-
@@ -296,7 +317,7 @@ MathJax.Hub.Config({
-
- - -
#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;
-}
--
@@ -317,7 +302,7 @@ MathJax.Hub.Config({
-
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;
+#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;
}
@@ -313,7 +338,7 @@ MathJax.Hub.Config({
- -
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)
+
+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;
+}
-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-")
-
@@ -345,7 +334,7 @@ pt.plot(it_array34
+
+
+
+And then as countor plot
+
+
+
+
+Find guesses
+
+
+
+
+Run it!
+
+
+
+
+What happened?
+
+
+
+
@@ -281,7 +366,7 @@ MathJax.Hub.Config({
-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:
-
-
@@ -309,7 +302,7 @@ $$
-Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
+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:
-
-It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by
+
@@ -301,7 +330,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
-Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
+Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
+
+
+It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by
$$
-\nabla_{\beta} C(\beta) = (\partial C(\beta) / \partial \beta_0, \partial C(\beta) / \partial \beta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
-\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
-\end{bmatrix} = 2X^T(X\beta - \mathbf{y}),
+X \equiv \begin{bmatrix}
+1 & x_1 \\
+\vdots & \vdots \\
+1 & x_{100} & \\
+\end{bmatrix}.
$$
-where \( X \) is the design matrix defined above.
+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.
@@ -291,7 +322,7 @@ where \( X \) is the design matrix defined above.
+Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
$$
-\hat{H} \equiv \begin{bmatrix}
-\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
-\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
-\end{bmatrix} = 2X^T X.
+\nabla_{\beta} C(\beta) = (\partial C(\beta) / \partial \beta_0, \partial C(\beta) / \partial \beta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
+\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
+\end{bmatrix} = 2X^T(X\beta - \mathbf{y}),
$$
-This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite.
+where \( X \) is the design matrix defined above.
@@ -290,7 +312,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
-We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
+
-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 \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \).
+This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite.
-
-And finally we can compare our solution for \( \beta \) with the analytic result given by
-\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).
-
-
-
-
@@ -317,7 +311,7 @@ beta_NE = np.40
-Another simple example is here
+We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
+$$
+\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots
+$$
+
+
+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
+\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).
-
@@ -325,7 +338,7 @@ plt.show()
+Another simple example is here
@@ -262,7 +285,10 @@ MathJax.Hub.Config({
from random import random, seed
import numpy as np
import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+from matplotlib.ticker import LinearLocator, FormatStrFormatter
+import sys
x = 2*np.random.rand(100,1)
y = 4+3*x+np.random.randn(100,1)
@@ -270,9 +296,29 @@ 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)
-sgdreg.fit(x,y.ravel())
-print(sgdreg.intercept_, sgdreg.coef_)
+beta = np.random.randn(2,1)
+
+eta = 0.1
+Niterations = 1000
+m = 100
+
+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()
@@ -300,7 +346,7 @@ sgdreg.fit(x,y.
-We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
-$$
-C_{\text{ridge}}(\beta) = ||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0.
-$$
-
-
-In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows
-$$
-\nabla_\beta C_{\text{ridge}}(\beta) = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
-\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
-\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta).
-$$
-
-
-We can now extend our program to minimize \( C_{\text{ridge}}(\beta) \) using gradient descent and compare with the analytical solution given by
-$$
-\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y},
-$$
-
-for \( \lambda = {0,1,10,50,100} \) (\( \lambda = 0 \) corresponds to ordinary least squares).
-We can then compute \( ||\beta_{\text{ridge}}|| \) for each \( \lambda \).
+
-
@@ -333,7 +321,7 @@ beta_ridge = np
+We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
$$
-f(x) = \sin\left(2\pi x + x^2\right)
+C_{\text{ridge}}(\beta) = ||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0.
$$
-which has the following derivative
+
+In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows
$$
-f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
+\nabla_\beta C_{\text{ridge}}(\beta) = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
+\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
+\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta).
$$
-Using autograd we have
+
+We can now extend our program to minimize \( C_{\text{ridge}}(\beta) \) using gradient descent and compare with the analytical solution given by
+$$
+\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y},
+$$
+
+for \( \lambda = {0,1,10,50,100} \) (\( \lambda = 0 \) corresponds to ordinary least squares).
+We can then compute \( ||\beta_{\text{ridge}}|| \) for each \( \lambda \).
-
@@ -331,7 +354,7 @@ plt.show()
-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.
+which has the following derivative
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
+$$
+
+Using autograd we have
@@ -309,7 +352,7 @@ grad_analytical = 45
-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.
+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.
-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.
-
@@ -326,7 +330,7 @@ Note that the grad function will not produce the true gradient of the function.
+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.
-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.
+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.
@@ -310,7 +347,7 @@ could expect form a gradient-evaluting function.
+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.
+
@@ -302,7 +331,7 @@ f4_grad_analytical = x48
@@ -299,7 +323,7 @@ x = 2.7
-
-
-
@@ -322,7 +320,7 @@ f6_grad_analytical = 50
-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.
+
+
@@ -314,7 +343,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
-Assigning a value to the variable being differentiated with respect to
+
-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.
+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.
@@ -302,7 +335,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
+Assigning a value to the variable being differentiated with respect to
-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:
+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.
-
-
-
-
@@ -318,7 +323,7 @@ x = np.a
-
+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:
+
+
+
+
+
@@ -289,7 +339,7 @@ a /=b
-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 \( \{\mathbf{x}_i\}_{i=1}^n \),
-$$
-C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
-\mathbf{\beta}).
-$$
+
+
@@ -294,7 +310,7 @@ $$
-This in turn means that the gradient can be
-computed as a sum over \( i \)-gradients
-$$
-\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
-\mathbf{\beta}).
-$$
+Stochastic gradient descent (SGD) and variants thereof address some of
+the shortcomings of the Gradient descent method discussed above.
-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
-\( k=1,\cdots,n/M \).
+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 \( \{\mathbf{x}_i\}_{i=1}^n \),
+$$
+C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
+\mathbf{\beta}).
+$$
@@ -296,7 +315,7 @@ minibatches. We denote these minibatches by \( B_k \) where
-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
+This in turn means that the gradient can be
+computed as a sum over \( i \)-gradients
$$
-\nabla_{\beta}
-C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
-\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
-c_i(\mathbf{x}_i, \mathbf{\beta}).
+\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}).
$$
+
+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
+\( k=1,\cdots,n/M \).
+
@@ -300,7 +317,7 @@ $$
-Thus a gradient descent step now looks like
+The idea is now to approximate the gradient by replacing the sum over
+all data points with a sum over the data points in one the minibatches
+picked at random in each gradient descent step
$$
-\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
-\mathbf{\beta})
+\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}).
$$
-
-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.
