot(b)
-
-f9_grad = grad(f9)
-
-x = np.array([1.0,0.0])
-
-print("The derivative of f9 is:",f9_grad(x))
-
    
-
    
-Here we are told that the 'dot' function does not belong to Autograd's
-version of a Numpy array.  To overcome this, an alternative syntax
-which also computed the dot product can be used:
+
    Momentum parameter 
    
 
 
    
+Notice that this equation is identical to previous one if we identify
+the position of the particle, \( \mathbf{w} \), with the parameters
+\( \boldsymbol{\theta} \). This allows us to identify the momentum
+parameter and learning rate with the mass of the particle and the
+viscous drag as:
 
-
-
    import autograd.numpy as np
-from autograd import grad
-def f9_alternative(x): # Assume a is an array with 2 elements
-    b = np.array([1.0,2.0])
-    return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
 
-f9_alternative_grad = grad(f9_alternative)
+
    
+Thus, as the name suggests, the momentum parameter is proportional to
+the mass of the particle and effectively provides inertia.
+Furthermore, in the large viscosity/small learning rate limit, our
+memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
 
-x = np.array([3.0,0.0])
+
    
+Why is momentum useful? SGD momentum helps the gradient descent
+algorithm gain speed in directions with persistent but small gradients
+even in the presence of stochasticity, while suppressing oscillations
+in high-curvature directions. This becomes especially important in
+situations where the landscape is shallow and flat in some directions
+and narrow and steep in others. It has been argued that first-order
+methods (with appropriate initial conditions) can perform comparable
+to more expensive second order methods, especially in the context of
+complex deep learning models.
 
-print("The gradient of f9 is:",f9_alternative_grad(x))
+
    
+These beneficial properties of momentum can sometimes become even more
+pronounced by using a slight modification of the classical momentum
+algorithm called Nesterov Accelerated Gradient (NAG).
+
+
    
+In the NAG algorithm, rather than calculating the gradient at the
+current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one
+calculates the gradient at the expected value of the parameters given
+our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma
+\mathbf{v}_{t-1}) \). This yields the NAG update rule
+
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\tag{3}
+\end{align}
+$$
+
+
    
+One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \).
 
-# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively
-# w.r.t x is (b_1, b_2).
-
    
 
    
 
    
 
@@ -347,7 +338,7 @@ x = np.a
   
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    • 
   
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    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs056.html b/doc/pub/week39/html/._week39-bs056.html
index 41f7c3c63..1603701ca 100644
--- a/doc/pub/week39/html/._week39-bs056.html
+++ b/doc/pub/week39/html/._week39-bs056.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,16 +259,31 @@ MathJax.Hub.Config({
 
 
 
-
    Recommended to avoid 
    
-The documentation recommends to avoid inplace operations such as
-
    
+
    Second moment of the gradient 
    
+
+
    
+In stochastic gradient descent, with and without momentum, we still
+have to specify a schedule for tuning the learning rates \( \eta_t \)
+as a function of time.  As discussed in the context of Newton's
+method, this presents a number of dilemmas. The learning rate is
+limited by the steepest direction which can change depending on the
+current position in the landscape. To circumvent this problem, ideally
+our algorithm would keep track of curvature and take large steps in
+shallow, flat directions and small steps in steep, narrow directions.
+Second-order methods accomplish this by calculating or approximating
+the Hessian and normalizing the learning rate by the
+curvature. However, this is very computationally expensive for
+extremely large models. Ideally, we would like to be able to
+adaptively change the step size to match the landscape without paying
+the steep computational price of calculating or approximating
+Hessians.
+
+
    
+Recently, a number of methods have been introduced that accomplish
+this by tracking not only the gradient, but also the second moment of
+the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and
+ADAM.
 
-
-
    a += b
-a -= b
-a*= b
-a /=b
-
    
 
    
 
    
 
@@ -318,7 +310,7 @@ a /=b
   
  • 65
    • 
   
  • 66
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs057.html b/doc/pub/week39/html/._week39-bs057.html
index f73ca1f77..e5787dc0e 100644
--- a/doc/pub/week39/html/._week39-bs057.html
+++ b/doc/pub/week39/html/._week39-bs057.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,38 +259,33 @@ MathJax.Hub.Config({
 
 
 
-
    Standard steepest descent 
    
+
    RMS prop 
    
 
 
    
-Before we proceed, we would like to discuss the approach called the
-standard Steepest descent, which again leads to us having to be able
-to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+In RMS prop, in addition to keeping a running average of the first
+moment of the gradient, we also keep track of the second moment
+denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule
+for RMS prop is given by
 
-
    
-The success of the CG method
-for finding solutions of non-linear problems is based on the theory
-of conjugate gradients for linear systems of equations. It belongs to
-the class of iterative methods for solving problems from linear
-algebra of the type 
 $$
-\begin{equation*} 
-\hat{A}\hat{x} = \hat{b}.
-\end{equation*} 
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\tag{4}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
 $$
 
 
    
-In the iterative process we end up with a problem like
-
-$$
-\begin{equation*}
-  \hat{r}= \hat{b}-\hat{A}\hat{x},
-\end{equation*}
-$$
-
-where \( \hat{r} \) is the so-called residual or error in the iterative process.
-
-
    
-When we have found the exact solution, \( \hat{r}=0 \).
+where \( \beta \) controls the averaging time of the second moment and is
+typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate
+typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8}  \) is a
+small regularization constant to prevent divergences. Multiplication
+and division by vectors is understood as an element-wise operation. It
+is clear from this formula that the learning rate is reduced in
+directions where the norm of the gradient is consistently large. This
+greatly speeds up the convergence by allowing us to use a larger
+learning rate for flat directions.
 
 
    
 
    
@@ -341,7 +313,7 @@ When we have found the exact solution, \( \hat{r}=0 \).
   
  • 66
    • 
   
  • 67
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs058.html b/doc/pub/week39/html/._week39-bs058.html
index c2a266507..d3c59b373 100644
--- a/doc/pub/week39/html/._week39-bs058.html
+++ b/doc/pub/week39/html/._week39-bs058.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,19 +259,51 @@ MathJax.Hub.Config({
 
 
 
-
    Gradient method 
    
+
    ADAM optimizer 
    
 
 
    
-The residual is zero when we reach the minimum of the quadratic equation
+A related algorithm is the ADAM optimizer. In ADAM, we keep a running
+average of both the first and second moment of the gradient and use
+this information to adaptively change the learning rate for different
+parameters. In addition to keeping a running average of the first and
+second moments of the gradient
+(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and
+\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM
+performs an additional bias correction to account for the fact that we
+are estimating the first two moments of the gradient using a running
+average (denoted by the hats in the update rule below). The update
+rule for ADAM is given by (where multiplication and division are once
+again understood to be element-wise operations below)
+
 $$
-\begin{equation*}
-  P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
-\end{equation*}
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) 
+\tag{5}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\hat{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\hat{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \hat{\mathbf{m}}_t \over \sqrt{\hat{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\tag{6}
+\end{align}
 $$
 
 
    
-with the constraint that the matrix \( \hat{A} \) is positive definite and
-symmetric.  This defines also the Hessian and we want it to be  positive definite.
+where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and
+second moment and are typically taken to be \( 0.9 \) and \( 0.99 \)
+respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop.
+
+
    
+Like in RMSprop, the effective step size of a parameter depends on the
+magnitude of its gradient squared.  To understand this better, let us
+rewrite this expression in terms of the variance
+\( \boldsymbol{\sigma}_t^2 = \hat{\mathbf{s}}_t -
+(\hat{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The
+update rule for this parameter is given by
+
+$$
+\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 +  m_t^2 }+\epsilon}.
+$$
 
 
    
 
    
@@ -322,7 +331,7 @@ symmetric.  This defines also the Hessian and we want it to be  positive definit
   
  • 67
    • 
   
  • 68
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs059.html b/doc/pub/week39/html/._week39-bs059.html
index 25240201f..6ab9e296c 100644
--- a/doc/pub/week39/html/._week39-bs059.html
+++ b/doc/pub/week39/html/._week39-bs059.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,25 +259,16 @@ MathJax.Hub.Config({
 
 
 
-
    Steepest descent  method 
    
+
    Practical tips 
    
 
-
    
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
+
      
+
    •  Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
      • 
+
    •  Transform your inputs. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the energy matrix is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.
      • 
+
    •  Monitor the out-of-sample performance. Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This early stopping significantly improves performance in many settings.
      • 
+
    •  Adaptive optimization methods don't always have good generalization. Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.
      • 
+
    
 
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
+Geron's text, see chapter 11, has several interesting discussions.
 
 
    
 
    
@@ -328,7 +296,7 @@ instead.
   
  • 68
    • 
   
  • 69
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs060.html b/doc/pub/week39/html/._week39-bs060.html
index 77f11031b..f19327a9b 100644
--- a/doc/pub/week39/html/._week39-bs060.html
+++ b/doc/pub/week39/html/._week39-bs060.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,34 +259,88 @@ MathJax.Hub.Config({
 
 
 
-
    Steepest descent  method 
    
-
    
-
    
-
     
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
-  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) 
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and 
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
    Automatic differentiation 
    
 
 
    
-
    
-
    
+Automatic differentiation (AD), 
+also called algorithmic
+differentiation or computational differentiation,is a set of
+techniques to numerically evaluate the derivative of a function
+specified by a computer program. AD exploits the fact that every
+computer program, no matter how complicated, executes a sequence of
+elementary arithmetic operations (addition, subtraction,
+multiplication, division, etc.) and elementary functions (exp, log,
+sin, cos, etc.). By applying the chain rule repeatedly to these
+operations, derivatives of arbitrary order can be computed
+automatically, accurately to working precision, and using at most a
+small constant factor more arithmetic operations than the original
+program.
+
+
    
+Automatic differentiation is neither:
+
+
      
+
    •  Symbolic differentiation, nor
      • 
+
    •  Numerical differentiation (the method of finite differences).
      • 
+
    
+
+Symbolic differentiation can lead to inefficient code and faces the
+difficulty of converting a computer program into a single expression,
+while numerical differentiation can introduce round-off errors in the
+discretization process and cancellation
+
+
    
+Python has tools for so-called automatic differentiation.
+Consider the following example
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
+
+which has the following derivative
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) 
+$$
+
+Using autograd we have
+
+
    
+
+
+
    import autograd.numpy as np
+
+# To do elementwise differentiation:
+from autograd import elementwise_grad as egrad 
+
+# To plot:
+import matplotlib.pyplot as plt 
 
 
+def f(x):
+    return np.sin(2*np.pi*x + x**2)
+
+def f_grad_analytic(x):
+    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)
+
+# Do the comparison:
+x = np.linspace(0,1,1000)
+
+f_grad = egrad(f)
+
+computed = f_grad(x)
+analytic = f_grad_analytic(x)
+
+plt.title('Derivative computed from Autograd compared with the analytical derivative')
+plt.plot(x,computed,label='autograd')
+plt.plot(x,analytic,label='analytic')
+
+plt.xlabel('x')
+plt.ylabel('y')
+plt.legend()
+
+plt.show()
+
+print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))
+
    
 
    
 
    
 
@@ -336,7 +367,7 @@ and
   
  • 69
    • 
   
  • 70
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs061.html b/doc/pub/week39/html/._week39-bs061.html
index a22244a09..4e319bd19 100644
--- a/doc/pub/week39/html/._week39-bs061.html
+++ b/doc/pub/week39/html/._week39-bs061.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -280,49 +257,38 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Final expressions 
    
-
    
-
    
-
     
-We can compute the residual iteratively as
-$$
-\begin{equation*}
-\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
- \end{equation*}
-$$
+
    Using autograd 
    
 
-which equals
-$$
-\begin{equation*}
-\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k),
- \end{equation*}
-$$
+
    
+Here we
+experiment with what kind of functions Autograd is capable
+of finding the gradient of. The following Python functions are just
+meant to illustrate what Autograd can do, but please feel free to
+experiment with other, possibly more complicated, functions as well.
 
-or
-$$
-\begin{equation*}
-(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k,
- \end{equation*}
-$$
+
    
 
-which gives
+
+
    import autograd.numpy as np
+from autograd import grad
 
-$$
-\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k}
-$$
+def f1(x):
+    return x**3 + 1
 
-leading to the iterative scheme
-$$
-\begin{equation*}
-\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k},
- \end{equation*}
-$$
-
    
-
    
+f1_grad = grad(f1)
 
+# Remember to send in float as argument to the computed gradient from Autograd!
+a = 1.0
 
+# See the evaluated gradient at a using autograd:
+print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
+
+# Compare with the analytical derivative, that is f1'(x) = 3*x**2 
+grad_analytical = 3*a**2
+print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
+
    
 
    
 
    
 
@@ -348,8 +314,6 @@ $$
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • ...
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs062.html b/doc/pub/week39/html/._week39-bs062.html
index 645ec4e3d..a308f130b 100644
--- a/doc/pub/week39/html/._week39-bs062.html
+++ b/doc/pub/week39/html/._week39-bs062.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,7 +259,52 @@ MathJax.Hub.Config({
 
 
 
-
    Code examples for steepest descent 
    
+
    Autograd with more complicated functions 
    
+
+
    
+To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+
+
    
+
+
+
    import autograd.numpy as np
+from autograd import grad
+def f2(x1,x2):
+    return 3*x1**3 + x2*(x1 - 5) + 1
+
+# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
+f2_grad_x1 = grad(f2,0)
+
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
+
+x1 = 1.0
+x2 = 3.0 
+
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
+
+# Compare with the analytical derivatives:
+
+# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
+f2_grad_x1_analytical = 9*x1**2 + x2
+
+# Derivative of f2 w.r.t x2 is: x1 - 5:
+f2_grad_x2_analytical = x1 - 5
+
+# See the evaluated derivations:
+print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+
+print()
+
+print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+
    
+
    
+Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
 
 
    
 
    
@@ -308,9 +330,6 @@ MathJax.Hub.Config({
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • ...
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs063.html b/doc/pub/week39/html/._week39-bs063.html
index 92d4f6d06..b374049de 100644
--- a/doc/pub/week39/html/._week39-bs063.html
+++ b/doc/pub/week39/html/._week39-bs063.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,43 +259,36 @@ MathJax.Hub.Config({
 
 
 
-
    Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come 
    
-
    
-
    
-
     
+
    More complicated functions using the elements of their arguments directly 
    
+
 
    
 
-
-
    #include <cmath>
-#include <iostream>
-#include <fstream>
-#include <iomanip>
-#include "vectormatrixclass.h"
-using namespace  std;
-//   Main function begins here
-int main(int  argc, char * argv[]){
-  int dim = 2;
-  Vector x(dim),xsd(dim), b(dim),x0(dim);
-  Matrix A(dim,dim);
+
+
    import autograd.numpy as np
+from autograd import grad
+def f3(x): # Assumes x is an array of length 5 or higher
+    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
 
-  // Set our initial guess
-  x0(0) = x0(1) = 0;
-  // Set the matrix
-  A(0,0) =  3;    A(1,0) =  2;   A(0,1) =  2;   A(1,1) =  6;
-  b(0) = 2; b(1) = -8;
-  cout << "The Matrix A that we are using: " << endl;
-  A.Print();
-  cout << endl;
-  xsd = SteepestDescent(A,b,x0);
-  cout << "The approximate solution using Steepest Descent is: " << endl;
-  xsd.Print();
-  cout << endl;
-}
+f3_grad = grad(f3)
+
+x = np.linspace(0,4,5)
+
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_grad(x))
+
+# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
+f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+
+# Print the analytical gradient:
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
 
    
 
    
-
    
-
    
-
+Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
 
 
    
 
    
@@ -343,10 +313,6 @@ MathJax.Hub.Config({
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • 73
    • 
-  
  • ...
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs064.html b/doc/pub/week39/html/._week39-bs064.html
index bfd71eecf..f55421c7a 100644
--- a/doc/pub/week39/html/._week39-bs064.html
+++ b/doc/pub/week39/html/._week39-bs064.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -280,42 +257,31 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
+
+
    Functions using mathematical functions from Numpy 
    
 
-
    The routine for the steepest descent method 
    
-
    
-
    
-
     
 
    
 
-
-
    Vector SteepestDescent(Matrix A, Vector b, Vector x0){
-  int IterMax, i;
-  int dim = x0.Dimension();
-  const double tolerance = 1.0e-14;
-  Vector x(dim),f(dim),z(dim);
-  double c,alpha,d;
-  IterMax = 30;
-  x = x0;
-  r = A*x-b;
-  i = 0;
-  while (i <= IterMax){
-    z = A*r;
-    c = dot(r,r);
-    alpha = c/dot(r,z);
-    x = x - alpha*r;
-    r =  A*x-b;
-    if(sqrt(dot(r,r)) < tolerance) break;
-    i++;
-  }
-  return x;
-}
+
+
    import autograd.numpy as np
+from autograd import grad
+def f4(x):
+    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
+
+f4_grad = grad(f4)
+
+x = 2.7
+
+# Print the computed derivative:
+print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+
+# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
+f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
+
+# Print the analytical gradient:
+print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
 
    
-
    
-
    
-
    
-
-
 
    
 
    
 
@@ -338,11 +304,6 @@ MathJax.Hub.Config({
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • 73
    • 
-  
  • 74
    • 
-  
  • ...
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs065.html b/doc/pub/week39/html/._week39-bs065.html
index ca4bb41cd..0b997d350 100644
--- a/doc/pub/week39/html/._week39-bs065.html
+++ b/doc/pub/week39/html/._week39-bs065.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,71 +259,25 @@ MathJax.Hub.Config({
 
 
 
-
    Steepest descent example 
    
+
    More autograd 
    
 
 
    
 
 
-
    import numpy as np
-import numpy.linalg as la
+
    import autograd.numpy as np
+from autograd import grad
+def f5(x):
+    if x >= 0:
+        return x**2
+    else:
+        return -3*x + 1
 
-import scipy.optimize as sopt
+f5_grad = grad(f5)
 
-import matplotlib.pyplot as pt
-from mpl_toolkits.mplot3d import axes3d
+x = 2.7
 
-def f(x):
-    return 0.5*x[0]**2 + 2.5*x[1]**2
-
-def df(x):
-    return np.array([x[0], 5*x[1]])
-
-fig = pt.figure()
-ax = fig.gca(projection="3d")
-
-xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
-fmesh = f(np.array([xmesh, ymesh]))
-ax.plot_surface(xmesh, ymesh, fmesh)
-
    
-
    
-And then as countor plot
-
    
-
-
-
    pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh)
-guesses = [np.array([2, 2./5])]
-
    
-
    
-Find guesses
-
    
-
-
-
    x = guesses[-1]
-s = -df(x)
-
    
-
    
-Run it!
-
    
-
-
-
    def f1d(alpha):
-    return f(x + alpha*s)
-
-alpha_opt = sopt.golden(f1d)
-next_guess = x + alpha_opt * s
-guesses.append(next_guess)
-print(next_guess)
-
    
-
    
-What happened?
-
    
-
-
-
    pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh, 50)
-it_array = np.array(guesses)
-pt.plot(it_array.T[0], it_array.T[1], "x-")
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
 
    
 
    
 
    
@@ -369,12 +300,6 @@ pt.plot(it_array69
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • 73
    • 
-  
  • 74
    • 
-  
  • 75
    • 
-  
  • ...
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs066.html b/doc/pub/week39/html/._week39-bs066.html
index e6ff07457..5ac54cf2f 100644
--- a/doc/pub/week39/html/._week39-bs066.html
+++ b/doc/pub/week39/html/._week39-bs066.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,34 +259,49 @@ MathJax.Hub.Config({
 
 
 
-
    Conjugate gradient method 
    
-
    
-
    
-
     
-In the CG method we define so-called conjugate directions and two vectors 
-\( \hat{s} \) and \( \hat{t} \)
-are said to be
-conjugate if
-$$
-\begin{equation*}
-\hat{s}^T\hat{A}\hat{t}= 0.
-\end{equation*}
-$$
+
    And  with loops 
    
 
-The philosophy of the CG method is to perform searches in various conjugate directions
-of our vectors \( \hat{x}_i \) obeying the above criterion, namely
-$$
-\begin{equation*}
-\hat{x}_i^T\hat{A}\hat{x}_j= 0.
-\end{equation*}
-$$
+
    
 
-Two vectors are conjugate if they are orthogonal with respect to 
-this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
-
    
