Optimization, the central part of any Machine Learning algortithm
-
-
-Almost every problem in machine learning and data science starts with
-a dataset \( X \), a model \( g(\beta) \), which is a function of the
-parameters \( \beta \) and a cost function \( C(X, g(\beta)) \) that allows
-us to judge how well the model \( g(\beta) \) explains the observations
-\( X \). The model is fit by finding the values of \( \beta \) that minimize
-the cost function. Ideally we would be able to solve for \( \beta \)
-analytically, however this is not possible in general and we must use
-some approximative/numerical method to compute the minimum.
+
Optimization problems, why?
@@ -299,7 +292,7 @@ some approximative/numerical method to compute the minimum.
Optimization, the central part of any Machine Learning algortithm
-In our discussion on Logistic Regression we studied the
-case of
-two classes, with \( y_i \) either
-\( 0 \) or \( 1 \). Furthermore we assumed also that we have only two
-parameters \( \beta \) in our fitting, that is we
-defined probabilities
-
-$$
-\begin{align*}
-p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
-p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}),
-\end{align*}
-$$
-
-where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
+Almost every problem in machine learning and data science starts with
+a dataset \( X \), a model \( g(\beta) \), which is a function of the
+parameters \( \beta \) and a cost function \( C(X, g(\beta)) \) that allows
+us to judge how well the model \( g(\beta) \) explains the observations
+\( X \). The model is fit by finding the values of \( \beta \) that minimize
+the cost function. Ideally we would be able to solve for \( \beta \)
+analytically, however this is not possible in general and we must use
+some approximative/numerical method to compute the minimum.
@@ -307,7 +303,7 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas
-Our compact equations used a definition of a vector \( \hat{y} \) with \( n \)
-elements \( y_i \), an \( n\times p \) matrix \( \hat{X} \) which contains the
-\( x_i \) values and a vector \( \hat{p} \) of fitted probabilities
-\( p(y_i\vert x_i,\hat{\beta}) \). We rewrote in a more compact form
-the first derivative of the cost function as
+In our discussion on Logistic Regression we studied the
+case of
+two classes, with \( y_i \) either
+\( 0 \) or \( 1 \). Furthermore we assumed also that we have only two
+parameters \( \beta \) in our fitting, that is we
+defined probabilities
$$
-\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right).
+\begin{align*}
+p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
+p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}),
+\end{align*}
$$
-
-If we in addition define a diagonal matrix \( \hat{W} \) with elements
-\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as
-
-$$
-\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}.
-$$
-
-This defines what is called the Hessian matrix.
+where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
@@ -312,7 +311,7 @@ This defines what is called the Hessian matrix.
-If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
+Our compact equations used a definition of a vector \( \hat{y} \) with \( n \)
+elements \( y_i \), an \( n\times p \) matrix \( \hat{X} \) which contains the
+\( x_i \) values and a vector \( \hat{p} \) of fitted probabilities
+\( p(y_i\vert x_i,\hat{\beta}) \). We rewrote in a more compact form
+the first derivative of the cost function as
+
+$$
+\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right).
+$$
-Our iterative scheme is then given by
+If we in addition define a diagonal matrix \( \hat{W} \) with elements
+\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as
$$
-\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T}\right)^{-1}_{\hat{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}}\right)_{\hat{\beta}^{\mathrm{old}}},
+\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}.
$$
-or in matrix form as
-
-$$
-\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\hat{X}^T\hat{W}\hat{X} \right)^{-1}\times \left(-\hat{X}^T(\hat{y}-\hat{p}) \right)_{\hat{\beta}^{\mathrm{old}}}.
-$$
-
-The right-hand side is computed with the old values of \( \beta \).
-
-
-If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.
+This defines what is called the Hessian matrix.
@@ -313,7 +316,7 @@ If we can compute these matrices, in particular the Hessian, the above is often
-Let us quickly remind ourselves how we derive the above method.
+If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
-Perhaps the most celebrated of all one-dimensional root-finding
-routines is Newton's method, also called the Newton-Raphson
-method. This method requires the evaluation of both the
-function \( f \) and its derivative \( f' \) at arbitrary points.
-If you can only calculate the derivative
-numerically and/or your function is not of the smooth type, we
-normally discourage the use of this method.
+Our iterative scheme is then given by
+
+$$
+\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T}\right)^{-1}_{\hat{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}}\right)_{\hat{\beta}^{\mathrm{old}}},
+$$
+
+or in matrix form as
+
+$$
+\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\hat{X}^T\hat{W}\hat{X} \right)^{-1}\times \left(-\hat{X}^T(\hat{y}-\hat{p}) \right)_{\hat{\beta}^{\mathrm{old}}}.
+$$
+
+The right-hand side is computed with the old values of \( \beta \).
+
+
+If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.
@@ -305,7 +317,7 @@ normally discourage the use of this method.
-The Newton-Raphson formula consists geometrically of extending the
-tangent line at a current point until it crosses zero, then setting
-the next guess to the abscissa of that zero-crossing. The mathematics
-behind this method is rather simple. Employing a Taylor expansion for
-\( x \) sufficiently close to the solution \( s \), we have
-
-$$
- f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
- \tag{1}
-$$
+Let us quickly remind ourselves how we derive the above method.
-For small enough values of the function and for well-behaved
-functions, the terms beyond linear are unimportant, hence we obtain
-
-$$
- f(x)+(s-x)f'(x)\approx 0,
-$$
-
-yielding
-$$
- s\approx x-\frac{f(x)}{f'(x)}.
-$$
-
-
-Having in mind an iterative procedure, it is natural to start iterating with
-$$
- x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
-$$
+Perhaps the most celebrated of all one-dimensional root-finding
+routines is Newton's method, also called the Newton-Raphson
+method. This method requires the evaluation of both the
+function \( f \) and its derivative \( f' \) at arbitrary points.
+If you can only calculate the derivative
+numerically and/or your function is not of the smooth type, we
+normally discourage the use of this method.
-The above is Newton-Raphson's method. It has a simple geometric
-interpretation, namely \( x_{n+1} \) is the point where the tangent from
-\( (x_n,f(x_n)) \) crosses the \( x \)-axis. Close to the solution,
-Newton-Raphson converges fast to the desired result. However, if we
-are far from a root, where the higher-order terms in the series are
-important, the Newton-Raphson formula can give grossly inaccurate
-results. For instance, the initial guess for the root might be so far
-from the true root as to let the search interval include a local
-maximum or minimum of the function. If an iteration places a trial
-guess near such a local extremum, so that the first derivative nearly
-vanishes, then Newton-Raphson may fail totally
+The Newton-Raphson formula consists geometrically of extending the
+tangent line at a current point until it crosses zero, then setting
+the next guess to the abscissa of that zero-crossing. The mathematics
+behind this method is rather simple. Employing a Taylor expansion for
+\( x \) sufficiently close to the solution \( s \), we have
+
+$$
+ f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
+ \tag{1}
+$$
+
+
+For small enough values of the function and for well-behaved
+functions, the terms beyond linear are unimportant, hence we obtain
+
+$$
+ f(x)+(s-x)f'(x)\approx 0,
+$$
+
+yielding
+$$
+ s\approx x-\frac{f(x)}{f'(x)}.
+$$
+
+
+Having in mind an iterative procedure, it is natural to start iterating with
+$$
+ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
+$$
@@ -308,7 +329,7 @@ vanishes, then Newton-Raphson may fail totally
-Newton's method can be generalized to systems of several non-linear equations
-and variables. Consider the case with two equations
-$$
- \begin{array}{cc} f_1(x_1,x_2) &=0\\
- f_2(x_1,x_2) &=0,\end{array}
-$$
-
-which we Taylor expand to obtain
-
-$$
- \begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1
- \partial f_1/\partial x_1+h_2
- \partial f_1/\partial x_2+\dots\\
- 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1
- \partial f_2/\partial x_1+h_2
- \partial f_2/\partial x_2+\dots
- \end{array}.
-$$
-
-Defining the Jacobian matrix \( {\bf \hat{J}} \) we have
-$$
- {\bf \hat{J}}=\left( \begin{array}{cc}
- \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\
- \partial f_2/\partial x_1 &\partial f_2/\partial x_2
- \end{array} \right),
-$$
-
-we can rephrase Newton's method as
-$$
-\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)=
-\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+
-\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right),
-$$
-
-where we have defined
-$$
- \left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
- -{\bf \hat{J}}^{-1}
- \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
-$$
-
-We need thus to compute the inverse of the Jacobian matrix and it
-is to understand that difficulties may
-arise in case \( {\bf \hat{J}} \) is nearly singular.
-
-
-It is rather straightforward to extend the above scheme to systems of
-more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.
+The above is Newton-Raphson's method. It has a simple geometric
+interpretation, namely \( x_{n+1} \) is the point where the tangent from
+\( (x_n,f(x_n)) \) crosses the \( x \)-axis. Close to the solution,
+Newton-Raphson converges fast to the desired result. However, if we
+are far from a root, where the higher-order terms in the series are
+important, the Newton-Raphson formula can give grossly inaccurate
+results. For instance, the initial guess for the root might be so far
+from the true root as to let the search interval include a local
+maximum or minimum of the function. If an iteration places a trial
+guess near such a local extremum, so that the first derivative nearly
+vanishes, then Newton-Raphson may fail totally
@@ -346,7 +312,7 @@ more than two non-linear equations. In our case, the Jacobian matrix is given by
-The basic idea of gradient descent is
-that a function \( F(\mathbf{x}) \),
-\( \mathbf{x} \equiv (x_1,\cdots,x_n) \), decreases fastest if one goes from \( \bf {x} \) in the
-direction of the negative gradient \( -\nabla F(\mathbf{x}) \).
-
-
-It can be shown that if
+Newton's method can be generalized to systems of several non-linear equations
+and variables. Consider the case with two equations
$$
-\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k),
+ \begin{array}{cc} f_1(x_1,x_2) &=0\\
+ f_2(x_1,x_2) &=0,\end{array}
$$
-with \( \gamma_k > 0 \).
+which we Taylor expand to obtain
+
+$$
+ \begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1
+ \partial f_1/\partial x_1+h_2
+ \partial f_1/\partial x_2+\dots\\
+ 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1
+ \partial f_2/\partial x_1+h_2
+ \partial f_2/\partial x_2+\dots
+ \end{array}.
+$$
+
+Defining the Jacobian matrix \( {\bf \hat{J}} \) we have
+$$
+ {\bf \hat{J}}=\left( \begin{array}{cc}
+ \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\
+ \partial f_2/\partial x_1 &\partial f_2/\partial x_2
+ \end{array} \right),
+$$
+
+we can rephrase Newton's method as
+$$
+\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)=
+\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+
+\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right),
+$$
+
+where we have defined
+$$
+ \left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
+ -{\bf \hat{J}}^{-1}
+ \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
+$$
+
+We need thus to compute the inverse of the Jacobian matrix and it
+is to understand that difficulties may
+arise in case \( {\bf \hat{J}} \) is nearly singular.
-For \( \gamma_k \) small enough, then \( F(\mathbf{x}_{k+1}) \leq
-F(\mathbf{x}_k) \). This means that for a sufficiently small \( \gamma_k \)
-we are always moving towards smaller function values, i.e a minimum.
+It is rather straightforward to extend the above scheme to systems of
+more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.
@@ -316,7 +350,7 @@ we are always moving towards smaller function values, i.e a minimum.
-The previous observation is the basis of the method of steepest
-descent, which is also referred to as just gradient descent (GD). One
-starts with an initial guess \( \mathbf{x}_0 \) for a minimum of \( F \) and
-computes new approximations according to
-
-$$
-\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0.
-$$
+The basic idea of gradient descent is
+that a function \( F(\mathbf{x}) \),
+\( \mathbf{x} \equiv (x_1,\cdots,x_n) \), decreases fastest if one goes from \( \bf {x} \) in the
+direction of the negative gradient \( -\nabla F(\mathbf{x}) \).
-The parameter \( \gamma_k \) is often referred to as the step length or
-the learning rate within the context of Machine Learning.
+It can be shown that if
+$$
+\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k),
+$$
+
+with \( \gamma_k > 0 \).
+
+
+For \( \gamma_k \) small enough, then \( F(\mathbf{x}_{k+1}) \leq
+F(\mathbf{x}_k) \). This means that for a sufficiently small \( \gamma_k \)
+we are always moving towards smaller function values, i.e a minimum.
@@ -312,7 +320,7 @@ the learning rate within the context of Machine Learning.
-Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
-minimum of the function \( F \). In general we do not know if we are in a
-global or local minimum. In the special case when \( F \) is a convex
-function, all local minima are also global minima, so in this case
-gradient descent can converge to the global solution. The advantage of
-this scheme is that it is conceptually simple and straightforward to
-implement. However the method in this form has some severe
-limitations:
+The previous observation is the basis of the method of steepest
+descent, which is also referred to as just gradient descent (GD). One
+starts with an initial guess \( \mathbf{x}_0 \) for a minimum of \( F \) and
+computes new approximations according to
+
+$$
+\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0.
+$$
-In machine learing we are often faced with non-convex high dimensional
-cost functions with many local minima. Since GD is deterministic we
-will get stuck in a local minimum, if the method converges, unless we
-have a very good intial guess. This also implies that the scheme is
-sensitive to the chosen initial condition.
-
-
-Note that the gradient is a function of \( \mathbf{x} =
-(x_1,\cdots,x_n) \) which makes it expensive to compute numerically.
+The parameter \( \gamma_k \) is often referred to as the step length or
+the learning rate within the context of Machine Learning.
@@ -319,7 +315,7 @@ Note that the gradient is a function of \( \mathbf{x} =
-The gradient descent method
-is sensitive to the choice of learning rate \( \gamma_k \). This is due
-to the fact that we are only guaranteed that \( F(\mathbf{x}_{k+1}) \leq
-F(\mathbf{x}_k) \) for sufficiently small \( \gamma_k \). The problem is to
-determine an optimal learning rate. If the learning rate is chosen too
-small the method will take a long time to converge and if it is too
-large we can experience erratic behavior.
+Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
+minimum of the function \( F \). In general we do not know if we are in a
+global or local minimum. In the special case when \( F \) is a convex
+function, all local minima are also global minima, so in this case
+gradient descent can converge to the global solution. The advantage of
+this scheme is that it is conceptually simple and straightforward to
+implement. However the method in this form has some severe
+limitations:
-Many of these shortcomings can be alleviated by introducing
-randomness. One such method is that of Stochastic Gradient Descent
-(SGD), see below.
+In machine learing we are often faced with non-convex high dimensional
+cost functions with many local minima. Since GD is deterministic we
+will get stuck in a local minimum, if the method converges, unless we
+have a very good intial guess. This also implies that the scheme is
+sensitive to the chosen initial condition.
+
+
+Note that the gradient is a function of \( \mathbf{x} =
+(x_1,\cdots,x_n) \) which makes it expensive to compute numerically.
@@ -312,7 +322,7 @@ randomness. One such method is that of Stochastic Gradient Descent
-Ideally we want our cost/loss function to be convex(concave).
+The gradient descent method
+is sensitive to the choice of learning rate \( \gamma_k \). This is due
+to the fact that we are only guaranteed that \( F(\mathbf{x}_{k+1}) \leq
+F(\mathbf{x}_k) \) for sufficiently small \( \gamma_k \). The problem is to
+determine an optimal learning rate. If the learning rate is chosen too
+small the method will take a long time to converge and if it is too
+large we can experience erratic behavior.
-First we give the definition of a convex set: A set \( C \) in
-\( \mathbb{R}^n \) is said to be convex if, for all \( x \) and \( y \) in \( C \) and
-all \( t \in (0,1) \) , the point \( (1 − t)x + ty \) also belongs to
-C. Geometrically this means that every point on the line segment
-connecting \( x \) and \( y \) is in \( C \) as discussed below.
-
-
-The convex subsets of \( \mathbb{R} \) are the intervals of
-\( \mathbb{R} \). Examples of convex sets of \( \mathbb{R}^2 \) are the
-regular polygons (triangles, rectangles, pentagons, etc...).
+Many of these shortcomings can be alleviated by introducing
+randomness. One such method is that of Stochastic Gradient Descent
+(SGD), see below.
-Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.
+Ideally we want our cost/loss function to be convex(concave).
+
+
+First we give the definition of a convex set: A set \( C \) in
+\( \mathbb{R}^n \) is said to be convex if, for all \( x \) and \( y \) in \( C \) and
+all \( t \in (0,1) \) , the point \( (1 − t)x + ty \) also belongs to
+C. Geometrically this means that every point on the line segment
+connecting \( x \) and \( y \) is in \( C \) as discussed below.
+
+
+The convex subsets of \( \mathbb{R} \) are the intervals of
+\( \mathbb{R} \). Examples of convex sets of \( \mathbb{R}^2 \) are the
+regular polygons (triangles, rectangles, pentagons, etc...).
-In the following we state first and second-order conditions which
-ensures convexity of a function \( f \). We write \( D_f \) to denote the
-domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more
-details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press.
-
-
-
-
-
-Suppose \( f \) is differentiable (i.e \( \nabla f(x) \) is well defined for
-all \( x \) in the domain of \( f \)). Then \( f \) is convex if and only if \( D_f \)
-is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
-for all \( x,y \in D_f \). This condition means that for a convex function
-the first order Taylor expansion (right hand side above) at any point
-a global under estimator of the function. To convince yourself you can
-make a drawing of \( f(x) = x^2+1 \) and draw the tangent line to \( f(x) \) and
-note that it is always below the graph.
-
-
-
-
-
-
-
-
-Assume that \( f \) is twice
-differentiable, i.e the Hessian matrix exists at each point in
-\( D_f \). Then \( f \) is convex if and only if \( D_f \) is a convex set and its
-Hessian is positive semi-definite for all \( x\in D_f \). For a
-single-variable function this reduces to \( f''(x) \geq 0 \). Geometrically this means that \( f \) has nonnegative curvature
-everywhere.
-
-
-
-
-
-This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
+Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.