-
@@ -295,7 +321,7 @@ the number of minibatches, as exemplified in the code below.
+Thus a gradient descent step now looks like
+$$
+\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta})
+$$
-
-
-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.
+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.
@@ -308,7 +316,7 @@ all \( n \) datapoints.
-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.
+
+
+
+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.
@@ -293,7 +329,7 @@ gave the lowest value.
-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.
+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.
-
-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.
-
-
-
-
-
@@ -324,7 +314,7 @@ j = 0
+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.
+
+
+
+
+
+
+
+
+
+
+
+
+
+Conjugate gradient
+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)
+
pt.axis("equal")
+pt.contour(xmesh, ymesh, fmesh)
+guesses = [np.array([2, 2./5])]
+
x = guesses[-1]
+s = -df(x)
+
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)
+
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-")
+
Revisiting our first homework
-
-
-
-
-We revisit the example from homework set 1 where we had
-$$
-y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100
-$$
-
-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
-$$
-\hat{y}_i = \beta_0 + \beta_1 x_i.
-$$
+Conjugate gradient
Gradient descent example
+Revisiting our first homework
+
+
+We revisit the example from homework set 1 where we had
$$
-X \equiv \begin{bmatrix}
-1 & x_1 \\
-\vdots & \vdots \\
-1 & x_{100} & \\
-\end{bmatrix}.
+y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100
$$
-The loss function is given by
+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
$$
-C(\beta) = ||X\beta-\mathbf{y}||^2 = ||X\beta||^2 - 2 \mathbf{y}^T X\beta + ||\mathbf{y}||^2 = \sum_{i=1}^{100} (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2
+h_\beta(x) = \hat{y} = \beta_0 + \beta_1 x,
$$
-and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
+such that
+$$
+\hat{y}_i = \beta_0 + \beta_1 x_i.
+$$
The derivative of the cost/loss function
+Gradient descent example
The Hessian matrix
-The Hessian matrix of \( C(\beta) \) is given by
+The derivative of the cost/loss function
+
+Simple program
-
-The Hessian matrix
+The Hessian matrix of \( C(\beta) \) is given by
$$
-\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots
+\hat{H} \equiv \begin{bmatrix}
+\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
+\end{bmatrix} = 2X^T X.
$$
-import numpy as np
-
-"""
-The following setup is just a suggestion, feel free to write it the way you like.
-"""
-
-#Setup problem described in the exercise
-N = 100 #Nr of datapoints
-M = 2 #Nr of features
-x = np.random.rand(N) #Uniformly generated x-values in [0,1]
-y = 5*x**2 + 0.1*np.random.randn(N)
-X = np.c_[np.ones(N),x] #Construct design matrix
-
-#Compute beta according to normal equations to compare with GD solution
-Xt_X_inv = np.linalg.inv(np.dot(X.T,X))
-Xt_y = np.dot(X.transpose(),y)
-beta_NE = np.dot(Xt_X_inv,Xt_y)
-print(beta_NE)
-
Gradient Descent Example
+Simple program
# 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
-x = 2*np.random.rand(100,1)
-y = 4+3*x+np.random.randn(100,1)
+"""
+The following setup is just a suggestion, feel free to write it the way you like.
+"""
-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)
-beta = np.random.randn(2,1)
+#Setup problem described in the exercise
+N = 100 #Nr of datapoints
+M = 2 #Nr of features
+x = np.random.rand(N) #Uniformly generated x-values in [0,1]
+y = 5*x**2 + 0.1*np.random.randn(N)
+X = np.c_[np.ones(N),x] #Construct design matrix
-eta = 0.1
-Niterations = 1000
-m = 100
-
-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()
+#Compute beta according to normal equations to compare with GD solution
+Xt_X_inv = np.linalg.inv(np.dot(X.T,X))
+Xt_y = np.dot(X.transpose(),y)
+beta_NE = np.dot(Xt_X_inv,Xt_y)
+print(beta_NE)
And a corresponding example using scikit-learn
+Gradient Descent Example
+Gradient descent and Ridge
-
-And a corresponding example using scikit-learn
import numpy as np
+
# Importing various packages
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import SGDRegressor
-"""
-The following setup is just a suggestion, feel free to write it the way you like.
-"""
+x = 2*np.random.rand(100,1)
+y = 4+3*x+np.random.randn(100,1)
-#Setup problem described in the exercise
-N = 100 #Nr of datapoints
-M = 2 #Nr of features
-x = np.random.rand(N)
-y = 5*x**2 + 0.1*np.random.randn(N)
-
-
-#Compute analytic beta for Ridge regression
-X = np.c_[np.ones(N),x]
-XT_X = np.dot(X.T,X)
-
-l = 0.1 #Ridge parameter lambda
-Id = np.eye(XT_X.shape[0])
-
-Z = np.linalg.inv(XT_X+l*Id)
-beta_ridge = np.dot(Z,np.dot(X.T,y))
-
-print(beta_ridge)
-print(np.linalg.norm(beta_ridge)) #||beta||
+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.fit(x,y.ravel())
+print(sgdreg.intercept_, sgdreg.coef_)
Automatic differentiation
-Python has tools for so-called automatic differentiation.
-Consider the following example
+Gradient descent and Ridge
+
+import autograd.numpy as np
+
import numpy as np
-# To do elementwise differentiation:
-from autograd import elementwise_grad as egrad
+"""
+The following setup is just a suggestion, feel free to write it the way you like.
+"""
-# To plot:
-import matplotlib.pyplot as plt
+#Setup problem described in the exercise
+N = 100 #Nr of datapoints
+M = 2 #Nr of features
+x = np.random.rand(N)
+y = 5*x**2 + 0.1*np.random.randn(N)
-def f(x):
- return np.sin(2*np.pi*x + x**2)
+#Compute analytic beta for Ridge regression
+X = np.c_[np.ones(N),x]
+XT_X = np.dot(X.T,X)
-def f_grad_analytic(x):
- return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)
+l = 0.1 #Ridge parameter lambda
+Id = np.eye(XT_X.shape[0])
-# Do the comparison:
-x = np.linspace(0,1,1000)
+Z = np.linalg.inv(XT_X+l*Id)
+beta_ridge = np.dot(Z,np.dot(X.T,y))
-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))))
+print(beta_ridge)
+print(np.linalg.norm(beta_ridge)) #||beta||
Using autograd
+Automatic differentiation
+Python has tools for so-called automatic differentiation.