-
    
+
+
    import autograd.numpy as np
+from autograd import grad
+def f6_for(x):
+    val = 0
+    for i in range(10):
+        val = val + x**i
+    return val
 
+def f6_while(x):
+    val = 0
+    i = 0
+    while i < 10:
+        val = val + x**i
+        i = i + 1
+    return val
 
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x = 0.5
+
+# Print the computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+
    
+
    
+
+
+
    import autograd.numpy as np
+from autograd import grad
+# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9
+# The analytical derivative is: sum(i*x**(i-1)) 
+f6_grad_analytical = 0
+for i in range(10):
+    f6_grad_analytical += i*x**(i-1)
+
+print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
+
    
 
    
 
    
 
@@ -330,11 +322,6 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • 73
    • 
-  
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    • 
-  
  • 75
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs067.html b/doc/pub/week39/html/._week39-bs067.html
index a9ff00f3a..8a590236d 100644
--- a/doc/pub/week39/html/._week39-bs067.html
+++ b/doc/pub/week39/html/._week39-bs067.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,21 +259,40 @@ MathJax.Hub.Config({
 
 
 
-
    Conjugate gradient method 
    
-
    
-
    
-
     
-An example is given by the eigenvectors of the matrix
-$$
-\begin{equation*}
-\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
-\end{equation*}
-$$
+
    Using recursion 
    
+
    
 
-which is zero unless \( i=j \).
-
    
-
    
+
+
    import autograd.numpy as np
+from autograd import grad
 
+def f7(n): # Assume that n is an integer
+    if n == 1 or n == 0:
+        return 1
+    else:
+        return n*f7(n-1)
+
+f7_grad = grad(f7)
+
+n = 2.0
+
+print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
+
+# The function f7 is an implementation of the factorial of n.
+# By using the product rule, one can find that the derivative is:
+
+f7_grad_analytical = 0
+for i in range(int(n)-1):
+    tmp = 1
+    for k in range(int(n)-1):
+        if k != i:
+            tmp *= (n - k)
+    f7_grad_analytical += tmp
+
+print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
+
    
+
    
+Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
 
 
    
 
    
@@ -317,11 +313,6 @@ which is zero unless \( i=j \).
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • 73
    • 
-  
  • 74
    • 
-  
  • 75
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs068.html b/doc/pub/week39/html/._week39-bs068.html
index 6c1e7919c..28975e8e2 100644
--- a/doc/pub/week39/html/._week39-bs068.html
+++ b/doc/pub/week39/html/._week39-bs068.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,30 +259,28 @@ MathJax.Hub.Config({
 
 
 
-
    Conjugate gradient method 
    
-
    
-
    
-
     
-Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
-\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
-$$
-\begin{equation*}
-\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. 
-\end{equation*}
-$$
+
    Unsupported functions 
    
+Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
 
-We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. 
-Then the \( \hat{p}_{i} \)  form a basis of \( R^n \) and we can expand the solution 
-$  \hat{A}\hat{x} = \hat{b}$ in this basis, namely
+
    
+Assigning a value to the variable being differentiated with respect to
+
    
 
-$$
-\begin{equation*}
-  \hat{x}  = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
-\end{equation*}
-$$
-
    
-
    
+
+
    import autograd.numpy as np
+from autograd import grad
+def f8(x): # Assume x is an array
+    x[2] = 3
+    return x*2
 
+f8_grad = grad(f8)
+
+x = 8.4
+
+print("The derivative of f8 is:",f8_grad(x))
+
    
+
    
+Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
 
 
    
 
    
@@ -325,11 +300,6 @@ $$
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • 73
    • 
-  
  • 74
    • 
-  
  • 75
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs069.html b/doc/pub/week39/html/._week39-bs069.html
index c9aa99a4d..c9a548193 100644
--- a/doc/pub/week39/html/._week39-bs069.html
+++ b/doc/pub/week39/html/._week39-bs069.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,36 +259,45 @@ MathJax.Hub.Config({
 
 
 
-
    Conjugate gradient method 
    
-
    
-
    
-
     
-The coefficients are given by
-$$
-\begin{equation*}
-    \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
-\end{equation*}
-$$
+
    The syntax a.dot(b) when finding the dot product 
    
+
    
 
-Multiplying with \( \hat{p}_k^T \)  from the left gives
+
+
    import autograd.numpy as np
+from autograd import grad
+def f9(a): # Assume a is an array with 2 elements
+    b = np.array([1.0,2.0])
+    return a.dot(b)
 
-$$
-\begin{equation*}
-  \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
-\end{equation*}
-$$
+f9_grad = grad(f9)
 
-and we can define the coefficients \( \alpha_k \) as
+x = np.array([1.0,0.0])
 
-$$
-\begin{equation*}
-    \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
-\end{equation*}
-$$
-
    
-
    
+print("The derivative of f9 is:",f9_grad(x))
+
    
+
    
+Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array.  To overcome this, an alternative syntax
+which also computed the dot product can be used:
 
+
    
 
+
+
    import autograd.numpy as np
+from autograd import grad
+def f9_alternative(x): # Assume a is an array with 2 elements
+    b = np.array([1.0,2.0])
+    return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2
+
+f9_alternative_grad = grad(f9_alternative)
+
+x = np.array([3.0,0.0])
+
+print("The gradient of f9 is:",f9_alternative_grad(x))
+
+# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively
+# w.r.t x is (b_1, b_2).
+
    
 
    
 
    
 
@@ -329,11 +315,6 @@ $$
   
  • 69
    • 
   
  • 70
    • 
   
  • 71
    • 
-  
  • 72
    • 
-  
  • 73
    • 
-  
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    • 
-  
  • 75
    • 
-  
  • 76
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/._week39-bs070.html b/doc/pub/week39/html/._week39-bs070.html
index 46944c9e0..1137f6c93 100644
--- a/doc/pub/week39/html/._week39-bs070.html
+++ b/doc/pub/week39/html/._week39-bs070.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -282,41 +259,18 @@ MathJax.Hub.Config({
 
 
 
-
    Conjugate gradient method and iterations 
    
-
    
-
    
-
     
-
+
    Recommended to avoid 
    
+The documentation recommends to avoid inplace operations such as
 
    
-If we choose the conjugate vectors \( \hat{p}_k \) carefully, 
-then we may not need all of them to obtain a good approximation to the solution 
-\( \hat{x} \). 
-We want to regard the conjugate gradient method as an iterative method. 
-This will us to solve systems where \( n \) is so large that the direct 
-method would take too much time.
 
+
+
    a += b
+a -= b
+a*= b
+a /=b
+
    
 
    
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
 
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
-
    
-
    
-
-
-
    
 
    
 
 
    
 
 
diff --git a/doc/pub/week39/html/week39-bs.html b/doc/pub/week39/html/week39-bs.html
index d1107c7b3..2e9c5fff5 100644
--- a/doc/pub/week39/html/week39-bs.html
+++ b/doc/pub/week39/html/week39-bs.html
@@ -67,94 +67,76 @@ Automatically generated HTML file from DocOnce source
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -211,62 +193,57 @@ MathJax.Hub.Config({
       
  • Conditions on convex functions
    • 
       
  • More on convex functions
    • 
       
  • Some simple problems
    • 
-      
  • Revisiting our first homework
    • 
-      
  • Gradient descent example
    • 
-      
  • The derivative of the cost/loss function
    • 
-      
  • The Hessian matrix
    • 
-      
  • Simple program
    • 
-      
  • Gradient Descent Example
    • 
-      
  • And a corresponding example using scikit-learn
    • 
-      
  • Gradient descent and Ridge
    • 
-      
  • Program example for gradient descent with Ridge Regression
    • 
-      
  • Using gradient descent methods, limitations
    • 
-      
  • Friday September 25
    • 
-      
  • Stochastic Gradient Descent
    • 
-      
  • Computation of gradients
    • 
-      
  • SGD example
    • 
-      
  • The gradient step
    • 
-      
  • Simple example code
    • 
-      
  • When do we stop?
    • 
-      
  • Slightly different approach
    • 
-      
  • Program for stochastic gradient
    • 
-      
  • Momentum based GD
    • 
-      
  • More on momentum based approaches
    • 
-      
  • Momentum parameter
    • 
-      
  • Second moment of the gradient
    • 
-      
  • RMS prop
    • 
-      
  • ADAM optimizer
    • 
-      
  • Practical tips
    • 
-      
  • Automatic differentiation
    • 
-      
  • Using autograd
    • 
-      
  • Autograd with more complicated functions
    • 
-      
  • More complicated functions using the elements of their arguments directly
    • 
-      
  • Functions using mathematical functions from Numpy
    • 
-      
  • More autograd
    • 
-      
  • And  with loops
    • 
-      
  • Using recursion
    • 
-      
  • Unsupported functions
    • 
-      
  • The syntax a.dot(b) when finding the dot product
    • 
-      
  • Recommended to avoid
    • 
-      
  • Standard steepest descent
    • 
-      
  • Gradient method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Steepest descent  method
    • 
-      
  • Final expressions
    • 
-      
  • Code examples for steepest descent
    • 
-      
  • Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
    • 
-      
  • The routine for the steepest descent method
    • 
-      
  • Steepest descent example
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method and iterations
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Conjugate gradient method
    • 
-      
  • Simple implementation of the Conjugate gradient algorithm
    • 
-      
  • Broyden–Fletcher–Goldfarb–Shanno algorithm
    • 
+      
  • Standard steepest descent
    • 
+      
  • Gradient method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Steepest descent  method
    • 
+      
  • Final expressions
    • 
+      
  • Steepest descent example
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method and iterations
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Conjugate gradient method
    • 
+      
  • Revisiting our first homework
    • 
+      
  • Gradient descent example
    • 
+      
  • The derivative of the cost/loss function
    • 
+      
  • The Hessian matrix
    • 
+      
  • Simple program
    • 
+      
  • Gradient Descent Example
    • 
+      
  • And a corresponding example using scikit-learn
    • 
+      
  • Gradient descent and Ridge
    • 
+      
  • Program example for gradient descent with Ridge Regression
    • 
+      
  • Using gradient descent methods, limitations
    • 
+      
  • Friday September 25
    • 
+      
  • Stochastic Gradient Descent
    • 
+      
  • Computation of gradients
    • 
+      
  • SGD example
    • 
+      
  • The gradient step
    • 
+      
  • Simple example code
    • 
+      
  • When do we stop?
    • 
+      
  • Slightly different approach
    • 
+      
  • Program for stochastic gradient
    • 
+      
  • Momentum based GD
    • 
+      
  • More on momentum based approaches
    • 
+      
  • Momentum parameter
    • 
+      
  • Second moment of the gradient
    • 
+      
  • RMS prop
    • 
+      
  • ADAM optimizer
    • 
+      
  • Practical tips
    • 
+      
  • Automatic differentiation
    • 
+      
  • Using autograd
    • 
+      
  • Autograd with more complicated functions
    • 
+      
  • More complicated functions using the elements of their arguments directly
    • 
+      
  • Functions using mathematical functions from Numpy
    • 
+      
  • More autograd
    • 
+      
  • And  with loops
    • 
+      
  • Using recursion
    • 
+      
  • Unsupported functions
    • 
+      
  • The syntax a.dot(b) when finding the dot product
    • 
+      
  • Recommended to avoid
    • 
 
         
       
@@ -325,7 +302,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 76
    • 
+  
  • 71
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week39/html/week39-reveal.html b/doc/pub/week39/html/week39-reveal.html
index 382f963eb..b973ae8ac 100644
--- a/doc/pub/week39/html/week39-reveal.html
+++ b/doc/pub/week39/html/week39-reveal.html
@@ -643,7 +643,514 @@ Using the definition of convexity, try to show that a function satisfying the pr
 
 
 
    
-
    Revisiting our first homework 
    
+
    Standard steepest descent 
    
+
+
    
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent, which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
    
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type 
+
     
    
+$$
+\begin{equation*} 
+\hat{A}\hat{x} = \hat{b}.
+\end{equation*} 
+$$
+
     
    
+
+
    
+In the iterative process we end up with a problem like
+
+
     
    
+$$
+\begin{equation*}
+  \hat{r}= \hat{b}-\hat{A}\hat{x},
+\end{equation*}
+$$
+
     
    
+
+where \( \hat{r} \) is the so-called residual or error in the iterative process.
+
+
    
+When we have found the exact solution, \( \hat{r}=0 \).
+
    
+
+
+
    
+
    Gradient method 
    
+
+
    
+The residual is zero when we reach the minimum of the quadratic equation
+
     
    
+$$
+\begin{equation*}
+  P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
+\end{equation*}
+$$
+
     
    
+
+
    
+with the constraint that the matrix \( \hat{A} \) is positive definite and
+symmetric.  This defines also the Hessian and we want it to be  positive definite.
+
    
+
+
+
    
+
    Steepest descent  method 
    
+
+
    
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
+We can assume without loss of generality that
+
     
    
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
     
    
+
+or consider the system
+
     
    
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
     
    
+
+instead.
+
    
+
+
+
    
+
    Steepest descent  method 
    
+
    
+
+
    
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+
     
    
+$$
+\begin{equation*}
+  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
+\end{equation*}
+$$
+
     
    
+
+This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) 
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
+which equals
+
     
    
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
     
    
+
+and 
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
+
+
    
+
    
+
+
+
    
+
    Final expressions 
    
+
    
+
+
    
+We can compute the residual iteratively as
+
     
    
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
+$$
+
     
    
+
+which equals
+
     
    
+$$
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k),
+ \end{equation*}
+$$
+
     
    
+
+or
+
     
    
+$$
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k,
+ \end{equation*}
+$$
+
     
    
+
+which gives
+
+
     
    
+$$
+\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k}
+$$
+
     
    
+
+leading to the iterative scheme
+
     
    
+$$
+\begin{equation*}
+\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k},
+ \end{equation*}
+$$
+
     
    
+
    
+
    
+
+
+
    
+
    Steepest descent example 
    
+
+
    
+
+
+
    import numpy as np
+import numpy.linalg as la
+
+import scipy.optimize as sopt
+
+import matplotlib.pyplot as pt
+from mpl_toolkits.mplot3d import axes3d
+
+def f(x):
+    return 0.5*x[0]**2 + 2.5*x[1]**2
+
+def df(x):
+    return np.array([x[0], 5*x[1]])
+
+fig = pt.figure()
+ax = fig.gca(projection="3d")
+
+xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
+fmesh = f(np.array([xmesh, ymesh]))
+ax.plot_surface(xmesh, ymesh, fmesh)
+
    
+
    
+And then as countor plot
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh)
+guesses = [np.array([2, 2./5])]
+
    
+
    
+Find guesses
+
    
+
+
+
    x = guesses[-1]
+s = -df(x)
+
    
+
    
+Run it!
+
    
+
+
+
    def f1d(alpha):
+    return f(x + alpha*s)
+
+alpha_opt = sopt.golden(f1d)
+next_guess = x + alpha_opt * s
+guesses.append(next_guess)
+print(next_guess)
+
    
+
    
+What happened?
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh, 50)
+it_array = np.array(guesses)
+pt.plot(it_array.T[0], it_array.T[1], "x-")
+
    
+
    
+
+
+
    
+
    Conjugate gradient method 
    
+
    
+
+
    
+In the CG method we define so-called conjugate directions and two vectors 
+\( \hat{s} \) and \( \hat{t} \)
+are said to be
+conjugate if
+
     
    
+$$
+\begin{equation*}
+\hat{s}^T\hat{A}\hat{t}= 0.
+\end{equation*}
+$$
+
     
    
+
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \hat{x}_i \) obeying the above criterion, namely
+
     
    
+$$
+\begin{equation*}
+\hat{x}_i^T\hat{A}\hat{x}_j= 0.
+\end{equation*}
+$$
+
     
    
+
+Two vectors are conjugate if they are orthogonal with respect to 
+this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
+
    
+
    
+
+
+
    
+
    Conjugate gradient method 
    
+
    
+
+
    
+An example is given by the eigenvectors of the matrix
+
     
    
+$$
+\begin{equation*}
+\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
+\end{equation*}
+$$
+
     
    
+
+which is zero unless \( i=j \).
+
    
+
    
+
+
+
    
+
    Conjugate gradient method 
    
+
    
+
+
    
+Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
+
     
    
+$$
+\begin{equation*}
+\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. 
+\end{equation*}
+$$
+
     
    
+
+We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. 
+Then the \( \hat{p}_{i} \)  form a basis of \( R^n \) and we can expand the solution 
+$  \hat{A}\hat{x} = \hat{b}$ in this basis, namely
+
+
     
    
+$$
+\begin{equation*}
+  \hat{x}  = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
+\end{equation*}
+$$
+
     
    
+
    
+
    
+
+
+
    
+
    Conjugate gradient method 
    
+
    
+
+
    
+The coefficients are given by
+
     
    
+$$
+\begin{equation*}
+    \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
+\end{equation*}
+$$
+
     
    
+
+Multiplying with \( \hat{p}_k^T \)  from the left gives
+
+
     
    
+$$
+\begin{equation*}
+  \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
+\end{equation*}
+$$
+
     
    
+
+and we can define the coefficients \( \alpha_k \) as
+
+
     
    
+$$
+\begin{equation*}
+    \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
+\end{equation*}
+$$
+
     
    
+
    
+
    
+
+
+
    
+
    Conjugate gradient method and iterations 
    
+
    
+
+
    
+If we choose the conjugate vectors \( \hat{p}_k \) carefully, 
+then we may not need all of them to obtain a good approximation to the solution 
+\( \hat{x} \). 
+We want to regard the conjugate gradient method as an iterative method. 
+This will us to solve systems where \( n \) is so large that the direct 
+method would take too much time.
+
+
    
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
+We can assume without loss of generality that
+
     
    
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
     
    
+
+or consider the system
+
     
    
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
     
    
+
+instead.
+
    
+
    
+
+
+
    
+
    Conjugate gradient method 
    
+
    
+
+
    
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+
     
    
+$$
+\begin{equation*}
+  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
+\end{equation*}
+$$
+
     
    
+
+This suggests taking the first basis vector \( \hat{p}_1 \) 
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
+which equals
+
     
    
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
     
    
+
+and 
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+The other vectors in the basis will be conjugate to the gradient, 
+hence the name conjugate gradient method.
+
    
+
    
+
+
+
    
+
    Conjugate gradient method 
    
+
    
+
+
    
+Let  \( \hat{r}_k \) be the residual at the \( k \)-th step:
+
     
    
+$$
+\begin{equation*}
+\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
+\end{equation*}
+$$
+
     
    
+
+Note that \( \hat{r}_k \) is the negative gradient of \( f \) at 
+\( \hat{x}=\hat{x}_k \), 
+so the gradient descent method would be to move in the direction \( \hat{r}_k \). 
+Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other, 
+so we take the direction closest to the gradient \( \hat{r}_k \)  
+under the conjugacy constraint. 
+This gives the following expression
+
     
    
+$$
+\begin{equation*}
+\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
+\end{equation*}
+$$
+
     
    
+
    
+
    
+
+
+
    
+
    Conjugate gradient method 
    
+
    
+
+
    
+We can also  compute the residual iteratively as
+
     
    
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
+$$
+
     
    
+
+which equals
+
     
    
+$$
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k),
+ \end{equation*}
+$$
+
     
    
+
+or
+
     
    
+$$
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k,
+ \end{equation*}
+$$
+
     
    
+
+which gives
+
+
     
    
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k},
+ \end{equation*}
+$$
+
     
    
+
    
+
    
+
+
+
    
+
    Revisiting our first homework 
    
 
 
    
 We will use linear regression as a case study for the gradient descent
@@ -683,7 +1190,7 @@ $$
 
 
 
    
-
    Gradient descent example 
    
+
    Gradient descent example 
    
 
 
    
 Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -693,9 +1200,9 @@ It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb
 
     
    
 $$
 X \equiv \begin{bmatrix}
-1 & x_1  \\
-\vdots & \vdots  \\
-1 & x_{100} &  \\
+1 &; x_1  \\
+\vdots &; \vdots  \\
+1 &; x_{100} &;  \\
 \end{bmatrix}.
 $$
 
     
    
@@ -712,7 +1219,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
 
 
 
    
-
    The derivative of the cost/loss function 
    
+
    The derivative of the cost/loss function 
    
 
 
    
 Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show  that the gradient can be written as
@@ -729,13 +1236,13 @@ where \( X \) is the design matrix defined above.
 