@@ -337,7 +304,7 @@ This condition is particularly useful since it gives us an procedure for determi
-The next result is of great importance to us and the reason why we are
-going on about convex functions. In machine learning we frequently
-have to minimize a loss/cost function in order to find the best
-parameters for the model we are considering.
-
-
-Ideally we want the
-global minimum (for high-dimensional models it is hard to know
-if we have local or global minimum). However, if the cost/loss function
-is convex the following result provides invaluable information:
+In the following we state first and second-order conditions which
+ensures convexity of a function \( f \). We write \( D_f \) to denote the
+domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more
+details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press.
-Consider the problem of finding \( x \in \mathbb{R}^n \) such that \( f(x) \)
-is minimal, where \( f \) is convex and differentiable. Then, any point
-\( x^* \) that satisfies \( \nabla f(x^*) = 0 \) is a global minimum.
+Suppose \( f \) is differentiable (i.e \( \nabla f(x) \) is well defined for
+all \( x \) in the domain of \( f \)). Then \( f \) is convex if and only if \( D_f \)
+is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
+for all \( x,y \in D_f \). This condition means that for a convex function
+the first order Taylor expansion (right hand side above) at any point
+a global under estimator of the function. To convince yourself you can
+make a drawing of \( f(x) = x^2+1 \) and draw the tangent line to \( f(x) \) and
+note that it is always below the graph.
-This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.
+
+
+
+Assume that \( f \) is twice
+differentiable, i.e the Hessian matrix exists at each point in
+\( D_f \). Then \( f \) is convex if and only if \( D_f \) is a convex set and its
+Hessian is positive semi-definite for all \( x\in D_f \). For a
+single-variable function this reduces to \( f''(x) \geq 0 \). Geometrically this means that \( f \) has nonnegative curvature
+everywhere.
+
+
+
+
+
+This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
@@ -324,7 +340,7 @@ This result means that if we know that the cost/loss function is convex and we a
Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
-
Using the second order condition show that the following functions are convex on the specified domain.
+
+The next result is of great importance to us and the reason why we are
+going on about convex functions. In machine learning we frequently
+have to minimize a loss/cost function in order to find the best
+parameters for the model we are considering.
-
-
\( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).
-
\( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).
-
+
+Ideally we want the
+global minimum (for high-dimensional models it is hard to know
+if we have local or global minimum). However, if the cost/loss function
+is convex the following result provides invaluable information:
-
Let \( f(x) = x^2 \) and \( g(x) = e^x \). Show that \( f(g(x)) \) and \( g(f(x)) \) is convex for \( x \in \mathbb{R} \). Also show that if \( f(x) \) is any convex function than \( h(x) = e^{f(x)} \) is convex.
-
A norm is any function that satisfy the following properties
+
+
+
+
+Consider the problem of finding \( x \in \mathbb{R}^n \) such that \( f(x) \)
+is minimal, where \( f \) is convex and differentiable. Then, any point
+\( x^* \) that satisfies \( \nabla f(x^*) = 0 \) is a global minimum.
+
+
-
-
\( f(\alpha x) = |\alpha| f(x) \) for all \( \alpha \in \mathbb{R} \).
-
\( f(x+y) \leq f(x) + f(y) \)
-
\( f(x) \leq 0 \) for all \( x \in \mathbb{R}^n \) with equality if and only if \( x = 0 \)
-
-
-
-Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
+
+This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.
@@ -320,7 +327,7 @@ Using the definition of convexity, try to show that a function satisfying the pr
-Before we proceed, we would like to discuss the approach called the
-standard Steepest descent, which again leads to us having to be able
-to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
+
Using the second order condition show that the following functions are convex on the specified domain.
-
-The success of the CG method
-for finding solutions of non-linear problems is based on the theory
-of conjugate gradients for linear systems of equations. It belongs to
-the class of iterative methods for solving problems from linear
-algebra of the type
-$$
-\begin{equation*}
-\hat{A}\hat{x} = \hat{b}.
-\end{equation*}
-$$
+
+
\( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).
+
\( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).
+
-
-In the iterative process we end up with a problem like
+
Let \( f(x) = x^2 \) and \( g(x) = e^x \). Show that \( f(g(x)) \) and \( g(f(x)) \) is convex for \( x \in \mathbb{R} \). Also show that if \( f(x) \) is any convex function than \( h(x) = e^{f(x)} \) is convex.
+
A norm is any function that satisfy the following properties
\( f(\alpha x) = |\alpha| f(x) \) for all \( \alpha \in \mathbb{R} \).
+
\( f(x+y) \leq f(x) + f(y) \)
+
\( f(x) \leq 0 \) for all \( x \in \mathbb{R}^n \) with equality if and only if \( x = 0 \)
+
-where \( \hat{r} \) is the so-called residual or error in the iterative process.
+
-
-When we have found the exact solution, \( \hat{r}=0 \).
+Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
@@ -329,7 +323,7 @@ When we have found the exact solution, \( \hat{r}=0 \).
-The residual is zero when we reach the minimum of the quadratic equation
+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:
+
+
+
An analytical solution (recall homework set 1).
+
The gradient can be computed analytically.
+
The cost function is convex which guarantees that gradient descent converges for small enough learning rates
+
+
+We revisit the example from homework set 1 where we had
$$
-\begin{equation*}
- P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
-\end{equation*}
+y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100
$$
-
-with the constraint that the matrix \( \hat{A} \) is positive definite and
-symmetric. This defines also the Hessian and we want it to be positive definite.
+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
+$$
+\hat{y}_i = \beta_0 + \beta_1 x_i.
+$$
@@ -310,7 +329,7 @@ symmetric. This defines also the Hessian and we want it to be positive definit
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
-We can assume without loss of generality that
+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
$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
+X \equiv \begin{bmatrix}
+1 & x_1 \\
+\vdots & \vdots \\
+1 & x_{100} & \\
+\end{bmatrix}.
$$
-or consider the system
+The loss function is given by
$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
+C(\beta) = ||X\beta-\mathbf{y}||^2 = ||X\beta||^2 - 2 \mathbf{y}^T X\beta + ||\mathbf{y}||^2 = \sum_{i=1}^{100} (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2
$$
-instead.
+and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
- f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
The derivative of the cost/loss function
-
-
+Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
+$$
+\nabla_{\beta} C(\beta) = (\partial C(\beta) / \partial \beta_0, \partial C(\beta) / \partial \beta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
+\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
+\end{bmatrix} = 2X^T(X\beta - \mathbf{y}),
+$$
+where \( X \) is the design matrix defined above.
+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
+\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).
+
+
+
+
importnumpyasnp
+
+"""
+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)
+
+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 \( \lambda \).
importnumpyasnp
-importnumpy.linalgasla
-importscipy.optimizeassopt
+"""
+The following setup is just a suggestion, feel free to write it the way you like.
+"""
-importmatplotlib.pyplotaspt
-frommpl_toolkits.mplot3dimport axes3d
+#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)
-deff(x):
- return0.5*x[0]**2+2.5*x[1]**2
-defdf(x):
- return np.array([x[0], 5*x[1]])
+#Compute analytic beta for Ridge regression
+X = np.c_[np.ones(N),x]
+XT_X = np.dot(X.T,X)
-fig = pt.figure()
-ax = fig.gca(projection="3d")
+l =0.1#Ridge parameter lambda
+Id = np.eye(XT_X.shape[0])
-xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
-fmesh = f(np.array([xmesh, ymesh]))
-ax.plot_surface(xmesh, ymesh, fmesh)
-
-In the CG method we define so-called conjugate directions and two vectors
-\( \hat{s} \) and \( \hat{t} \)
-are said to be
-conjugate if
-$$
-\begin{equation*}
-\hat{s}^T\hat{A}\hat{t}= 0.
-\end{equation*}
-$$
+
Stochastic Gradient Descent
-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 \( \hat{s} \).
-
-
+
+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 \( \{\mathbf{x}_i\}_{i=1}^n \),
+$$
+C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
+\mathbf{\beta}).
+$$
@@ -324,7 +314,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is
-An example is given by the eigenvectors of the matrix
+
Computation of gradients
+
+
+This in turn means that the gradient can be
+computed as a sum over \( i \)-gradients
$$
-\begin{equation*}
-\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
-\end{equation*}
+\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}).
$$
-which is zero unless \( i=j \).
-
-
-
+
+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
+\( k=1,\cdots,n/M \).
@@ -312,7 +316,7 @@ 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
+
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
+\( B_k = \mathbf{x}_k \).
+
+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
$$
-\begin{equation*}
- \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
-\end{equation*}
+\nabla_{\beta}
+C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
+c_i(\mathbf{x}_i, \mathbf{\beta}).
$$
-
+Thus a gradient descent step now looks like
$$
-\begin{equation*}
- \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
-\end{equation*}
+\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
+\mathbf{\beta})
$$
-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
-
-$$
-\begin{equation*}
- \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
-\end{equation*}
-$$
-
-
-
+
+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.
-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.
+
+
importnumpyasnp
+
+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 inrange(1,n_epochs+1):
+ for i inrange(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
+
-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.
-
-
-
+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.
-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.
-
-
+
When do we stop?
+
+A natural question is when do we stop the search for a new minimum?
+One possibility is to compute the full gradient after a given number
+of epochs and check if the norm of the gradient is smaller than some
+threshold and stop if true. However, the condition that the gradient
+is zero is valid also for local minima, so this would only tell us
+that we are close to a local/global minimum. However, we could also
+evaluate the cost function at this point, store the result and
+continue the search. If the test kicks in at a later stage we can
+compare the values of the cost function and keep the \( \beta \) that
+gave the lowest value.
@@ -324,7 +313,7 @@ hence the name conjugate gradient method.
-Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
-$$
-\begin{equation*}
-\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
-\end{equation*}
-$$
+
Slightly different approach
-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
-$$
-\begin{equation*}
-\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
-\end{equation*}
-$$
-
-
+
+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.
+
+
+
+
+
importnumpyasnp
+
+defstep_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 inrange(1,n_epochs+1):
+ for i inrange(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))
+
Simple implementation of the Conjugate gradient algorithm
-
-
-
-
+
Using gradient descent methods, limitations
-
-
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
- int dim = x0.Dimension();
- constdouble tolerance =1.0e-14;
- Vector x(dim),r(dim),v(dim),z(dim);
- double c,t,d;
+
+
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.
+
- 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.
+
Momentum based GD
-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
+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
$$
-B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
+\tag{2}
+\end{align}
$$
-
-where \( B_{k} \) 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
+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
$$
-f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),
+\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
$$
-over the scalar \( \alpha > 0 \).
-
-
-
-
-
+where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).
@@ -329,7 +319,7 @@ over the scalar \( \alpha > 0 \).
-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:
-
-
-
An analytical solution (recall homework set 1).
-
The gradient can be computed analytically.
-
The cost function is convex which guarantees that gradient descent converges for small enough learning rates
-
-
-We revisit the example from homework set 1 where we had
+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
$$
-y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100
+m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
$$
-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
+We can discretize this equation in the usual way to get
$$
-h_\beta(x) = \hat{y} = \beta_0 + \beta_1 x,
+m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
$$
-such that
+Rearranging this equation, we can rewrite this as
$$
-\hat{y}_i = \beta_0 + \beta_1 x_i.
+\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
$$
+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 \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
-Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
+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.
-It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by
+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
$$
-X \equiv \begin{bmatrix}
-1 & x_1 \\
-\vdots & \vdots \\
-1 & x_{100} & \\
-\end{bmatrix}.
+\begin{align}
+\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
+\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
+\tag{3}
+\end{align}
$$
-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.
+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 \( \gamma \).
@@ -318,7 +326,7 @@ 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
-$$
-\nabla_{\beta} C(\beta) = (\partial C(\beta) / \partial \beta_0, \partial C(\beta) / \partial \beta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
-\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
-\end{bmatrix} = 2X^T(X\beta - \mathbf{y}),
-$$
+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.
-where \( X \) is the design matrix defined above.
+
+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.
@@ -308,7 +321,7 @@ where \( X \) is the design matrix defined above.
-The Hessian matrix of \( C(\beta) \) is given by
+
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
$$
-\hat{H} \equiv \begin{bmatrix}
-\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
-\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
-\end{bmatrix} = 2X^T X.
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\tag{4}\\
+\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
+\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
+\end{align}
$$
-This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite.
+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.
@@ -307,7 +314,7 @@ 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
+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)
$$
-\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots
+\begin{align}
+\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
+\tag{5}\\
+\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
+\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
+\hat{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
+\hat{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
+\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \hat{\mathbf{m}}_t \over \sqrt{\hat{\mathbf{s}}_t} +\epsilon}, \nonumber \\
+\tag{6}
+\end{align}
$$
-
-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 \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \).
+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.
-And finally we can compare our solution for \( \beta \) with the analytic result given by
-\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).
-
+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
+$$
+\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
+$$
-
-
importnumpyasnp
-
-"""
-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)
-
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.
+Python has tools for so-called automatic differentiation.
+Consider the following example
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
+
+which has the following derivative
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
+$$
+
+Using autograd we have
-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 \( \lambda \).
+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.
-
importnumpyasnp
+
importautograd.numpyasnp
+fromautogradimport grad
-"""
-The following setup is just a suggestion, feel free to write it the way you like.
-"""
+deff1(x):
+ return x**3+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)
+f1_grad = grad(f1)
+# Remember to send in float as argument to the computed gradient from Autograd!
+a =1.0
-#Compute analytic beta for Ridge regression
-X = np.c_[np.ones(N),x]
-XT_X = np.dot(X.T,X)
+# 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)))
-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||
+# 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))
-Python has tools for so-called automatic differentiation.
-Consider the following example
-$$
-f(x) = \sin\left(2\pi x + x^2\right)
-$$
+
Autograd with more complicated functions
-which has the following derivative
-$$
-f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
-$$
-
-Using autograd we have
+
+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.
importautograd.numpyasnp
+fromautogradimport grad
+deff2(x1,x2):
+ return3*x1**3+ x2*(x1 -5) +1
-# To do elementwise differentiation:
-fromautogradimport elementwise_grad as egrad
+# 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)
-# To plot:
-importmatplotlib.pyplotasplt
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
+x1 =1.0
+x2 =3.0
-deff(x):
- return np.sin(2*np.pi*x + x**2)
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
-deff_grad_analytic(x):
- return np.cos(2*np.pi*x + x**2)*(2*np.pi +2*x)
+# Compare with the analytical derivatives:
-# Do the comparison:
-x = np.linspace(0,1,1000)
+# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
+f2_grad_x1_analytical =9*x1**2+ x2
-f_grad = egrad(f)
+# Derivative of f2 w.r.t x2 is: x1 - 5:
+f2_grad_x2_analytical = x1 -5
-computed = f_grad(x)
-analytic = f_grad_analytic(x)
+# 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) ))
-plt.title('Derivative computed from Autograd compared with the analytical derivative')
-plt.plot(x,computed,label='autograd')
-plt.plot(x,analytic,label='analytic')
+print()
-plt.xlabel('x')
-plt.ylabel('y')
-plt.legend()
-
-plt.show()
-
-print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))
+print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+
+Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
+
-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.
+
More complicated functions using the elements of their arguments directly
importautograd.numpyasnpfromautogradimport grad
+deff3(x): # Assumes x is an array of length 5 or higher
+ return2*x[0] +3*x[1] +5*x[2] +7*x[3] +11*x[4]**2
-deff1(x):
- return x**3+1
+f3_grad = grad(f3)
-f1_grad = grad(f1)
+x = np.linspace(0,4,5)
-# Remember to send in float as argument to the computed gradient from Autograd!
-a =1.0
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_grad(x))
-# 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)))
+# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
+f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
-# Compare with the analytical derivative, that is f1'(x) = 3*x**2
-grad_analytical =3*a**2
-print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
+# Print the analytical gradient:
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
+
+Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+
-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.
+
Functions using mathematical functions from Numpy
importautograd.numpyasnpfromautogradimport grad
-deff2(x1,x2):
- return3*x1**3+ x2*(x1 -5) +1
+deff4(x):
+ return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
-# 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)
+f4_grad = grad(f4)
-# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
-f2_grad_x2 = grad(f2,1)
+x =2.7
-x1 =1.0
-x2 =3.0
+# Print the computed derivative:
+print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
-print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
-print("-"*30)
+# 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
-# 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) ))
+# Print the analytical gradient:
+print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
-
-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.
-
@@ -343,7 +322,7 @@ Note that the grad function will not produce the true gradient of the function.
More complicated functions using the elements of their arguments directly
+
More autograd
importautograd.numpyasnpfromautogradimport grad
-deff3(x): # Assumes x is an array of length 5 or higher
- return2*x[0] +3*x[1] +5*x[2] +7*x[3] +11*x[4]**2
+deff5(x):
+ if x >=0:
+ return x**2
+ else:
+ return-3*x +1
-f3_grad = grad(f3)
+f5_grad = grad(f5)
-x = np.linspace(0,4,5)
+x =2.7
-# Print the computed gradient:
-print("The computed gradient of f3 is: ", f3_grad(x))
-
-# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
-f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
-
-# Print the analytical gradient:
-print("The analytical gradient of f3 is: ", f3_grad_analytical)
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
-
-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.