+Consider the following example
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
-import autograd.numpy as np
-from autograd import grad
-def f1(x):
- return x**3 + 1
+# To do elementwise differentiation:
+from autograd import elementwise_grad as egrad
-f1_grad = grad(f1)
+# To plot:
+import matplotlib.pyplot as plt
-# 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)))
+def f(x):
+ return np.sin(2*np.pi*x + x**2)
-# 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))
+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))))
Autograd with more complicated functions
+Using autograd
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)
+def f1(x):
+ return x**3 + 1
-# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
-f2_grad_x2 = grad(f2,1)
+f1_grad = grad(f1)
-x1 = 1.0
-x2 = 3.0
+# Remember to send in float as argument to the computed gradient from Autograd!
+a = 1.0
-print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
-print("-"*30)
+# 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 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) ))
+# 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))
More complicated functions using the elements of their arguments directly
+Autograd with more complicated functions
+
+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
+def f2(x1,x2):
+ return 3*x1**3 + x2*(x1 - 5) + 1
-f3_grad = grad(f3)
+# 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)
-x = np.linspace(0,4,5)
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
-# Print the computed gradient:
-print("The computed gradient of f3 is: ", f3_grad(x))
+x1 = 1.0
+x2 = 3.0
-# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
-f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
-# Print the analytical gradient:
-print("The analytical gradient of f3 is: ", f3_grad_analytical)
+# 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) ))
Functions using mathematical functions from Numpy
+More complicated functions using the elements of their arguments directly
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)
+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
-f4_grad = grad(f4)
+f3_grad = grad(f3)
-x = 2.7
+x = np.linspace(0,4,5)
-# Print the computed derivative:
-print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_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
+# 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 f4 at x = %g is: %g"%(x,f4_grad_analytical))
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
More autograd
+Functions using mathematical functions from Numpy
import autograd.numpy as np
from autograd import grad
-def f5(x):
- if x >= 0:
- return x**2
- else:
- return -3*x + 1
+def f4(x):
+ return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
-f5_grad = grad(f5)
+f4_grad = grad(f4)
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 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))
And with loops
+More autograd
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 f5(x):
+ if x >= 0:
+ return x**2
+ else:
+ return -3*x + 1
-def f6_while(x):
- val = 0
- i = 0
- while i < 10:
- val = val + x**i
- i = i + 1
- return val
+f5_grad = grad(f5)
-f6_for_grad = grad(f6_for)
-f6_while_grad = grad(f6_while)
+x = 2.7
-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))
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
Using recursion
+And with loops
+
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 f7(n): # Assume that n is an integer
- if n == 1 or n == 0:
- return 1
- else:
- return n*f7(n-1)
+def f6_while(x):
+ val = 0
+ i = 0
+ while i < 10:
+ val = val + x**i
+ i = i + 1
+ return val
-f7_grad = grad(f7)
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
-n = 2.0
+x = 0.5
-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))
+# 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))
+
Unsupported functions
-Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
-
-Using recursion
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)
+def f7(n): # Assume that n is an integer
+ if n == 1 or n == 0:
+ return 1
+ else:
+ return n*f7(n-1)
-x = 8.4
+f7_grad = grad(f7)
-print("The derivative of f8 is:",f8_grad(x))
+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))
The syntax a.dot(b) when finding the dot product
+Unsupported functions
+Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
+
+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)
+def f8(x): # Assume x is an array
+ x[2] = 3
+ return x*2
-f9_grad = grad(f9)
+f8_grad = grad(f8)
-x = np.array([1.0,0.0])
+x = 8.4
-print("The derivative of f9 is:",f9_grad(x))
+print("The derivative of f8 is:",f8_grad(x))
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).
-
Recommended to avoid
-The documentation recommends to avoid inplace operations such as
+The syntax a.dot(b) when finding the dot product
a += b
-a -= b
-a*= b
-a /=b
+
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))
+
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).
Stochastic Gradient Descent
-
+Recommended to avoid
+The documentation recommends to avoid inplace operations such as
a += b
+a -= b
+a*= b
+a /=b
+
Computation of gradients
+Stochastic Gradient Descent
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
-\( B_k = \mathbf{x}_k \).
+Computation of gradients
The gradient step
+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
+\( B_k = \mathbf{x}_k \).
Simple example code
+The gradient step
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
-
When do we stop?
+Simple example code
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
+
Slightly different approach
+When do we stop?
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))
-
Slightly different approach
+
+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))
+
Program for stochastic gradient
+
+# 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
+
+x = 2*np.random.rand(100,1)
+y = 4+3*x+np.random.randn(100,1)
+
+xb = np.c_[np.ones((100,1)), x]
+theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
+print("Own inversion")
+print(theta_linreg)
+sgdreg = SGDRegressor(n_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
+m = 100
+
+for iter in range(Niterations):
+ gradients = 2.0/m*xb.T.dot(xb.dot(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
+m = 100
+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.dot(xi.dot(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()
+
+ + + + +
+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 \( \hat{s} \). +
+
+ +
+ + +
+ + + + +
+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 \( i=j \). +
+
+ +
+ + +
+ + + + +
+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 + +$$ +\begin{equation*} + \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i. +\end{equation*} +$$ +
+
+ +
+ + +
+ + + + +
+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 + +$$ +\begin{equation*} + \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k} +\end{equation*} +$$ +
+
+ +
+ + +
+ + + + +
+ +
+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. +
+
+ +
+ + +
+ + + + +
+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. +
+
+ +
+ + +
+ + + + +
+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 +$$ +\begin{equation*} +\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k. +\end{equation*} +$$ +
+
+ +
+ + +
+ + + + +
+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 + +$$ +\begin{equation*} +\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k}, + \end{equation*} +$$ +
+
+ +
+ + +
+ + + + +
+
+ + +
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;
+}
++
+
+ +
+ + +
+ + + + +
+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 \( \alpha > 0 \). + +
+
+
+ +
+ + +
+ + + + +
+ +
+ + +
+ + + + +
+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 +$$ +\begin{align} +\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\ +\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}, +\tag{2} +\end{align} +$$ + +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 \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \). + +
+
+ +
+ + +
+ + + + +
+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 +$$ +m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}). +$$ + +Rearranging this equation, we can rewrite this as +$$ +\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t. +$$ + +
+
+ +
+ + +
+ + + + +
+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 +$$ +\begin{align} +\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\ +\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}. +\tag{3} +\end{align} +$$ + +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 \( \gamma \). + +
+
+ +
+ + +
+ + + + +
+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. + +
+
+ +
+ + +
+ + + + +
+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. + +
+
+ +
+ + +
+ + + + +
+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 +$$ +\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}. +$$ + +
+
+ +
+ + +
+ + + + +
+ +
+ + +-
@@ -296,7 +317,7 @@ MathJax.Hub.Config({
-
@@ -793,7 +793,12 @@ $$
@@ -831,7 +836,7 @@ $$
@@ -865,7 +870,7 @@ $$
@@ -935,12 +940,12 @@ pt.plot(it_array.T[0], it_array.T[
-
We will use linear regression as a case study for the gradient descent
@@ -980,7 +985,7 @@ $$
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -1009,7 +1014,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
@@ -1026,7 +1031,7 @@ where \( X \) is the design matrix defined above.