 
 
    
-
    The Hessian matrix 
    
+
    The Hessian matrix 
    
 The Hessian matrix of \( C(\beta) \) is given by 
 
     
    
 $$
 \hat{H} \equiv \begin{bmatrix}
-\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1}  \\
-\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &  \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0^2} &; \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1}  \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} &; \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &;  \\
 \end{bmatrix} = 2X^T X.
 $$
 
     
    
@@ -745,7 +1252,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
 
 
 
    
-
    Simple program 
    
+
    Simple program 
    
 
 
    
 We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to 
@@ -767,7 +1274,7 @@ And finally we can compare our solution for \( \beta \) with the analytic result
 
 
 
    
-
    Gradient Descent Example 
    
+
    Gradient Descent Example 
    
 
 
    
 Here our simple example
@@ -818,7 +1325,7 @@ plt.show()
 
 
 
    
-
    And a corresponding example using scikit-learn 
    
+
    And a corresponding example using scikit-learn 
    
 
 
    
 
@@ -843,7 +1350,7 @@ sgdreg.fit(x,y.ravel())
 
 
 
    
-
    Gradient descent and Ridge 
    
+
    Gradient descent and Ridge 
    
 
 
    
 We have also discussed Ridge regression where the loss function contains a regularized term given by the \( L_2 \) norm of \( \beta \), 
@@ -874,7 +1381,7 @@ $$
 
 
 
    
-
    Program example for gradient descent with Ridge Regression 
    
+
    Program example for gradient descent with Ridge Regression 
    
 
    
 
 
@@ -926,7 +1433,7 @@ plt.show()
 
 
 
    
-
    Using gradient descent methods, limitations 
    
+
    Using gradient descent methods, limitations 
    
 
 
      
 
    •  Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our energy function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
      • 
@@ -940,12 +1447,12 @@ plt.show()
 
 
 
      
-
      Friday September 25 
      
+
      Friday September 25 
      
 
      
 
 
 
      
-
      Stochastic Gradient Descent 
      
+
      Stochastic Gradient Descent 
      
 
 
      
 Stochastic gradient descent (SGD) and variants thereof address some of
@@ -965,7 +1472,7 @@ $$
 
 
 
      
-
      Computation of gradients 
      
+
      Computation of gradients 
      
 
 
      
 This in turn means that the gradient can be
@@ -987,7 +1494,7 @@ minibatches. We denote these minibatches by \( B_k \) where
 
 
 
      
-
      SGD example 
      
+
      SGD example 
      
 As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
 and we choose to have \( M=5 \) minibathces,
 then each minibatch contains two data points. In particular we have
@@ -1013,7 +1520,7 @@ $$
 
 
 
      
-
      The gradient step 
      
+
      The gradient step 
      
 
 
      
 Thus a gradient descent step now looks like 
@@ -1034,7 +1541,7 @@ the number of minibatches, as exemplified in the code below.
 
 
 
      
-
      Simple example code 
      
+
      Simple example code 
      
 
 
      
 
@@ -1066,7 +1573,7 @@ all \( n \) datapoints.
 
 
 
      
-
      When do we stop? 
      
+
      When do we stop? 
      
 
 
      
 A natural question is when do we stop the search for a new minimum?
@@ -1083,7 +1590,7 @@ gave the lowest value.
 
 
 
      
-
      Slightly different approach 
      
+
      Slightly different approach 
      
 
 
      
 Another approach is to let the step length \( \gamma_j \) depend on the
@@ -1134,7 +1641,7 @@ j = 0
 
 
 
      
-
      Program for stochastic gradient 
      
+
      Program for stochastic gradient 
      
 
 
      
 
@@ -1210,7 +1717,7 @@ plt.show()
 
 
 
      
-
      Momentum based GD 
      
+
      Momentum based GD 
      
 
 
      
 The stochastic gradient descent (SGD) is almost always used with a
@@ -1251,7 +1758,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
 
 
 
      
-
      More on momentum based approaches 
      
+
      More on momentum based approaches 
      
 
 
      
 Let us try to get more intuition from these equations. It is helpful
@@ -1287,7 +1794,7 @@ $$
 
 
 
      
-
      Momentum parameter 
      
+
      Momentum parameter 
      
 
 
      
 Notice that this equation is identical to previous one if we identify
@@ -1347,7 +1854,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
 
 
 
      
-
      Second moment of the gradient 
      
+
      Second moment of the gradient 
      
 
 
      
 In stochastic gradient descent, with and without momentum, we still
@@ -1375,7 +1882,7 @@ ADAM.
 
 
 
      
-
      RMS prop 
      
+
      RMS prop 
      
 
 
      
 In RMS prop, in addition to keeping a running average of the first
@@ -1408,7 +1915,7 @@ learning rate for flat directions.
 
 
 
      
-
      ADAM optimizer 
      
+
      ADAM optimizer 
      
 
 
      
 A related algorithm is the ADAM optimizer. In ADAM, we keep a running
@@ -1461,7 +1968,7 @@ $$
 
 
 
      
-
      Practical tips 
      
+
      Practical tips 
      
 
 
        
 
      •  Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
        • 
@@ -1476,7 +1983,7 @@ Geron's text, see chapter 11, has several interesting discussions.
 
 
 
        
-
        Automatic differentiation 
        
+
        Automatic differentiation 
        
 
 
        
 Automatic differentiation (AD), 
@@ -1567,7 +2074,7 @@ plt.show()
 
 
 
        
-
        Using autograd 
        
+
        Using autograd 
        
 
 
        
 Here we
@@ -1601,7 +2108,7 @@ grad_analytical = 3*a**Autograd with more complicated functions 
+
        Autograd with more complicated functions 
        
 
 
        
 To differentiate with respect to two (or more) arguments of a Python
@@ -1651,7 +2158,7 @@ Note that the grad function will not produce the true gradient of the function.
 
 
 
        
-
        More complicated functions using the elements of their arguments directly 
        
+
        More complicated functions using the elements of their arguments directly 
        
 
 
        
 
@@ -1685,7 +2192,7 @@ could expect form a gradient-evaluting function.
 
 
 
        
-
        Functions using mathematical functions from Numpy 
        
+
        Functions using mathematical functions from Numpy 
        
 
 
        
 
@@ -1712,7 +2219,7 @@ f4_grad_analytical = x/np.sqrt(1 + x**
-
        More autograd 
        
+
        More autograd 
        
 
 
        
 
@@ -1736,7 +2243,7 @@ x = 2.7
 
 
 
        
-
        And  with loops 
        
+
        And  with loops 
        
 
 
        
 
@@ -1783,7 +2290,7 @@ f6_grad_analytical = 0
 
 
 
        
-
        Using recursion 
        
+
        Using recursion 
        
 
        
 
 
@@ -1821,7 +2328,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
 
 
 
        
-
        Unsupported functions 
        
+
        Unsupported functions 
        
 Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
 
 
        
@@ -1847,7 +2354,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
 
 
 
        
-
        The syntax a.dot(b) when finding the dot product 
        
+
        The syntax a.dot(b) when finding the dot product 
        
 
        
 
 
@@ -1890,7 +2397,7 @@ x = np.array([3.0,Recommended to avoid 
+
        Recommended to avoid 
        
 The documentation recommends to avoid inplace operations such as
 
        
 
@@ -1903,665 +2410,6 @@ a /=b
 
        
 
 
-
        
-
        Standard steepest descent 
        
-
-
        
-Before we proceed, we would like to discuss the approach called the
-standard Steepest descent, which again leads to us having to be able
-to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
-
-
        
-The success of the CG method
-for finding solutions of non-linear problems is based on the theory
-of conjugate gradients for linear systems of equations. It belongs to
-the class of iterative methods for solving problems from linear
-algebra of the type 
-
         
        
-$$
-\begin{equation*} 
-\hat{A}\hat{x} = \hat{b}.
-\end{equation*} 
-$$
-
         
        
-
-
        
-In the iterative process we end up with a problem like
-
-
         
        
-$$
-\begin{equation*}
-  \hat{r}= \hat{b}-\hat{A}\hat{x},
-\end{equation*}
-$$
-
         
        
-
-where \( \hat{r} \) is the so-called residual or error in the iterative process.
-
-
        
-When we have found the exact solution, \( \hat{r}=0 \).
-
        
-
-
-
        
-
        Gradient method 
        
-
-
        
-The residual is zero when we reach the minimum of the quadratic equation
-
         
        
-$$
-\begin{equation*}
-  P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
-\end{equation*}
-$$
-
         
        
-
-
        
-with the constraint that the matrix \( \hat{A} \) is positive definite and
-symmetric.  This defines also the Hessian and we want it to be  positive definite.
-
        
-
-
-
        
-
        Steepest descent  method 
        
-
-
        
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
-We can assume without loss of generality that
-
         
        
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
         
        
-
-or consider the system
-
         
        
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
         
        
-
-instead.
-
        
-
-
-
        
-
        Steepest descent  method 
        
-
        
-
-
        
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-
         
        
-$$
-\begin{equation*}
-  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
-\end{equation*}
-$$
-
         
        
-
-This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) 
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
-which equals
-
         
        
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
         
        
-
-and 
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-
-
-
        
-
        
-
-
-
        
-
        Final expressions 
        
-
        
-
-
        
-We can compute the residual iteratively as
-
         
        
-$$
-\begin{equation*}
-\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
- \end{equation*}
-$$
-
         
        
-
-which equals
-
         
        
-$$
-\begin{equation*}
-\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k),
- \end{equation*}
-$$
-
         
        
-
-or
-
         
        
-$$
-\begin{equation*}
-(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k,
- \end{equation*}
-$$
-
         
        
-
-which gives
-
-
         
        
-$$
-\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k}
-$$
-
         
        
-
-leading to the iterative scheme
-
         
        
-$$
-\begin{equation*}
-\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k},
- \end{equation*}
-$$
-
         
        
-
        
-
        
-
-
-
        
-
        Code examples for steepest descent 
        
-
        
-
-
-
        
-
        Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come 
        
-
        
-
-
        
-
-
-
        #include <cmath>
-#include <iostream>
-#include <fstream>
-#include <iomanip>
-#include "vectormatrixclass.h"
-using namespace  std;
-//   Main function begins here
-int main(int  argc, char * argv[]){
-  int dim = 2;
-  Vector x(dim),xsd(dim), b(dim),x0(dim);
-  Matrix A(dim,dim);
-
-  // Set our initial guess
-  x0(0) = x0(1) = 0;
-  // Set the matrix
-  A(0,0) =  3;    A(1,0) =  2;   A(0,1) =  2;   A(1,1) =  6;
-  b(0) = 2; b(1) = -8;
-  cout << "The Matrix A that we are using: " << endl;
-  A.Print();
-  cout << endl;
-  xsd = SteepestDescent(A,b,x0);
-  cout << "The approximate solution using Steepest Descent is: " << endl;
-  xsd.Print();
-  cout << endl;
-}
-
        
-
-
        
-
        
-
-
-
        
-
        The routine for the steepest descent method 
        
-
        
-
-
        
-
-
-
        Vector SteepestDescent(Matrix A, Vector b, Vector x0){
-  int IterMax, i;
-  int dim = x0.Dimension();
-  const double tolerance = 1.0e-14;
-  Vector x(dim),f(dim),z(dim);
-  double c,alpha,d;
-  IterMax = 30;
-  x = x0;
-  r = A*x-b;
-  i = 0;
-  while (i <= IterMax){
-    z = A*r;
-    c = dot(r,r);
-    alpha = c/dot(r,z);
-    x = x - alpha*r;
-    r =  A*x-b;
-    if(sqrt(dot(r,r)) < tolerance) break;
-    i++;
-  }
-  return x;
-}
-
        
-
-
        
-
        
-
-
-
        
-
        Steepest descent example 
        
-
-
        
-
-
-
        import numpy as np
-import numpy.linalg as la
-
-import scipy.optimize as sopt
-
-import matplotlib.pyplot as pt
-from mpl_toolkits.mplot3d import axes3d
-
-def f(x):
-    return 0.5*x[0]**2 + 2.5*x[1]**2
-
-def df(x):
-    return np.array([x[0], 5*x[1]])
-
-fig = pt.figure()
-ax = fig.gca(projection="3d")
-
-xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
-fmesh = f(np.array([xmesh, ymesh]))
-ax.plot_surface(xmesh, ymesh, fmesh)
-
        
-
        
-And then as countor plot
-
        
-
-
-
        pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh)
-guesses = [np.array([2, 2./5])]
-
        
-
        
-Find guesses
-
        
-
-
-
        x = guesses[-1]
-s = -df(x)
-
        
-
        
-Run it!
-
        
-
-
-
        def f1d(alpha):
-    return f(x + alpha*s)
-
-alpha_opt = sopt.golden(f1d)
-next_guess = x + alpha_opt * s
-guesses.append(next_guess)
-print(next_guess)
-
        
-
        
-What happened?
-
        
-
-
-
        pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh, 50)
-it_array = np.array(guesses)
-pt.plot(it_array.T[0], it_array.T[1], "x-")
-
        
-
        
-
-
-
        
-
        Conjugate gradient method 
        
-
        
-
-
        
-In the CG method we define so-called conjugate directions and two vectors 
-\( \hat{s} \) and \( \hat{t} \)
-are said to be
-conjugate if
-
         
        
-$$
-\begin{equation*}
-\hat{s}^T\hat{A}\hat{t}= 0.
-\end{equation*}
-$$
-
         
        
-
-The philosophy of the CG method is to perform searches in various conjugate directions
-of our vectors \( \hat{x}_i \) obeying the above criterion, namely
-
         
        
-$$
-\begin{equation*}
-\hat{x}_i^T\hat{A}\hat{x}_j= 0.
-\end{equation*}
-$$
-
         
        
-
-Two vectors are conjugate if they are orthogonal with respect to 
-this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
-
        
-
        
-
-
-
        
-
        Conjugate gradient method 
        
-
        
-
-
        
-An example is given by the eigenvectors of the matrix
-
         
        
-$$
-\begin{equation*}
-\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
-\end{equation*}
-$$
-
         
        
-
-which is zero unless \( i=j \).
-
        
-
        
-
-
-
        
-
        Conjugate gradient method 
        
-
        
-
-
        
-Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
-\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
-
         
        
-$$
-\begin{equation*}
-\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. 
-\end{equation*}
-$$
-
         
        
-
-We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. 
-Then the \( \hat{p}_{i} \)  form a basis of \( R^n \) and we can expand the solution 
-$  \hat{A}\hat{x} = \hat{b}$ in this basis, namely
-
-
         
        
-$$
-\begin{equation*}
-  \hat{x}  = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
-\end{equation*}
-$$
-
         
        
-
        
-
        
-
-
-
        
-
        Conjugate gradient method 
        
-
        
-
-
        
-The coefficients are given by
-
         
        
-$$
-\begin{equation*}
-    \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
-\end{equation*}
-$$
-
         
        
-
-Multiplying with \( \hat{p}_k^T \)  from the left gives
-
-
         
        
-$$
-\begin{equation*}
-  \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
-\end{equation*}
-$$
-
         
        
-
-and we can define the coefficients \( \alpha_k \) as
-
-
         
        
-$$
-\begin{equation*}
-    \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
-\end{equation*}
-$$
-
         
        
-
        
-
        
-
-
-
        
-
        Conjugate gradient method and iterations 
        
-
        
-
-
        
-If we choose the conjugate vectors \( \hat{p}_k \) carefully, 
-then we may not need all of them to obtain a good approximation to the solution 
-\( \hat{x} \). 
-We want to regard the conjugate gradient method as an iterative method. 
-This will us to solve systems where \( n \) is so large that the direct 
-method would take too much time.
-
-
        
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
-We can assume without loss of generality that
-
         
        
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
         
        
-
-or consider the system
-
         
        
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
         
        
-
-instead.
-
        
-
        
-
-
-
        
-
        Conjugate gradient method 
        
-
        
-
-
        
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-
         
        
-$$
-\begin{equation*}
-  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
-\end{equation*}
-$$
-
         
        
-
-This suggests taking the first basis vector \( \hat{p}_1 \) 
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
-which equals
-
         
        
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
         
        
-
-and 
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-The other vectors in the basis will be conjugate to the gradient, 
-hence the name conjugate gradient method.
-
        
-
        
-
-
-
        
-
        Conjugate gradient method 
        
-
        
-
-
        
-Let  \( \hat{r}_k \) be the residual at the \( k \)-th step:
-
         
        
-$$
-\begin{equation*}
-\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
-\end{equation*}
-$$
-
         
        
-
-Note that \( \hat{r}_k \) is the negative gradient of \( f \) at 
-\( \hat{x}=\hat{x}_k \), 
-so the gradient descent method would be to move in the direction \( \hat{r}_k \). 
-Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other, 
-so we take the direction closest to the gradient \( \hat{r}_k \)  
-under the conjugacy constraint. 
-This gives the following expression
-
         
        
-$$
-\begin{equation*}
-\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
-\end{equation*}
-$$
-
         
        
-
        
-
        
-
-
-
        
-
        Conjugate gradient method 
        
-
        
-
-
        
-We can also  compute the residual iteratively as
-
         
        
-$$
-\begin{equation*}
-\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
- \end{equation*}
-$$
-
         
        
-
-which equals
-
         
        
-$$
-\begin{equation*}
-\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k),
- \end{equation*}
-$$
-
         
        
-
-or
-
         
        
-$$
-\begin{equation*}
-(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k,
- \end{equation*}
-$$
-
         
        
-
-which gives
-
-
         
        
-$$
-\begin{equation*}
-\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k},
- \end{equation*}
-$$
-
         
        
-
        
-
        
-
-
-
        
-
        Simple implementation of the Conjugate gradient algorithm 
        
-
        
-
-
        
-
-
-
          Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
-  int dim = x0.Dimension();
-  const double tolerance = 1.0e-14;
-  Vector x(dim),r(dim),v(dim),z(dim);
-  double c,t,d;
-
-  x = x0;
-  r = b - A*x;
-  v = r;
-  c = dot(r,r);
-  int i = 0; IterMax = dim;
-  while(i <= IterMax){
-    z = A*v;
-    t = c/dot(v,z);
-    x = x + t*v;
-    r = r - t*z;
-    d = dot(r,r);
-    if(sqrt(d) < tolerance)
-      break;
-    v = r + (d/c)*v;
-    c = d;  i++;
-  }
-  return x;
-} 
-
        
-
-
        
-
        
-
-
-
        
-
        Broyden–Fletcher–Goldfarb–Shanno algorithm 
        
-
        
-
-
        
-The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \)  is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that  \( \mathbf {x} \)  can take.
-
-
        
-The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
-
-
        
-The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
-
         
        
-$$
-B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
-$$
-
         
        
-
-
        
-where \( B_{k} \) is an approximation to the Hessian matrix, which is
-updated iteratively at each stage, and \( \nabla f(\mathbf {x} _{k}) \)
-is the gradient of the function
-evaluated at \( x_k \). 
-A line search in the direction \( p_k \) is then used to
-find the next point \( x_{k+1} \) by minimising 
-
         
        
-$$
-f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),
-$$
-
         
        
-
-over the scalar \( \alpha > 0 \).
-
-
-
        
-
        
-
-
 
 
     
 
     
diff --git a/doc/pub/week39/html/week39-solarized.html b/doc/pub/week39/html/week39-solarized.html
index b6409b568..8c85728a9 100644
--- a/doc/pub/week39/html/week39-solarized.html
+++ b/doc/pub/week39/html/week39-solarized.html
@@ -87,94 +87,76 @@ div { text-align: justify; text-justify: inter-word; }
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -668,10 +650,468 @@ This result means that if we know that the cost/loss function is convex and we a
 
 Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
 
+
    
+









    
+
+
    Standard steepest descent 
    
+
+
    
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent, which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
    
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type 
+$$
+\begin{equation*} 
+\hat{A}\hat{x} = \hat{b}.
+\end{equation*} 
+$$
+
+
    
+In the iterative process we end up with a problem like
+
+$$
+\begin{equation*}
+  \hat{r}= \hat{b}-\hat{A}\hat{x},
+\end{equation*}
+$$
+
+where \( \hat{r} \) is the so-called residual or error in the iterative process.
+
+
    
+When we have found the exact solution, \( \hat{r}=0 \).
+
+
    
+









    
+
+
    Gradient method 
    
+
+
    
+The residual is zero when we reach the minimum of the quadratic equation
+$$
+\begin{equation*}
+  P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
+\end{equation*}
+$$
+
+
    