-
@@ -327,7 +319,7 @@ could expect form a gradient-evaluting function.
importautograd.numpyasnpfromautogradimport grad
-deff4(x):
- return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
+deff6_for(x):
+ val =0
+ for i inrange(10):
+ val = val + x**i
+ return val
-f4_grad = grad(f4)
+deff6_while(x):
+ val =0
+ i =0
+ while i <10:
+ val = val + x**i
+ i = i +1
+ return val
-x =2.7
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
-# Print the computed derivative:
-print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+x =0.5
-# 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 computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+
+
-# Print the analytical gradient:
-print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
+
+
importautograd.numpyasnp
+fromautogradimport 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 inrange(10):
+ f6_grad_analytical += i*x**(i-1)
+
+print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
importautograd.numpyasnpfromautogradimport grad
-deff5(x):
- if x >=0:
- return x**2
+
+deff7(n): # Assume that n is an integer
+ if n ==1or n ==0:
+ return1else:
- return-3*x +1
+ return n*f7(n-1)
-f5_grad = grad(f5)
+f7_grad = grad(f7)
-x =2.7
+n =2.0
-# Print the computed derivative:
-print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
+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 inrange(int(n)-1):
+ tmp =1
+ for k inrange(int(n)-1):
+ if k != i:
+ tmp *= (n - k)
+ f7_grad_analytical += tmp
+
+print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
+
+Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
+
+Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
+
+Assigning a value to the variable being differentiated with respect to
importautograd.numpyasnpfromautogradimport grad
-deff6_for(x):
- val =0
- for i inrange(10):
- val = val + x**i
- return val
+deff8(x): # Assume x is an array
+ x[2] =3
+ return x*2
-deff6_while(x):
- val =0
- i =0
- while i <10:
- val = val + x**i
- i = i +1
- return val
+f8_grad = grad(f8)
-f6_for_grad = grad(f6_for)
-f6_while_grad = grad(f6_while)
+x =8.4
-x =0.5
-
-# Print the computed derivaties of f6_for and f6_while
-print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
-print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+print("The derivative of f8 is:",f8_grad(x))
+Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
-
-
importautograd.numpyasnp
-fromautogradimport 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 inrange(10):
- f6_grad_analytical += i*x**(i-1)
-
-print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
-
importautograd.numpyasnpfromautogradimport grad
+deff9(a): # Assume a is an array with 2 elements
+ b = np.array([1.0,2.0])
+ return a.dot(b)
-deff7(n): # Assume that n is an integer
- if n ==1or n ==0:
- return1
- else:
- return n*f7(n-1)
+f9_grad = grad(f9)
-f7_grad = grad(f7)
+x = np.array([1.0,0.0])
-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 inrange(int(n)-1):
- tmp =1
- for k inrange(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 derivative of f9 is:",f9_grad(x))
-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.
+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:
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff9_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).
+
@@ -331,7 +338,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
-Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
-
-
-Assigning a value to the variable being differentiated with respect to
+
Recommended to avoid
+The documentation recommends to avoid inplace operations such as
-
importautograd.numpyasnp
-fromautogradimport grad
-deff8(x): # Assume x is an array
- x[2] =3
- return x*2
-
-f8_grad = grad(f8)
-
-x =8.4
-
-print("The derivative of f8 is:",f8_grad(x))
+
a += b
+a -= b
+a*= b
+a /=b
-
-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.
-
@@ -319,7 +309,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
importautograd.numpyasnp
-fromautogradimport grad
-deff9(a): # Assume a is an array with 2 elements
- b = np.array([1.0,2.0])
- return a.dot(b)
-
-f9_grad = grad(f9)
-
-x = np.array([1.0,0.0])
-
-print("The derivative of f9 is:",f9_grad(x))
-
-
-Here we are told that the 'dot' function does not belong to Autograd's
-version of a Numpy array. To overcome this, an alternative syntax
-which also computed the dot product can be used:
+
Standard steepest descent
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent, which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff9_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
+
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type
+$$
+\begin{equation*}
+\hat{A}\hat{x} = \hat{b}.
+\end{equation*}
+$$
-f9_alternative_grad = grad(f9_alternative)
+
+In the iterative process we end up with a problem like
-x = np.array([3.0,0.0])
+$$
+\begin{equation*}
+ \hat{r}= \hat{b}-\hat{A}\hat{x},
+\end{equation*}
+$$
-print("The gradient of f9 is:",f9_alternative_grad(x))
+where \( \hat{r} \) is the so-called residual or error in the iterative process.
+
+
+When we have found the exact solution, \( \hat{r}=0 \).
-# 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).
-
-The documentation recommends to avoid inplace operations such as
-
+
Gradient method
+
+
+The residual is zero when we reach the minimum of the quadratic equation
+$$
+\begin{equation*}
+ P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
+\end{equation*}
+$$
+
+
+with the constraint that the matrix \( \hat{A} \) is positive definite and
+symmetric. This defines also the Hessian and we want it to be positive definite.
-
-
-Stochastic gradient descent (SGD) and variants thereof address some of
-the shortcomings of the Gradient descent method discussed above.
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
-
-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 \( \{\mathbf{x}_i\}_{i=1}^n \),
+or consider the system
$$
-C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
-\mathbf{\beta}).
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
$$
+instead.
+
-This in turn means that the gradient can be
-computed as a sum over \( i \)-gradients
+
Steepest descent method
+
+
+
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
$$
-\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
-\mathbf{\beta}).
+\begin{equation*}
+ f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
+\end{equation*}
$$
+This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
-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
-\( k=1,\cdots,n/M \).
+
+
+
@@ -313,7 +327,7 @@ minibatches. We denote these minibatches by \( B_k \) where
-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
-\( B_k = \mathbf{x}_k \).
+
Final expressions
+
+
+
+We can compute the residual iteratively as
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
+$$
-
-The idea is now to approximate the gradient by replacing the sum over
-all data points with a sum over the data points in one the minibatches
-picked at random in each gradient descent step
+which equals
$$
-\nabla_{\beta}
-C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
-\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
-c_i(\mathbf{x}_i, \mathbf{\beta}).
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{r}_k),
+ \end{equation*}
$$
+or
+$$
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{r}_k,
+ \end{equation*}
+$$
+
+which gives
+
+$$
+\alpha_k = \frac{\hat{r}_k^T\hat{r}_k}{\hat{r}_k^T\hat{A}\hat{r}_k}
+$$
+
+leading to the iterative scheme
+$$
+\begin{equation*}
+\hat{x}_{k+1}=\hat{x}_k-\alpha_k\hat{r}_{k},
+ \end{equation*}
+$$
+
-Thus a gradient descent step now looks like
-$$
-\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
-\mathbf{\beta})
-$$
-
-
-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.
+
Code examples for steepest descent
@@ -312,7 +301,7 @@ the number of minibatches, as exemplified in the code below.
Simple codes for steepest descent and conjugate gradient using a \( 2\times 2 \) matrix, in c++, Python code to come
+
+
+
-
-
importnumpyasnp
+
+
#include<cmath>
+#include<iostream>
+#include<fstream>
+#include<iomanip>
+#include"vectormatrixclass.h"
+usingnamespace std;
+// Main function begins here
+intmain(int argc, char* argv[]){
+ int dim =2;
+ Vector x(dim),xsd(dim), b(dim),x0(dim);
+ Matrix A(dim,dim);
-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 inrange(1,n_epochs+1):
- for i inrange(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
+ // 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;
+}
-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.
+
-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.
+
+
+
Vector SteepestDescent(Matrix A, Vector b, Vector x0){
+ int IterMax, i;
+ int dim = x0.Dimension();
+ constdouble 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;
+}
+
-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.
+
Steepest descent example
-
importnumpyasnp
+
importnumpyasnp
+importnumpy.linalgasla
-defstep_length(t,t0,t1):
- return t0/(t+t1)
+importscipy.optimizeassopt
-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
+importmatplotlib.pyplotaspt
+frommpl_toolkits.mplot3dimport axes3d
-gamma_j = t0/t1
-j =0
-for epoch inrange(1,n_epochs+1):
- for i inrange(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
+deff(x):
+ return0.5*x[0]**2+2.5*x[1]**2
-print("gamma_j after %d epochs: %g"% (n_epochs,gamma_j))
+defdf(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)
+
+In the CG method we define so-called conjugate directions and two vectors
+\( \hat{s} \) and \( \hat{t} \)
+are said to be
+conjugate if
+$$
+\begin{equation*}
+\hat{s}^T\hat{A}\hat{t}= 0.
+\end{equation*}
+$$
-
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \hat{x}_i \) obeying the above criterion, namely
+$$
+\begin{equation*}
+\hat{x}_i^T\hat{A}\hat{x}_j= 0.
+\end{equation*}
+$$
-
-
# Importing various packages
-frommathimport exp, sqrt
-fromrandomimport random, seed
-importnumpyasnp
-importmatplotlib.pyplotasplt
-fromsklearn.linear_modelimport 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_)
+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
+$$
+\begin{equation*}
+\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
+\end{equation*}
+$$
-
-
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.
-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
+
Conjugate gradient method
+
+
+
+Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
$$
-\begin{align}
-\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\
-\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t},
-\tag{2}
-\end{align}
+\begin{equation*}
+\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}.
+\end{equation*}
$$
-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
-$$
-\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t),
-$$
+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
+
+$$
+\begin{equation*}
+ \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
+\end{equation*}
+$$
+
+
-where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).
@@ -312,6 +320,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
-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
+
Conjugate gradient method
+
+
+
+The coefficients are given by
$$
-m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}).
+\begin{equation*}
+ \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
+\end{equation*}
$$
-We can discretize this equation in the usual way to get
+Multiplying with \( \hat{p}_k^T \) from the left gives
+
$$
-m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
+\begin{equation*}
+ \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
+\end{equation*}
$$
-Rearranging this equation, we can rewrite this as
+and we can define the coefficients \( \alpha_k \) as
+
$$
-\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
+\begin{equation*}
+ \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
+\end{equation*}
$$
+
-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 \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \).
+
Conjugate gradient method and iterations
+
+
+
-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.
+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.
-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
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
+We can assume without loss of generality that
$$
-\begin{align}
-\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\
-\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}.
-\tag{3}
-\end{align}
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
$$
-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 \( \gamma \).
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
+
@@ -317,6 +327,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+ f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
+\end{equation*}
+$$
-
-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.
+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.
+
+
-
-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.
@@ -311,6 +320,7 @@ Recently, a number of methods have been introduced that accomplish this by track
-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
+
Conjugate gradient method
+
+
+
+Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
$$
-\begin{align}
-\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
-\tag{4}\\
-\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\
-\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber
-\end{align}
+\begin{equation*}
+\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
+\end{equation*}
$$
-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.
+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
+$$
+\begin{equation*}
+\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
+\end{equation*}
+$$
+
+
+
@@ -303,6 +318,7 @@ where \( \beta \) controls the averaging time of the second moment and is typica
-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)
+
Conjugate gradient method
+
+
+
+We can also compute the residual iteratively as
$$
-\begin{align}
-\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta})
-\tag{5}\\
-\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\
-\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\
-\hat{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\
-\hat{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\
-\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \hat{\mathbf{m}}_t \over \sqrt{\hat{\mathbf{s}}_t} +\epsilon}, \nonumber \\
-\tag{6}
-\end{align}
+\begin{equation*}
+\hat{r}_{k+1}=\hat{b}-\hat{A}\hat{x}_{k+1},
+ \end{equation*}
$$
-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.
+which equals
+$$
+\begin{equation*}
+\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k),
+ \end{equation*}
+$$
-
-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
+or
$$
-\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
+\begin{equation*}
+(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k,
+ \end{equation*}
$$
+which gives
+
+$$
+\begin{equation*}
+\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k},
+ \end{equation*}
+$$
+
Simple implementation of the Conjugate gradient algorithm
+
+
+
+
-
-
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.
-
+
+
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
+ int dim = x0.Dimension();
+ constdouble tolerance =1.0e-14;
+ Vector x(dim),r(dim),v(dim),z(dim);
+ double c,t,d;
-Geron's text, see chapter 11, has several interesting discussions.
+ 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;
+}
+
+
+
+
+
+
@@ -296,6 +326,8 @@ Geron's text, see chapter 11, has several interesting discussions.
+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.
-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.
+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.
-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
+The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
$$
-\Delta \theta_{t+1}= -\eta_t { \hat{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}.
+B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
$$
+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 \( \alpha > 0 \).
+
+
Optimization, the central part of any Machine Learning algortithm
+
Optimization problems, why?
+
+
+
+
+
Optimization, the central part of any Machine Learning algortithm
Almost every problem in machine learning and data science starts with
@@ -174,7 +179,7 @@ some approximative/numerical method to compute the minimum.
-
Revisiting our Logistic Regression case
+
Revisiting our Logistic Regression case
In our discussion on Logistic Regression we studied the
@@ -198,7 +203,7 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas
-
The equations to solve
+
The equations to solve
Our compact equations used a definition of a vector \( \hat{y} \) with \( n \)
@@ -228,7 +233,7 @@ This defines what is called the Hessian matrix.
-
Solving using Newton-Raphson's method
+
Solving using Newton-Raphson's method
If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
@@ -258,7 +263,7 @@ If we can compute these matrices, in particular the Hessian, the above is often
-
Brief reminder on Newton-Raphson's method
+
Brief reminder on Newton-Raphson's method
Let us quickly remind ourselves how we derive the above method.
@@ -275,7 +280,7 @@ normally discourage the use of this method.
-
The equations
+
The equations
The Newton-Raphson formula consists geometrically of extending the
@@ -319,7 +324,7 @@ $$
-
Simple geometric interpretation
+
Simple geometric interpretation
The above is Newton-Raphson's method. It has a simple geometric
@@ -337,7 +342,7 @@ vanishes, then Newton-Raphson may fail totally
-
Extending to more than one variable
+
Extending to more than one variable
Newton's method can be generalized to systems of several non-linear equations
@@ -402,7 +407,7 @@ more than two non-linear equations. In our case, the Jacobian matrix is given by
-
Steepest descent
+
Steepest descent
The basic idea of gradient descent is
@@ -428,7 +433,7 @@ we are always moving towards smaller function values, i.e a minimum.
-
More on Steepest descent
+
More on Steepest descent
The previous observation is the basis of the method of steepest
@@ -449,7 +454,7 @@ the learning rate within the context of Machine Learning.
-
The ideal
+
The ideal
Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
@@ -475,7 +480,7 @@ Note that the gradient is a function of \( \mathbf{x} =
-
The sensitiveness of the gradient descent
+
The sensitiveness of the gradient descent
The gradient descent method
@@ -494,7 +499,7 @@ randomness. One such method is that of Stochastic Gradient Descent
-
Convex functions
+
Convex functions
Ideally we want our cost/loss function to be convex(concave).
@@ -514,7 +519,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).
-
Convex function
+
Convex function
Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if
In the following we state first and second-order conditions which
@@ -566,7 +571,7 @@ This condition is particularly useful since it gives us an procedure for determi
-
More on convex functions
+
More on convex functions
The next result is of great importance to us and the reason why we are
@@ -595,7 +600,7 @@ This result means that if we know that the cost/loss function is convex and we a
-
Some simple problems
+
Some simple problems
Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
@@ -622,666 +627,7 @@ Using the definition of convexity, try to show that a function satisfying the pr
-
Standard steepest descent
-
-
-Before we proceed, we would like to discuss the approach called the
-standard Steepest descent, which again leads to us having to be able
-to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
-
-
-The success of the CG method
-for finding solutions of non-linear problems is based on the theory
-of conjugate gradients for linear systems of equations. It belongs to
-the class of iterative methods for solving problems from linear
-algebra of the type
-
-with the constraint that the matrix \( \hat{A} \) is positive definite and
-symmetric. This defines also the Hessian and we want it to be positive definite.
-
-
-
-
-
Steepest descent method
-
-
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
-We can assume without loss of generality that
-
-
-This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
-which equals
-
-
-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
-
-
-Two vectors are conjugate if they are orthogonal with respect to
-this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-An example is given by the eigenvectors of the matrix
-
-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
-
-
-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
-
-
-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
-
-
-and
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-The other vectors in the basis will be conjugate to the gradient,
-hence the name conjugate gradient method.
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
-
-
-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
-
Simple implementation of the Conjugate gradient algorithm
-
-
-
-
-
-
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
- int dim = x0.Dimension();
- constdouble tolerance = 1.0e-14;
- Vector x(dim),r(dim),v(dim),z(dim);
- double c,t,d;
-
- x = x0;
- r = b - A*x;
- v = r;
- c = dot(r,r);
- int i = 0; IterMax = dim;
- while(i <= IterMax){
- z = A*v;
- t = c/dot(v,z);
- x = x + t*v;
- r = r - t*z;
- d = dot(r,r);
- if(sqrt(d) < tolerance)
- break;
- v = r + (d/c)*v;
- c = d; i++;
- }
- return x;
-}
-
-
-
-
-
-
-
-
Broyden–Fletcher–Goldfarb–Shanno algorithm
-
-
-
-The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \) is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that \( \mathbf {x} \) can take.
-
-
-The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
-
-
-The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
-
-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
-
We will use linear regression as a case study for the gradient descent
@@ -1321,7 +667,7 @@ $$
-
Gradient descent example
+
Gradient descent example
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -1350,7 +696,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
-
The derivative of the cost/loss function
+
The derivative of the cost/loss function
Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
@@ -1367,7 +713,7 @@ where \( X \) is the design matrix defined above.
-
The Hessian matrix
+
The Hessian matrix
The Hessian matrix of \( C(\beta) \) is given by
$$
@@ -1383,7 +729,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
-
Simple program
+
Simple program
We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
@@ -1427,7 +773,7 @@ beta_NE = np.dot(Xt_X_inv,Xt_y)
-
Gradient Descent Example
+
Gradient Descent Example
Another simple example is here
@@ -1477,7 +823,7 @@ plt.show()
-
And a corresponding example using scikit-learn
+
And a corresponding example using scikit-learn
@@ -1502,7 +848,7 @@ sgdreg.fit(x,y.ravel())
-
Gradient descent and Ridge
+
Gradient descent and Ridge
We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
@@ -1566,404 +912,7 @@ beta_ridge = np.dot(Z,np.dot(X.T,y))
-
Automatic differentiation
-Python has tools for so-called automatic differentiation.
-Consider the following example
-
importautograd.numpyasnp
-
-# To do elementwise differentiation:
-fromautogradimport elementwise_grad as egrad
-
-# To plot:
-importmatplotlib.pyplotasplt
-
-
-deff(x):
- return np.sin(2*np.pi*x + x**2)
-
-deff_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))))
-
-
-
-
-
-
Using autograd
-
-
-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.
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-
-deff1(x):
- return x**3 + 1
-
-f1_grad = grad(f1)
-
-# Remember to send in float as argument to the computed gradient from Autograd!