We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
@@ -1086,7 +1091,7 @@ beta_NE = np.dot(Xt_X_inv,Xt_y)
Another simple example is here
@@ -1136,7 +1141,7 @@ plt.show()
@@ -1161,7 +1166,7 @@ sgdreg.fit(x,y.ravel())
We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
@@ -1225,7 +1230,7 @@ beta_ridge = np.dot(Z,np.dot(X.T,y))
Here we
@@ -1319,7 +1324,7 @@ grad_analytical = 3*a**Autograd with more complicated functions
+
To differentiate with respect to two (or more) arguments of a Python
@@ -1369,7 +1374,7 @@ Note that the grad function will not produce the true gradient of the function.
@@ -1403,7 +1408,7 @@ could expect form a gradient-evaluting function.
@@ -1430,7 +1435,7 @@ f4_grad_analytical = x/np.sqrt(1 + x**
-
@@ -1454,7 +1459,7 @@ x = 2.7
@@ -1501,7 +1506,7 @@ f6_grad_analytical = 0
@@ -1539,7 +1544,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
@@ -1565,7 +1570,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
@@ -1608,7 +1613,7 @@ x = np.array([3.0,Recommended to avoid
+
@@ -1622,7 +1627,7 @@ a /=b
Stochastic gradient descent (SGD) and variants thereof address some of
@@ -1642,7 +1647,7 @@ $$
This in turn means that the gradient can be
@@ -1664,7 +1669,7 @@ minibatches. We denote these minibatches by \( B_k \) where
Thus a gradient descent step now looks like
@@ -1711,7 +1716,7 @@ the number of minibatches, as exemplified in the code below.
@@ -1743,7 +1748,7 @@ all \( n \) datapoints.
A natural question is when do we stop the search for a new minimum?
@@ -1760,7 +1765,7 @@ gave the lowest value.
Another approach is to let the step length \( \gamma_j \) depend on the
@@ -1811,7 +1816,7 @@ j = 0
@@ -1891,12 +1896,12 @@ plt.show()
@@ -1929,7 +1934,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
@@ -1948,7 +1953,7 @@ which is zero unless \( i=j \).
@@ -1978,7 +1983,7 @@ $$
@@ -2015,7 +2020,7 @@ $$
@@ -2052,7 +2057,7 @@ instead.
@@ -2085,7 +2090,7 @@ hence the name conjugate gradient method.
@@ -2117,7 +2122,7 @@ $$
@@ -2162,7 +2167,7 @@ $$
@@ -2199,7 +2204,7 @@ $$
@@ -2236,6 +2241,201 @@ over the scalar \( \alpha > 0 \).
+
+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
+
+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
+
+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 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.
+
+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
+
+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)
+
+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
+
+
+Geron's text, see chapter 11, has several interesting discussions.
+
-
+
@@ -822,7 +839,7 @@ $$
@@ -858,7 +875,7 @@ $$
@@ -927,12 +944,12 @@ pt.plot(it_array.T[0], it_array.T[
-
We will use linear regression as a case study for the gradient descent
@@ -965,7 +982,7 @@ $$
-
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -990,7 +1007,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
@@ -1005,7 +1022,7 @@ where \( X \) is the design matrix defined above.
We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
@@ -1060,7 +1077,7 @@ beta_NE = np.dot(Xt_X_inv,Xt_y)
Another simple example is here
@@ -1109,7 +1126,7 @@ plt.show()
@@ -1133,7 +1150,7 @@ sgdreg.fit(x,y.ravel())
-
We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
@@ -1190,7 +1207,7 @@ beta_ridge = np.dot(Z,np.dot(X.T,y))
-
Here we
@@ -1278,7 +1295,7 @@ grad_analytical = 3*a**Autograd with more complicated functions
+
To differentiate with respect to two (or more) arguments of a Python
@@ -1328,7 +1345,7 @@ Note that the grad function will not produce the true gradient of the function.
@@ -1362,7 +1379,7 @@ could expect form a gradient-evaluting function.
-
@@ -1388,7 +1405,7 @@ f4_grad_analytical = x/np.sqrt(1 + x**
@@ -1411,7 +1428,7 @@ x = 2.7
@@ -1457,7 +1474,7 @@ f6_grad_analytical = 0
@@ -1495,7 +1512,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
@@ -1521,7 +1538,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
@@ -1563,7 +1580,7 @@ x = np.array([3.0,Recommended to avoid
+
@@ -1576,7 +1593,7 @@ a /=b
Stochastic gradient descent (SGD) and variants thereof address some of
@@ -1594,7 +1611,7 @@ $$
This in turn means that the gradient can be
@@ -1614,7 +1631,7 @@ minibatches. We denote these minibatches by \( B_k \) where
Thus a gradient descent step now looks like
@@ -1657,7 +1674,7 @@ the number of minibatches, as exemplified in the code below.
@@ -1689,7 +1706,7 @@ all \( n \) datapoints.
A natural question is when do we stop the search for a new minimum?
@@ -1706,7 +1723,7 @@ gave the lowest value.
Another approach is to let the step length \( \gamma_j \) depend on the
@@ -1754,7 +1771,7 @@ j = 0
@@ -1833,12 +1850,12 @@ plt.show()
@@ -1868,7 +1885,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
@@ -1886,7 +1903,7 @@ which is zero unless \( i=j \).
@@ -1913,7 +1930,7 @@ $$
@@ -1945,7 +1962,7 @@ $$
@@ -1981,7 +1998,7 @@ instead.
@@ -2011,7 +2028,7 @@ hence the name conjugate gradient method.
@@ -2040,7 +2057,7 @@ $$
@@ -2078,7 +2095,7 @@ $$
@@ -2117,7 +2134,7 @@ $$
@@ -2150,6 +2167,177 @@ over the scalar \( \alpha > 0 \).
+
+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
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\label{_auto1}
+\end{align}
+$$
+
+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 \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).
+
+
+
+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
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
+
+Rearranging this equation, we can rewrite this as
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
+
+
+
+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
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\label{_auto2}
+\end{align}
+$$
+
+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 \( \gamma \).
+
+
+
+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.
+
+
+
+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
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\label{_auto3}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
+$$
+
+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.