+with the constraint that the matrix \( \hat{A} \) is positive definite and
+symmetric.  This defines also the Hessian and we want it to be  positive definite.
+
+
    
+









    
+
+
    Steepest descent  method 
    
+
+
    
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
    
+









    
+
+
    Steepest descent  method 
    
+
    
+
+
    
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) 
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and 
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
+
+
    
+
+
+
    
+









    
+
+
    Final expressions 
    
+
    
+
+
    
+We can compute the residual iteratively as
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
+$$
+
+which equals
+$$
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k),
+ \end{equation*}
+$$
+
+or
+$$
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k,
+ \end{equation*}
+$$
+
+which gives
+
+$$
+\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k}
+$$
+
+leading to the iterative scheme
+$$
+\begin{equation*}
+\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k},
+ \end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Steepest descent example 
    
+
+
    
+
+
+
    import numpy as np
+import numpy.linalg as la
+
+import scipy.optimize as sopt
+
+import matplotlib.pyplot as pt
+from mpl_toolkits.mplot3d import axes3d
+
+def f(x):
+    return 0.5*x[0]**2 + 2.5*x[1]**2
+
+def df(x):
+    return np.array([x[0], 5*x[1]])
+
+fig = pt.figure()
+ax = fig.gca(projection="3d")
+
+xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
+fmesh = f(np.array([xmesh, ymesh]))
+ax.plot_surface(xmesh, ymesh, fmesh)
+
    
+
    
+And then as countor plot
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh)
+guesses = [np.array([2, 2./5])]
+
    
+
    
+Find guesses
+
    
+
+
+
    x = guesses[-1]
+s = -df(x)
+
    
+
    
+Run it!
+
    
+
+
+
    def f1d(alpha):
+    return f(x + alpha*s)
+
+alpha_opt = sopt.golden(f1d)
+next_guess = x + alpha_opt * s
+guesses.append(next_guess)
+print(next_guess)
+
    
+
    
+What happened?
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh, 50)
+it_array = np.array(guesses)
+pt.plot(it_array.T[0], it_array.T[1], "x-")
+
    
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+In the CG method we define so-called conjugate directions and two vectors 
+\( \hat{s} \) and \( \hat{t} \)
+are said to be
+conjugate if
+$$
+\begin{equation*}
+\hat{s}^T\hat{A}\hat{t}= 0.
+\end{equation*}
+$$
+
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \hat{x}_i \) obeying the above criterion, namely
+$$
+\begin{equation*}
+\hat{x}_i^T\hat{A}\hat{x}_j= 0.
+\end{equation*}
+$$
+
+Two vectors are conjugate if they are orthogonal with respect to 
+this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+An example is given by the eigenvectors of the matrix
+$$
+\begin{equation*}
+\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
+\end{equation*}
+$$
+
+which is zero unless \( i=j \).
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
+$$
+\begin{equation*}
+\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. 
+\end{equation*}
+$$
+
+We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. 
+Then the \( \hat{p}_{i} \)  form a basis of \( R^n \) and we can expand the solution 
+$  \hat{A}\hat{x} = \hat{b}$ in this basis, namely
+
+$$
+\begin{equation*}
+  \hat{x}  = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
+\end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+The coefficients are given by
+$$
+\begin{equation*}
+    \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
+\end{equation*}
+$$
+
+Multiplying with \( \hat{p}_k^T \)  from the left gives
+
+$$
+\begin{equation*}
+  \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
+\end{equation*}
+$$
+
+and we can define the coefficients \( \alpha_k \) as
+
+$$
+\begin{equation*}
+    \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
+\end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method and iterations 
    
+
    
+
+
    
+
+
    
+If we choose the conjugate vectors \( \hat{p}_k \) carefully, 
+then we may not need all of them to obtain a good approximation to the solution 
+\( \hat{x} \). 
+We want to regard the conjugate gradient method as an iterative method. 
+This will us to solve systems where \( n \) is so large that the direct 
+method would take too much time.
+
+
    
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{p}_1 \) 
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and 
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+The other vectors in the basis will be conjugate to the gradient, 
+hence the name conjugate gradient method.
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+Let  \( \hat{r}_k \) be the residual at the \( k \)-th step:
+$$
+\begin{equation*}
+\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
+\end{equation*}
+$$
+
+Note that \( \hat{r}_k \) is the negative gradient of \( f \) at 
+\( \hat{x}=\hat{x}_k \), 
+so the gradient descent method would be to move in the direction \( \hat{r}_k \). 
+Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other, 
+so we take the direction closest to the gradient \( \hat{r}_k \)  
+under the conjugacy constraint. 
+This gives the following expression
+$$
+\begin{equation*}
+\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
+\end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+We can also  compute the residual iteratively as
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
+$$
+
+which equals
+$$
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k),
+ \end{equation*}
+$$
+
+or
+$$
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k,
+ \end{equation*}
+$$
+
+which gives
+
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k},
+ \end{equation*}
+$$
+
    
+
+
 
    
 
 
-
    Revisiting our first homework 
    
+
    Revisiting our first homework 
    
 
 
    
 We will use linear regression as a case study for the gradient descent
@@ -704,7 +1144,7 @@ $$
 
    
 
 
-
    Gradient descent example 
    
+
    Gradient descent example 
    
 
 
    
 Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -713,9 +1153,9 @@ Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdo
 It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2}  \) is the design matrix given by
 $$
 X \equiv \begin{bmatrix}
-1 & x_1  \\
-\vdots & \vdots  \\
-1 & x_{100} &  \\
+1 &; x_1  \\
+\vdots &; \vdots  \\
+1 &; x_{100} &;  \\
 \end{bmatrix}.
 $$
 
@@ -729,7 +1169,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
 
    
 









    
 
-
    The derivative of the cost/loss function 
    
+
    The derivative of the cost/loss function 
    
 
 
    
 Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show  that the gradient can be written as
@@ -744,12 +1184,12 @@ where \( X \) is the design matrix defined above.
 
    
 









    
 
-
    The Hessian matrix 
    
+
    The Hessian matrix 
    
 The Hessian matrix of \( C(\beta) \) is given by 
 $$
 \hat{H} \equiv \begin{bmatrix}
-\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1}  \\
-\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &  \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0^2} &; \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1}  \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} &; \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &;  \\
 \end{bmatrix} = 2X^T X.
 $$
 
@@ -758,7 +1198,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
 
    
 









    
 
-
    Simple program 
    
+
    Simple program 
    
 
 
    
 We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to 
@@ -778,7 +1218,7 @@ And finally we can compare our solution for \( \beta \) with the analytic result
 
    
 









    
 
-
    Gradient Descent Example 
    
+
    Gradient Descent Example 
    
 
 
    
 Here our simple example
@@ -828,7 +1268,7 @@ plt.show()
 
    
 









    
 
-
    And a corresponding example using scikit-learn 
    
+
    And a corresponding example using scikit-learn 
    
 
 
    
 
@@ -852,7 +1292,7 @@ sgdreg.fit(x,y.ravel())
 
    
 
 
-
    Gradient descent and Ridge 
    
+
    Gradient descent and Ridge 
    
 
 
    
 We have also discussed Ridge regression where the loss function contains a regularized term given by the \( L_2 \) norm of \( \beta \), 
@@ -877,7 +1317,7 @@ $$
 
    
 









    
 
-
    Program example for gradient descent with Ridge Regression 
    
+
    Program example for gradient descent with Ridge Regression 
    
 
    
 
 
@@ -928,7 +1368,7 @@ plt.show()
 
    
 









    
 
-
    Using gradient descent methods, limitations 
    
+
    Using gradient descent methods, limitations 
    
 
 
      
 
    •  Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our energy function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
      • 
@@ -941,12 +1381,12 @@ plt.show()
 
 









      
 
-
      Friday September 25 
      
+
      Friday September 25 
      
 
 
      
 









      
 
-
      Stochastic Gradient Descent 
      
+
      Stochastic Gradient Descent 
      
 
 
      
 Stochastic gradient descent (SGD) and variants thereof address some of
@@ -964,7 +1404,7 @@ $$
 
      
 









      
 
-
      Computation of gradients 
      
+
      Computation of gradients 
      
 
 
      
 This in turn means that the gradient can be
@@ -984,7 +1424,7 @@ minibatches. We denote these minibatches by \( B_k \) where
 
      
 









      
 
-
      SGD example 
      
+
      SGD example 
      
 As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
 and we choose to have \( M=5 \) minibathces,
 then each minibatch contains two data points. In particular we have
@@ -1008,7 +1448,7 @@ $$
 
      
 









      
 
-
      The gradient step 
      
+
      The gradient step 
      
 
 
      
 Thus a gradient descent step now looks like 
@@ -1027,7 +1467,7 @@ the number of minibatches, as exemplified in the code below.
 
      
 









      
 
-
      Simple example code 
      
+
      Simple example code 
      
 
 
      
 
@@ -1059,7 +1499,7 @@ all \( n \) datapoints.
 
      
 









      
 
-
      When do we stop? 
      
+
      When do we stop? 
      
 
 
      
 A natural question is when do we stop the search for a new minimum?
@@ -1076,7 +1516,7 @@ gave the lowest value.
 
      
 









      
 
-
      Slightly different approach 
      
+
      Slightly different approach 
      
 
 
      
 Another approach is to let the step length \( \gamma_j \) depend on the
@@ -1124,7 +1564,7 @@ j = 0
 
      
 









      
 
-
      Program for stochastic gradient 
      
+
      Program for stochastic gradient 
      
 
 
      
 
@@ -1200,7 +1640,7 @@ plt.show()
 
      
 









      
 
-
      Momentum based GD 
      
+
      Momentum based GD 
      
 
 
      
 The stochastic gradient descent (SGD) is almost always used with a
@@ -1237,7 +1677,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
 
      
 









      
 
-
      More on momentum based approaches 
      
+
      More on momentum based approaches 
      
 
 
      
 Let us try to get more intuition from these equations. It is helpful
@@ -1267,7 +1707,7 @@ $$
 
      
 









      
 
-
      Momentum parameter 
      
+
      Momentum parameter 
      
 
 
      
 Notice that this equation is identical to previous one if we identify
@@ -1323,7 +1763,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
 
      
 









      
 
-
      Second moment of the gradient 
      
+
      Second moment of the gradient 
      
 
 
      
 In stochastic gradient descent, with and without momentum, we still
@@ -1351,7 +1791,7 @@ ADAM.
 
      
 









      
 
-
      RMS prop 
      
+
      RMS prop 
      
 
 
      
 In RMS prop, in addition to keeping a running average of the first
@@ -1382,7 +1822,7 @@ learning rate for flat directions.
 
      
 









      
 
-
      ADAM optimizer 
      
+
      ADAM optimizer 
      
 
 
      
 A related algorithm is the ADAM optimizer. In ADAM, we keep a running
@@ -1431,7 +1871,7 @@ $$
 
      
 









      
 
-
      Practical tips 
      
+
      Practical tips 
      
 
 
        
 
      •  Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
        • 
@@ -1445,7 +1885,7 @@ Geron's text, see chapter 11, has several interesting discussions.
 
        
 









        
 
-
        Automatic differentiation 
        
+
        Automatic differentiation 
        
 
 
        
 Automatic differentiation (AD), 
@@ -1530,7 +1970,7 @@ plt.show()
 
        
 
 
-
        Using autograd 
        
+
        Using autograd 
        
 
 
        
 Here we
@@ -1563,7 +2003,7 @@ grad_analytical = 3*a**Autograd with more complicated functions 
+
        Autograd with more complicated functions 
        
 
 
        
 To differentiate with respect to two (or more) arguments of a Python
@@ -1613,7 +2053,7 @@ Note that the grad function will not produce the true gradient of the function.
 
        
 









        
 
-
        More complicated functions using the elements of their arguments directly 
        
+
        More complicated functions using the elements of their arguments directly 
        
 
 
        
 
@@ -1647,7 +2087,7 @@ could expect form a gradient-evaluting function.
 
        
 
 
-
        Functions using mathematical functions from Numpy 
        
+
        Functions using mathematical functions from Numpy 
        
 
 
        
 
@@ -1673,7 +2113,7 @@ f4_grad_analytical = x/np.sqrt(1 + x**
 









        
 
-
        More autograd 
        
+
        More autograd 
        
 
 
        
 
@@ -1696,7 +2136,7 @@ x = 2.7
 
        
 









        
 
-
        And  with loops 
        
+
        And  with loops 
        
 
 
        
 
@@ -1742,7 +2182,7 @@ f6_grad_analytical = 0
 
        
 









        
 
-
        Using recursion 
        
+
        Using recursion 
        
 
        
 
 
@@ -1780,7 +2220,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
 
        
 









        
 
-
        Unsupported functions 
        
+
        Unsupported functions 
        
 Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
 
 
        
@@ -1806,7 +2246,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
 
        
 









        
 
-
        The syntax a.dot(b) when finding the dot product 
        
+
        The syntax a.dot(b) when finding the dot product 
        
 
        
 
 
@@ -1848,7 +2288,7 @@ x = np.array([3.0,Recommended to avoid 
+
        Recommended to avoid 
        
 The documentation recommends to avoid inplace operations such as
 
        
 
@@ -1858,619 +2298,6 @@ a -= b
 a*= b
 a /=b
 
    
-
    
-









    
-
-
    Standard steepest descent 
    
-
-
    
-Before we proceed, we would like to discuss the approach called the
-standard Steepest descent, which again leads to us having to be able
-to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
-
-
    
-The success of the CG method
-for finding solutions of non-linear problems is based on the theory
-of conjugate gradients for linear systems of equations. It belongs to
-the class of iterative methods for solving problems from linear
-algebra of the type 
-$$
-\begin{equation*} 
-\hat{A}\hat{x} = \hat{b}.
-\end{equation*} 
-$$
-
-
    
-In the iterative process we end up with a problem like
-
-$$
-\begin{equation*}
-  \hat{r}= \hat{b}-\hat{A}\hat{x},
-\end{equation*}
-$$
-
-where \( \hat{r} \) is the so-called residual or error in the iterative process.
-
-
    
-When we have found the exact solution, \( \hat{r}=0 \).
-
-
    
-









    
-
-
    Gradient method 
    
-
-
    
-The residual is zero when we reach the minimum of the quadratic equation
-$$
-\begin{equation*}
-  P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
-\end{equation*}
-$$
-
-
    
-with the constraint that the matrix \( \hat{A} \) is positive definite and
-symmetric.  This defines also the Hessian and we want it to be  positive definite.
-
-
    
-









    
-
-
    Steepest descent  method 
    
-
-
    
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
-
-
    
-









    
-
-
    Steepest descent  method 
    
-
    
-
-
    
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
-  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) 
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and 
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-
-
-
    
-
-
-
    
-









    
-
-
    Final expressions 
    
-
    
-
-
    
-We can compute the residual iteratively as
-$$
-\begin{equation*}
-\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
- \end{equation*}
-$$
-
-which equals
-$$
-\begin{equation*}
-\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k),
- \end{equation*}
-$$
-
-or
-$$
-\begin{equation*}
-(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k,
- \end{equation*}
-$$
-
-which gives
-
-$$
-\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k}
-$$
-
-leading to the iterative scheme
-$$
-\begin{equation*}
-\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k},
- \end{equation*}
-$$
-
    
-
-
-
    
-









    
-
-
    Code examples for steepest descent 
    
-
-
    
-









    
-
-
    Simple codes for  steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come 
    
-
    
-
-
    
-
    
-
-
-
    #include <cmath>
-#include <iostream>
-#include <fstream>
-#include <iomanip>
-#include "vectormatrixclass.h"
-using namespace  std;
-//   Main function begins here
-int main(int  argc, char * argv[]){
-  int dim = 2;
-  Vector x(dim),xsd(dim), b(dim),x0(dim);
-  Matrix A(dim,dim);
-
-  // Set our initial guess
-  x0(0) = x0(1) = 0;
-  // Set the matrix
-  A(0,0) =  3;    A(1,0) =  2;   A(0,1) =  2;   A(1,1) =  6;
-  b(0) = 2; b(1) = -8;
-  cout << "The Matrix A that we are using: " << endl;
-  A.Print();
-  cout << endl;
-  xsd = SteepestDescent(A,b,x0);
-  cout << "The approximate solution using Steepest Descent is: " << endl;
-  xsd.Print();
-  cout << endl;
-}
-
    
-
-
    
-
-
-
    
-









    
-
-
    The routine for the steepest descent method 
    
-
    
-
-
    
-
    
-
-
-
    Vector SteepestDescent(Matrix A, Vector b, Vector x0){
-  int IterMax, i;
-  int dim = x0.Dimension();
-  const double tolerance = 1.0e-14;
-  Vector x(dim),f(dim),z(dim);
-  double c,alpha,d;
-  IterMax = 30;
-  x = x0;
-  r = A*x-b;
-  i = 0;
-  while (i <= IterMax){
-    z = A*r;
-    c = dot(r,r);
-    alpha = c/dot(r,z);
-    x = x - alpha*r;
-    r =  A*x-b;
-    if(sqrt(dot(r,r)) < tolerance) break;
-    i++;
-  }
-  return x;
-}
-
    
-
-
    
-
-
-
    
-









    
-
-
    Steepest descent example 
    
-
-
    
-
-
-
    import numpy as np
-import numpy.linalg as la
-
-import scipy.optimize as sopt
-
-import matplotlib.pyplot as pt
-from mpl_toolkits.mplot3d import axes3d
-
-def f(x):
-    return 0.5*x[0]**2 + 2.5*x[1]**2
-
-def df(x):
-    return np.array([x[0], 5*x[1]])
-
-fig = pt.figure()
-ax = fig.gca(projection="3d")
-
-xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
-fmesh = f(np.array([xmesh, ymesh]))
-ax.plot_surface(xmesh, ymesh, fmesh)
-
    
-
    
-And then as countor plot
-
    
-
-
-
    pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh)
-guesses = [np.array([2, 2./5])]
-
    
-
    
-Find guesses
-
    
-
-
-
    x = guesses[-1]
-s = -df(x)
-
    
-
    
-Run it!
-
    
-
-
-
    def f1d(alpha):
-    return f(x + alpha*s)
-
-alpha_opt = sopt.golden(f1d)
-next_guess = x + alpha_opt * s
-guesses.append(next_guess)
-print(next_guess)
-
    
-
    
-What happened?
-
    
-
-
-
    pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh, 50)
-it_array = np.array(guesses)
-pt.plot(it_array.T[0], it_array.T[1], "x-")
-
    
-
    
-









    
-
-
    Conjugate gradient method 
    
-
    
-
-
    
-In the CG method we define so-called conjugate directions and two vectors 
-\( \hat{s} \) and \( \hat{t} \)
-are said to be
-conjugate if
-$$
-\begin{equation*}
-\hat{s}^T\hat{A}\hat{t}= 0.
-\end{equation*}
-$$
-
-The philosophy of the CG method is to perform searches in various conjugate directions
-of our vectors \( \hat{x}_i \) obeying the above criterion, namely
-$$
-\begin{equation*}
-\hat{x}_i^T\hat{A}\hat{x}_j= 0.
-\end{equation*}
-$$
-
-Two vectors are conjugate if they are orthogonal with respect to 
-this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
-
    
-
-
-
    
-









    
-
-
    Conjugate gradient method 
    
-
    
-
-
    
-An example is given by the eigenvectors of the matrix
-$$
-\begin{equation*}
-\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
-\end{equation*}
-$$
-
-which is zero unless \( i=j \).
-
    
-
-
-
    
-









    
-
-
    Conjugate gradient method 
    
-
    
-
-
    
-Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
-\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
-$$
-\begin{equation*}
-\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. 
-\end{equation*}
-$$
-
-We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. 
-Then the \( \hat{p}_{i} \)  form a basis of \( R^n \) and we can expand the solution 
-$  \hat{A}\hat{x} = \hat{b}$ in this basis, namely
-
-$$
-\begin{equation*}
-  \hat{x}  = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
-\end{equation*}
-$$
-
    
-
-
-
    
-









    
-
-
    Conjugate gradient method 
    
-
    
-
-
    
-The coefficients are given by
-$$
-\begin{equation*}
-    \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
-\end{equation*}
-$$
-
-Multiplying with \( \hat{p}_k^T \)  from the left gives
-
-$$
-\begin{equation*}
-  \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
-\end{equation*}
-$$
-
-and we can define the coefficients \( \alpha_k \) as
-
-$$
-\begin{equation*}
-    \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
-\end{equation*}
-$$
-
    