-a = 1.0
-
-# See the evaluated gradient at a using autograd:
-print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
-
-# Compare with the analytical derivative, that is f1'(x) = 3*x**2
-grad_analytical = 3*a**2
-print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
-
-
-
-
-
-
Autograd with more complicated functions
-
-
-To differentiate with respect to two (or more) arguments of a Python
-function, Autograd need to know at which variable the function if
-being differentiated with respect to.
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff2(x1,x2):
- return3*x1**3 + x2*(x1 - 5) + 1
-
-# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
-f2_grad_x1 = grad(f2,0)
-
-# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
-f2_grad_x2 = grad(f2,1)
-
-x1 = 1.0
-x2 = 3.0
-
-print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
-print("-"*30)
-
-# Compare with the analytical derivatives:
-
-# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
-f2_grad_x1_analytical = 9*x1**2 + x2
-
-# Derivative of f2 w.r.t x2 is: x1 - 5:
-f2_grad_x2_analytical = x1 - 5
-
-# See the evaluated derivations:
-print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-
-print()
-
-print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-
-
-Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
-
-
-
-
-
More complicated functions using the elements of their arguments directly
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff3(x): # Assumes x is an array of length 5 or higher
- return2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
-
-f3_grad = grad(f3)
-
-x = np.linspace(0,4,5)
-
-# Print the computed gradient:
-print("The computed gradient of f3 is: ", f3_grad(x))
-
-# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
-f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
-
-# Print the analytical gradient:
-print("The analytical gradient of f3 is: ", f3_grad_analytical)
-
-
-Note that in this case, when sending an array as input argument, the
-output from Autograd is another array. This is the true gradient of
-the function, as opposed to the function in the previous example. By
-using arrays to represent the variables, the output from Autograd
-might be easier to work with, as the output is closer to what one
-could expect form a gradient-evaluting function.
-
-
-
-
-
Functions using mathematical functions from Numpy
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff4(x):
- return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
-
-f4_grad = grad(f4)
-
-x = 2.7
-
-# Print the computed derivative:
-print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
-
-# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
-f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
-
-# Print the analytical gradient:
-print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
-
-
-
-
-
-
More autograd
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff5(x):
- if x >= 0:
- return x**2
- else:
- return -3*x + 1
-
-f5_grad = grad(f5)
-
-x = 2.7
-
-# Print the computed derivative:
-print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
-
-
-
-
-
-
And with loops
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff6_for(x):
- val = 0
- for i inrange(10):
- val = val + x**i
- return val
-
-deff6_while(x):
- val = 0
- i = 0
- while i < 10:
- val = val + x**i
- i = i + 1
- return val
-
-f6_for_grad = grad(f6_for)
-f6_while_grad = grad(f6_while)
-
-x = 0.5
-
-# Print the computed derivaties of f6_for and f6_while
-print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
-print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport 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 inrange(10):
- f6_grad_analytical += i*x**(i-1)
-
-print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
-
-
-
-
-
-
Using recursion
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-
-deff7(n): # Assume that n is an integer
- if n == 1or n == 0:
- return1
- else:
- return n*f7(n-1)
-
-f7_grad = grad(f7)
-
-n = 2.0
-
-print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
-
-# The function f7 is an implementation of the factorial of n.
-# By using the product rule, one can find that the derivative is:
-
-f7_grad_analytical = 0
-for i inrange(int(n)-1):
- tmp = 1
- for k inrange(int(n)-1):
- if k != i:
- tmp *= (n - k)
- f7_grad_analytical += tmp
-
-print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
-
-
-Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
-
-
-
-
-
Unsupported functions
-Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
-
-
-Assigning a value to the variable being differentiated with respect to
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff8(x): # Assume x is an array
- x[2] = 3
- return x*2
-
-f8_grad = grad(f8)
-
-x = 8.4
-
-print("The derivative of f8 is:",f8_grad(x))
-
-
-Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
-
-
-
-
-
The syntax a.dot(b) when finding the dot product
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff9(a): # Assume a is an array with 2 elements
- b = np.array([1.0,2.0])
- return a.dot(b)
-
-f9_grad = grad(f9)
-
-x = np.array([1.0,0.0])
-
-print("The derivative of f9 is:",f9_grad(x))
-
-
-Here we are told that the 'dot' function does not belong to Autograd's
-version of a Numpy array. To overcome this, an alternative syntax
-which also computed the dot product can be used:
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff9_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
-
-
-
-
a += b
-a -= b
-a*= b
-a /=b
-
-
-
-
-
-
Stochastic Gradient Descent
+
Stochastic Gradient Descent
Stochastic gradient descent (SGD) and variants thereof address some of
@@ -1983,7 +932,7 @@ $$
-
Computation of gradients
+
Computation of gradients
This in turn means that the gradient can be
@@ -2005,7 +954,7 @@ minibatches. We denote these minibatches by \( B_k \) where
-
SGD example
+
SGD example
As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \)
and we choose to have \( M=5 \) minibathces,
then each minibatch contains two data points. In particular we have
@@ -2031,7 +980,7 @@ $$
-
The gradient step
+
The gradient step
Thus a gradient descent step now looks like
@@ -2052,7 +1001,7 @@ the number of minibatches, as exemplified in the code below.
-
Simple example code
+
Simple example code
@@ -2084,7 +1033,7 @@ all \( n \) datapoints.
-
When do we stop?
+
When do we stop?
A natural question is when do we stop the search for a new minimum?
@@ -2101,7 +1050,7 @@ gave the lowest value.
-
Slightly different approach
+
Slightly different approach
Another approach is to let the step length \( \gamma_j \) depend on the
@@ -2152,7 +1101,7 @@ j = 0
-
Program for stochastic gradient
+
Program for stochastic gradient
@@ -2232,7 +1181,7 @@ plt.show()
-
Using gradient descent methods, limitations
+
Using gradient descent methods, limitations
Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our energy function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
@@ -2246,7 +1195,7 @@ plt.show()
-
Momentum based GD
+
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
@@ -2273,7 +1222,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
-
More on momentum based approaches
+
More on momentum based approaches
Let us try to get more intuition from these equations. It is helpful to consider a simple physical analogy with a particle of mass \( m \) moving in a viscous medium with drag coefficient \( \mu \) and potential
@@ -2301,7 +1250,7 @@ $$
-
Momentum parameter
+
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:
@@ -2335,7 +1284,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
-
Second moment of the gradient
+
Second moment of the gradient
In stochastic gradient descent, with and without momentum, we still
@@ -2360,7 +1309,7 @@ Recently, a number of methods have been introduced that accomplish this by track
-
RMS prop
+
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
@@ -2380,7 +1329,7 @@ where \( \beta \) controls the averaging time of the second moment and is typica
-
ADAM optimizer
+
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)
@@ -2412,7 +1361,7 @@ $$
-
Practical tips
+
Practical tips
Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
@@ -2426,6 +1375,1062 @@ Geron's text, see chapter 11, has several interesting discussions.
+
+
Automatic differentiation
+Python has tools for so-called automatic differentiation.
+Consider the following example
+
importautograd.numpyasnp
+
+# To do elementwise differentiation:
+fromautogradimport elementwise_grad as egrad
+
+# To plot:
+importmatplotlib.pyplotasplt
+
+
+deff(x):
+ return np.sin(2*np.pi*x + x**2)
+
+deff_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))))
+
+
+
+
+
+
Using autograd
+
+
+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.
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+
+deff1(x):
+ return x**3 + 1
+
+f1_grad = grad(f1)
+
+# Remember to send in float as argument to the computed gradient from Autograd!
+a = 1.0
+
+# See the evaluated gradient at a using autograd:
+print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
+
+# Compare with the analytical derivative, that is f1'(x) = 3*x**2
+grad_analytical = 3*a**2
+print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
+
+
+
+
+
+
Autograd with more complicated functions
+
+
+To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff2(x1,x2):
+ return3*x1**3 + x2*(x1 - 5) + 1
+
+# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
+f2_grad_x1 = grad(f2,0)
+
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
+
+x1 = 1.0
+x2 = 3.0
+
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
+
+# Compare with the analytical derivatives:
+
+# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
+f2_grad_x1_analytical = 9*x1**2 + x2
+
+# Derivative of f2 w.r.t x2 is: x1 - 5:
+f2_grad_x2_analytical = x1 - 5
+
+# See the evaluated derivations:
+print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+
+print()
+
+print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+
+
+Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
+
+
+
+
+
More complicated functions using the elements of their arguments directly
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff3(x): # Assumes x is an array of length 5 or higher
+ return2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
+
+f3_grad = grad(f3)
+
+x = np.linspace(0,4,5)
+
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_grad(x))
+
+# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
+f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+
+# Print the analytical gradient:
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
+
+
+Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+
+
+
+
+
Functions using mathematical functions from Numpy
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff4(x):
+ return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
+
+f4_grad = grad(f4)
+
+x = 2.7
+
+# Print the computed derivative:
+print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+
+# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
+f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
+
+# Print the analytical gradient:
+print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
+
+
+
+
+
+
More autograd
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff5(x):
+ if x >= 0:
+ return x**2
+ else:
+ return -3*x + 1
+
+f5_grad = grad(f5)
+
+x = 2.7
+
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
+
+
+
+
+
+
And with loops
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff6_for(x):
+ val = 0
+ for i inrange(10):
+ val = val + x**i
+ return val
+
+deff6_while(x):
+ val = 0
+ i = 0
+ while i < 10:
+ val = val + x**i
+ i = i + 1
+ return val
+
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x = 0.5
+
+# Print the computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport 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 inrange(10):
+ f6_grad_analytical += i*x**(i-1)
+
+print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
+
+
+
+
+
+
Using recursion
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+
+deff7(n): # Assume that n is an integer
+ if n == 1or n == 0:
+ return1
+ else:
+ return n*f7(n-1)
+
+f7_grad = grad(f7)
+
+n = 2.0
+
+print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
+
+# The function f7 is an implementation of the factorial of n.
+# By using the product rule, one can find that the derivative is:
+
+f7_grad_analytical = 0
+for i inrange(int(n)-1):
+ tmp = 1
+ for k inrange(int(n)-1):
+ if k != i:
+ tmp *= (n - k)
+ f7_grad_analytical += tmp
+
+print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
+
+
+Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
+
+
+
+
+
Unsupported functions
+Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
+
+
+Assigning a value to the variable being differentiated with respect to
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff8(x): # Assume x is an array
+ x[2] = 3
+ return x*2
+
+f8_grad = grad(f8)
+
+x = 8.4
+
+print("The derivative of f8 is:",f8_grad(x))
+
+
+Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
+
+
+
+
+
The syntax a.dot(b) when finding the dot product
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff9(a): # Assume a is an array with 2 elements
+ b = np.array([1.0,2.0])
+ return a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([1.0,0.0])
+
+print("The derivative of f9 is:",f9_grad(x))
+
+
+Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array. To overcome this, an alternative syntax
+which also computed the dot product can be used:
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff9_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
+
+
+
+
a += b
+a -= b
+a*= b
+a /=b
+
+
+
+
+
+
Standard steepest descent
+
+
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent, which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type
+
+with the constraint that the matrix \( \hat{A} \) is positive definite and
+symmetric. This defines also the Hessian and we want it to be positive definite.
+
+
+
+
+
Steepest descent method
+
+
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
+We can assume without loss of generality that
+
+
+This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
+which equals
+
+
+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
+
+
+Two vectors are conjugate if they are orthogonal with respect to
+this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+An example is given by the eigenvectors of the matrix
+
+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
+
+
+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
+
+
+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
+
+
+and
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+The other vectors in the basis will be conjugate to the gradient,
+hence the name conjugate gradient method.
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
+
+
+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
+
Simple implementation of the Conjugate gradient algorithm
+
+
+
+
+
+
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
+ int dim = x0.Dimension();
+ constdouble tolerance = 1.0e-14;
+ Vector x(dim),r(dim),v(dim),z(dim);
+ double c,t,d;
+
+ x = x0;
+ r = b - A*x;
+ v = r;
+ c = dot(r,r);
+ int i = 0; IterMax = dim;
+ while(i <= IterMax){
+ z = A*v;
+ t = c/dot(v,z);
+ x = x + t*v;
+ r = r - t*z;
+ d = dot(r,r);
+ if(sqrt(d) < tolerance)
+ break;
+ v = r + (d/c)*v;
+ c = d; i++;
+ }
+ return x;
+}
+
+
+
+
+
+
+
+
Broyden–Fletcher–Goldfarb–Shanno algorithm
+
+
+
+The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \) is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that \( \mathbf {x} \) can take.
+
+
+The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
+
+
+The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
+
+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
+
diff --git a/doc/pub/Splines/html/Splines-solarized.html b/doc/pub/Splines/html/Splines-solarized.html
index 380b5b670..ca6b66717 100644
--- a/doc/pub/Splines/html/Splines-solarized.html
+++ b/doc/pub/Splines/html/Splines-solarized.html
@@ -6,6 +6,7 @@ Automatically generated HTML file from DocOnce source
+
Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods
@@ -60,113 +61,114 @@ div { text-align: justify; text-justify: inter-word; }
@@ -208,12 +210,17 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
-
Oct 18, 2018
+
Sep 19, 2019
-
Optimization, the central part of any Machine Learning algortithm
+
Optimization problems, why?
+
+
+
+
+
Optimization, the central part of any Machine Learning algortithm
Almost every problem in machine learning and data science starts with
@@ -228,7 +235,7 @@ some approximative/numerical method to compute the minimum.
-
Revisiting our Logistic Regression case
+
Revisiting our Logistic Regression case
In our discussion on Logistic Regression we studied the
@@ -250,7 +257,7 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas
-
The equations to solve
+
The equations to solve
Our compact equations used a definition of a vector \( \hat{y} \) with \( n \)
@@ -276,7 +283,7 @@ This defines what is called the Hessian matrix.
-
Solving using Newton-Raphson's method
+
Solving using Newton-Raphson's method
If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
@@ -302,7 +309,7 @@ If we can compute these matrices, in particular the Hessian, the above is often
-
Brief reminder on Newton-Raphson's method
+
Brief reminder on Newton-Raphson's method
Let us quickly remind ourselves how we derive the above method.
@@ -319,7 +326,7 @@ normally discourage the use of this method.
-
The equations
+
The equations
The Newton-Raphson formula consists geometrically of extending the
@@ -355,7 +362,7 @@ $$
-
Simple geometric interpretation
+
Simple geometric interpretation
The above is Newton-Raphson's method. It has a simple geometric
@@ -373,7 +380,7 @@ vanishes, then Newton-Raphson may fail totally
-
Extending to more than one variable
+
Extending to more than one variable
Newton's method can be generalized to systems of several non-linear equations
@@ -428,7 +435,7 @@ more than two non-linear equations. In our case, the Jacobian matrix is given by
-
Steepest descent
+
Steepest descent
The basic idea of gradient descent is
@@ -452,7 +459,7 @@ we are always moving towards smaller function values, i.e a minimum.
-
More on Steepest descent
+
More on Steepest descent
The previous observation is the basis of the method of steepest
@@ -471,7 +478,7 @@ the learning rate within the context of Machine Learning.
-
The ideal
+
The ideal
Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
@@ -497,7 +504,7 @@ Note that the gradient is a function of \( \mathbf{x} =
-
The sensitiveness of the gradient descent
+
The sensitiveness of the gradient descent
The gradient descent method
@@ -516,7 +523,7 @@ randomness. One such method is that of Stochastic Gradient Descent
-
Convex functions
+
Convex functions
Ideally we want our cost/loss function to be convex(concave).
@@ -536,7 +543,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).
-
Convex function
+
Convex function
Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.
@@ -544,7 +551,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).
-
Conditions on convex functions
+
Conditions on convex functions
In the following we state first and second-order conditions which
@@ -586,7 +593,7 @@ This condition is particularly useful since it gives us an procedure for determi
-
More on convex functions
+
More on convex functions
The next result is of great importance to us and the reason why we are
@@ -616,7 +623,7 @@ This result means that if we know that the cost/loss function is convex and we a
-
Some simple problems
+
Some simple problems
Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
@@ -640,623 +647,10 @@ This result means that if we know that the cost/loss function is convex and we a
Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
-
-
-
-
Standard steepest descent
-
-
-Before we proceed, we would like to discuss the approach called the
-standard Steepest descent, which again leads to us having to be able
-to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
-
-
-The success of the CG method
-for finding solutions of non-linear problems is based on the theory
-of conjugate gradients for linear systems of equations. It belongs to
-the class of iterative methods for solving problems from linear
-algebra of the type
-$$
-\begin{equation*}
-\hat{A}\hat{x} = \hat{b}.
-\end{equation*}
-$$
-
-
-In the iterative process we end up with a problem like
-
-$$
-\begin{equation*}
- \hat{r}= \hat{b}-\hat{A}\hat{x},
-\end{equation*}
-$$
-
-where \( \hat{r} \) is the so-called residual or error in the iterative process.
-
-
-When we have found the exact solution, \( \hat{r}=0 \).
-
-
-
-
-
Gradient method
-
-
-The residual is zero when we reach the minimum of the quadratic equation
-$$
-\begin{equation*}
- P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
-\end{equation*}
-$$
-
-
-with the constraint that the matrix \( \hat{A} \) is positive definite and
-symmetric. This defines also the Hessian and we want it to be positive definite.
-
-
-
-
-
Steepest descent method
-
-
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
-
-
-
-
-
Steepest descent method
-
-
-
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
- f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-
-
-
-In the CG method we define so-called conjugate directions and two vectors
-\( \hat{s} \) and \( \hat{t} \)
-are said to be
-conjugate if
-$$
-\begin{equation*}
-\hat{s}^T\hat{A}\hat{t}= 0.
-\end{equation*}
-$$
-
-The philosophy of the CG method is to perform searches in various conjugate directions
-of our vectors \( \hat{x}_i \) obeying the above criterion, namely
-$$
-\begin{equation*}
-\hat{x}_i^T\hat{A}\hat{x}_j= 0.
-\end{equation*}
-$$
-
-Two vectors are conjugate if they are orthogonal with respect to
-this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-An example is given by the eigenvectors of the matrix
-$$
-\begin{equation*}
-\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
-\end{equation*}
-$$
-
-which is zero unless \( i=j \).