+
+
+
+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)
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\label{_auto4}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\hat{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\hat{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \hat{\mathbf{m}}_t \over \sqrt{\hat{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\label{_auto5}
+\end{align}
+$$
+
+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
+$$
+\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
+$$
+
+
+
-
+
@@ -827,7 +844,7 @@ $$
@@ -863,7 +880,7 @@ $$
@@ -932,12 +949,12 @@ pt.plot(it_array
-
We will use linear regression as a case study for the gradient descent
@@ -970,7 +987,7 @@ $$
-
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -995,7 +1012,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
@@ -1010,7 +1027,7 @@ where \( X \) is the design matrix defined above.
We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
@@ -1065,7 +1082,7 @@ beta_NE = np.
Another simple example is here
@@ -1114,7 +1131,7 @@ plt.show()
@@ -1138,7 +1155,7 @@ sgdreg.fit(x,y.
-
We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
@@ -1195,7 +1212,7 @@ beta_ridge = np
-
Here we
@@ -1283,7 +1300,7 @@ grad_analytical = Autograd with more complicated functions
+
To differentiate with respect to two (or more) arguments of a Python
@@ -1333,7 +1350,7 @@ Note that the grad function will not produce the true gradient of the function.
@@ -1367,7 +1384,7 @@ could expect form a gradient-evaluting function.
-
@@ -1393,7 +1410,7 @@ f4_grad_analytical = xMore autograd
+
@@ -1416,7 +1433,7 @@ x = 2.7
@@ -1462,7 +1479,7 @@ f6_grad_analytical = Using recursion
+
@@ -1500,7 +1517,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
@@ -1526,7 +1543,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
@@ -1568,7 +1585,7 @@ x = np.a
@@ -1581,7 +1598,7 @@ a /=b
Stochastic gradient descent (SGD) and variants thereof address some of
@@ -1599,7 +1616,7 @@ $$
This in turn means that the gradient can be
@@ -1619,7 +1636,7 @@ minibatches. We denote these minibatches by \( B_k \) where
Thus a gradient descent step now looks like
@@ -1662,7 +1679,7 @@ the number of minibatches, as exemplified in the code below.
@@ -1694,7 +1711,7 @@ all \( n \) datapoints.
A natural question is when do we stop the search for a new minimum?
@@ -1711,7 +1728,7 @@ gave the lowest value.
Another approach is to let the step length \( \gamma_j \) depend on the
@@ -1759,7 +1776,7 @@ j = 0
@@ -1838,12 +1855,12 @@ plt.show()
@@ -1873,7 +1890,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
@@ -1891,7 +1908,7 @@ which is zero unless \( i=j \).
@@ -1918,7 +1935,7 @@ $$
@@ -1950,7 +1967,7 @@ $$
@@ -1986,7 +2003,7 @@ instead.
@@ -2016,7 +2033,7 @@ hence the name conjugate gradient method.
@@ -2045,7 +2062,7 @@ $$
@@ -2083,7 +2100,7 @@ $$
@@ -2122,7 +2139,7 @@ $$
@@ -2155,6 +2172,177 @@ over the scalar \( \alpha > 0 \).
+
+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
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\label{_auto1}
+\end{align}
+$$
+
+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 \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).
+
+
+
+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
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
+
+Rearranging this equation, we can rewrite this as
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
+
+
+
+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
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\label{_auto2}
+\end{align}
+$$
+
+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 \( \gamma \).
+
+
+
+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.
+
+
+
+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
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\label{_auto3}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
+$$
+
+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.
+
+
+
+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)
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\label{_auto4}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\hat{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\hat{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \hat{\mathbf{m}}_t \over \sqrt{\hat{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\label{_auto5}
+\end{align}
+$$
+
+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
+$$
+\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
+$$
+
+
+Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
+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
+The routine for the steepest descent method
Steepest descent example
+Steepest descent example
Conjugate gradient
+Conjugate gradient
Revisiting our first homework
+Revisiting our first homework
Gradient descent example
+Gradient descent example
The derivative of the cost/loss function
+The derivative of the cost/loss function
The Hessian matrix
+The Hessian matrix
The Hessian matrix of \( C(\beta) \) is given by
$$
@@ -1042,7 +1047,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
Simple program
+Simple program
Gradient Descent Example
+Gradient Descent Example
And a corresponding example using scikit-learn
+And a corresponding example using scikit-learn
Gradient descent and Ridge
+Gradient descent and Ridge
Automatic differentiation
+Automatic differentiation
Python has tools for so-called automatic differentiation.
Consider the following example
@@ -1285,7 +1290,7 @@ plt.show()
Using autograd
+Using autograd
Autograd with more complicated functions
More complicated functions using the elements of their arguments directly
+More complicated functions using the elements of their arguments directly
Functions using mathematical functions from Numpy
+Functions using mathematical functions from Numpy
More autograd
+More autograd
And with loops
+And with loops
Using recursion
+Using recursion
Unsupported functions
+Unsupported functions
Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
The syntax a.dot(b) when finding the dot product
+The syntax a.dot(b) when finding the dot product
Recommended to avoid
The documentation recommends to avoid inplace operations such as
Stochastic Gradient Descent
+Stochastic Gradient Descent
Computation of gradients
+Computation of gradients
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
@@ -1690,7 +1695,7 @@ $$
The gradient step
+The gradient step
Simple example code
+Simple example code
When do we stop?
+When do we stop?
Slightly different approach
+Slightly different approach
Program for stochastic gradient
+Program for stochastic gradient
Momentum based methods
+Momentum based methods
Conjugate gradient method
+Conjugate gradient method
Conjugate gradient method
+Conjugate gradient method
Conjugate gradient method
+Conjugate gradient method
Conjugate gradient method
+Conjugate gradient method
Conjugate gradient method and iterations
+Conjugate gradient method and iterations
Conjugate gradient method
+Conjugate gradient method
Conjugate gradient method
+Conjugate gradient method
Conjugate gradient method
+Conjugate gradient method
Simple implementation of the Conjugate gradient algorithm
+Simple implementation of the Conjugate gradient algorithm
Broyden–Fletcher–Goldfarb–Shanno algorithm
+Broyden–Fletcher–Goldfarb–Shanno algorithm
Using gradient descent methods, limitations
+
+
+
+Momentum based GD
+
+
+$$
+\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}
+$$
+
+
+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 \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).
+More on momentum based approaches
+
+
+$$
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+$$
+
+
+We can discretize this equation in the usual way to get
+
+$$
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+$$
+
+
+Rearranging this equation, we can rewrite this as
+
+$$
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+$$
+
+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:
+
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
+
+
+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 \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+
+
+$$
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\tag{3}
+\end{align}
+$$
+
+
+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 \( \gamma \).