-
-
-
    
-









    
-
-
    Conjugate gradient method and iterations 
    
-
    
-
-
    
-
-
    
-If we choose the conjugate vectors \( \hat{p}_k \) carefully, 
-then we may not need all of them to obtain a good approximation to the solution 
-\( \hat{x} \). 
-We want to regard the conjugate gradient method as an iterative method. 
-This will us to solve systems where \( n \) is so large that the direct 
-method would take too much time.
-
-
    
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
-
    
-
-
-
    
-









    
-
-
    Conjugate gradient method 
    
-
    
-
-
    
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
-  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{p}_1 \) 
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and 
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-The other vectors in the basis will be conjugate to the gradient, 
-hence the name conjugate gradient method.
-
    
-
-
-
    
-









    
-
-
    Conjugate gradient method 
    
-
    
-
-
    
-Let  \( \hat{r}_k \) be the residual at the \( k \)-th step:
-$$
-\begin{equation*}
-\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
-\end{equation*}
-$$
-
-Note that \( \hat{r}_k \) is the negative gradient of \( f \) at 
-\( \hat{x}=\hat{x}_k \), 
-so the gradient descent method would be to move in the direction \( \hat{r}_k \). 
-Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other, 
-so we take the direction closest to the gradient \( \hat{r}_k \)  
-under the conjugacy constraint. 
-This gives the following expression
-$$
-\begin{equation*}
-\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
-\end{equation*}
-$$
-
    
-
-
-
    
-









    
-
-
    Conjugate gradient method 
    
-
    
-
-
    
-We can also  compute the residual iteratively as
-$$
-\begin{equation*}
-\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
- \end{equation*}
-$$
-
-which equals
-$$
-\begin{equation*}
-\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k),
- \end{equation*}
-$$
-
-or
-$$
-\begin{equation*}
-(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k,
- \end{equation*}
-$$
-
-which gives
-
-$$
-\begin{equation*}
-\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k},
- \end{equation*}
-$$
-
    
-
-
-
    
-









    
-
-
    Simple implementation of the Conjugate gradient algorithm 
    
-
    
-
-
    
-
    
-
-
-
      Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
-  int dim = x0.Dimension();
-  const double tolerance = 1.0e-14;
-  Vector x(dim),r(dim),v(dim),z(dim);
-  double c,t,d;
-
-  x = x0;
-  r = b - A*x;
-  v = r;
-  c = dot(r,r);
-  int i = 0; IterMax = dim;
-  while(i <= IterMax){
-    z = A*v;
-    t = c/dot(v,z);
-    x = x + t*v;
-    r = r - t*z;
-    d = dot(r,r);
-    if(sqrt(d) < tolerance)
-      break;
-    v = r + (d/c)*v;
-    c = d;  i++;
-  }
-  return x;
-} 
-
    
-
-
    
-
-
-
    
-









    
-
-
    Broyden–Fletcher–Goldfarb–Shanno algorithm 
    
-
    
-
-
    
-The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \)  is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that  \( \mathbf {x} \)  can take.
-
-
    
-The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
-
-
    
-The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
-$$
-B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
-$$
-
-
    
-where \( B_{k} \) is an approximation to the Hessian matrix, which is
-updated iteratively at each stage, and \( \nabla f(\mathbf {x} _{k}) \)
-is the gradient of the function
-evaluated at \( x_k \). 
-A line search in the direction \( p_k \) is then used to
-find the next point \( x_{k+1} \) by minimising 
-$$
-f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),
-$$
-
-over the scalar \( \alpha > 0 \).
-
-
-
    
-
-
 
    
 
 
diff --git a/doc/pub/week39/html/week39.html b/doc/pub/week39/html/week39.html
index 23b999f95..45ab68aa8 100644
--- a/doc/pub/week39/html/week39.html
+++ b/doc/pub/week39/html/week39.html
@@ -92,94 +92,76 @@ div { text-align: justify; text-justify: inter-word; }
               ('Conditions on convex functions', 2, None, '___sec16'),
               ('More on convex functions', 2, None, '___sec17'),
               ('Some simple problems', 2, None, '___sec18'),
-              ('Revisiting our first homework', 2, None, '___sec19'),
-              ('Gradient descent example', 2, None, '___sec20'),
-              ('The derivative of the cost/loss function', 2, None, '___sec21'),
-              ('The Hessian matrix', 2, None, '___sec22'),
-              ('Simple program', 2, None, '___sec23'),
-              ('Gradient Descent Example', 2, None, '___sec24'),
+              ('Standard steepest descent', 2, None, '___sec19'),
+              ('Gradient method', 2, None, '___sec20'),
+              ('Steepest descent  method', 2, None, '___sec21'),
+              ('Steepest descent  method', 2, None, '___sec22'),
+              ('Final expressions', 2, None, '___sec23'),
+              ('Steepest descent example', 2, None, '___sec24'),
+              ('Conjugate gradient method', 2, None, '___sec25'),
+              ('Conjugate gradient method', 2, None, '___sec26'),
+              ('Conjugate gradient method', 2, None, '___sec27'),
+              ('Conjugate gradient method', 2, None, '___sec28'),
+              ('Conjugate gradient method and iterations', 2, None, '___sec29'),
+              ('Conjugate gradient method', 2, None, '___sec30'),
+              ('Conjugate gradient method', 2, None, '___sec31'),
+              ('Conjugate gradient method', 2, None, '___sec32'),
+              ('Revisiting our first homework', 2, None, '___sec33'),
+              ('Gradient descent example', 2, None, '___sec34'),
+              ('The derivative of the cost/loss function', 2, None, '___sec35'),
+              ('The Hessian matrix', 2, None, '___sec36'),
+              ('Simple program', 2, None, '___sec37'),
+              ('Gradient Descent Example', 2, None, '___sec38'),
               ('And a corresponding example using _scikit-learn_',
                2,
                None,
-               '___sec25'),
-              ('Gradient descent and Ridge', 2, None, '___sec26'),
+               '___sec39'),
+              ('Gradient descent and Ridge', 2, None, '___sec40'),
               ('Program example for gradient descent with Ridge Regression',
                2,
                None,
-               '___sec27'),
+               '___sec41'),
               ('Using gradient descent methods, limitations',
                2,
                None,
-               '___sec28'),
-              ('Friday September 25', 2, None, '___sec29'),
-              ('Stochastic Gradient Descent', 2, None, '___sec30'),
-              ('Computation of gradients', 2, None, '___sec31'),
-              ('SGD example', 2, None, '___sec32'),
-              ('The gradient step', 2, None, '___sec33'),
-              ('Simple example code', 2, None, '___sec34'),
-              ('When do we stop?', 2, None, '___sec35'),
-              ('Slightly different approach', 2, None, '___sec36'),
-              ('Program for stochastic gradient', 2, None, '___sec37'),
-              ('Momentum based GD', 2, None, '___sec38'),
-              ('More on momentum based approaches', 2, None, '___sec39'),
-              ('Momentum parameter', 2, None, '___sec40'),
-              ('Second moment of the gradient', 2, None, '___sec41'),
-              ('RMS prop', 2, None, '___sec42'),
-              ('ADAM optimizer', 2, None, '___sec43'),
-              ('Practical tips', 2, None, '___sec44'),
-              ('Automatic differentiation', 2, None, '___sec45'),
-              ('Using autograd', 2, None, '___sec46'),
-              ('Autograd with more complicated functions', 2, None, '___sec47'),
+               '___sec42'),
+              ('Friday September 25', 2, None, '___sec43'),
+              ('Stochastic Gradient Descent', 2, None, '___sec44'),
+              ('Computation of gradients', 2, None, '___sec45'),
+              ('SGD example', 2, None, '___sec46'),
+              ('The gradient step', 2, None, '___sec47'),
+              ('Simple example code', 2, None, '___sec48'),
+              ('When do we stop?', 2, None, '___sec49'),
+              ('Slightly different approach', 2, None, '___sec50'),
+              ('Program for stochastic gradient', 2, None, '___sec51'),
+              ('Momentum based GD', 2, None, '___sec52'),
+              ('More on momentum based approaches', 2, None, '___sec53'),
+              ('Momentum parameter', 2, None, '___sec54'),
+              ('Second moment of the gradient', 2, None, '___sec55'),
+              ('RMS prop', 2, None, '___sec56'),
+              ('ADAM optimizer', 2, None, '___sec57'),
+              ('Practical tips', 2, None, '___sec58'),
+              ('Automatic differentiation', 2, None, '___sec59'),
+              ('Using autograd', 2, None, '___sec60'),
+              ('Autograd with more complicated functions', 2, None, '___sec61'),
               ('More complicated functions using the elements of their '
                'arguments directly',
                2,
                None,
-               '___sec48'),
+               '___sec62'),
               ('Functions using mathematical functions from Numpy',
                2,
                None,
-               '___sec49'),
-              ('More autograd', 2, None, '___sec50'),
-              ('And  with loops', 2, None, '___sec51'),
-              ('Using recursion', 2, None, '___sec52'),
-              ('Unsupported functions', 2, None, '___sec53'),
+               '___sec63'),
+              ('More autograd', 2, None, '___sec64'),
+              ('And  with loops', 2, None, '___sec65'),
+              ('Using recursion', 2, None, '___sec66'),
+              ('Unsupported functions', 2, None, '___sec67'),
               ('The syntax a.dot(b) when finding the dot product',
                2,
                None,
-               '___sec54'),
-              ('Recommended to avoid', 2, None, '___sec55'),
-              ('Standard steepest descent', 2, None, '___sec56'),
-              ('Gradient method', 2, None, '___sec57'),
-              ('Steepest descent  method', 2, None, '___sec58'),
-              ('Steepest descent  method', 2, None, '___sec59'),
-              ('Final expressions', 2, None, '___sec60'),
-              ('Code examples for steepest descent', 2, None, '___sec61'),
-              ('Simple codes for  steepest descent and conjugate gradient '
-               'using a $2\\times 2$ matrix, in c++, Python code to come',
-               2,
-               None,
-               '___sec62'),
-              ('The routine for the steepest descent method',
-               2,
-               None,
-               '___sec63'),
-              ('Steepest descent example', 2, None, '___sec64'),
-              ('Conjugate gradient method', 2, None, '___sec65'),
-              ('Conjugate gradient method', 2, None, '___sec66'),
-              ('Conjugate gradient method', 2, None, '___sec67'),
-              ('Conjugate gradient method', 2, None, '___sec68'),
-              ('Conjugate gradient method and iterations', 2, None, '___sec69'),
-              ('Conjugate gradient method', 2, None, '___sec70'),
-              ('Conjugate gradient method', 2, None, '___sec71'),
-              ('Conjugate gradient method', 2, None, '___sec72'),
-              ('Simple implementation of the Conjugate gradient algorithm',
-               2,
-               None,
-               '___sec73'),
-              ('Broyden–Fletcher–Goldfarb–Shanno algorithm',
-               2,
-               None,
-               '___sec74')]}
+               '___sec68'),
+              ('Recommended to avoid', 2, None, '___sec69')]}
 end of tocinfo -->
 
 
@@ -673,10 +655,468 @@ This result means that if we know that the cost/loss function is convex and we a
 
 Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
 
+
    
+









    
+
+
    Standard steepest descent 
    
+
+
    
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent, which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
    
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type 
+$$
+\begin{equation*} 
+\hat{A}\hat{x} = \hat{b}.
+\end{equation*} 
+$$
+
+
    
+In the iterative process we end up with a problem like
+
+$$
+\begin{equation*}
+  \hat{r}= \hat{b}-\hat{A}\hat{x},
+\end{equation*}
+$$
+
+where \( \hat{r} \) is the so-called residual or error in the iterative process.
+
+
    
+When we have found the exact solution, \( \hat{r}=0 \).
+
+
    
+









    
+
+
    Gradient method 
    
+
+
    
+The residual is zero when we reach the minimum of the quadratic equation
+$$
+\begin{equation*}
+  P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
+\end{equation*}
+$$
+
+
    
+with the constraint that the matrix \( \hat{A} \) is positive definite and
+symmetric.  This defines also the Hessian and we want it to be  positive definite.
+
+
    
+









    
+
+
    Steepest descent  method 
    
+
+
    
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
    
+









    
+
+
    Steepest descent  method 
    
+
    
+
+
    
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) 
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and 
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
+
+
    
+
+
+
    
+









    
+
+
    Final expressions 
    
+
    
+
+
    
+We can compute the residual iteratively as
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
+$$
+
+which equals
+$$
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k),
+ \end{equation*}
+$$
+
+or
+$$
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k,
+ \end{equation*}
+$$
+
+which gives
+
+$$
+\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k}
+$$
+
+leading to the iterative scheme
+$$
+\begin{equation*}
+\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k},
+ \end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Steepest descent example 
    
+
+
    
+
+
+
    import numpy as np
+import numpy.linalg as la
+
+import scipy.optimize as sopt
+
+import matplotlib.pyplot as pt
+from mpl_toolkits.mplot3d import axes3d
+
+def f(x):
+    return 0.5*x[0]**2 + 2.5*x[1]**2
+
+def df(x):
+    return np.array([x[0], 5*x[1]])
+
+fig = pt.figure()
+ax = fig.gca(projection="3d")
+
+xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
+fmesh = f(np.array([xmesh, ymesh]))
+ax.plot_surface(xmesh, ymesh, fmesh)
+
    
+
    
+And then as countor plot
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh)
+guesses = [np.array([2, 2./5])]
+
    
+
    
+Find guesses
+
    
+
+
+
    x = guesses[-1]
+s = -df(x)
+
    
+
    
+Run it!
+
    
+
+
+
    def f1d(alpha):
+    return f(x + alpha*s)
+
+alpha_opt = sopt.golden(f1d)
+next_guess = x + alpha_opt * s
+guesses.append(next_guess)
+print(next_guess)
+
    
+
    
+What happened?
+
    
+
+
+
    pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh, 50)
+it_array = np.array(guesses)
+pt.plot(it_array.T[0], it_array.T[1], "x-")
+
    
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+In the CG method we define so-called conjugate directions and two vectors 
+\( \hat{s} \) and \( \hat{t} \)
+are said to be
+conjugate if
+$$
+\begin{equation*}
+\hat{s}^T\hat{A}\hat{t}= 0.
+\end{equation*}
+$$
+
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \hat{x}_i \) obeying the above criterion, namely
+$$
+\begin{equation*}
+\hat{x}_i^T\hat{A}\hat{x}_j= 0.
+\end{equation*}
+$$
+
+Two vectors are conjugate if they are orthogonal with respect to 
+this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+An example is given by the eigenvectors of the matrix
+$$
+\begin{equation*}
+\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
+\end{equation*}
+$$
+
+which is zero unless \( i=j \).
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
+$$
+\begin{equation*}
+\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. 
+\end{equation*}
+$$
+
+We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. 
+Then the \( \hat{p}_{i} \)  form a basis of \( R^n \) and we can expand the solution 
+$  \hat{A}\hat{x} = \hat{b}$ in this basis, namely
+
+$$
+\begin{equation*}
+  \hat{x}  = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
+\end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+The coefficients are given by
+$$
+\begin{equation*}
+    \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
+\end{equation*}
+$$
+
+Multiplying with \( \hat{p}_k^T \)  from the left gives
+
+$$
+\begin{equation*}
+  \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
+\end{equation*}
+$$
+
+and we can define the coefficients \( \alpha_k \) as
+
+$$
+\begin{equation*}
+    \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
+\end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method and iterations 
    
+
    
+
+
    
+
+
    
+If we choose the conjugate vectors \( \hat{p}_k \) carefully, 
+then we may not need all of them to obtain a good approximation to the solution 
+\( \hat{x} \). 
+We want to regard the conjugate gradient method as an iterative method. 
+This will us to solve systems where \( n \) is so large that the direct 
+method would take too much time.
+
+
    
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). 
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+  f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. 
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{p}_1 \) 
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), 
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and 
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+The other vectors in the basis will be conjugate to the gradient, 
+hence the name conjugate gradient method.
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+Let  \( \hat{r}_k \) be the residual at the \( k \)-th step:
+$$
+\begin{equation*}
+\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
+\end{equation*}
+$$
+
+Note that \( \hat{r}_k \) is the negative gradient of \( f \) at 
+\( \hat{x}=\hat{x}_k \), 
+so the gradient descent method would be to move in the direction \( \hat{r}_k \). 
+Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other, 
+so we take the direction closest to the gradient \( \hat{r}_k \)  
+under the conjugacy constraint. 
+This gives the following expression
+$$
+\begin{equation*}
+\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
+\end{equation*}
+$$
+
    
+
+
+
    
+









    
+
+
    Conjugate gradient method 
    
+
    
+
+
    
+We can also  compute the residual iteratively as
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
+$$
+
+which equals
+$$
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k),
+ \end{equation*}
+$$
+
+or
+$$
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k,
+ \end{equation*}
+$$
+
+which gives
+
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k},
+ \end{equation*}
+$$
+
    
+
+
 
    
 
 
-
    Revisiting our first homework 
    
+
    Revisiting our first homework 
    
 
 
    
 We will use linear regression as a case study for the gradient descent
@@ -709,7 +1149,7 @@ $$
 
    
 
 
-
    Gradient descent example 
    
+
    Gradient descent example 
    
 
 
    
 Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -718,9 +1158,9 @@ Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdo
 It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2}  \) is the design matrix given by
 $$
 X \equiv \begin{bmatrix}
-1 & x_1  \\
-\vdots & \vdots  \\
-1 & x_{100} &  \\
+1 &; x_1  \\
+\vdots &; \vdots  \\
+1 &; x_{100} &;  \\
 \end{bmatrix}.
 $$
 
@@ -734,7 +1174,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
 
    
 









    
 
-
    The derivative of the cost/loss function 
    
+
    The derivative of the cost/loss function 
    
 
 
    
 Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show  that the gradient can be written as
@@ -749,12 +1189,12 @@ where \( X \) is the design matrix defined above.
 
    
 









    
 
-
    The Hessian matrix 
    
+
    The Hessian matrix 
    
 The Hessian matrix of \( C(\beta) \) is given by 
 $$
 \hat{H} \equiv \begin{bmatrix}
-\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1}  \\
-\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &  \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0^2} &; \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1}  \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} &; \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &;  \\
 \end{bmatrix} = 2X^T X.
 $$
 
@@ -763,7 +1203,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
 
    
 









    
 
-
    Simple program 
    
+
    Simple program 
    
 
 
    
 We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to 
@@ -783,7 +1223,7 @@ And finally we can compare our solution for \( \beta \) with the analytic result
 
    
 









    
 
-
    Gradient Descent Example 
    
+
    Gradient Descent Example 
    
 
 
    
 Here our simple example
@@ -833,7 +1273,7 @@ plt.show()
 
    
 









    
 
-
    And a corresponding example using scikit-learn 
    
+
    And a corresponding example using scikit-learn 
    
 
 
    
 
@@ -857,7 +1297,7 @@ sgdreg.fit(x,y.
 
    
 
 
-
    Gradient descent and Ridge 
    
+
    Gradient descent and Ridge 
    
 
 
    
 We have also discussed Ridge regression where the loss function contains a regularized term given by the \( L_2 \) norm of \( \beta \), 
@@ -882,7 +1322,7 @@ $$
 
    
 









    
 
-
    Program example for gradient descent with Ridge Regression 
    
+
    Program example for gradient descent with Ridge Regression 
    
 
    
 
 
@@ -933,7 +1373,7 @@ plt.show()
 
    
 









    
 
-
    Using gradient descent methods, limitations 
    
+
    Using gradient descent methods, limitations 
    
 
 
      
 
    •  Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our energy function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
      • 
@@ -946,12 +1386,12 @@ plt.show()
 
 









      
 
-
      Friday September 25 
      
+
      Friday September 25 
      
 
 
      
 









      
 
-
      Stochastic Gradient Descent 
      
+
      Stochastic Gradient Descent 
      
 
 
      
 Stochastic gradient descent (SGD) and variants thereof address some of
@@ -969,7 +1409,7 @@ $$
 
      
 









      
 
-
      Computation of gradients 
      
+
      Computation of gradients 
      
 
 
      
 This in turn means that the gradient can be
@@ -989,7 +1429,7 @@ minibatches. We denote these minibatches by \( B_k \) where
 
      
 









      
 
-
      SGD example 
      
+
      SGD example 
      
 As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
 and we choose to have \( M=5 \) minibathces,
 then each minibatch contains two data points. In particular we have
@@ -1013,7 +1453,7 @@ $$
 
      
 









      
 
-
      The gradient step 
      
+
      The gradient step 
      
 
 
      
 Thus a gradient descent step now looks like 
@@ -1032,7 +1472,7 @@ the number of minibatches, as exemplified in the code below.
 