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
-\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
-$$
-\begin{equation*}
-\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}.
-\end{equation*}
-$$
-
-We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
-Then the \( \hat{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
-$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely
-
-$$
-\begin{equation*}
- \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
-\end{equation*}
-$$
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-The coefficients are given by
-$$
-\begin{equation*}
- \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
-\end{equation*}
-$$
-
-Multiplying with \( \hat{p}_k^T \) from the left gives
-
-$$
-\begin{equation*}
- \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
-\end{equation*}
-$$
-
-and we can define the coefficients \( \alpha_k \) as
-
-$$
-\begin{equation*}
- \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
-\end{equation*}
-$$
-
-
-
-
-
-
-
Conjugate gradient method and iterations
-
-
-
-
-
-If we choose the conjugate vectors \( \hat{p}_k \) carefully,
-then we may not need all of them to obtain a good approximation to the solution
-\( \hat{x} \).
-We want to regard the conjugate gradient method as an iterative method.
-This will us to solve systems where \( n \) is so large that the direct
-method would take too much time.
-
-
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
- f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{p}_1 \)
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-The other vectors in the basis will be conjugate to the gradient,
-hence the name conjugate gradient method.
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
-$$
-\begin{equation*}
-\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
-\end{equation*}
-$$
-
-Note that \( \hat{r}_k \) is the negative gradient of \( f \) at
-\( \hat{x}=\hat{x}_k \),
-so the gradient descent method would be to move in the direction \( \hat{r}_k \).
-Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other,
-so we take the direction closest to the gradient \( \hat{r}_k \)
-under the conjugacy constraint.
-This gives the following expression
-$$
-\begin{equation*}
-\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
-\end{equation*}
-$$
-
Simple implementation of the Conjugate gradient algorithm
-
-
-
-
-
-
-
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
- int dim = x0.Dimension();
- constdouble tolerance = 1.0e-14;
- Vector x(dim),r(dim),v(dim),z(dim);
- double c,t,d;
-
- x = x0;
- r = b - A*x;
- v = r;
- c = dot(r,r);
- int i = 0; IterMax = dim;
- while(i <= IterMax){
- z = A*v;
- t = c/dot(v,z);
- x = x + t*v;
- r = r - t*z;
- d = dot(r,r);
- if(sqrt(d) < tolerance)
- break;
- v = r + (d/c)*v;
- c = d; i++;
- }
- return x;
-}
-
-
-
-
-
-
-
-
-
Broyden–Fletcher–Goldfarb–Shanno algorithm
-
-
-
-The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \) is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that \( \mathbf {x} \) can take.
-
-
-The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
-
-
-The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
-$$
-B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
-$$
-
-
-where \( B_{k} \) is an approximation to the Hessian matrix, which is
-updated iteratively at each stage, and \( \nabla f(\mathbf {x} _{k}) \)
-is the gradient of the function
-evaluated at \( x_k \).
-A line search in the direction \( p_k \) is then used to
-find the next point \( x_{k+1} \) by minimising
-$$
-f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),
-$$
-
-over the scalar \( \alpha > 0 \).
-
-
-
-
-
-
Revisiting our first homework
+
Revisiting our first homework
We will use linear regression as a case study for the gradient descent
@@ -1289,7 +683,7 @@ $$
-
Gradient descent example
+
Gradient descent example
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -1314,7 +708,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
-
The derivative of the cost/loss function
+
The derivative of the cost/loss function
Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
@@ -1329,7 +723,7 @@ where \( X \) is the design matrix defined above.
-
The Hessian matrix
+
The Hessian matrix
The Hessian matrix of \( C(\beta) \) is given by
$$
\hat{H} \equiv \begin{bmatrix}
@@ -1343,7 +737,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
-
Simple program
+
Simple program
We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
@@ -1384,7 +778,7 @@ beta_NE = np.dot(Xt_X_inv,Xt_y)
-
Gradient Descent Example
+
Gradient Descent Example
Another simple example is here
@@ -1433,7 +827,7 @@ plt.show()
-
And a corresponding example using scikit-learn
+
And a corresponding example using scikit-learn
@@ -1457,7 +851,7 @@ sgdreg.fit(x,y.ravel())
-
Gradient descent and Ridge
+
Gradient descent and Ridge
We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
@@ -1514,393 +908,7 @@ beta_ridge = np.dot(Z,np.dot(X.T,y))
-
Automatic differentiation
-Python has tools for so-called automatic differentiation.
-Consider the following example
-$$
-f(x) = \sin\left(2\pi x + x^2\right)
-$$
-
-which has the following derivative
-$$
-f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
-$$
-
-Using autograd we have
-
-
-
-
-
importautograd.numpyasnp
-
-# To do elementwise differentiation:
-fromautogradimport elementwise_grad as egrad
-
-# To plot:
-importmatplotlib.pyplotasplt
-
-
-deff(x):
- return np.sin(2*np.pi*x + x**2)
-
-deff_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))))
-
-
-
-
-
Using autograd
-
-
-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.
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-
-deff1(x):
- return x**3 + 1
-
-f1_grad = grad(f1)
-
-# Remember to send in float as argument to the computed gradient from Autograd!
-a = 1.0
-
-# See the evaluated gradient at a using autograd:
-print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
-
-# Compare with the analytical derivative, that is f1'(x) = 3*x**2
-grad_analytical = 3*a**2
-print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
-
-
-
-
-
Autograd with more complicated functions
-
-
-To differentiate with respect to two (or more) arguments of a Python
-function, Autograd need to know at which variable the function if
-being differentiated with respect to.
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff2(x1,x2):
- return3*x1**3 + x2*(x1 - 5) + 1
-
-# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
-f2_grad_x1 = grad(f2,0)
-
-# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
-f2_grad_x2 = grad(f2,1)
-
-x1 = 1.0
-x2 = 3.0
-
-print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
-print("-"*30)
-
-# Compare with the analytical derivatives:
-
-# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
-f2_grad_x1_analytical = 9*x1**2 + x2
-
-# Derivative of f2 w.r.t x2 is: x1 - 5:
-f2_grad_x2_analytical = x1 - 5
-
-# See the evaluated derivations:
-print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-
-print()
-
-print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-
-
-Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
-
-
-
-
-
More complicated functions using the elements of their arguments directly
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff3(x): # Assumes x is an array of length 5 or higher
- return2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
-
-f3_grad = grad(f3)
-
-x = np.linspace(0,4,5)
-
-# Print the computed gradient:
-print("The computed gradient of f3 is: ", f3_grad(x))
-
-# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
-f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
-
-# Print the analytical gradient:
-print("The analytical gradient of f3 is: ", f3_grad_analytical)
-
-
-Note that in this case, when sending an array as input argument, the
-output from Autograd is another array. This is the true gradient of
-the function, as opposed to the function in the previous example. By
-using arrays to represent the variables, the output from Autograd
-might be easier to work with, as the output is closer to what one
-could expect form a gradient-evaluting function.
-
-
-
-
-
Functions using mathematical functions from Numpy
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff4(x):
- return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
-
-f4_grad = grad(f4)
-
-x = 2.7
-
-# Print the computed derivative:
-print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
-
-# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
-f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
-
-# Print the analytical gradient:
-print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
-
-
-
-
-
More autograd
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff5(x):
- if x >= 0:
- return x**2
- else:
- return -3*x + 1
-
-f5_grad = grad(f5)
-
-x = 2.7
-
-# Print the computed derivative:
-print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
-
-
-
-
-
And with loops
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff6_for(x):
- val = 0
- for i inrange(10):
- val = val + x**i
- return val
-
-deff6_while(x):
- val = 0
- i = 0
- while i < 10:
- val = val + x**i
- i = i + 1
- return val
-
-f6_for_grad = grad(f6_for)
-f6_while_grad = grad(f6_while)
-
-x = 0.5
-
-# Print the computed derivaties of f6_for and f6_while
-print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
-print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport 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 inrange(10):
- f6_grad_analytical += i*x**(i-1)
-
-print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
-
-
-
-
-
Using recursion
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-
-deff7(n): # Assume that n is an integer
- if n == 1or n == 0:
- return1
- else:
- return n*f7(n-1)
-
-f7_grad = grad(f7)
-
-n = 2.0
-
-print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
-
-# The function f7 is an implementation of the factorial of n.
-# By using the product rule, one can find that the derivative is:
-
-f7_grad_analytical = 0
-for i inrange(int(n)-1):
- tmp = 1
- for k inrange(int(n)-1):
- if k != i:
- tmp *= (n - k)
- f7_grad_analytical += tmp
-
-print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
-
-
-Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
-
-
-
-
-
Unsupported functions
-Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
-
-
-Assigning a value to the variable being differentiated with respect to
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff8(x): # Assume x is an array
- x[2] = 3
- return x*2
-
-f8_grad = grad(f8)
-
-x = 8.4
-
-print("The derivative of f8 is:",f8_grad(x))
-
-
-Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
-
-
-
-
-
The syntax a.dot(b) when finding the dot product
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff9(a): # Assume a is an array with 2 elements
- b = np.array([1.0,2.0])
- return a.dot(b)
-
-f9_grad = grad(f9)
-
-x = np.array([1.0,0.0])
-
-print("The derivative of f9 is:",f9_grad(x))
-
-
-Here we are told that the 'dot' function does not belong to Autograd's
-version of a Numpy array. To overcome this, an alternative syntax
-which also computed the dot product can be used:
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff9_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
-
-
-
-
a += b
-a -= b
-a*= b
-a /=b
-
-
-
-
-
Stochastic Gradient Descent
+
Stochastic Gradient Descent
Stochastic gradient descent (SGD) and variants thereof address some of
@@ -1918,7 +926,7 @@ $$
-
Computation of gradients
+
Computation of gradients
This in turn means that the gradient can be
@@ -1938,7 +946,7 @@ minibatches. We denote these minibatches by \( B_k \) where
-
SGD example
+
SGD example
As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \)
and we choose to have \( M=5 \) minibathces,
then each minibatch contains two data points. In particular we have
@@ -1962,7 +970,7 @@ $$
-
The gradient step
+
The gradient step
Thus a gradient descent step now looks like
@@ -1981,7 +989,7 @@ the number of minibatches, as exemplified in the code below.
-
Simple example code
+
Simple example code
@@ -2013,7 +1021,7 @@ all \( n \) datapoints.
-
When do we stop?
+
When do we stop?
A natural question is when do we stop the search for a new minimum?
@@ -2030,7 +1038,7 @@ gave the lowest value.
-
Slightly different approach
+
Slightly different approach
Another approach is to let the step length \( \gamma_j \) depend on the
@@ -2078,7 +1086,7 @@ j = 0
-
Program for stochastic gradient
+
Program for stochastic gradient
@@ -2157,7 +1165,7 @@ plt.show()
-
Using gradient descent methods, limitations
+
Using gradient descent methods, limitations
Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our energy function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
@@ -2170,7 +1178,7 @@ plt.show()
-
Momentum based GD
+
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
@@ -2193,7 +1201,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
-
More on momentum based approaches
+
More on momentum based approaches
Let us try to get more intuition from these equations. It is helpful to consider a simple physical analogy with a particle of mass \( m \) moving in a viscous medium with drag coefficient \( \mu \) and potential
@@ -2215,7 +1223,7 @@ $$
-
Momentum parameter
+
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:
$$
@@ -2245,7 +1253,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
-
Second moment of the gradient
+
Second moment of the gradient
In stochastic gradient descent, with and without momentum, we still
@@ -2270,7 +1278,7 @@ Recently, a number of methods have been introduced that accomplish this by track
-
RMS prop
+
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
@@ -2288,7 +1296,7 @@ where \( \beta \) controls the averaging time of the second moment and is typica
-
ADAM optimizer
+
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)
@@ -2316,7 +1324,7 @@ $$
-
Practical tips
+
Practical tips
Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
@@ -2327,11 +1335,1012 @@ $$
Geron's text, see chapter 11, has several interesting discussions.
+
+
+
+
Automatic differentiation
+Python has tools for so-called automatic differentiation.
+Consider the following example
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
+
+which has the following derivative
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
+$$
+
+Using autograd we have
+
+
+
+
+
importautograd.numpyasnp
+
+# To do elementwise differentiation:
+fromautogradimport elementwise_grad as egrad
+
+# To plot:
+importmatplotlib.pyplotasplt
+
+
+deff(x):
+ return np.sin(2*np.pi*x + x**2)
+
+deff_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))))
+
+
+
+
+
Using autograd
+
+
+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.
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+
+deff1(x):
+ return x**3 + 1
+
+f1_grad = grad(f1)
+
+# Remember to send in float as argument to the computed gradient from Autograd!
+a = 1.0
+
+# See the evaluated gradient at a using autograd:
+print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
+
+# Compare with the analytical derivative, that is f1'(x) = 3*x**2
+grad_analytical = 3*a**2
+print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
+
+
+
+
+
Autograd with more complicated functions
+
+
+To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff2(x1,x2):
+ return3*x1**3 + x2*(x1 - 5) + 1
+
+# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
+f2_grad_x1 = grad(f2,0)
+
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
+
+x1 = 1.0
+x2 = 3.0
+
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
+
+# Compare with the analytical derivatives:
+
+# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
+f2_grad_x1_analytical = 9*x1**2 + x2
+
+# Derivative of f2 w.r.t x2 is: x1 - 5:
+f2_grad_x2_analytical = x1 - 5
+
+# See the evaluated derivations:
+print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+
+print()
+
+print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+
+
+Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
+
+
+
+
+
More complicated functions using the elements of their arguments directly
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff3(x): # Assumes x is an array of length 5 or higher
+ return2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
+
+f3_grad = grad(f3)
+
+x = np.linspace(0,4,5)
+
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_grad(x))
+
+# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
+f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+
+# Print the analytical gradient:
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
+
+
+Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+
+
+
+
+
Functions using mathematical functions from Numpy
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff4(x):
+ return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
+
+f4_grad = grad(f4)
+
+x = 2.7
+
+# Print the computed derivative:
+print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+
+# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
+f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
+
+# Print the analytical gradient:
+print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
+
+
+
+
+
More autograd
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff5(x):
+ if x >= 0:
+ return x**2
+ else:
+ return -3*x + 1
+
+f5_grad = grad(f5)
+
+x = 2.7
+
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
+
+
+
+
+
And with loops
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff6_for(x):
+ val = 0
+ for i inrange(10):
+ val = val + x**i
+ return val
+
+deff6_while(x):
+ val = 0
+ i = 0
+ while i < 10:
+ val = val + x**i
+ i = i + 1
+ return val
+
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x = 0.5
+
+# Print the computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport 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 inrange(10):
+ f6_grad_analytical += i*x**(i-1)
+
+print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
+
+
+
+
+
Using recursion
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+
+deff7(n): # Assume that n is an integer
+ if n == 1or n == 0:
+ return1
+ else:
+ return n*f7(n-1)
+
+f7_grad = grad(f7)
+
+n = 2.0
+
+print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
+
+# The function f7 is an implementation of the factorial of n.
+# By using the product rule, one can find that the derivative is:
+
+f7_grad_analytical = 0
+for i inrange(int(n)-1):
+ tmp = 1
+ for k inrange(int(n)-1):
+ if k != i:
+ tmp *= (n - k)
+ f7_grad_analytical += tmp
+
+print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
+
+
+Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
+
+
+
+
+
Unsupported functions
+Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
+
+
+Assigning a value to the variable being differentiated with respect to
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff8(x): # Assume x is an array
+ x[2] = 3
+ return x*2
+
+f8_grad = grad(f8)
+
+x = 8.4
+
+print("The derivative of f8 is:",f8_grad(x))
+
+
+Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
+
+
+
+
+
The syntax a.dot(b) when finding the dot product
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff9(a): # Assume a is an array with 2 elements
+ b = np.array([1.0,2.0])
+ return a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([1.0,0.0])
+
+print("The derivative of f9 is:",f9_grad(x))
+
+
+Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array. To overcome this, an alternative syntax
+which also computed the dot product can be used:
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff9_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
+
+
+
+
a += b
+a -= b
+a*= b
+a /=b
+
+
+
+
+
Standard steepest descent
+
+
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent, which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type
+$$
+\begin{equation*}
+\hat{A}\hat{x} = \hat{b}.
+\end{equation*}
+$$
+
+
+In the iterative process we end up with a problem like
+
+$$
+\begin{equation*}
+ \hat{r}= \hat{b}-\hat{A}\hat{x},
+\end{equation*}
+$$
+
+where \( \hat{r} \) is the so-called residual or error in the iterative process.
+
+
+When we have found the exact solution, \( \hat{r}=0 \).
+
+
+
+
+
Gradient method
+
+
+The residual is zero when we reach the minimum of the quadratic equation
+$$
+\begin{equation*}
+ P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
+\end{equation*}
+$$
+
+
+with the constraint that the matrix \( \hat{A} \) is positive definite and
+symmetric. This defines also the Hessian and we want it to be positive definite.
+
+
+
+
+
Steepest descent method
+
+
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
+
+
+
Steepest descent method
+
+
+
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+ f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
+
+
+In the CG method we define so-called conjugate directions and two vectors
+\( \hat{s} \) and \( \hat{t} \)
+are said to be
+conjugate if
+$$
+\begin{equation*}
+\hat{s}^T\hat{A}\hat{t}= 0.
+\end{equation*}
+$$
+
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \hat{x}_i \) obeying the above criterion, namely
+$$
+\begin{equation*}
+\hat{x}_i^T\hat{A}\hat{x}_j= 0.
+\end{equation*}
+$$
+
+Two vectors are conjugate if they are orthogonal with respect to
+this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+An example is given by the eigenvectors of the matrix
+$$
+\begin{equation*}
+\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
+\end{equation*}
+$$
+
+which is zero unless \( i=j \).
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
+$$
+\begin{equation*}
+\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}.