+Second moment of the gradient
+
+RMS prop
+
+
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\tag{4}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
+$$
+
+
+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.
+ADAM optimizer
+
+
+$$
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\tag{5}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\hat{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\hat{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \hat{\mathbf{m}}_t \over \sqrt{\hat{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\tag{6}
+\end{align}
+$$
+
+
+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.
+
+
+$$
+\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
+$$
+
+Practical tips
+
+
+
+
Oct 6, 2018
Oct 12, 2018
@@ -782,7 +794,12 @@ $$
-Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
+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
+The routine for the steepest descent method
-Steepest descent example
+Steepest descent example
-Conjugate gradient
+Conjugate gradient
Revisiting our first homework
+Revisiting our first homework
Gradient descent example
+Gradient descent example
-The derivative of the cost/loss function
+The derivative of the cost/loss function
-The Hessian matrix
+The Hessian matrix
The Hessian matrix of \( C(\beta) \) is given by
$$
\hat{H} \equiv \begin{bmatrix}
@@ -1019,7 +1036,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
-Simple program
+Simple program
-Gradient Descent Example
+Gradient Descent Example
-And a corresponding example using scikit-learn
+And a corresponding example using scikit-learn
Gradient descent and Ridge
+Gradient descent and Ridge
-Automatic differentiation
+Automatic differentiation
Python has tools for so-called automatic differentiation.
Consider the following example
$$
@@ -1245,7 +1262,7 @@ plt.show()
Using autograd
+Using autograd
Autograd with more complicated functions
-More complicated functions using the elements of their arguments directly
+More complicated functions using the elements of their arguments directly
Functions using mathematical functions from Numpy
+Functions using mathematical functions from Numpy
-More autograd
+More autograd
-And with loops
+And with loops
-Using recursion
+Using recursion
-Unsupported functions
+Unsupported functions
Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
-The syntax a.dot(b) when finding the dot product
+The syntax a.dot(b) when finding the dot product
Recommended to avoid
The documentation recommends to avoid inplace operations such as
-Stochastic Gradient Descent
+Stochastic Gradient Descent
-Computation of gradients
+Computation of gradients
-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
@@ -1638,7 +1655,7 @@ $$
-The gradient step
+The gradient step
-Simple example code
+Simple example code
-When do we stop?
+When do we stop?
-Slightly different approach
+Slightly different approach
-Program for stochastic gradient
+Program for stochastic gradient
-Momentum based methods
+Momentum based methods
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method and iterations
+Conjugate gradient method and iterations
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Simple implementation of the Conjugate gradient algorithm
+Simple implementation of the Conjugate gradient algorithm
-Broyden–Fletcher–Goldfarb–Shanno algorithm
+Broyden–Fletcher–Goldfarb–Shanno algorithm
+
+Using gradient descent methods, limitations
+
+
+
+
+
+
+Momentum based GD
+
+
+
+More on momentum based approaches
+
+
+
+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:
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
+
+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 \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+
+
+
+Second moment of the gradient
+
+
+
+RMS prop
+
+
+
+ADAM optimizer
+
+
+
+Practical tips
+
+
+
+
+Geron's text, see chapter 11, has several interesting discussions.
diff --git a/doc/pub/Splines/html/Splines.html b/doc/pub/Splines/html/Splines.html
index 31883185d..6f8548d8b 100644
--- a/doc/pub/Splines/html/Splines.html
+++ b/doc/pub/Splines/html/Splines.html
@@ -94,74 +94,86 @@ div { text-align: justify; text-justify: inter-word; }
('Steepest descent method', 2, None, '___sec19'),
('Steepest descent method', 2, None, '___sec20'),
('Final expressions', 2, None, '___sec21'),
+ ('Code examples for steepest descent', 2, None, '___sec22'),
('Simple codes for steepest descent and conjugate gradient '
'using a $2\\times 2$ matrix, in c++, Python code to come',
2,
None,
- '___sec22'),
+ '___sec23'),
('The routine for the steepest descent method',
2,
None,
- '___sec23'),
- ('Steepest descent example', 2, None, '___sec24'),
- ('Conjugate gradient', 2, None, '___sec25'),
- ('Revisiting our first homework', 2, None, '___sec26'),
- ('Gradient descent example', 2, None, '___sec27'),
- ('The derivative of the cost/loss function', 2, None, '___sec28'),
- ('The Hessian matrix', 2, None, '___sec29'),
- ('Simple program', 2, None, '___sec30'),
- ('Gradient Descent Example', 2, None, '___sec31'),
+ '___sec24'),
+ ('Steepest descent example', 2, None, '___sec25'),
+ ('Conjugate gradient', 2, None, '___sec26'),
+ ('Revisiting our first homework', 2, None, '___sec27'),
+ ('Gradient descent example', 2, None, '___sec28'),
+ ('The derivative of the cost/loss function', 2, None, '___sec29'),
+ ('The Hessian matrix', 2, None, '___sec30'),
+ ('Simple program', 2, None, '___sec31'),
+ ('Gradient Descent Example', 2, None, '___sec32'),
('And a corresponding example using _scikit-learn_',
2,
None,
- '___sec32'),
- ('Gradient descent and Ridge', 2, None, '___sec33'),
- ('Automatic differentiation', 2, None, '___sec34'),
- ('Using autograd', 2, None, '___sec35'),
- ('Autograd with more complicated functions', 2, None, '___sec36'),
+ '___sec33'),
+ ('Gradient descent and Ridge', 2, None, '___sec34'),
+ ('Automatic differentiation', 2, None, '___sec35'),
+ ('Using autograd', 2, None, '___sec36'),
+ ('Autograd with more complicated functions', 2, None, '___sec37'),
('More complicated functions using the elements of their '
'arguments directly',
2,
None,
- '___sec37'),
+ '___sec38'),
('Functions using mathematical functions from Numpy',
2,
None,
- '___sec38'),
- ('More autograd', 2, None, '___sec39'),
- ('And with loops', 2, None, '___sec40'),
- ('Using recursion', 2, None, '___sec41'),
- ('Unsupported functions', 2, None, '___sec42'),