      
 









      
 
-
      Simple example code 
      
+
      Simple example code 
      
 
 
      
 
@@ -1064,7 +1504,7 @@ all \( n \) datapoints.
 
      
 









      
 
-
      When do we stop? 
      
+
      When do we stop? 
      
 
 
      
 A natural question is when do we stop the search for a new minimum?
@@ -1081,7 +1521,7 @@ gave the lowest value.
 
      
 









      
 
-
      Slightly different approach 
      
+
      Slightly different approach 
      
 
 
      
 Another approach is to let the step length \( \gamma_j \) depend on the
@@ -1129,7 +1569,7 @@ j = 0
 
      
 









      
 
-
      Program for stochastic gradient 
      
+
      Program for stochastic gradient 
      
 
 
      
 
@@ -1205,7 +1645,7 @@ plt.show()
 
      
 









      
 
-
      Momentum based GD 
      
+
      Momentum based GD 
      
 
 
      
 The stochastic gradient descent (SGD) is almost always used with a
@@ -1242,7 +1682,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
 
      
 









      
 
-
      More on momentum based approaches 
      
+
      More on momentum based approaches 
      
 
 
      
 Let us try to get more intuition from these equations. It is helpful
@@ -1272,7 +1712,7 @@ $$
 
      
 









      
 
-
      Momentum parameter 
      
+
      Momentum parameter 
      
 
 
      
 Notice that this equation is identical to previous one if we identify
@@ -1328,7 +1768,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
 
      
 









      
 
-
      Second moment of the gradient 
      
+
      Second moment of the gradient 
      
 
 
      
 In stochastic gradient descent, with and without momentum, we still
@@ -1356,7 +1796,7 @@ ADAM.
 
      
 









      
 
-
      RMS prop 
      
+
      RMS prop 
      
 
 
      
 In RMS prop, in addition to keeping a running average of the first
@@ -1387,7 +1827,7 @@ learning rate for flat directions.
 
      
 









      
 
-
      ADAM optimizer 
      
+
      ADAM optimizer 
      
 
 
      
 A related algorithm is the ADAM optimizer. In ADAM, we keep a running
@@ -1436,7 +1876,7 @@ $$
 
      
 









      
 
-
      Practical tips 
      
+
      Practical tips 
      
 
 
        
 
      •  Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
        • 
@@ -1450,7 +1890,7 @@ Geron's text, see chapter 11, has several interesting discussions.
 
        
 









        
 
-
        Automatic differentiation 
        
+
        Automatic differentiation 
        
 
 
        
 Automatic differentiation (AD), 
@@ -1535,7 +1975,7 @@ plt.show()
 
        
 
 
-
        Using autograd 
        
+
        Using autograd 
        
 
 
        
 Here we
@@ -1568,7 +2008,7 @@ grad_analytical = Autograd with more complicated functions 
+
        Autograd with more complicated functions 
        
 
 
        
 To differentiate with respect to two (or more) arguments of a Python
@@ -1618,7 +2058,7 @@ Note that the grad function will not produce the true gradient of the function.
 
        
 









        
 
-
        More complicated functions using the elements of their arguments directly 
        
+
        More complicated functions using the elements of their arguments directly 
        
 
 
        
 
@@ -1652,7 +2092,7 @@ could expect form a gradient-evaluting function.
 
        
 
 
-
        Functions using mathematical functions from Numpy 
        
+
        Functions using mathematical functions from Numpy 
        
 
 
        
 
@@ -1678,7 +2118,7 @@ f4_grad_analytical = xMore autograd 
+
        More autograd 
        
 
 
        
 
@@ -1701,7 +2141,7 @@ x = 2.7
 
        
 









        
 
-
        And  with loops 
        
+
        And  with loops 
        
 
 
        
 
@@ -1747,7 +2187,7 @@ f6_grad_analytical = Using recursion 
+
        Using recursion 
        
 
        
 
 
@@ -1785,7 +2225,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
 
        
 









        
 
-
        Unsupported functions 
        
+
        Unsupported functions 
        
 Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
 
 
        
@@ -1811,7 +2251,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
 
        
 









        
 
-
        The syntax a.dot(b) when finding the dot product 
        
+
        The syntax a.dot(b) when finding the dot product 
        
 
        
 
 
@@ -1853,7 +2293,7 @@ x = np.a
 
        
 









        
 
-
        Recommended to avoid 
        
+
        Recommended to avoid 
        
 The documentation recommends to avoid inplace operations such as
 
        
 
@@ -1863,619 +2303,6 @@ a -= b
 a*= b
 a /=b
    -

    -









    - -

    Standard steepest descent

    - -

    -Before we proceed, we would like to discuss the approach called the -standard Steepest descent, which again leads to us having to be able -to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). - -

    -The success of the CG method -for finding solutions of non-linear problems is based on the theory -of conjugate gradients for linear systems of equations. It belongs to -the class of iterative methods for solving problems from linear -algebra of the type -$$ -\begin{equation*} -\hat{A}\hat{x} = \hat{b}. -\end{equation*} -$$ - -

    -In the iterative process we end up with a problem like - -$$ -\begin{equation*} - \hat{r}= \hat{b}-\hat{A}\hat{x}, -\end{equation*} -$$ - -where \( \hat{r} \) is the so-called residual or error in the iterative process. - -

    -When we have found the exact solution, \( \hat{r}=0 \). - -

    -









    - -

    Gradient method

    - -

    -The residual is zero when we reach the minimum of the quadratic equation -$$ -\begin{equation*} - P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b}, -\end{equation*} -$$ - -

    -with the constraint that the matrix \( \hat{A} \) is positive definite and -symmetric. This defines also the Hessian and we want it to be positive definite. - -

    -









    - -

    Steepest descent method

    - -

    -We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). -We can assume without loss of generality that -$$ -\begin{equation*} -\hat{x}_0=0, -\end{equation*} -$$ - -or consider the system -$$ -\begin{equation*} -\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, -\end{equation*} -$$ - -instead. - -

    -









    - -

    Steepest descent method

    -
    - -

    -One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form -$$ -\begin{equation*} - f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. -\end{equation*} -$$ - -This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition) -to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), -which equals -$$ -\begin{equation*} -\hat{A}\hat{x}_0-\hat{b}, -\end{equation*} -$$ - -and -\( \hat{x}_0=0 \) it is equal \( -\hat{b} \). - - -

    - - -

    -









    - -

    Final expressions

    -
    - -

    -We can compute the residual iteratively as -$$ -\begin{equation*} -\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, - \end{equation*} -$$ - -which equals -$$ -\begin{equation*} -\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k), - \end{equation*} -$$ - -or -$$ -\begin{equation*} -(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k, - \end{equation*} -$$ - -which gives - -$$ -\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k} -$$ - -leading to the iterative scheme -$$ -\begin{equation*} -\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k}, - \end{equation*} -$$ -

    - - -

    -









    - -

    Code examples for steepest descent

    - -

    -









    - -

    Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come

    -
    - -

    -

    - - -

    #include <cmath>
-#include <iostream>
-#include <fstream>
-#include <iomanip>
-#include "vectormatrixclass.h"
-using namespace  std;
-//   Main function begins here
-int main(int  argc, char * argv[]){
-  int dim = 2;
-  Vector x(dim),xsd(dim), b(dim),x0(dim);
-  Matrix A(dim,dim);
-
-  // Set our initial guess
-  x0(0) = x0(1) = 0;
-  // Set the matrix
-  A(0,0) =  3;    A(1,0) =  2;   A(0,1) =  2;   A(1,1) =  6;
-  b(0) = 2; b(1) = -8;
-  cout << "The Matrix A that we are using: " << endl;
-  A.Print();
-  cout << endl;
-  xsd = SteepestDescent(A,b,x0);
-  cout << "The approximate solution using Steepest Descent is: " << endl;
-  xsd.Print();
-  cout << endl;
-}
-
    - -
    - - -

    -









    - -

    The routine for the steepest descent method

    -
    - -

    -

    - - -

    Vector SteepestDescent(Matrix A, Vector b, Vector x0){
-  int IterMax, i;
-  int dim = x0.Dimension();
-  const double tolerance = 1.0e-14;
-  Vector x(dim),f(dim),z(dim);
-  double c,alpha,d;
-  IterMax = 30;
-  x = x0;
-  r = A*x-b;
-  i = 0;
-  while (i <= IterMax){
-    z = A*r;
-    c = dot(r,r);
-    alpha = c/dot(r,z);
-    x = x - alpha*r;
-    r =  A*x-b;
-    if(sqrt(dot(r,r)) < tolerance) break;
-    i++;
-  }
-  return x;
-}
-
    - -
    - - -

    -









    - -

    Steepest descent example

    - -

    - - -

    import numpy as np
-import numpy.linalg as la
-
-import scipy.optimize as sopt
-
-import matplotlib.pyplot as pt
-from mpl_toolkits.mplot3d import axes3d
-
-def f(x):
-    return 0.5*x[0]**2 + 2.5*x[1]**2
-
-def df(x):
-    return np.array([x[0], 5*x[1]])
-
-fig = pt.figure()
-ax = fig.gca(projection="3d")
-
-xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
-fmesh = f(np.array([xmesh, ymesh]))
-ax.plot_surface(xmesh, ymesh, fmesh)
-
    -

    -And then as countor plot -

    - - -

    pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh)
-guesses = [np.array([2, 2./5])]
-
    -

    -Find guesses -

    - - -

    x = guesses[-1]
-s = -df(x)
-
    -

    -Run it! -

    - - -

    def f1d(alpha):
-    return f(x + alpha*s)
-
-alpha_opt = sopt.golden(f1d)
-next_guess = x + alpha_opt * s
-guesses.append(next_guess)
-print(next_guess)
-
    -

    -What happened? -

    - - -

    pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh, 50)
-it_array = np.array(guesses)
-pt.plot(it_array.T[0], it_array.T[1], "x-")
-
    -

    -









    - -

    Conjugate gradient method

    -
    - -

    -In the CG method we define so-called conjugate directions and two vectors -\( \hat{s} \) and \( \hat{t} \) -are said to be -conjugate if -$$ -\begin{equation*} -\hat{s}^T\hat{A}\hat{t}= 0. -\end{equation*} -$$ - -The philosophy of the CG method is to perform searches in various conjugate directions -of our vectors \( \hat{x}_i \) obeying the above criterion, namely -$$ -\begin{equation*} -\hat{x}_i^T\hat{A}\hat{x}_j= 0. -\end{equation*} -$$ - -Two vectors are conjugate if they are orthogonal with respect to -this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \). -

    - - -

    -









    - -

    Conjugate gradient method

    -
    - -

    -An example is given by the eigenvectors of the matrix -$$ -\begin{equation*} -\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j, -\end{equation*} -$$ - -which is zero unless \( i=j \). -

    - - -

    -









    - -

    Conjugate gradient method

    -
    - -

    -Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size -\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector -$$ -\begin{equation*} -\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. -\end{equation*} -$$ - -We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions. -Then the \( \hat{p}_{i} \) form a basis of \( R^n \) and we can expand the solution -$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely - -$$ -\begin{equation*} - \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i. -\end{equation*} -$$ -

    - - -

    -









    - -

    Conjugate gradient method

    -
    - -

    -The coefficients are given by -$$ -\begin{equation*} - \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. -\end{equation*} -$$ - -Multiplying with \( \hat{p}_k^T \) from the left gives - -$$ -\begin{equation*} - \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b}, -\end{equation*} -$$ - -and we can define the coefficients \( \alpha_k \) as - -$$ -\begin{equation*} - \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k} -\end{equation*} -$$ -

    - - -

    -









    - -

    Conjugate gradient method and iterations

    -
    - -

    - -

    -If we choose the conjugate vectors \( \hat{p}_k \) carefully, -then we may not need all of them to obtain a good approximation to the solution -\( \hat{x} \). -We want to regard the conjugate gradient method as an iterative method. -This will us to solve systems where \( n \) is so large that the direct -method would take too much time. - -

    -We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \). -We can assume without loss of generality that -$$ -\begin{equation*} -\hat{x}_0=0, -\end{equation*} -$$ - -or consider the system -$$ -\begin{equation*} -\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, -\end{equation*} -$$ - -instead. -

    - - -

    -









    - -

    Conjugate gradient method

    -
    - -

    -One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form -$$ -\begin{equation*} - f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. -\end{equation*} -$$ - -This suggests taking the first basis vector \( \hat{p}_1 \) -to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \), -which equals -$$ -\begin{equation*} -\hat{A}\hat{x}_0-\hat{b}, -\end{equation*} -$$ - -and -\( \hat{x}_0=0 \) it is equal \( -\hat{b} \). -The other vectors in the basis will be conjugate to the gradient, -hence the name conjugate gradient method. -

    - - -

    -









    - -

    Conjugate gradient method

    -
    - -

    -Let \( \hat{r}_k \) be the residual at the \( k \)-th step: -$$ -\begin{equation*} -\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k. -\end{equation*} -$$ - -Note that \( \hat{r}_k \) is the negative gradient of \( f \) at -\( \hat{x}=\hat{x}_k \), -so the gradient descent method would be to move in the direction \( \hat{r}_k \). -Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other, -so we take the direction closest to the gradient \( \hat{r}_k \) -under the conjugacy constraint. -This gives the following expression -$$ -\begin{equation*} -\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k. -\end{equation*} -$$ -

    - - -

    -









    - -

    Conjugate gradient method

    -
    - -

    -We can also compute the residual iteratively as -$$ -\begin{equation*} -\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, - \end{equation*} -$$ - -which equals -$$ -\begin{equation*} -\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k), - \end{equation*} -$$ - -or -$$ -\begin{equation*} -(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k, - \end{equation*} -$$ - -which gives - -$$ -\begin{equation*} -\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k}, - \end{equation*} -$$ -

    - - -

    -









    - -

    Simple implementation of the Conjugate gradient algorithm

    -
    - -

    -

    - - -

      Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
-  int dim = x0.Dimension();
-  const double tolerance = 1.0e-14;
-  Vector x(dim),r(dim),v(dim),z(dim);
-  double c,t,d;
-
-  x = x0;
-  r = b - A*x;
-  v = r;
-  c = dot(r,r);
-  int i = 0; IterMax = dim;
-  while(i <= IterMax){
-    z = A*v;
-    t = c/dot(v,z);
-    x = x + t*v;
-    r = r - t*z;
-    d = dot(r,r);
-    if(sqrt(d) < tolerance)
-      break;
-    v = r + (d/c)*v;
-    c = d;  i++;
-  }
-  return x;
-} 
-
    - -
    - - -

    -









    - -

    Broyden–Fletcher–Goldfarb–Shanno algorithm

    -
    - -

    -The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \) is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that \( \mathbf {x} \) can take. - -

    -The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage. - -

    -The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation -$$ -B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}), -$$ - -

    -where \( B_{k} \) is an approximation to the Hessian matrix, which is -updated iteratively at each stage, and \( \nabla f(\mathbf {x} _{k}) \) -is the gradient of the function -evaluated at \( x_k \). -A line search in the direction \( p_k \) is then used to -find the next point \( x_{k+1} \) by minimising -$$ -f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}), -$$ - -over the scalar \( \alpha > 0 \). - - -