+\end{equation*}
+$$
+
+We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
+Then the \( \hat{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
+$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely
+
+$$
+\begin{equation*}
+ \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
+\end{equation*}
+$$
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+The coefficients are given by
+$$
+\begin{equation*}
+ \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
+\end{equation*}
+$$
+
+Multiplying with \( \hat{p}_k^T \) from the left gives
+
+$$
+\begin{equation*}
+ \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
+\end{equation*}
+$$
+
+and we can define the coefficients \( \alpha_k \) as
+
+$$
+\begin{equation*}
+ \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
+\end{equation*}
+$$
+
+
+
+
+
+
+
Conjugate gradient method and iterations
+
+
+
+
+
+If we choose the conjugate vectors \( \hat{p}_k \) carefully,
+then we may not need all of them to obtain a good approximation to the solution
+\( \hat{x} \).
+We want to regard the conjugate gradient method as an iterative method.
+This will us to solve systems where \( n \) is so large that the direct
+method would take too much time.
+
+
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+ f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{p}_1 \)
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+The other vectors in the basis will be conjugate to the gradient,
+hence the name conjugate gradient method.
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
+$$
+\begin{equation*}
+\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
+\end{equation*}
+$$
+
+Note that \( \hat{r}_k \) is the negative gradient of \( f \) at
+\( \hat{x}=\hat{x}_k \),
+so the gradient descent method would be to move in the direction \( \hat{r}_k \).
+Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other,
+so we take the direction closest to the gradient \( \hat{r}_k \)
+under the conjugacy constraint.
+This gives the following expression
+$$
+\begin{equation*}
+\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
+\end{equation*}
+$$
+
Simple implementation of the Conjugate gradient algorithm
+
+
+
+
+
+
+
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
+ int dim = x0.Dimension();
+ constdouble tolerance = 1.0e-14;
+ Vector x(dim),r(dim),v(dim),z(dim);
+ double c,t,d;
+
+ x = x0;
+ r = b - A*x;
+ v = r;
+ c = dot(r,r);
+ int i = 0; IterMax = dim;
+ while(i <= IterMax){
+ z = A*v;
+ t = c/dot(v,z);
+ x = x + t*v;
+ r = r - t*z;
+ d = dot(r,r);
+ if(sqrt(d) < tolerance)
+ break;
+ v = r + (d/c)*v;
+ c = d; i++;
+ }
+ return x;
+}
+
+
+
+
+
+
+
+
+
Broyden–Fletcher–Goldfarb–Shanno algorithm
+
+
+
+The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \) is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that \( \mathbf {x} \) can take.
+
+
+The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
+
+
+The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
+$$
+B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
+$$
+
+
+where \( B_{k} \) is an approximation to the Hessian matrix, which is
+updated iteratively at each stage, and \( \nabla f(\mathbf {x} _{k}) \)
+is the gradient of the function
+evaluated at \( x_k \).
+A line search in the direction \( p_k \) is then used to
+find the next point \( x_{k+1} \) by minimising
+$$
+f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),
+$$
+
+over the scalar \( \alpha > 0 \).
+
+
+
diff --git a/doc/pub/Splines/html/Splines.html b/doc/pub/Splines/html/Splines.html
index f8973a018..299f5c123 100644
--- a/doc/pub/Splines/html/Splines.html
+++ b/doc/pub/Splines/html/Splines.html
@@ -6,6 +6,7 @@ Automatically generated HTML file from DocOnce source
+
Data Analysis and Machine Learning Lectures: Optimization and Gradient Methods
@@ -65,113 +66,114 @@ div { text-align: justify; text-justify: inter-word; }
@@ -213,12 +215,17 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
-
Oct 18, 2018
+
Sep 19, 2019
-
Optimization, the central part of any Machine Learning algortithm
+
Optimization problems, why?
+
+
+
+
+
Optimization, the central part of any Machine Learning algortithm
Almost every problem in machine learning and data science starts with
@@ -233,7 +240,7 @@ some approximative/numerical method to compute the minimum.
-
Revisiting our Logistic Regression case
+
Revisiting our Logistic Regression case
In our discussion on Logistic Regression we studied the
@@ -255,7 +262,7 @@ where \( \hat{\beta} \) are the weights we wish to extract from data, in our cas
-
The equations to solve
+
The equations to solve
Our compact equations used a definition of a vector \( \hat{y} \) with \( n \)
@@ -281,7 +288,7 @@ This defines what is called the Hessian matrix.
-
Solving using Newton-Raphson's method
+
Solving using Newton-Raphson's method
If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
@@ -307,7 +314,7 @@ If we can compute these matrices, in particular the Hessian, the above is often
-
Brief reminder on Newton-Raphson's method
+
Brief reminder on Newton-Raphson's method
Let us quickly remind ourselves how we derive the above method.
@@ -324,7 +331,7 @@ normally discourage the use of this method.
-
The equations
+
The equations
The Newton-Raphson formula consists geometrically of extending the
@@ -360,7 +367,7 @@ $$
-
Simple geometric interpretation
+
Simple geometric interpretation
The above is Newton-Raphson's method. It has a simple geometric
@@ -378,7 +385,7 @@ vanishes, then Newton-Raphson may fail totally
-
Extending to more than one variable
+
Extending to more than one variable
Newton's method can be generalized to systems of several non-linear equations
@@ -433,7 +440,7 @@ more than two non-linear equations. In our case, the Jacobian matrix is given by
-
Steepest descent
+
Steepest descent
The basic idea of gradient descent is
@@ -457,7 +464,7 @@ we are always moving towards smaller function values, i.e a minimum.
-
More on Steepest descent
+
More on Steepest descent
The previous observation is the basis of the method of steepest
@@ -476,7 +483,7 @@ the learning rate within the context of Machine Learning.
-
The ideal
+
The ideal
Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
@@ -502,7 +509,7 @@ Note that the gradient is a function of \( \mathbf{x} =
-
The sensitiveness of the gradient descent
+
The sensitiveness of the gradient descent
The gradient descent method
@@ -521,7 +528,7 @@ randomness. One such method is that of Stochastic Gradient Descent
-
Convex functions
+
Convex functions
Ideally we want our cost/loss function to be convex(concave).
@@ -541,7 +548,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).
-
Convex function
+
Convex function
Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.
@@ -549,7 +556,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).
-
Conditions on convex functions
+
Conditions on convex functions
In the following we state first and second-order conditions which
@@ -591,7 +598,7 @@ This condition is particularly useful since it gives us an procedure for determi
-
More on convex functions
+
More on convex functions
The next result is of great importance to us and the reason why we are
@@ -621,7 +628,7 @@ This result means that if we know that the cost/loss function is convex and we a
-
Some simple problems
+
Some simple problems
Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
@@ -645,623 +652,10 @@ This result means that if we know that the cost/loss function is convex and we a
Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
-
-
-
-
Standard steepest descent
-
-
-Before we proceed, we would like to discuss the approach called the
-standard Steepest descent, which again leads to us having to be able
-to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
-
-
-The success of the CG method
-for finding solutions of non-linear problems is based on the theory
-of conjugate gradients for linear systems of equations. It belongs to
-the class of iterative methods for solving problems from linear
-algebra of the type
-$$
-\begin{equation*}
-\hat{A}\hat{x} = \hat{b}.
-\end{equation*}
-$$
-
-
-In the iterative process we end up with a problem like
-
-$$
-\begin{equation*}
- \hat{r}= \hat{b}-\hat{A}\hat{x},
-\end{equation*}
-$$
-
-where \( \hat{r} \) is the so-called residual or error in the iterative process.
-
-
-When we have found the exact solution, \( \hat{r}=0 \).
-
-
-
-
-
Gradient method
-
-
-The residual is zero when we reach the minimum of the quadratic equation
-$$
-\begin{equation*}
- P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
-\end{equation*}
-$$
-
-
-with the constraint that the matrix \( \hat{A} \) is positive definite and
-symmetric. This defines also the Hessian and we want it to be positive definite.
-
-
-
-
-
Steepest descent method
-
-
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
-
-
-
-
-
Steepest descent method
-
-
-
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
- f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-
-
-
-In the CG method we define so-called conjugate directions and two vectors
-\( \hat{s} \) and \( \hat{t} \)
-are said to be
-conjugate if
-$$
-\begin{equation*}
-\hat{s}^T\hat{A}\hat{t}= 0.
-\end{equation*}
-$$
-
-The philosophy of the CG method is to perform searches in various conjugate directions
-of our vectors \( \hat{x}_i \) obeying the above criterion, namely
-$$
-\begin{equation*}
-\hat{x}_i^T\hat{A}\hat{x}_j= 0.
-\end{equation*}
-$$
-
-Two vectors are conjugate if they are orthogonal with respect to
-this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-An example is given by the eigenvectors of the matrix
-$$
-\begin{equation*}
-\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
-\end{equation*}
-$$
-
-which is zero unless \( i=j \).
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
-\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
-$$
-\begin{equation*}
-\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}.
-\end{equation*}
-$$
-
-We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
-Then the \( \hat{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
-$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely
-
-$$
-\begin{equation*}
- \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
-\end{equation*}
-$$
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-The coefficients are given by
-$$
-\begin{equation*}
- \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
-\end{equation*}
-$$
-
-Multiplying with \( \hat{p}_k^T \) from the left gives
-
-$$
-\begin{equation*}
- \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
-\end{equation*}
-$$
-
-and we can define the coefficients \( \alpha_k \) as
-
-$$
-\begin{equation*}
- \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
-\end{equation*}
-$$
-
-
-
-
-
-
-
Conjugate gradient method and iterations
-
-
-
-
-
-If we choose the conjugate vectors \( \hat{p}_k \) carefully,
-then we may not need all of them to obtain a good approximation to the solution
-\( \hat{x} \).
-We want to regard the conjugate gradient method as an iterative method.
-This will us to solve systems where \( n \) is so large that the direct
-method would take too much time.
-
-
-We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
-We can assume without loss of generality that
-$$
-\begin{equation*}
-\hat{x}_0=0,
-\end{equation*}
-$$
-
-or consider the system
-$$
-\begin{equation*}
-\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
-\end{equation*}
-$$
-
-instead.
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
-$$
-\begin{equation*}
- f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
-\end{equation*}
-$$
-
-This suggests taking the first basis vector \( \hat{p}_1 \)
-to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
-which equals
-$$
-\begin{equation*}
-\hat{A}\hat{x}_0-\hat{b},
-\end{equation*}
-$$
-
-and
-\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
-The other vectors in the basis will be conjugate to the gradient,
-hence the name conjugate gradient method.
-
-
-
-
-
-
-
Conjugate gradient method
-
-
-
-Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
-$$
-\begin{equation*}
-\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
-\end{equation*}
-$$
-
-Note that \( \hat{r}_k \) is the negative gradient of \( f \) at
-\( \hat{x}=\hat{x}_k \),
-so the gradient descent method would be to move in the direction \( \hat{r}_k \).
-Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other,
-so we take the direction closest to the gradient \( \hat{r}_k \)
-under the conjugacy constraint.
-This gives the following expression
-$$
-\begin{equation*}
-\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
-\end{equation*}
-$$
-
Simple implementation of the Conjugate gradient algorithm
-
-
-
-
-
-
-
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
- int dim = x0.Dimension();
- constdouble tolerance =1.0e-14;
- Vector x(dim),r(dim),v(dim),z(dim);
- double c,t,d;
-
- x = x0;
- r = b - A*x;
- v = r;
- c = dot(r,r);
- int i =0; IterMax = dim;
- while(i <= IterMax){
- z = A*v;
- t = c/dot(v,z);
- x = x + t*v;
- r = r - t*z;
- d = dot(r,r);
- if(sqrt(d) < tolerance)
- break;
- v = r + (d/c)*v;
- c = d; i++;
- }
- return x;
-}
-
-
-
-
-
-
-
-
-
Broyden–Fletcher–Goldfarb–Shanno algorithm
-
-
-
-The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \) is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that \( \mathbf {x} \) can take.
-
-
-The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
-
-
-The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
-$$
-B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
-$$
-
-
-where \( B_{k} \) is an approximation to the Hessian matrix, which is
-updated iteratively at each stage, and \( \nabla f(\mathbf {x} _{k}) \)
-is the gradient of the function
-evaluated at \( x_k \).
-A line search in the direction \( p_k \) is then used to
-find the next point \( x_{k+1} \) by minimising
-$$
-f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),
-$$
-
-over the scalar \( \alpha > 0 \).
-
-
-
-
-
-
Revisiting our first homework
+
Revisiting our first homework
We will use linear regression as a case study for the gradient descent
@@ -1294,7 +688,7 @@ $$
-
Gradient descent example
+
Gradient descent example
Let \( \mathbf{y} = (y_1,\cdots,y_n)^T \), \( \mathbf{\hat{y}} = (\hat{y}_1,\cdots,\hat{y}_n)^T \) and \( \beta = (\beta_0, \beta_1)^T \)
@@ -1319,7 +713,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.
-
The derivative of the cost/loss function
+
The derivative of the cost/loss function
Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
@@ -1334,7 +728,7 @@ where \( X \) is the design matrix defined above.
-
The Hessian matrix
+
The Hessian matrix
The Hessian matrix of \( C(\beta) \) is given by
$$
\hat{H} \equiv \begin{bmatrix}
@@ -1348,7 +742,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(
-
Simple program
+
Simple program
We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
@@ -1389,7 +783,7 @@ beta_NE = np.
-
Gradient Descent Example
+
Gradient Descent Example
Another simple example is here
@@ -1438,7 +832,7 @@ plt.show()
-
And a corresponding example using scikit-learn
+
And a corresponding example using scikit-learn
@@ -1462,7 +856,7 @@ sgdreg.fit(x,y.
-
Gradient descent and Ridge
+
Gradient descent and Ridge
We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \),
@@ -1519,393 +913,7 @@ beta_ridge = np
-
Automatic differentiation
-Python has tools for so-called automatic differentiation.
-Consider the following example
-$$
-f(x) = \sin\left(2\pi x + x^2\right)
-$$
-
-which has the following derivative
-$$
-f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
-$$
-
-Using autograd we have
-
-
-
-
-
importautograd.numpyasnp
-
-# To do elementwise differentiation:
-fromautogradimport elementwise_grad as egrad
-
-# To plot:
-importmatplotlib.pyplotasplt
-
-
-deff(x):
- return np.sin(2*np.pi*x + x**2)
-
-deff_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))))
-
-
-
-
-
Using autograd
-
-
-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.
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-
-deff1(x):
- return x**3+1
-
-f1_grad = grad(f1)
-
-# Remember to send in float as argument to the computed gradient from Autograd!
-a =1.0
-
-# See the evaluated gradient at a using autograd:
-print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
-
-# Compare with the analytical derivative, that is f1'(x) = 3*x**2
-grad_analytical =3*a**2
-print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
-
-
-
-
-
Autograd with more complicated functions
-
-
-To differentiate with respect to two (or more) arguments of a Python
-function, Autograd need to know at which variable the function if
-being differentiated with respect to.
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff2(x1,x2):
- return3*x1**3+ x2*(x1 -5) +1
-
-# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
-f2_grad_x1 = grad(f2,0)
-
-# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
-f2_grad_x2 = grad(f2,1)
-
-x1 =1.0
-x2 =3.0
-
-print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
-print("-"*30)
-
-# Compare with the analytical derivatives:
-
-# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
-f2_grad_x1_analytical =9*x1**2+ x2
-
-# Derivative of f2 w.r.t x2 is: x1 - 5:
-f2_grad_x2_analytical = x1 -5
-
-# See the evaluated derivations:
-print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-
-print()
-
-print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-
-
-Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
-
-
-
-
-
More complicated functions using the elements of their arguments directly
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff3(x): # Assumes x is an array of length 5 or higher
- return2*x[0] +3*x[1] +5*x[2] +7*x[3] +11*x[4]**2
-
-f3_grad = grad(f3)
-
-x = np.linspace(0,4,5)
-
-# Print the computed gradient:
-print("The computed gradient of f3 is: ", f3_grad(x))
-
-# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
-f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
-
-# Print the analytical gradient:
-print("The analytical gradient of f3 is: ", f3_grad_analytical)
-
-
-Note that in this case, when sending an array as input argument, the
-output from Autograd is another array. This is the true gradient of
-the function, as opposed to the function in the previous example. By
-using arrays to represent the variables, the output from Autograd
-might be easier to work with, as the output is closer to what one
-could expect form a gradient-evaluting function.
-
-
-
-
-
Functions using mathematical functions from Numpy
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff4(x):
- return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
-
-f4_grad = grad(f4)
-
-x =2.7
-
-# Print the computed derivative:
-print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
-
-# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
-f4_grad_analytical = x/np.sqrt(1+ x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
-
-# Print the analytical gradient:
-print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
-
-
-
-
-
More autograd
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff5(x):
- if x >=0:
- return x**2
- else:
- return-3*x +1
-
-f5_grad = grad(f5)
-
-x =2.7
-
-# Print the computed derivative:
-print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
-
-
-
-
-
And with loops
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff6_for(x):
- val =0
- for i inrange(10):
- val = val + x**i
- return val
-
-deff6_while(x):
- val =0
- i =0
- while i <10:
- val = val + x**i
- i = i +1
- return val
-
-f6_for_grad = grad(f6_for)
-f6_while_grad = grad(f6_while)
-
-x =0.5
-
-# Print the computed derivaties of f6_for and f6_while
-print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
-print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport 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 inrange(10):
- f6_grad_analytical += i*x**(i-1)
-
-print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
-
-
-
-
-
Using recursion
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-
-deff7(n): # Assume that n is an integer
- if n ==1or n ==0:
- return1
- else:
- return n*f7(n-1)
-
-f7_grad = grad(f7)
-
-n =2.0
-
-print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
-
-# The function f7 is an implementation of the factorial of n.