+ '___sec39'),
+ ('More autograd', 2, None, '___sec40'),
+ ('And with loops', 2, None, '___sec41'),
+ ('Using recursion', 2, None, '___sec42'),
+ ('Unsupported functions', 2, None, '___sec43'),
('The syntax a.dot(b) when finding the dot product',
2,
None,
- '___sec43'),
- ('Recommended to avoid', 2, None, '___sec44'),
- ('Stochastic Gradient Descent', 2, None, '___sec45'),
- ('Computation of gradients', 2, None, '___sec46'),
- ('SGD example', 2, None, '___sec47'),
- ('The gradient step', 2, None, '___sec48'),
- ('Simple example code', 2, None, '___sec49'),
- ('When do we stop?', 2, None, '___sec50'),
- ('Slightly different approach', 2, None, '___sec51'),
- ('Program for stochastic gradient', 2, None, '___sec52'),
- ('Momentum based methods', 2, None, '___sec53'),
- ('Conjugate gradient method', 2, None, '___sec54'),
+ '___sec44'),
+ ('Recommended to avoid', 2, None, '___sec45'),
+ ('Stochastic Gradient Descent', 2, None, '___sec46'),
+ ('Computation of gradients', 2, None, '___sec47'),
+ ('SGD example', 2, None, '___sec48'),
+ ('The gradient step', 2, None, '___sec49'),
+ ('Simple example code', 2, None, '___sec50'),
+ ('When do we stop?', 2, None, '___sec51'),
+ ('Slightly different approach', 2, None, '___sec52'),
+ ('Program for stochastic gradient', 2, None, '___sec53'),
+ ('Momentum based methods', 2, None, '___sec54'),
('Conjugate gradient method', 2, None, '___sec55'),
('Conjugate gradient method', 2, None, '___sec56'),
('Conjugate gradient method', 2, None, '___sec57'),
- ('Conjugate gradient method and iterations', 2, None, '___sec58'),
- ('Conjugate gradient method', 2, None, '___sec59'),
+ ('Conjugate gradient method', 2, None, '___sec58'),
+ ('Conjugate gradient method and iterations', 2, None, '___sec59'),
('Conjugate gradient method', 2, None, '___sec60'),
('Conjugate gradient method', 2, None, '___sec61'),
+ ('Conjugate gradient method', 2, None, '___sec62'),
('Simple implementation of the Conjugate gradient algorithm',
2,
None,
- '___sec62'),
+ '___sec63'),
('Broyden–Fletcher–Goldfarb–Shanno algorithm',
2,
None,
- '___sec63')]}
+ '___sec64'),
+ ('Using gradient descent methods, limitations',
+ 2,
+ None,
+ '___sec65'),
+ ('Momentum based GD', 2, None, '___sec66'),
+ ('More on momentum based approaches', 2, None, '___sec67'),
+ ('Momentum parameter', 2, None, '___sec68'),
+ ('Second moment of the gradient', 2, None, '___sec69'),
+ ('RMS prop', 2, None, '___sec70'),
+ ('ADAM optimizer', 2, None, '___sec71'),
+ ('Practical tips', 2, None, '___sec72')]}
end of tocinfo -->
@@ -203,7 +215,7 @@ MathJax.Hub.Config({
Oct 6, 2018
Oct 12, 2018
@@ -787,7 +799,12 @@ $$
-Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
+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
+The routine for the steepest descent method
-Steepest descent example
+Steepest descent example
-Conjugate gradient
+Conjugate gradient
Revisiting our first homework
+Revisiting our first homework
Gradient descent example
+Gradient descent example
-The derivative of the cost/loss function
+The derivative of the cost/loss function
-The Hessian matrix
+The Hessian matrix
The Hessian matrix of \( C(\beta) \) is given by
$$
\hat{H} \equiv \begin{bmatrix}
@@ -1024,7 +1041,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
-Simple program
+Simple program
-Gradient Descent Example
+Gradient Descent Example
-And a corresponding example using scikit-learn
+And a corresponding example using scikit-learn
Gradient descent and Ridge
+Gradient descent and Ridge
-Automatic differentiation
+Automatic differentiation
Python has tools for so-called automatic differentiation.
Consider the following example
$$
@@ -1250,7 +1267,7 @@ plt.show()
Using autograd
+Using autograd
Autograd with more complicated functions
-More complicated functions using the elements of their arguments directly
+More complicated functions using the elements of their arguments directly
Functions using mathematical functions from Numpy
+Functions using mathematical functions from Numpy
More autograd
-And with loops
+And with loops
Using recursion
-Unsupported functions
+Unsupported functions
Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
-The syntax a.dot(b) when finding the dot product
+The syntax a.dot(b) when finding the dot product
-Recommended to avoid
+Recommended to avoid
The documentation recommends to avoid inplace operations such as
-Stochastic Gradient Descent
+Stochastic Gradient Descent
-Computation of gradients
+Computation of gradients
-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
@@ -1643,7 +1660,7 @@ $$
-The gradient step
+The gradient step
-Simple example code
+Simple example code
-When do we stop?
+When do we stop?
-Slightly different approach
+Slightly different approach
-Program for stochastic gradient
+Program for stochastic gradient
-Momentum based methods
+Momentum based methods
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method and iterations
+Conjugate gradient method and iterations
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Conjugate gradient method
+Conjugate gradient method
-Simple implementation of the Conjugate gradient algorithm
+Simple implementation of the Conjugate gradient algorithm
-Broyden–Fletcher–Goldfarb–Shanno algorithm
+Broyden–Fletcher–Goldfarb–Shanno algorithm
+
+Using gradient descent methods, limitations
+
+
+
+
+
+
+Momentum based GD
+
+
+
+More on momentum based approaches
+
+
+
+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:
+$$
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+$$
+
+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 \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+
+
+
+Second moment of the gradient
+
+
+
+RMS prop
+
+
+
+ADAM optimizer
+
+
+
+Practical tips
+
+
+
+
+Geron's text, see chapter 11, has several interesting discussions.
diff --git a/doc/pub/Splines/ipynb/Splines.ipynb b/doc/pub/Splines/ipynb/Splines.ipynb
index 48e3c51fa..a4aeb3112 100644
--- a/doc/pub/Splines/ipynb/Splines.ipynb
+++ b/doc/pub/Splines/ipynb/Splines.ipynb
@@ -10,7 +10,7 @@
" \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: **Oct 6, 2018**\n",
+ "Date: **Oct 12, 2018**\n",
"\n",
"Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
@@ -766,6 +766,8 @@
"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"
]
},
@@ -2595,7 +2597,365 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "over the scalar $\\alpha > 0$."