    - -

    diff --git a/doc/pub/week39/ipynb/ipynb-week39-src.tar.gz b/doc/pub/week39/ipynb/ipynb-week39-src.tar.gz index e1e5227affc6fc1eee30898c3706695c6fa4bd50..49d930831c737ba37b53c88b6f024cf0c544237e 100644 GIT binary patch literal 192 zcmV;x06+g9iwFQ`CTw2-1MSaC3c@fD2H>uHia9|^nm%wT*o6y0#0#W!YGZ3slN9ak z?E`eBxG5s!+x!eO3^Rvpz1d}fy<2ZFgpeePV9Yd$rzGKAPbiInvXn6rG8&~oC}T18 zfGl^?OJ^+C!ztAnr9t_2ZWt@e4}0cU;F*8oSP28$eQzrzK`M*6QZ?KVYqknR+shmZ ug=TDl#%m`u0+&7Tq7YUn$s51b>ZJL_1pcp|aU92SzV-l=1pBlA2mk<}U{jm` literal 191 zcmV;w06_mAiwFQ1B5Yp(1MSaC3c@fD2H>uHia9}{g};8r^E}V{+5`F=LG1tt007*_SVRB- diff --git a/doc/pub/week39/ipynb/week39.ipynb b/doc/pub/week39/ipynb/week39.ipynb index 68c959a54..db37703e7 100644 --- a/doc/pub/week39/ipynb/week39.ipynb +++ b/doc/pub/week39/ipynb/week39.ipynb @@ -548,8 +548,700 @@ "\n", "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).\n", "\n", + "## Standard steepest descent\n", "\n", "\n", + "Before we proceed, we would like to discuss the approach called the\n", + "**standard Steepest descent**, which again leads to us having to be able\n", + "to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).\n", + "\n", + "[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)\n", + "for finding solutions of non-linear problems is based on the theory\n", + "of conjugate gradients for linear systems of equations. It belongs to\n", + "the class of iterative methods for solving problems from linear\n", + "algebra of the type" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{A}\\hat{x} = \\hat{b}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the iterative process we end up with a problem like" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{r}= \\hat{b}-\\hat{A}\\hat{x},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\hat{r}$ is the so-called residual or error in the iterative process.\n", + "\n", + "When we have found the exact solution, $\\hat{r}=0$.\n", + "\n", + "## Gradient method\n", + "\n", + "The residual is zero when we reach the minimum of the quadratic equation" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "P(\\hat{x})=\\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T\\hat{b},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with the constraint that the matrix $\\hat{A}$ is positive definite and\n", + "symmetric. This defines also the Hessian and we want it to be positive definite. \n", + "\n", + "\n", + "## Steepest descent method\n", + "\n", + "We denote the initial guess for $\\hat{x}$ as $\\hat{x}_0$. \n", + "We can assume without loss of generality that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{x}_0=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or consider the system" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{A}\\hat{z} = \\hat{b}-\\hat{A}\\hat{x}_0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "instead.\n", + "\n", + "\n", + "## Steepest descent method\n", + "One can show that the solution $\\hat{x}$ is also the unique minimizer of the quadratic form" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f(\\hat{x}) = \\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T \\hat{x} , \\quad \\hat{x}\\in\\mathbf{R}^n.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This suggests taking the first basis vector $\\hat{r}_1$ (see below for definition) \n", + "to be the gradient of $f$ at $\\hat{x}=\\hat{x}_0$, \n", + "which equals" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{A}\\hat{x}_0-\\hat{b},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and \n", + "$\\hat{x}_0=0$ it is equal $-\\hat{b}$.\n", + "\n", + "\n", + "\n", + "## Final expressions\n", + "We can compute the residual iteratively as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{r}_{k+1}=\\hat{b}-\\hat{A}\\hat{x}_{k+1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which equals" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{b}-\\hat{A}(\\hat{x}_k+\\alpha_k\\hat{r}_k),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(\\hat{b}-\\hat{A}\\hat{x}_k)-\\alpha_k\\hat{A}\\hat{r}_k,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which gives" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\alpha_k = \\frac{\\hat{r}_k^T\\hat{r}_k}{\\hat{r}_k^T\\hat{A}\\hat{r}_k}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "leading to the iterative scheme" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{x}_{k+1}=\\hat{x}_k-\\alpha_k\\hat{r}_{k},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Steepest descent example" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import numpy.linalg as la\n", + "\n", + "import scipy.optimize as sopt\n", + "\n", + "import matplotlib.pyplot as pt\n", + "from mpl_toolkits.mplot3d import axes3d\n", + "\n", + "def f(x):\n", + " return 0.5*x[0]**2 + 2.5*x[1]**2\n", + "\n", + "def df(x):\n", + " return np.array([x[0], 5*x[1]])\n", + "\n", + "fig = pt.figure()\n", + "ax = fig.gca(projection=\"3d\")\n", + "\n", + "xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]\n", + "fmesh = f(np.array([xmesh, ymesh]))\n", + "ax.plot_surface(xmesh, ymesh, fmesh)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And then as countor plot" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "pt.axis(\"equal\")\n", + "pt.contour(xmesh, ymesh, fmesh)\n", + "guesses = [np.array([2, 2./5])]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Find guesses" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "x = guesses[-1]\n", + "s = -df(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Run it!" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def f1d(alpha):\n", + " return f(x + alpha*s)\n", + "\n", + "alpha_opt = sopt.golden(f1d)\n", + "next_guess = x + alpha_opt * s\n", + "guesses.append(next_guess)\n", + "print(next_guess)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "What happened?" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "pt.axis(\"equal\")\n", + "pt.contour(xmesh, ymesh, fmesh, 50)\n", + "it_array = np.array(guesses)\n", + "pt.plot(it_array.T[0], it_array.T[1], \"x-\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Conjugate gradient method\n", + "In the CG method we define so-called conjugate directions and two vectors \n", + "$\\hat{s}$ and $\\hat{t}$\n", + "are said to be\n", + "conjugate if" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{s}^T\\hat{A}\\hat{t}= 0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The philosophy of the CG method is to perform searches in various conjugate directions\n", + "of our vectors $\\hat{x}_i$ obeying the above criterion, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{x}_i^T\\hat{A}\\hat{x}_j= 0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Two vectors are conjugate if they are orthogonal with respect to \n", + "this inner product. Being conjugate is a symmetric relation: if $\\hat{s}$ is conjugate to $\\hat{t}$, then $\\hat{t}$ is conjugate to $\\hat{s}$.\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "An example is given by the eigenvectors of the matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{v}_i^T\\hat{A}\\hat{v}_j= \\lambda\\hat{v}_i^T\\hat{v}_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which is zero unless $i=j$.\n", + "\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "Assume now that we have a symmetric positive-definite matrix $\\hat{A}$ of size\n", + "$n\\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{x}_{i+1}=\\hat{x}_{i}+\\alpha_i\\hat{p}_{i}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We assume that $\\hat{p}_{i}$ is a sequence of $n$ mutually conjugate directions. \n", + "Then the $\\hat{p}_{i}$ form a basis of $R^n$ and we can expand the solution \n", + "$ \\hat{A}\\hat{x} = \\hat{b}$ in this basis, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{x} = \\sum^{n}_{i=1} \\alpha_i \\hat{p}_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Conjugate gradient method\n", + "The coefficients are given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{A}\\mathbf{x} = \\sum^{n}_{i=1} \\alpha_i \\mathbf{A} \\mathbf{p}_i = \\mathbf{b}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Multiplying with $\\hat{p}_k^T$ from the left gives" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{p}_k^T \\hat{A}\\hat{x} = \\sum^{n}_{i=1} \\alpha_i\\hat{p}_k^T \\hat{A}\\hat{p}_i= \\hat{p}_k^T \\hat{b},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and we can define the coefficients $\\alpha_k$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\alpha_k = \\frac{\\hat{p}_k^T \\hat{b}}{\\hat{p}_k^T \\hat{A} \\hat{p}_k}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Conjugate gradient method and iterations\n", + "\n", + "If we choose the conjugate vectors $\\hat{p}_k$ carefully, \n", + "then we may not need all of them to obtain a good approximation to the solution \n", + "$\\hat{x}$. \n", + "We want to regard the conjugate gradient method as an iterative method. \n", + "This will us to solve systems where $n$ is so large that the direct \n", + "method would take too much time.\n", + "\n", + "We denote the initial guess for $\\hat{x}$ as $\\hat{x}_0$. \n", + "We can assume without loss of generality that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{x}_0=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or consider the system" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{A}\\hat{z} = \\hat{b}-\\hat{A}\\hat{x}_0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "instead.\n", + "\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "One can show that the solution $\\hat{x}$ is also the unique minimizer of the quadratic form" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f(\\hat{x}) = \\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T \\hat{x} , \\quad \\hat{x}\\in\\mathbf{R}^n.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This suggests taking the first basis vector $\\hat{p}_1$ \n", + "to be the gradient of $f$ at $\\hat{x}=\\hat{x}_0$, \n", + "which equals" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{A}\\hat{x}_0-\\hat{b},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and \n", + "$\\hat{x}_0=0$ it is equal $-\\hat{b}$.\n", + "The other vectors in the basis will be conjugate to the gradient, \n", + "hence the name conjugate gradient method.\n", + "\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "Let $\\hat{r}_k$ be the residual at the $k$-th step:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{r}_k=\\hat{b}-\\hat{A}\\hat{x}_k.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that $\\hat{r}_k$ is the negative gradient of $f$ at \n", + "$\\hat{x}=\\hat{x}_k$, \n", + "so the gradient descent method would be to move in the direction $\\hat{r}_k$. \n", + "Here, we insist that the directions $\\hat{p}_k$ are conjugate to each other, \n", + "so we take the direction closest to the gradient $\\hat{r}_k$ \n", + "under the conjugacy constraint. \n", + "This gives the following expression" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{p}_{k+1}=\\hat{r}_k-\\frac{\\hat{p}_k^T \\hat{A}\\hat{r}_k}{\\hat{p}_k^T\\hat{A}\\hat{p}_k} \\hat{p}_k.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Conjugate gradient method\n", + "We can also compute the residual iteratively as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{r}_{k+1}=\\hat{b}-\\hat{A}\\hat{x}_{k+1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which equals" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{b}-\\hat{A}(\\hat{x}_k+\\alpha_k\\hat{p}_k),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(\\hat{b}-\\hat{A}\\hat{x}_k)-\\alpha_k\\hat{A}\\hat{p}_k,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which gives" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{r}_{k+1}=\\hat{r}_k-\\hat{A}\\hat{p}_{k},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "\n", "## Revisiting our first homework\n", "\n", @@ -627,9 +1319,9 @@ "source": [ "$$\n", "X \\equiv \\begin{bmatrix}\n", - "1 & x_1 \\\\\n", - "\\vdots & \\vdots \\\\\n", - "1 & x_{100} & \\\\\n", + "1 &; x_1 \\\\\n", + "\\vdots &; \\vdots \\\\\n", + "1 &; x_{100} &; \\\\\n", "\\end{bmatrix}.\n", "$$" ] @@ -688,8 +1380,8 @@ "source": [ "$$\n", "\\hat{H} \\equiv \\begin{bmatrix}\n", - "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0^2} & \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} \\\\\n", - "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} & \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_1^2} & \\\\\n", + "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0^2} &; \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} \\\\\n", + "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} &; \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_1^2} &; \\\\\n", "\\end{bmatrix} = 2X^T X.\n", "$$" ] @@ -735,14 +1427,12 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ - "%matplotlib inline\n", - "\n", "\n", "# Importing various packages\n", "from random import random, seed\n", @@ -794,7 +1484,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 7, "metadata": { "collapsed": false }, @@ -879,7 +1569,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -1062,7 +1752,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 9, "metadata": { "collapsed": false }, @@ -1126,7 +1816,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -1167,7 +1857,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -1722,7 +2412,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -1780,7 +2470,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -1818,7 +2508,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -1871,7 +2561,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -1913,7 +2603,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -1947,7 +2637,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -1981,7 +2671,7 @@ "metadata": {}, "source": [ "1\n", - "2\n", + "7\n", " \n", "<\n", "<\n", @@ -2009,7 +2699,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -2035,7 +2725,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -2084,7 +2774,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -2114,7 +2804,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 21, "metadata": { "collapsed": false }, @@ -2144,7 +2834,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 22, "metadata": { "collapsed": false }, @@ -2176,7 +2866,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -2187,858 +2877,6 @@ "a*= b\n", "a /=b" ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Standard steepest descent\n", - "\n", - "\n", - "Before we proceed, we would like to discuss the approach called the\n", - "**standard Steepest descent**, which again leads to us having to be able\n", - "to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).\n", - "\n", - "[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)\n", - "for finding solutions of non-linear problems is based on the theory\n", - "of conjugate gradients for linear systems of equations. It belongs to\n", - "the class of iterative methods for solving problems from linear\n", - "algebra of the type" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{A}\\hat{x} = \\hat{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the iterative process we end up with a problem like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{r}= \\hat{b}-\\hat{A}\\hat{x},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\hat{r}$ is the so-called residual or error in the iterative process.\n", - "\n", - "When we have found the exact solution, $\\hat{r}=0$.\n", - "\n", - "## Gradient method\n", - "\n", - "The residual is zero when we reach the minimum of the quadratic equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(\\hat{x})=\\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T\\hat{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the constraint that the matrix $\\hat{A}$ is positive definite and\n", - "symmetric. This defines also the Hessian and we want it to be positive definite. \n", - "\n", - "\n", - "## Steepest descent method\n", - "\n", - "We denote the initial guess for $\\hat{x}$ as $\\hat{x}_0$. \n", - "We can assume without loss of generality that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{x}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or consider the system" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{A}\\hat{z} = \\hat{b}-\\hat{A}\\hat{x}_0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "instead.\n", - "\n", - "\n", - "## Steepest descent method\n", - "One can show that the solution $\\hat{x}$ is also the unique minimizer of the quadratic form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(\\hat{x}) = \\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T \\hat{x} , \\quad \\hat{x}\\in\\mathbf{R}^n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This suggests taking the first basis vector $\\hat{r}_1$ (see below for definition) \n", - "to be the gradient of $f$ at $\\hat{x}=\\hat{x}_0$, \n", - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{A}\\hat{x}_0-\\hat{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and \n", - "$\\hat{x}_0=0$ it is equal $-\\hat{b}$.\n", - "\n", - "\n", - "\n", - "## Final expressions\n", - "We can compute the residual iteratively as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{r}_{k+1}=\\hat{b}-\\hat{A}\\hat{x}_{k+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{b}-\\hat{A}(\\hat{x}_k+\\alpha_k\\hat{r}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\hat{b}-\\hat{A}\\hat{x}_k)-\\alpha_k\\hat{A}\\hat{r}_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\alpha_k = \\frac{\\hat{r}_k^T\\hat{r}_k}{\\hat{r}_k^T\\hat{A}\\hat{r}_k}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "leading to the iterative scheme" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{x}_{k+1}=\\hat{x}_k-\\alpha_k\\hat{r}_{k},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Code examples for steepest descent\n", - "\n", - "## Simple codes for steepest descent and conjugate gradient using a $2\\times 2$ matrix, in c++, Python code to come" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \"vectormatrixclass.h\"\n", - " using namespace std;\n", - " // Main function begins here\n", - " int main(int argc, char * argv[]){\n", - " int dim = 2;\n", - " Vector x(dim),xsd(dim), b(dim),x0(dim);\n", - " Matrix A(dim,dim);\n", - " \n", - " // Set our initial guess\n", - " x0(0) = x0(1) = 0;\n", - " // Set the matrix\n", - " A(0,0) = 3; A(1,0) = 2; A(0,1) = 2; A(1,1) = 6;\n", - " b(0) = 2; b(1) = -8;\n", - " cout << \"The Matrix A that we are using: \" << endl;\n", - " A.Print();\n", - " cout << endl;\n", - " xsd = SteepestDescent(A,b,x0);\n", - " cout << \"The approximate solution using Steepest Descent is: \" << endl;\n", - " xsd.Print();\n", - " cout << endl;\n", - " }\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The routine for the steepest descent method" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " Vector SteepestDescent(Matrix A, Vector b, Vector x0){\n", - " int IterMax, i;\n", - " int dim = x0.Dimension();\n", - " const double tolerance = 1.0e-14;\n", - " Vector x(dim),f(dim),z(dim);\n", - " double c,alpha,d;\n", - " IterMax = 30;\n", - " x = x0;\n", - " r = A*x-b;\n", - " i = 0;\n", - " while (i <= IterMax){\n", - " z = A*r;\n", - " c = dot(r,r);\n", - " alpha = c/dot(r,z);\n", - " x = x - alpha*r;\n", - " r = A*x-b;\n", - " if(sqrt(dot(r,r)) < tolerance) break;\n", - " i++;\n", - " }\n", - " return x;\n", - " }\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Steepest descent example" - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import numpy.linalg as la\n", - "\n", - "import scipy.optimize as sopt\n", - "\n", - "import matplotlib.pyplot as pt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "def f(x):\n", - " return 0.5*x[0]**2 + 2.5*x[1]**2\n", - "\n", - "def df(x):\n", - " return np.array([x[0], 5*x[1]])\n", - "\n", - "fig = pt.figure()\n", - "ax = fig.gca(projection=\"3d\")\n", - "\n", - "xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]\n", - "fmesh = f(np.array([xmesh, ymesh]))\n", - "ax.plot_surface(xmesh, ymesh, fmesh)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And then as countor plot" - ] - }, - { - "cell_type": "code", - "execution_count": 20, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "pt.axis(\"equal\")\n", - "pt.contour(xmesh, ymesh, fmesh)\n", - "guesses = [np.array([2, 2./5])]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Find guesses" - ] - }, - { - "cell_type": "code", - "execution_count": 21, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "x = guesses[-1]\n", - "s = -df(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Run it!" - ] - }, - { - "cell_type": "code", - "execution_count": 22, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "def f1d(alpha):\n", - " return f(x + alpha*s)\n", - "\n", - "alpha_opt = sopt.golden(f1d)\n", - "next_guess = x + alpha_opt * s\n", - "guesses.append(next_guess)\n", - "print(next_guess)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "What happened?" - ] - }, - { - "cell_type": "code", - "execution_count": 23, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "pt.axis(\"equal\")\n", - "pt.contour(xmesh, ymesh, fmesh, 50)\n", - "it_array = np.array(guesses)\n", - "pt.plot(it_array.T[0], it_array.T[1], \"x-\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "In the CG method we define so-called conjugate directions and two vectors \n", - "$\\hat{s}$ and $\\hat{t}$\n", - "are said to be\n", - "conjugate if" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{s}^T\\hat{A}\\hat{t}= 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The philosophy of the CG method is to perform searches in various conjugate directions\n", - "of our vectors $\\hat{x}_i$ obeying the above criterion, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{x}_i^T\\hat{A}\\hat{x}_j= 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Two vectors are conjugate if they are orthogonal with respect to \n", - "this inner product. Being conjugate is a symmetric relation: if $\\hat{s}$ is conjugate to $\\hat{t}$, then $\\hat{t}$ is conjugate to $\\hat{s}$.\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "An example is given by the eigenvectors of the matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{v}_i^T\\hat{A}\\hat{v}_j= \\lambda\\hat{v}_i^T\\hat{v}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is zero unless $i=j$.\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "Assume now that we have a symmetric positive-definite matrix $\\hat{A}$ of size\n", - "$n\\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{x}_{i+1}=\\hat{x}_{i}+\\alpha_i\\hat{p}_{i}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We assume that $\\hat{p}_{i}$ is a sequence of $n$ mutually conjugate directions. \n", - "Then the $\\hat{p}_{i}$ form a basis of $R^n$ and we can expand the solution \n", - "$ \\hat{A}\\hat{x} = \\hat{b}$ in this basis, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{x} = \\sum^{n}_{i=1} \\alpha_i \\hat{p}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "The coefficients are given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}\\mathbf{x} = \\sum^{n}_{i=1} \\alpha_i \\mathbf{A} \\mathbf{p}_i = \\mathbf{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Multiplying with $\\hat{p}_k^T$ from the left gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{p}_k^T \\hat{A}\\hat{x} = \\sum^{n}_{i=1} \\alpha_i\\hat{p}_k^T \\hat{A}\\hat{p}_i= \\hat{p}_k^T \\hat{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we can define the coefficients $\\alpha_k$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\alpha_k = \\frac{\\hat{p}_k^T \\hat{b}}{\\hat{p}_k^T \\hat{A} \\hat{p}_k}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method and iterations\n", - "\n", - "If we choose the conjugate vectors $\\hat{p}_k$ carefully, \n", - "then we may not need all of them to obtain a good approximation to the solution \n", - "$\\hat{x}$. \n", - "We want to regard the conjugate gradient method as an iterative method. \n", - "This will us to solve systems where $n$ is so large that the direct \n", - "method would take too much time.\n", - "\n", - "We denote the initial guess for $\\hat{x}$ as $\\hat{x}_0$. \n", - "We can assume without loss of generality that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{x}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or consider the system" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{A}\\hat{z} = \\hat{b}-\\hat{A}\\hat{x}_0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "instead.\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "One can show that the solution $\\hat{x}$ is also the unique minimizer of the quadratic form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(\\hat{x}) = \\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T \\hat{x} , \\quad \\hat{x}\\in\\mathbf{R}^n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This suggests taking the first basis vector $\\hat{p}_1$ \n", - "to be the gradient of $f$ at $\\hat{x}=\\hat{x}_0$, \n", - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{A}\\hat{x}_0-\\hat{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and \n", - "$\\hat{x}_0=0$ it is equal $-\\hat{b}$.