-# By using the product rule, one can find that the derivative is:
-
-f7_grad_analytical =0
-for i inrange(int(n)-1):
- tmp =1
- for k inrange(int(n)-1):
- if k != i:
- tmp *= (n - k)
- f7_grad_analytical += tmp
-
-print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
-
-
-Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
-
-
-
-
-
Unsupported functions
-Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
-
-
-Assigning a value to the variable being differentiated with respect to
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff8(x): # Assume x is an array
- x[2] =3
- return x*2
-
-f8_grad = grad(f8)
-
-x =8.4
-
-print("The derivative of f8 is:",f8_grad(x))
-
-
-Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
-
-
-
-
-
The syntax a.dot(b) when finding the dot product
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff9(a): # Assume a is an array with 2 elements
- b = np.array([1.0,2.0])
- return a.dot(b)
-
-f9_grad = grad(f9)
-
-x = np.array([1.0,0.0])
-
-print("The derivative of f9 is:",f9_grad(x))
-
-
-Here we are told that the 'dot' function does not belong to Autograd's
-version of a Numpy array. To overcome this, an alternative syntax
-which also computed the dot product can be used:
-
-
-
-
-
importautograd.numpyasnp
-fromautogradimport grad
-deff9_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
-
-
-
-
a += b
-a -= b
-a*= b
-a /=b
-
-
-
-
-
Stochastic Gradient Descent
+
Stochastic Gradient Descent
Stochastic gradient descent (SGD) and variants thereof address some of
@@ -1923,7 +931,7 @@ $$
-
Computation of gradients
+
Computation of gradients
This in turn means that the gradient can be
@@ -1943,7 +951,7 @@ minibatches. We denote these minibatches by \( B_k \) where
-
SGD example
+
SGD example
As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \)
and we choose to have \( M=5 \) minibathces,
then each minibatch contains two data points. In particular we have
@@ -1967,7 +975,7 @@ $$
-
The gradient step
+
The gradient step
Thus a gradient descent step now looks like
@@ -1986,7 +994,7 @@ the number of minibatches, as exemplified in the code below.
-
Simple example code
+
Simple example code
@@ -2018,7 +1026,7 @@ all \( n \) datapoints.
-
When do we stop?
+
When do we stop?
A natural question is when do we stop the search for a new minimum?
@@ -2035,7 +1043,7 @@ gave the lowest value.
-
Slightly different approach
+
Slightly different approach
Another approach is to let the step length \( \gamma_j \) depend on the
@@ -2083,7 +1091,7 @@ j =0
-
Program for stochastic gradient
+
Program for stochastic gradient
@@ -2162,7 +1170,7 @@ plt.show()
-
Using gradient descent methods, limitations
+
Using gradient descent methods, limitations
Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our energy function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
@@ -2175,7 +1183,7 @@ plt.show()
-
Momentum based GD
+
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
@@ -2198,7 +1206,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\
-
More on momentum based approaches
+
More on momentum based approaches
Let us try to get more intuition from these equations. It is helpful to consider a simple physical analogy with a particle of mass \( m \) moving in a viscous medium with drag coefficient \( \mu \) and potential
@@ -2220,7 +1228,7 @@ $$
-
Momentum parameter
+
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:
$$
@@ -2250,7 +1258,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea
-
Second moment of the gradient
+
Second moment of the gradient
In stochastic gradient descent, with and without momentum, we still
@@ -2275,7 +1283,7 @@ Recently, a number of methods have been introduced that accomplish this by track
-
RMS prop
+
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
@@ -2293,7 +1301,7 @@ where \( \beta \) controls the averaging time of the second moment and is typica
-
ADAM optimizer
+
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)
@@ -2321,7 +1329,7 @@ $$
-
Practical tips
+
Practical tips
Randomize the data when making mini-batches. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.
@@ -2332,11 +1340,1012 @@ $$
Geron's text, see chapter 11, has several interesting discussions.
+
+
+
+
Automatic differentiation
+Python has tools for so-called automatic differentiation.
+Consider the following example
+$$
+f(x) = \sin\left(2\pi x + x^2\right)
+$$
+
+which has the following derivative
+$$
+f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right)
+$$
+
+Using autograd we have
+
+
+
+
+
importautograd.numpyasnp
+
+# To do elementwise differentiation:
+fromautogradimport elementwise_grad as egrad
+
+# To plot:
+importmatplotlib.pyplotasplt
+
+
+deff(x):
+ return np.sin(2*np.pi*x + x**2)
+
+deff_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))))
+
+
+
+
+
Using autograd
+
+
+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.
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+
+deff1(x):
+ return x**3+1
+
+f1_grad = grad(f1)
+
+# Remember to send in float as argument to the computed gradient from Autograd!
+a =1.0
+
+# See the evaluated gradient at a using autograd:
+print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
+
+# Compare with the analytical derivative, that is f1'(x) = 3*x**2
+grad_analytical =3*a**2
+print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
+
+
+
+
+
Autograd with more complicated functions
+
+
+To differentiate with respect to two (or more) arguments of a Python
+function, Autograd need to know at which variable the function if
+being differentiated with respect to.
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff2(x1,x2):
+ return3*x1**3+ x2*(x1 -5) +1
+
+# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
+f2_grad_x1 = grad(f2,0)
+
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
+
+x1 =1.0
+x2 =3.0
+
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
+
+# Compare with the analytical derivatives:
+
+# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
+f2_grad_x1_analytical =9*x1**2+ x2
+
+# Derivative of f2 w.r.t x2 is: x1 - 5:
+f2_grad_x2_analytical = x1 -5
+
+# See the evaluated derivations:
+print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+
+print()
+
+print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+
+
+Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.
+
+
+
+
+
More complicated functions using the elements of their arguments directly
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff3(x): # Assumes x is an array of length 5 or higher
+ return2*x[0] +3*x[1] +5*x[2] +7*x[3] +11*x[4]**2
+
+f3_grad = grad(f3)
+
+x = np.linspace(0,4,5)
+
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_grad(x))
+
+# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
+f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+
+# Print the analytical gradient:
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
+
+
+Note that in this case, when sending an array as input argument, the
+output from Autograd is another array. This is the true gradient of
+the function, as opposed to the function in the previous example. By
+using arrays to represent the variables, the output from Autograd
+might be easier to work with, as the output is closer to what one
+could expect form a gradient-evaluting function.
+
+
+
+
+
Functions using mathematical functions from Numpy
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff4(x):
+ return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
+
+f4_grad = grad(f4)
+
+x =2.7
+
+# Print the computed derivative:
+print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+
+# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
+f4_grad_analytical = x/np.sqrt(1+ x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
+
+# Print the analytical gradient:
+print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
+
+
+
+
+
More autograd
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff5(x):
+ if x >=0:
+ return x**2
+ else:
+ return-3*x +1
+
+f5_grad = grad(f5)
+
+x =2.7
+
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
+
+
+
+
+
And with loops
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff6_for(x):
+ val =0
+ for i inrange(10):
+ val = val + x**i
+ return val
+
+deff6_while(x):
+ val =0
+ i =0
+ while i <10:
+ val = val + x**i
+ i = i +1
+ return val
+
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
+
+x =0.5
+
+# Print the computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport 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 inrange(10):
+ f6_grad_analytical += i*x**(i-1)
+
+print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
+
+
+
+
+
Using recursion
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+
+deff7(n): # Assume that n is an integer
+ if n ==1or n ==0:
+ return1
+ else:
+ return n*f7(n-1)
+
+f7_grad = grad(f7)
+
+n =2.0
+
+print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
+
+# The function f7 is an implementation of the factorial of n.
+# By using the product rule, one can find that the derivative is:
+
+f7_grad_analytical =0
+for i inrange(int(n)-1):
+ tmp =1
+ for k inrange(int(n)-1):
+ if k != i:
+ tmp *= (n - k)
+ f7_grad_analytical += tmp
+
+print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
+
+
+Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.
+
+
+
+
+
Unsupported functions
+Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.
+
+
+Assigning a value to the variable being differentiated with respect to
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff8(x): # Assume x is an array
+ x[2] =3
+ return x*2
+
+f8_grad = grad(f8)
+
+x =8.4
+
+print("The derivative of f8 is:",f8_grad(x))
+
+
+Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
+
+
+
+
+
The syntax a.dot(b) when finding the dot product
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff9(a): # Assume a is an array with 2 elements
+ b = np.array([1.0,2.0])
+ return a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([1.0,0.0])
+
+print("The derivative of f9 is:",f9_grad(x))
+
+
+Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array. To overcome this, an alternative syntax
+which also computed the dot product can be used:
+
+
+
+
+
importautograd.numpyasnp
+fromautogradimport grad
+deff9_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
+
+
+
+
a += b
+a -= b
+a*= b
+a /=b
+
+
+
+
+
Standard steepest descent
+
+
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent, which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type
+$$
+\begin{equation*}
+\hat{A}\hat{x} = \hat{b}.
+\end{equation*}
+$$
+
+
+In the iterative process we end up with a problem like
+
+$$
+\begin{equation*}
+ \hat{r}= \hat{b}-\hat{A}\hat{x},
+\end{equation*}
+$$
+
+where \( \hat{r} \) is the so-called residual or error in the iterative process.
+
+
+When we have found the exact solution, \( \hat{r}=0 \).
+
+
+
+
+
Gradient method
+
+
+The residual is zero when we reach the minimum of the quadratic equation
+$$
+\begin{equation*}
+ P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b},
+\end{equation*}
+$$
+
+
+with the constraint that the matrix \( \hat{A} \) is positive definite and
+symmetric. This defines also the Hessian and we want it to be positive definite.
+
+
+
+
+
Steepest descent method
+
+
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
+
+
+
Steepest descent method
+
+
+
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+ f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{r}_1 \) (see below for definition)
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+
+
+
+In the CG method we define so-called conjugate directions and two vectors
+\( \hat{s} \) and \( \hat{t} \)
+are said to be
+conjugate if
+$$
+\begin{equation*}
+\hat{s}^T\hat{A}\hat{t}= 0.
+\end{equation*}
+$$
+
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \hat{x}_i \) obeying the above criterion, namely
+$$
+\begin{equation*}
+\hat{x}_i^T\hat{A}\hat{x}_j= 0.
+\end{equation*}
+$$
+
+Two vectors are conjugate if they are orthogonal with respect to
+this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is conjugate to \( \hat{t} \), then \( \hat{t} \) is conjugate to \( \hat{s} \).
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+An example is given by the eigenvectors of the matrix
+$$
+\begin{equation*}
+\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j,
+\end{equation*}
+$$
+
+which is zero unless \( i=j \).
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
+$$
+\begin{equation*}
+\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}.
+\end{equation*}
+$$
+
+We assume that \( \hat{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
+Then the \( \hat{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
+$ \hat{A}\hat{x} = \hat{b}$ in this basis, namely
+
+$$
+\begin{equation*}
+ \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i.
+\end{equation*}
+$$
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+The coefficients are given by
+$$
+\begin{equation*}
+ \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
+\end{equation*}
+$$
+
+Multiplying with \( \hat{p}_k^T \) from the left gives
+
+$$
+\begin{equation*}
+ \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b},
+\end{equation*}
+$$
+
+and we can define the coefficients \( \alpha_k \) as
+
+$$
+\begin{equation*}
+ \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k}
+\end{equation*}
+$$
+
+
+
+
+
+
+
Conjugate gradient method and iterations
+
+
+
+
+
+If we choose the conjugate vectors \( \hat{p}_k \) carefully,
+then we may not need all of them to obtain a good approximation to the solution
+\( \hat{x} \).
+We want to regard the conjugate gradient method as an iterative method.
+This will us to solve systems where \( n \) is so large that the direct
+method would take too much time.
+
+
+We denote the initial guess for \( \hat{x} \) as \( \hat{x}_0 \).
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\hat{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\hat{A}\hat{z} = \hat{b}-\hat{A}\hat{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+ f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n.
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \hat{p}_1 \)
+to be the gradient of \( f \) at \( \hat{x}=\hat{x}_0 \),
+which equals
+$$
+\begin{equation*}
+\hat{A}\hat{x}_0-\hat{b},
+\end{equation*}
+$$
+
+and
+\( \hat{x}_0=0 \) it is equal \( -\hat{b} \).
+The other vectors in the basis will be conjugate to the gradient,
+hence the name conjugate gradient method.
+
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+Let \( \hat{r}_k \) be the residual at the \( k \)-th step:
+$$
+\begin{equation*}
+\hat{r}_k=\hat{b}-\hat{A}\hat{x}_k.
+\end{equation*}
+$$
+
+Note that \( \hat{r}_k \) is the negative gradient of \( f \) at
+\( \hat{x}=\hat{x}_k \),
+so the gradient descent method would be to move in the direction \( \hat{r}_k \).
+Here, we insist that the directions \( \hat{p}_k \) are conjugate to each other,
+so we take the direction closest to the gradient \( \hat{r}_k \)
+under the conjugacy constraint.
+This gives the following expression
+$$
+\begin{equation*}
+\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k.
+\end{equation*}
+$$
+
Simple implementation of the Conjugate gradient algorithm
+
+
+
+
+
+
+
Vector ConjugateGradient(Matrix A, Vector b, Vector x0){
+ int dim = x0.Dimension();
+ constdouble tolerance =1.0e-14;
+ Vector x(dim),r(dim),v(dim),z(dim);
+ double c,t,d;
+
+ x = x0;
+ r = b - A*x;
+ v = r;
+ c = dot(r,r);
+ int i =0; IterMax = dim;
+ while(i <= IterMax){
+ z = A*v;
+ t = c/dot(v,z);
+ x = x + t*v;
+ r = r - t*z;
+ d = dot(r,r);
+ if(sqrt(d) < tolerance)
+ break;
+ v = r + (d/c)*v;
+ c = d; i++;
+ }
+ return x;
+}
+
+
+
+
+
+
+
+
+
Broyden–Fletcher–Goldfarb–Shanno algorithm
+
+
+
+The optimization problem is to minimize \( f(\mathbf {x} ) \) where \( \mathbf {x} \) is a vector in \( R^{n} \), and \( f \) is a differentiable scalar function. There are no constraints on the values that \( \mathbf {x} \) can take.
+
+
+The algorithm begins at an initial estimate for the optimal value \( \mathbf {x}_{0} \) and proceeds iteratively to get a better estimate at each stage.
+
+
+The search direction \( p_k \) at stage \( k \) is given by the solution of the analogue of the Newton equation
+$$
+B_{k}\mathbf {p} _{k}=-\nabla f(\mathbf {x}_{k}),
+$$
+
+
+where \( B_{k} \) is an approximation to the Hessian matrix, which is
+updated iteratively at each stage, and \( \nabla f(\mathbf {x} _{k}) \)
+is the gradient of the function
+evaluated at \( x_k \).
+A line search in the direction \( p_k \) is then used to
+find the next point \( x_{k+1} \) by minimising
+$$
+f(\mathbf {x}_{k}+\alpha \mathbf {p}_{k}),
+$$
+
+over the scalar \( \alpha > 0 \).
+
+
+
diff --git a/doc/pub/Splines/html/reveal.js/.gitignore b/doc/pub/Splines/html/reveal.js/.gitignore
index e7b4f216a..a5df3133d 100644
--- a/doc/pub/Splines/html/reveal.js/.gitignore
+++ b/doc/pub/Splines/html/reveal.js/.gitignore
@@ -1,8 +1,3 @@
-.idea/
-*.iml
-*.iws
-*.eml
-out/
.DS_Store
.svn
log/*.log
@@ -10,4 +5,4 @@ tmp/**
node_modules/
.sass-cache
css/reveal.min.css
-js/reveal.min.js
\ No newline at end of file
+js/reveal.min.js
diff --git a/doc/pub/Splines/html/reveal.js/.travis.yml b/doc/pub/Splines/html/reveal.js/.travis.yml
index ec3b27d5d..165d9ae9f 100644
--- a/doc/pub/Splines/html/reveal.js/.travis.yml
+++ b/doc/pub/Splines/html/reveal.js/.travis.yml
@@ -1,7 +1,5 @@
language: node_js
node_js:
- - 4
+ - 0.10
before_script:
- - npm install -g grunt-cli
-after_script:
- - grunt retire
+ - npm install -g grunt-cli
\ No newline at end of file
diff --git a/doc/pub/Splines/html/reveal.js/LICENSE b/doc/pub/Splines/html/reveal.js/LICENSE
index c3e6e5fd6..09623076f 100644
--- a/doc/pub/Splines/html/reveal.js/LICENSE
+++ b/doc/pub/Splines/html/reveal.js/LICENSE
@@ -1,4 +1,4 @@
-Copyright (C) 2017 Hakim El Hattab, http://hakim.se, and reveal.js contributors
+Copyright (C) 2015 Hakim El Hattab, http://hakim.se
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
diff --git a/doc/pub/Splines/html/reveal.js/README.md b/doc/pub/Splines/html/reveal.js/README.md
index f2ab6ca88..573b19597 100644
--- a/doc/pub/Splines/html/reveal.js/README.md
+++ b/doc/pub/Splines/html/reveal.js/README.md
@@ -1,58 +1,12 @@
-# reveal.js [](https://travis-ci.org/hakimel/reveal.js)
+# reveal.js [](https://travis-ci.org/hakimel/reveal.js)
-A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://revealjs.com/).
+A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://lab.hakim.se/reveal-js/).
-reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). There's also a fully featured visual editor and platform for sharing reveal.js presentations at [slides.com](https://slides.com?ref=github).
+reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). It's best viewed in a modern browser but [fallbacks](https://github.com/hakimel/reveal.js/wiki/Browser-Support) are available to make sure your presentation can still be viewed elsewhere.