+ "over the scalar $\\alpha > 0$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "## Using gradient descent methods, limitations\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "* **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. \n",
+ "\n",
+ "* 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.\n",
+ "\n",
+ "## Momentum based GD\n",
+ "\n",
+ "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 \n",
+ "implemented as follows"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t) \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation} \n",
+ "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t},\n",
+ "\\label{_auto1} \\tag{2}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "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"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\Delta \\boldsymbol{\\theta}_{t+1} = \\gamma \\Delta \\boldsymbol{\\theta}_t -\\ \\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where we have defined $\\Delta \\boldsymbol{\\theta}_{t}= \\boldsymbol{\\theta}_t-\\boldsymbol{\\theta}_{t-1}$.\n",
+ "\n",
+ "## More on momentum based approaches\n",
+ "\n",
+ "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 \n",
+ "$E(\\mathbf{w})$. If we denote the particle's position by $\\mathbf{w}$, then its motion is described by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "m {d^2 \\mathbf{w} \\over dt^2} + \\mu {d \\mathbf{w} \\over dt }= -\\nabla_w E(\\mathbf{w}).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We can discretize this equation in the usual way to get"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "m { \\mathbf{w}_{t+\\Delta t}-2 \\mathbf{w}_{t} +\\mathbf{w}_{t-\\Delta t} \\over (\\Delta t)^2}+\\mu {\\mathbf{w}_{t+\\Delta t}- \\mathbf{w}_{t} \\over \\Delta t} = -\\nabla_w E(\\mathbf{w}).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Rearranging this equation, we can rewrite this as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\Delta \\mathbf{w}_{t +\\Delta t}= - { (\\Delta t)^2 \\over m +\\mu \\Delta t} \\nabla_w E(\\mathbf{w})+ {m \\over m +\\mu \\Delta t} \\Delta \\mathbf{w}_t.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Momentum parameter\n",
+ "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\n",
+ "us to identify the momentum parameter and learning rate with the mass of the particle and the viscous drag as:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\gamma= {m \\over m +\\mu \\Delta t }, \\qquad \\eta = {(\\Delta t)^2 \\over m +\\mu \\Delta t}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "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 $(1-\\gamma)^{-1} \\approx m/(\\mu \\Delta t)$.\n",
+ "\n",
+ "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. \n",
+ "\n",
+ "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). \n",
+ "\n",
+ "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"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma \\mathbf{v}_{t-1}) \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation} \n",
+ "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t}.\n",
+ "\\label{_auto2} \\tag{3}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "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 $\\gamma$.\n",
+ "\n",
+ "\n",
+ "## Second moment of the gradient\n",
+ "\n",
+ "\n",
+ "In stochastic gradient descent, with and without momentum, we still\n",
+ "have to specify a schedule for tuning the learning rates $\\eta_t$\n",
+ "as a function of time. As discussed in the context of Newton's\n",
+ "method, this presents a number of dilemmas. The learning rate is\n",
+ "limited by the steepest direction which can change depending on the\n",
+ "current position in the landscape. To circumvent this problem, ideally\n",
+ "our algorithm would keep track of curvature and take large steps in\n",
+ "shallow, flat directions and small steps in steep, narrow directions.\n",
+ "Second-order methods accomplish this by calculating or approximating\n",
+ "the Hessian and normalizing the learning rate by the\n",
+ "curvature. However, this is very computationally expensive for\n",
+ "extremely large models. Ideally, we would like to be able to\n",
+ "adaptively change the step size to match the landscape without paying\n",
+ "the steep computational price of calculating or approximating\n",
+ "Hessians.\n",
+ "\n",
+ "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.\n",
+ "\n",
+ "## RMS prop\n",
+ "\n",
+ "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"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n",
+ "\\label{_auto3} \\tag{4}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mathbf{s}_t =\\beta \\mathbf{s}_{t-1} +(1-\\beta)\\mathbf{g}_t^2 \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\mathbf{g}_t \\over \\sqrt{\\mathbf{s}_t +\\epsilon}}, \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "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.\n",
+ "\n",
+ "\n",
+ "## ADAM optimizer\n",
+ "\n",
+ "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)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n",
+ "\\label{_auto4} \\tag{5}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mathbf{m}_t = \\beta_1 \\mathbf{m}_{t-1} + (1-\\beta_1) \\mathbf{g}_t \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mathbf{s}_t =\\beta_2 \\mathbf{s}_{t-1} +(1-\\beta_2)\\mathbf{g}_t^2 \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\mathbf{m}}_t={\\mathbf{m}_t \\over 1-\\beta_1^t} \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\mathbf{s}}_t ={\\mathbf{s}_t \\over1-\\beta_2^t} \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\hat{\\mathbf{m}}_t \\over \\sqrt{\\hat{\\mathbf{s}}_t} +\\epsilon}, \\nonumber\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation} \n",
+ "\\label{_auto5} \\tag{6}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "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.\n",
+ "\n",
+ "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"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\Delta \\theta_{t+1}= -\\eta_t { \\hat{m}_t \\over \\sqrt{\\sigma_t^2 + m_t^2 }+\\epsilon}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Practical tips\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "* **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.\n",
+ "\n",
+ "Geron's text, see chapter 11, has several interesting discussions."
]
}
],
diff --git a/doc/pub/Splines/ipynb/ipynb-Splines-src.tar.gz b/doc/pub/Splines/ipynb/ipynb-Splines-src.tar.gz
index 7b7bb3026..dc7646280 100644
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diff --git a/doc/pub/Splines/pdf/Splines-minted.pdf b/doc/pub/Splines/pdf/Splines-minted.pdf
index 4e31571bd..3c43ab626 100644
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diff --git a/doc/src/Splines/Splines.do.txt b/doc/src/Splines/Splines.do.txt
index 8e8a86700..c95505d03 100644
--- a/doc/src/Splines/Splines.do.txt
+++ b/doc/src/Splines/Splines.do.txt
@@ -1712,3 +1712,162 @@ over the scalar $\alpha > 0$.
!eblock
+
+!split
+===== 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.
+
+* _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.
+
+
+!split
+===== 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
+!bt
+\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},
+\end{align}
+!et
+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
+!bt
+\[
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
+\]
+!et
+where we have defined $\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}$.
+
+!split
+===== More on momentum based approaches =====
+
+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
+!bt
+\[
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+\]
+!et
+We can discretize this equation in the usual way to get
+!bt
+\[
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+\]
+!et
+Rearranging this equation, we can rewrite this as
+!bt
+\[
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+\]
+!et
+
+!split
+===== 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:
+!bt
+\[
+\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}.
+\]
+!et
+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
+!bt
+\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}.
+\end{align}
+!et
+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 $\gamma$.
+
+
+!split
+===== 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.
+
+!split
+===== RMS prop =====
+
+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
+!bt
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\
+\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}
+!et
+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.
+
+
+!split
+===== 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)
+!bt
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\
+\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 \\
+\end{align}
+!et
+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
+!bt
+\[
+\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
+\]
+!et
+
+
+
+
+!split
+===== 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.
+
+* _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.
+
+Geron's text, see chapter 11, has several interesting discussions.
+