\n", - "The other vectors in the basis will be conjugate to the gradient, \n", - "hence the name conjugate gradient method.\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "Let $\\hat{r}_k$ be the residual at the $k$-th step:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{r}_k=\\hat{b}-\\hat{A}\\hat{x}_k.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that $\\hat{r}_k$ is the negative gradient of $f$ at \n", - "$\\hat{x}=\\hat{x}_k$, \n", - "so the gradient descent method would be to move in the direction $\\hat{r}_k$. \n", - "Here, we insist that the directions $\\hat{p}_k$ are conjugate to each other, \n", - "so we take the direction closest to the gradient $\\hat{r}_k$ \n", - "under the conjugacy constraint. \n", - "This gives the following expression" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{p}_{k+1}=\\hat{r}_k-\\frac{\\hat{p}_k^T \\hat{A}\\hat{r}_k}{\\hat{p}_k^T\\hat{A}\\hat{p}_k} \\hat{p}_k.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "We can also compute the residual iteratively as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{r}_{k+1}=\\hat{b}-\\hat{A}\\hat{x}_{k+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{b}-\\hat{A}(\\hat{x}_k+\\alpha_k\\hat{p}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\hat{b}-\\hat{A}\\hat{x}_k)-\\alpha_k\\hat{A}\\hat{p}_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{r}_{k+1}=\\hat{r}_k-\\hat{A}\\hat{p}_{k},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Simple implementation of the Conjugate gradient algorithm" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " Vector ConjugateGradient(Matrix A, Vector b, Vector x0){\n", - " int dim = x0.Dimension();\n", - " const double tolerance = 1.0e-14;\n", - " Vector x(dim),r(dim),v(dim),z(dim);\n", - " double c,t,d;\n", - " \n", - " x = x0;\n", - " r = b - A*x;\n", - " v = r;\n", - " c = dot(r,r);\n", - " int i = 0; IterMax = dim;\n", - " while(i <= IterMax){\n", - " z = A*v;\n", - " t = c/dot(v,z);\n", - " x = x + t*v;\n", - " r = r - t*z;\n", - " d = dot(r,r);\n", - " if(sqrt(d) < tolerance)\n", - " break;\n", - " v = r + (d/c)*v;\n", - " c = d; i++;\n", - " }\n", - " return x;\n", - " } \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Broyden–Fletcher–Goldfarb–Shanno algorithm\n", - "The optimization problem is to minimize $f(\\mathbf {x} )$ where $\\mathbf {x}$ is a vector in $R^{n}$, and $f$ is a differentiable scalar function. There are no constraints on the values that $\\mathbf {x}$ can take.\n", - "\n", - "The algorithm begins at an initial estimate for the optimal value $\\mathbf {x}_{0}$ and proceeds iteratively to get a better estimate at each stage.\n", - "\n", - "The search direction $p_k$ at stage $k$ is given by the solution of the analogue of the Newton equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "B_{k}\\mathbf {p} _{k}=-\\nabla f(\\mathbf {x}_{k}),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $B_{k}$ is an approximation to the Hessian matrix, which is\n", - "updated iteratively at each stage, and $\\nabla f(\\mathbf {x} _{k})$\n", - "is the gradient of the function\n", - "evaluated at $x_k$. \n", - "A line search in the direction $p_k$ is then used to\n", - "find the next point $x_{k+1}$ by minimising" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(\\mathbf {x}_{k}+\\alpha \\mathbf {p}_{k}),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "over the scalar $\\alpha > 0$." - ] } ], "metadata": {}, diff --git a/doc/src/week39/week39.do.txt b/doc/src/week39/week39.do.txt index 9ed997293..75535d279 100644 --- a/doc/src/week39/week39.do.txt +++ b/doc/src/week39/week39.do.txt @@ -386,6 +386,372 @@ o A norm is any function that satisfy the following properties Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this). +!split +===== Standard steepest descent ===== + + +Before we proceed, we would like to discuss the approach called the +_standard Steepest descent_, which again leads to us having to be able +to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). + +"The success of the CG method":"https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf" +for finding solutions of non-linear problems is based on the theory +of conjugate gradients for linear systems of equations. It belongs to +the class of iterative methods for solving problems from linear +algebra of the type +!bt +\begin{equation*} +\hat{A}\hat{x} = \hat{b}. +\end{equation*} +!et + +In the iterative process we end up with a problem like + +!bt +\begin{equation*} + \hat{r}= \hat{b}-\hat{A}\hat{x}, +\end{equation*} +!et +where $\hat{r}$ is the so-called residual or error in the iterative process. + +When we have found the exact solution, $\hat{r}=0$. + +!split +===== Gradient method ===== + +The residual is zero when we reach the minimum of the quadratic equation +!bt +\begin{equation*} + P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b}, +\end{equation*} +!et + +with the constraint that the matrix $\hat{A}$ is positive definite and +symmetric. This defines also the Hessian and we want it to be positive definite. + + +!split +===== Steepest descent method ===== + +We denote the initial guess for $\hat{x}$ as $\hat{x}_0$. +We can assume without loss of generality that +!bt +\begin{equation*} +\hat{x}_0=0, +\end{equation*} +!et +or consider the system +!bt +\begin{equation*} +\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, +\end{equation*} +!et +instead. + + +!split +===== Steepest descent method ===== +!bblock +One can show that the solution $\hat{x}$ is also the unique minimizer of the quadratic form +!bt +\begin{equation*} + f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. +\end{equation*} +!et +This suggests taking the first basis vector $\hat{r}_1$ (see below for definition) +to be the gradient of $f$ at $\hat{x}=\hat{x}_0$, +which equals +!bt +\begin{equation*} +\hat{A}\hat{x}_0-\hat{b}, +\end{equation*} +!et +and +$\hat{x}_0=0$ it is equal $-\hat{b}$. + +!eblock + +!split +===== Final expressions ===== +!bblock +We can compute the residual iteratively as +!bt +\begin{equation*} +\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, + \end{equation*} +!et +which equals +!bt +\begin{equation*} +\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k), + \end{equation*} +!et +or +!bt +\begin{equation*} +(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k, + \end{equation*} +!et +which gives + +!bt +\[ +\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k} +\] +!et +leading to the iterative scheme +!bt +\begin{equation*} +\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k}, + \end{equation*} +!et +!eblock + + + +!split +===== Steepest descent example ===== + +!bc pycod +import numpy as np +import numpy.linalg as la + +import scipy.optimize as sopt + +import matplotlib.pyplot as pt +from mpl_toolkits.mplot3d import axes3d + +def f(x): + return 0.5*x[0]**2 + 2.5*x[1]**2 + +def df(x): + return np.array([x[0], 5*x[1]]) + +fig = pt.figure() +ax = fig.gca(projection="3d") + +xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] +fmesh = f(np.array([xmesh, ymesh])) +ax.plot_surface(xmesh, ymesh, fmesh) +!ec +And then as countor plot +!bc pycod +pt.axis("equal") +pt.contour(xmesh, ymesh, fmesh) +guesses = [np.array([2, 2./5])] +!ec +Find guesses +!bc pycod +x = guesses[-1] +s = -df(x) +!ec +Run it! +!bc pycod +def f1d(alpha): + return f(x + alpha*s) + +alpha_opt = sopt.golden(f1d) +next_guess = x + alpha_opt * s +guesses.append(next_guess) +print(next_guess) +!ec +What happened? +!bc pycod +pt.axis("equal") +pt.contour(xmesh, ymesh, fmesh, 50) +it_array = np.array(guesses) +pt.plot(it_array.T[0], it_array.T[1], "x-") +!ec + +!split +===== Conjugate gradient method ===== +!bblock +In the CG method we define so-called conjugate directions and two vectors +$\hat{s}$ and $\hat{t}$ +are said to be +conjugate if +!bt +\begin{equation*} +\hat{s}^T\hat{A}\hat{t}= 0. +\end{equation*} +!et +The philosophy of the CG method is to perform searches in various conjugate directions +of our vectors $\hat{x}_i$ obeying the above criterion, namely +!bt +\begin{equation*} +\hat{x}_i^T\hat{A}\hat{x}_j= 0. +\end{equation*} +!et +Two vectors are conjugate if they are orthogonal with respect to +this inner product. Being conjugate is a symmetric relation: if $\hat{s}$ is conjugate to $\hat{t}$, then $\hat{t}$ is conjugate to $\hat{s}$. +!eblock + +!split +===== Conjugate gradient method ===== +!bblock +An example is given by the eigenvectors of the matrix +!bt +\begin{equation*} +\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j, +\end{equation*} +!et +which is zero unless $i=j$. +!eblock + + +!split +===== Conjugate gradient method ===== +!bblock +Assume now that we have a symmetric positive-definite matrix $\hat{A}$ of size +$n\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector +!bt +\begin{equation*} +\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. +\end{equation*} +!et +We assume that $\hat{p}_{i}$ is a sequence of $n$ mutually conjugate directions. +Then the $\hat{p}_{i}$ form a basis of $R^n$ and we can expand the solution +$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely + +!bt +\begin{equation*} + \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i. +\end{equation*} +!et +!eblock + +!split +===== Conjugate gradient method ===== +!bblock +The coefficients are given by +!bt +\begin{equation*} + \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. +\end{equation*} +!et +Multiplying with $\hat{p}_k^T$ from the left gives + +!bt +\begin{equation*} + \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b}, +\end{equation*} +!et +and we can define the coefficients $\alpha_k$ as + +!bt +\begin{equation*} + \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k} +\end{equation*} +!et +!eblock + +!split +===== Conjugate gradient method and iterations ===== +!bblock + +If we choose the conjugate vectors $\hat{p}_k$ carefully, +then we may not need all of them to obtain a good approximation to the solution +$\hat{x}$. +We want to regard the conjugate gradient method as an iterative method. +This will us to solve systems where $n$ is so large that the direct +method would take too much time. + +We denote the initial guess for $\hat{x}$ as $\hat{x}_0$. +We can assume without loss of generality that +!bt +\begin{equation*} +\hat{x}_0=0, +\end{equation*} +!et +or consider the system +!bt +\begin{equation*} +\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, +\end{equation*} +!et +instead. +!eblock + + +!split +===== Conjugate gradient method ===== +!bblock +One can show that the solution $\hat{x}$ is also the unique minimizer of the quadratic form +!bt +\begin{equation*} + f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. +\end{equation*} +!et +This suggests taking the first basis vector $\hat{p}_1$ +to be the gradient of $f$ at $\hat{x}=\hat{x}_0$, +which equals +!bt +\begin{equation*} +\hat{A}\hat{x}_0-\hat{b}, +\end{equation*} +!et +and +$\hat{x}_0=0$ it is equal $-\hat{b}$. +The other vectors in the basis will be conjugate to the gradient, +hence the name conjugate gradient method. +!eblock + + +!split +===== Conjugate gradient method ===== +!bblock +Let $\hat{r}_k$ be the residual at the $k$-th step: +!bt +\begin{equation*} +\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k. +\end{equation*} +!et +Note that $\hat{r}_k$ is the negative gradient of $f$ at +$\hat{x}=\hat{x}_k$, +so the gradient descent method would be to move in the direction $\hat{r}_k$. +Here, we insist that the directions $\hat{p}_k$ are conjugate to each other, +so we take the direction closest to the gradient $\hat{r}_k$ +under the conjugacy constraint. +This gives the following expression +!bt +\begin{equation*} +\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k. +\end{equation*} +!et +!eblock + +!split +===== Conjugate gradient method ===== +!bblock +We can also compute the residual iteratively as +!bt +\begin{equation*} +\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, + \end{equation*} +!et +which equals +!bt +\begin{equation*} +\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k), + \end{equation*} +!et +or +!bt +\begin{equation*} +(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k, + \end{equation*} +!et +which gives + +!bt +\begin{equation*} +\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k}, + \end{equation*} +!et +!eblock + + + !split @@ -429,9 +795,9 @@ It is convenient to write $\mathbf{\hat{y}} = X\beta$ where $X \in \mathbb{R}^{1 !bt \[ X \equiv \begin{bmatrix} -1 & x_1 \\ -\vdots & \vdots \\ -1 & x_{100} & \\ +1 &; x_1 \\ +\vdots &; \vdots \\ +1 &; x_{100} &; \\ \end{bmatrix}. \] !et @@ -462,8 +828,8 @@ The Hessian matrix of $C(\beta)$ is given by !bt \[ \hat{H} \equiv \begin{bmatrix} -\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ -\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ +\frac{\partial^2 C(\beta)}{\partial \beta_0^2} &; \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ +\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} &; \frac{\partial^2 C(\beta)}{\partial \beta_1^2} &; \\ \end{bmatrix} = 2X^T X. \] !et @@ -1497,498 +1863,3 @@ a /=b !ec -!split -===== Standard steepest descent ===== - - -Before we proceed, we would like to discuss the approach called the -_standard Steepest descent_, which again leads to us having to be able -to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). - -"The success of the CG method":"https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf" -for finding solutions of non-linear problems is based on the theory -of conjugate gradients for linear systems of equations. It belongs to -the class of iterative methods for solving problems from linear -algebra of the type -!bt -\begin{equation*} -\hat{A}\hat{x} = \hat{b}. -\end{equation*} -!et - -In the iterative process we end up with a problem like - -!bt -\begin{equation*} - \hat{r}= \hat{b}-\hat{A}\hat{x}, -\end{equation*} -!et -where $\hat{r}$ is the so-called residual or error in the iterative process. - -When we have found the exact solution, $\hat{r}=0$. - -!split -===== Gradient method ===== - -The residual is zero when we reach the minimum of the quadratic equation -!bt -\begin{equation*} - P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b}, -\end{equation*} -!et - -with the constraint that the matrix $\hat{A}$ is positive definite and -symmetric. This defines also the Hessian and we want it to be positive definite. - - -!split -===== Steepest descent method ===== - -We denote the initial guess for $\hat{x}$ as $\hat{x}_0$. -We can assume without loss of generality that -!bt -\begin{equation*} -\hat{x}_0=0, -\end{equation*} -!et -or consider the system -!bt -\begin{equation*} -\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, -\end{equation*} -!et -instead. - - -!split -===== Steepest descent method ===== -!bblock -One can show that the solution $\hat{x}$ is also the unique minimizer of the quadratic form -!bt -\begin{equation*} - f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. -\end{equation*} -!et -This suggests taking the first basis vector $\hat{r}_1$ (see below for definition) -to be the gradient of $f$ at $\hat{x}=\hat{x}_0$, -which equals -!bt -\begin{equation*} -\hat{A}\hat{x}_0-\hat{b}, -\end{equation*} -!et -and -$\hat{x}_0=0$ it is equal $-\hat{b}$. - -!eblock - -!split -===== Final expressions ===== -!bblock -We can compute the residual iteratively as -!bt -\begin{equation*} -\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, - \end{equation*} -!et -which equals -!bt -\begin{equation*} -\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k), - \end{equation*} -!et -or -!bt -\begin{equation*} -(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k, - \end{equation*} -!et -which gives - -!bt -\[ -\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k} -\] -!et -leading to the iterative scheme -!bt -\begin{equation*} -\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k}, - \end{equation*} -!et -!eblock - - -!split -===== Code examples for steepest descent ===== - -!split -===== Simple codes for steepest descent and conjugate gradient using a $2\times 2$ matrix, in c++, Python code to come ===== -!bblock -!bc cppcod -#include -#include -#include -#include -#include "vectormatrixclass.h" -using namespace std; -// Main function begins here -int main(int argc, char * argv[]){ - int dim = 2; - Vector x(dim),xsd(dim), b(dim),x0(dim); - Matrix A(dim,dim); - - // Set our initial guess - x0(0) = x0(1) = 0; - // Set the matrix - A(0,0) = 3; A(1,0) = 2; A(0,1) = 2; A(1,1) = 6; - b(0) = 2; b(1) = -8; - cout << "The Matrix A that we are using: " << endl; - A.Print(); - cout << endl; - xsd = SteepestDescent(A,b,x0); - cout << "The approximate solution using Steepest Descent is: " << endl; - xsd.Print(); - cout << endl; -} -!ec -!eblock - -!split -===== The routine for the steepest descent method ===== -!bblock -!bc cppcod -Vector SteepestDescent(Matrix A, Vector b, Vector x0){ - int IterMax, i; - int dim = x0.Dimension(); - const double tolerance = 1.0e-14; - Vector x(dim),f(dim),z(dim); - double c,alpha,d; - IterMax = 30; - x = x0; - r = A*x-b; - i = 0; - while (i <= IterMax){ - z = A*r; - c = dot(r,r); - alpha = c/dot(r,z); - x = x - alpha*r; - r = A*x-b; - if(sqrt(dot(r,r)) < tolerance) break; - i++; - } - return x; -} -!ec -!eblock - - -!split -===== Steepest descent example ===== - -!bc pycod -import numpy as np -import numpy.linalg as la - -import scipy.optimize as sopt - -import matplotlib.pyplot as pt -from mpl_toolkits.mplot3d import axes3d - -def f(x): - return 0.5*x[0]**2 + 2.5*x[1]**2 - -def df(x): - return np.array([x[0], 5*x[1]]) - -fig = pt.figure() -ax = fig.gca(projection="3d") - -xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] -fmesh = f(np.array([xmesh, ymesh])) -ax.plot_surface(xmesh, ymesh, fmesh) -!ec -And then as countor plot -!bc pycod -pt.axis("equal") -pt.contour(xmesh, ymesh, fmesh) -guesses = [np.array([2, 2./5])] -!ec -Find guesses -!bc pycod -x = guesses[-1] -s = -df(x) -!ec -Run it! -!bc pycod -def f1d(alpha): - return f(x + alpha*s) - -alpha_opt = sopt.golden(f1d) -next_guess = x + alpha_opt * s -guesses.append(next_guess) -print(next_guess) -!ec -What happened? -!bc pycod -pt.axis("equal") -pt.contour(xmesh, ymesh, fmesh, 50) -it_array = np.array(guesses) -pt.plot(it_array.T[0], it_array.T[1], "x-") -!ec - -!split -===== Conjugate gradient method ===== -!bblock -In the CG method we define so-called conjugate directions and two vectors -$\hat{s}$ and $\hat{t}$ -are said to be -conjugate if -!bt -\begin{equation*} -\hat{s}^T\hat{A}\hat{t}= 0. -\end{equation*} -!et -The philosophy of the CG method is to perform searches in various conjugate directions -of our vectors $\hat{x}_i$ obeying the above criterion, namely -!bt -\begin{equation*} -\hat{x}_i^T\hat{A}\hat{x}_j= 0. -\end{equation*} -!et -Two vectors are conjugate if they are orthogonal with respect to -this inner product. Being conjugate is a symmetric relation: if $\hat{s}$ is conjugate to $\hat{t}$, then $\hat{t}$ is conjugate to $\hat{s}$. -!eblock - -!split -===== Conjugate gradient method ===== -!bblock -An example is given by the eigenvectors of the matrix -!bt -\begin{equation*} -\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j, -\end{equation*} -!et -which is zero unless $i=j$. -!eblock - - -!split -===== Conjugate gradient method ===== -!bblock -Assume now that we have a symmetric positive-definite matrix $\hat{A}$ of size -$n\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector -!bt -\begin{equation*} -\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. -\end{equation*} -!et -We assume that $\hat{p}_{i}$ is a sequence of $n$ mutually conjugate directions. -Then the $\hat{p}_{i}$ form a basis of $R^n$ and we can expand the solution -$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely - -!bt -\begin{equation*} - \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i. -\end{equation*} -!et -!eblock - -!split -===== Conjugate gradient method ===== -!bblock -The coefficients are given by -!bt -\begin{equation*} - \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. -\end{equation*} -!et -Multiplying with $\hat{p}_k^T$ from the left gives - -!bt -\begin{equation*} - \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b}, -\end{equation*} -!et -and we can define the coefficients $\alpha_k$ as - -!bt -\begin{equation*} - \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k} -\end{equation*} -!et -!eblock - -!split -===== Conjugate gradient method and iterations ===== -!bblock - -If we choose the conjugate vectors $\hat{p}_k$ carefully, -then we may not need all of them to obtain a good approximation to the solution -$\hat{x}$. -We want to regard the conjugate gradient method as an iterative method. -This will us to solve systems where $n$ is so large that the direct -method would take too much time. - -We denote the initial guess for $\hat{x}$ as $\hat{x}_0$. -We can assume without loss of generality that -!bt -\begin{equation*} -\hat{x}_0=0, -\end{equation*} -!et -or consider the system -!bt -\begin{equation*} -\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0, -\end{equation*} -!et -instead. -!eblock - - -!split -===== Conjugate gradient method ===== -!bblock -One can show that the solution $\hat{x}$ is also the unique minimizer of the quadratic form -!bt -\begin{equation*} - f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. -\end{equation*} -!et -This suggests taking the first basis vector $\hat{p}_1$ -to be the gradient of $f$ at $\hat{x}=\hat{x}_0$, -which equals -!bt -\begin{equation*} -\hat{A}\hat{x}_0-\hat{b}, -\end{equation*} -!et -and -$\hat{x}_0=0$ it is equal $-\hat{b}$. -The other vectors in the basis will be conjugate to the gradient, -hence the name conjugate gradient method. -!eblock - - -!split -===== Conjugate gradient method ===== -!bblock -Let $\hat{r}_k$ be the residual at the $k$-th step: -!bt -\begin{equation*} -\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k. -\end{equation*} -!et -Note that $\hat{r}_k$ is the negative gradient of $f$ at -$\hat{x}=\hat{x}_k$, -so the gradient descent method would be to move in the direction $\hat{r}_k$. -Here, we insist that the directions $\hat{p}_k$ are conjugate to each other, -so we take the direction closest to the gradient $\hat{r}_k$ -under the conjugacy constraint. -This gives the following expression -!bt -\begin{equation*} -\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k. -\end{equation*} -!et -!eblock - -!split -===== Conjugate gradient method ===== -!bblock -We can also compute the residual iteratively as -!bt -\begin{equation*} -\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1}, - \end{equation*} -!et -which equals -!bt -\begin{equation*} -\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k), - \end{equation*} -!et -or -!bt -\begin{equation*} -(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k, - \end{equation*} -!et -which gives - -!bt -\begin{equation*} -\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k}, - \end{equation*} -!et -!eblock - - - -!split -===== Simple implementation of the Conjugate gradient algorithm ===== -!bblock -!bc cppcod - Vector ConjugateGradient(Matrix A, Vector b, Vector x0){ - int dim = x0.Dimension(); - const double tolerance = 1.0e-14; - Vector x(dim),r(dim),v(dim),z(dim); - double c,t,d; - - x = x0; - r = b - A*x; - v = r; - c = dot(r,r); - int i = 0; IterMax = dim; - while(i <= IterMax){ - z = A*v; - t = c/dot(v,z); - x = x + t*v; - r = r - t*z; - d = dot(r,r); - if(sqrt(d) < tolerance) - break; - v = r + (d/c)*v; - c = d; i++; - } - return x; -} -!ec -!eblock - - -!split -===== Broyden–Fletcher–Goldfarb–Shanno algorithm ===== -!bblock -The optimization problem is to minimize $f(\mathbf {x} )$ where $\mathbf {x}$ is a vector in $R^{n}$, and $f$ is a differentiable scalar function. There are no constraints on the values that $\mathbf {x}$ can take. - -The algorithm begins at an initial estimate for the optimal value $\mathbf {x}_{0}$ and proceeds iteratively to get a better estimate at each stage. - -The search direction $p_k$ at stage $k$ is given by the solution of the analogue of the Newton equation -!bt -\[ -B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}), -\] -!et - -where $B_{k}$ is an approximation to the Hessian matrix, which is -updated iteratively at each stage, and $\nabla f(\mathbf {x} _{k})$ -is the gradient of the function -evaluated at $x_k$. -A line search in the direction $p_k$ is then used to -find the next point $x_{k+1}$ by minimising -!bt -\[ -f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}), -\] -!et -over the scalar $\alpha > 0$. - -!eblock - - - - - -