-## Table of contents
-- [Online Editor](#online-editor)
-- [Instructions](#instructions)
- - [Markup](#markup)
- - [Markdown](#markdown)
- - [Element Attributes](#element-attributes)
- - [Slide Attributes](#slide-attributes)
-- [Configuration](#configuration)
-- [Presentation Size](#presentation-size)
-- [Dependencies](#dependencies)
-- [Ready Event](#ready-event)
-- [Auto-sliding](#auto-sliding)
-- [Keyboard Bindings](#keyboard-bindings)
-- [Touch Navigation](#touch-navigation)
-- [Lazy Loading](#lazy-loading)
-- [API](#api)
- - [Slide Changed Event](#slide-changed-event)
- - [Presentation State](#presentation-state)
- - [Slide States](#slide-states)
- - [Slide Backgrounds](#slide-backgrounds)
- - [Parallax Background](#parallax-background)
- - [Slide Transitions](#slide-transitions)
- - [Internal links](#internal-links)
- - [Fragments](#fragments)
- - [Fragment events](#fragment-events)
- - [Code syntax highlighting](#code-syntax-highlighting)
- - [Slide number](#slide-number)
- - [Overview mode](#overview-mode)
- - [Fullscreen mode](#fullscreen-mode)
- - [Embedded media](#embedded-media)
- - [Stretching elements](#stretching-elements)
- - [postMessage API](#postmessage-api)
-- [PDF Export](#pdf-export)
-- [Theming](#theming)
-- [Speaker Notes](#speaker-notes)
- - [Share and Print Speaker Notes](#share-and-print-speaker-notes)
- - [Server Side Speaker Notes](#server-side-speaker-notes)
-- [Multiplexing](#multiplexing)
- - [Master presentation](#master-presentation)
- - [Client presentation](#client-presentation)
- - [Socket.io server](#socketio-server)
-- [MathJax](#mathjax)
-- [Installation](#installation)
- - [Basic setup](#basic-setup)
- - [Full setup](#full-setup)
- - [Folder Structure](#folder-structure)
-- [License](#license)
-#### More reading
+#### More reading:
+- [Installation](#installation): Step-by-step instructions for getting reveal.js running on your computer.
- [Changelog](https://github.com/hakimel/reveal.js/releases): Up-to-date version history.
- [Examples](https://github.com/hakimel/reveal.js/wiki/Example-Presentations): Presentations created with reveal.js, add your own!
- [Browser Support](https://github.com/hakimel/reveal.js/wiki/Browser-Support): Explanation of browser support and fallbacks.
@@ -60,36 +14,14 @@ reveal.js comes with a broad range of features including [nested slides](https:/
## Online Editor
-Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [https://slides.com](https://slides.com?ref=github).
+Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [http://slides.com](http://slides.com).
## Instructions
### Markup
-Here's a barebones example of a fully working reveal.js presentation:
-```html
-
-
-
-
-
-
-
-
- Slide 1
- Slide 2
-
-
-
-
-
-
-```
-
-The presentation markup hierarchy needs to be `.reveal > .slides > section` where the `section` represents one slide and can be repeated indefinitely. If you place multiple `section` elements inside of another `section` they will be shown as vertical slides. The first of the vertical slides is the "root" of the others (at the top), and will be included in the horizontal sequence. For example:
+Markup hierarchy needs to be ``
`` where the ```` represents one slide and can be repeated indefinitely. If you place multiple ````'s inside of another ```` they will be shown as vertical slides. The first of the vertical slides is the "root" of the others (at the top), and it will be included in the horizontal sequence. For example:
```html
@@ -105,36 +37,32 @@ The presentation markup hierarchy needs to be `.reveal > .slides > section` wher
### Markdown
-It's possible to write your slides using Markdown. To enable Markdown, add the `data-markdown` attribute to your `` elements and wrap the contents in a `
+
```
#### External Markdown
-You can write your content as a separate file and have reveal.js load it at runtime. Note the separator arguments which determine how slides are delimited in the external file: the `data-separator` attribute defines a regular expression for horizontal slides (defaults to `^\r?\n---\r?\n$`, a newline-bounded horizontal rule) and `data-separator-vertical` defines vertical slides (disabled by default). The `data-separator-notes` attribute is a regular expression for specifying the beginning of the current slide's speaker notes (defaults to `note:`). The `data-charset` attribute is optional and specifies which charset to use when loading the external file.
+You can write your content as a separate file and have reveal.js load it at runtime. Note the separator arguments which determine how slides are delimited in the external file. The ```data-charset``` attribute is optional and specifies which charset to use when loading the external file.
-When used locally, this feature requires that reveal.js [runs from a local web server](#full-setup). The following example customises all available options:
+When used locally, this feature requires that reveal.js [runs from a local web server](#full-setup).
```html
-
-
```
@@ -164,19 +92,6 @@ Special syntax (in html comment) is available for adding attributes to the slide
```
-#### Configuring *marked*
-
-We use [marked](https://github.com/chjj/marked) to parse Markdown. To customise marked's rendering, you can pass in options when [configuring Reveal](#configuration):
-
-```javascript
-Reveal.initialize({
- // Options which are passed into marked
- // See https://github.com/chjj/marked#options-1
- markdown: {
- smartypants: true
- }
-});
-```
### Configuration
@@ -185,26 +100,12 @@ At the end of your page you need to initialize reveal by running the following c
```javascript
Reveal.initialize({
- // Display presentation control arrows
+ // Display controls in the bottom right corner
controls: true,
- // Help the user learn the controls by providing hints, for example by
- // bouncing the down arrow when they first encounter a vertical slide
- controlsTutorial: true,
-
- // Determines where controls appear, "edges" or "bottom-right"
- controlsLayout: 'bottom-right',
-
- // Visibility rule for backwards navigation arrows; "faded", "hidden"
- // or "visible"
- controlsBackArrows: 'faded',
-
// Display a presentation progress bar
progress: true,
- // Set default timing of 2 minutes per slide
- defaultTiming: 120,
-
// Display the page number of the current slide
slideNumber: false,
@@ -229,9 +130,6 @@ Reveal.initialize({
// Change the presentation direction to be RTL
rtl: false,
- // Randomizes the order of slides each time the presentation loads
- shuffle: false,
-
// Turns fragments on and off globally
fragments: true,
@@ -243,15 +141,6 @@ Reveal.initialize({
// key is pressed
help: true,
- // Flags if speaker notes should be visible to all viewers
- showNotes: false,
-
- // Global override for autoplaying embedded media (video/audio/iframe)
- // - null: Media will only autoplay if data-autoplay is present
- // - true: All media will autoplay, regardless of individual setting
- // - false: No media will autoplay, regardless of individual setting
- autoPlayMedia: null,
-
// Number of milliseconds between automatically proceeding to the
// next slide, disabled when set to 0, this value can be overwritten
// by using a data-autoslide attribute on your slides
@@ -260,9 +149,6 @@ Reveal.initialize({
// Stop auto-sliding after user input
autoSlideStoppable: true,
- // Use this method for navigation when auto-sliding
- autoSlideMethod: Reveal.navigateNext,
-
// Enable slide navigation via mouse wheel
mouseWheel: false,
@@ -270,18 +156,16 @@ Reveal.initialize({
hideAddressBar: true,
// Opens links in an iframe preview overlay
- // Add `data-preview-link` and `data-preview-link="false"` to customise each link
- // individually
previewLinks: false,
// Transition style
- transition: 'slide', // none/fade/slide/convex/concave/zoom
+ transition: 'default', // none/fade/slide/convex/concave/zoom
// Transition speed
transitionSpeed: 'default', // default/fast/slow
// Transition style for full page slide backgrounds
- backgroundTransition: 'fade', // none/fade/slide/convex/concave/zoom
+ backgroundTransition: 'default', // none/fade/slide/convex/concave/zoom
// Number of slides away from the current that are visible
viewDistance: 3,
@@ -292,14 +176,10 @@ Reveal.initialize({
// Parallax background size
parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px"
- // Number of pixels to move the parallax background per slide
- // - Calculated automatically unless specified
- // - Set to 0 to disable movement along an axis
- parallaxBackgroundHorizontal: null,
- parallaxBackgroundVertical: null,
-
- // The display mode that will be used to show slides
- display: 'block'
+ // Amount to move parallax background (horizontal and vertical) on slide change
+ // Number, e.g. 100
+ parallaxBackgroundHorizontal: '',
+ parallaxBackgroundVertical: ''
});
```
@@ -316,6 +196,56 @@ Reveal.configure({ autoSlide: 5000 });
```
+### Dependencies
+
+Reveal.js doesn't _rely_ on any third party scripts to work but a few optional libraries are included by default. These libraries are loaded as dependencies in the order they appear, for example:
+
+```javascript
+Reveal.initialize({
+ dependencies: [
+ // Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/
+ { src: 'lib/js/classList.js', condition: function() { return !document.body.classList; } },
+
+ // Interpret Markdown in elements
+ { src: 'plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
+ { src: 'plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
+
+ // Syntax highlight for elements
+ { src: 'plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } },
+
+ // Zoom in and out with Alt+click
+ { src: 'plugin/zoom-js/zoom.js', async: true },
+
+ // Speaker notes
+ { src: 'plugin/notes/notes.js', async: true },
+
+ // Remote control your reveal.js presentation using a touch device
+ { src: 'plugin/remotes/remotes.js', async: true },
+
+ // MathJax
+ { src: 'plugin/math/math.js', async: true }
+ ]
+});
+```
+
+You can add your own extensions using the same syntax. The following properties are available for each dependency object:
+- **src**: Path to the script to load
+- **async**: [optional] Flags if the script should load after reveal.js has started, defaults to false
+- **callback**: [optional] Function to execute when the script has loaded
+- **condition**: [optional] Function which must return true for the script to be loaded
+
+
+### Ready Event
+
+A 'ready' event is fired when reveal.js has loaded all non-async dependencies and is ready to start navigating. To check if reveal.js is already 'ready' you can call `Reveal.isReady()`.
+
+```javascript
+Reveal.addEventListener( 'ready', function( event ) {
+ // event.currentSlide, event.indexh, event.indexv
+} );
+```
+
+
### Presentation Size
All presentations have a normal size, that is the resolution at which they are authored. The framework will automatically scale presentations uniformly based on this size to ensure that everything fits on any given display or viewport.
@@ -343,69 +273,6 @@ Reveal.initialize({
});
```
-If you wish to disable this behavior and do your own scaling (e.g. using media queries), try these settings:
-
-```javascript
-Reveal.initialize({
-
- ...
-
- width: "100%",
- height: "100%",
- margin: 0,
- minScale: 1,
- maxScale: 1
-});
-```
-
-### Dependencies
-
-Reveal.js doesn't _rely_ on any third party scripts to work but a few optional libraries are included by default. These libraries are loaded as dependencies in the order they appear, for example:
-
-```javascript
-Reveal.initialize({
- dependencies: [
- // Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/
- { src: 'lib/js/classList.js', condition: function() { return !document.body.classList; } },
-
- // Interpret Markdown in elements
- { src: 'plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
- { src: 'plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
-
- // Syntax highlight for elements
- { src: 'plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } },
-
- // Zoom in and out with Alt+click
- { src: 'plugin/zoom-js/zoom.js', async: true },
-
- // Speaker notes
- { src: 'plugin/notes/notes.js', async: true },
-
- // MathJax
- { src: 'plugin/math/math.js', async: true }
- ]
-});
-```
-
-You can add your own extensions using the same syntax. The following properties are available for each dependency object:
-- **src**: Path to the script to load
-- **async**: [optional] Flags if the script should load after reveal.js has started, defaults to false
-- **callback**: [optional] Function to execute when the script has loaded
-- **condition**: [optional] Function which must return true for the script to be loaded
-
-To load these dependencies, reveal.js requires [head.js](http://headjs.com/) *(a script loading library)* to be loaded before reveal.js.
-
-### Ready Event
-
-A 'ready' event is fired when reveal.js has loaded all non-async dependencies and is ready to start navigating. To check if reveal.js is already 'ready' you can call `Reveal.isReady()`.
-
-```javascript
-Reveal.addEventListener( 'ready', function( event ) {
- // event.currentSlide, event.indexh, event.indexv
-} );
-```
-
-Note that we also add a `.ready` class to the `.reveal` element so that you can hook into this with CSS.
### Auto-sliding
@@ -429,8 +296,6 @@ You can also override the slide duration for individual slides and fragments by
```
-To override the method used for navigation when auto-sliding, you can specify the ```autoSlideMethod``` setting. To only navigate along the top layer and ignore vertical slides, set this to ```Reveal.navigateRight```.
-
Whenever the auto-slide mode is resumed or paused the ```autoslideresumed``` and ```autoslidepaused``` events are fired.
@@ -448,13 +313,6 @@ Reveal.configure({
});
```
-### Touch Navigation
-
-You can swipe to navigate through a presentation on any touch-enabled device. Horizontal swipes change between horizontal slides, vertical swipes change between vertical slides. If you wish to disable this you can set the `touch` config option to false when initializing reveal.js.
-
-If there's some part of your content that needs to remain accessible to touch events you'll need to highlight this by adding a `data-prevent-swipe` attribute to the element. One common example where this is useful is elements that need to be scrolled.
-
-
### Lazy Loading
When working on presentation with a lot of media or iframe content it's important to load lazily. Lazy loading means that reveal.js will only load content for the few slides nearest to the current slide. The number of slides that are preloaded is determined by the `viewDistance` configuration option.
@@ -489,18 +347,11 @@ Reveal.next();
Reveal.prevFragment();
Reveal.nextFragment();
-// Randomize the order of slides
-Reveal.shuffle();
-
// Toggle presentation states, optionally pass true/false to force on/off
Reveal.toggleOverview();
Reveal.togglePause();
Reveal.toggleAutoSlide();
-// Shows a help overlay with keyboard shortcuts, optionally pass true/false
-// to force on/off
-Reveal.toggleHelp();
-
// Change a config value at runtime
Reveal.configure({ controls: true });
@@ -514,14 +365,9 @@ Reveal.getScale();
Reveal.getPreviousSlide();
Reveal.getCurrentSlide();
-Reveal.getIndices(); // { h: 0, v: 0 } }
-Reveal.getPastSlideCount();
-Reveal.getProgress(); // (0 == first slide, 1 == last slide)
-Reveal.getSlides(); // Array of all slides
-Reveal.getTotalSlides(); // total number of slides
-
-// Returns the speaker notes for the current slide
-Reveal.getSlideNotes();
+Reveal.getIndices(); // { h: 0, v: 0 } }
+Reveal.getProgress(); // 0-1
+Reveal.getTotalSlides();
// State checks
Reveal.isFirstSlide();
@@ -574,59 +420,26 @@ Reveal.addEventListener( 'somestate', function() {
### Slide Backgrounds
-Slides are contained within a limited portion of the screen by default to allow them to fit any display and scale uniformly. You can apply full page backgrounds outside of the slide area by adding a ```data-background``` attribute to your `````` elements. Four different types of backgrounds are supported: color, image, video and iframe.
+Slides are contained within a limited portion of the screen by default to allow them to fit any display and scale uniformly. You can apply full page backgrounds outside of the slide area by adding a ```data-background``` attribute to your `````` elements. Four different types of backgrounds are supported: color, image, video and iframe. Below are a few examples.
-#### Color Backgrounds
-All CSS color formats are supported, like rgba() or hsl().
```html
-
-
Color
+
+
All CSS color formats are supported, like rgba() or hsl().
+
+
+
This slide will have a full-size background image.
+
+
+
This background image will be sized to 100px and repeated.
+
+
+
Video. Multiple sources can be defined using a comma separated list. Video will loop when the data-background-video-loop attribute is provided.
+
+
+
Embeds a web page as a background. Note that the page won't be interactive.
```
-#### Image Backgrounds
-By default, background images are resized to cover the full page. Available options:
-
-| Attribute | Default | Description |
-| :--------------------------- | :--------- | :---------- |
-| data-background-image | | URL of the image to show. GIFs restart when the slide opens. |
-| data-background-size | cover | See [background-size](https://developer.mozilla.org/docs/Web/CSS/background-size) on MDN. |
-| data-background-position | center | See [background-position](https://developer.mozilla.org/docs/Web/CSS/background-position) on MDN. |
-| data-background-repeat | no-repeat | See [background-repeat](https://developer.mozilla.org/docs/Web/CSS/background-repeat) on MDN. |
-```html
-
-
Image
-
-
-
This background image will be sized to 100px and repeated
-
-```
-
-#### Video Backgrounds
-Automatically plays a full size video behind the slide.
-
-| Attribute | Default | Description |
-| :--------------------------- | :------ | :---------- |
-| data-background-video | | A single video source, or a comma separated list of video sources. |
-| data-background-video-loop | false | Flags if the video should play repeatedly. |
-| data-background-video-muted | false | Flags if the audio should be muted. |
-| data-background-size | cover | Use `cover` for full screen and some cropping or `contain` for letterboxing. |
-
-```html
-
-
Video
-
-```
-
-#### Iframe Backgrounds
-Embeds a web page as a slide background that covers 100% of the reveal.js width and height. The iframe is in the background layer, behind your slides, and as such it's not possible to interact with it by default. To make your background interactive, you can add the `data-background-interactive` attribute.
-```html
-
-
Iframe
-
-```
-
-#### Background Transitions
Backgrounds transition using a fade animation by default. This can be changed to a linear sliding transition by passing ```backgroundTransition: 'slide'``` to the ```Reveal.initialize()``` call. Alternatively you can set ```data-background-transition``` on any section with a background to override that specific transition.
@@ -643,16 +456,16 @@ Reveal.initialize({
// Parallax background size
parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" - currently only pixels are supported (don't use % or auto)
- // Number of pixels to move the parallax background per slide
- // - Calculated automatically unless specified
- // - Set to 0 to disable movement along an axis
+ // Amount of pixels to move the parallax background per slide step,
+ // a value of 0 disables movement along the given axis
+ // These are optional, if they aren't specified they'll be calculated automatically
parallaxBackgroundHorizontal: 200,
parallaxBackgroundVertical: 50
});
```
-Make sure that the background size is much bigger than screen size to allow for some scrolling. [View example](http://revealjs.com/?parallaxBackgroundImage=https%3A%2F%2Fs3.amazonaws.com%2Fhakim-static%2Freveal-js%2Freveal-parallax-1.jpg¶llaxBackgroundSize=2100px%20900px).
+Make sure that the background size is much bigger than screen size to allow for some scrolling. [View example](http://lab.hakim.se/reveal-js/?parallaxBackgroundImage=https%3A%2F%2Fs3.amazonaws.com%2Fhakim-static%2Freveal-js%2Freveal-parallax-1.jpg¶llaxBackgroundSize=2100px%20900px).
@@ -673,15 +486,15 @@ You can also use different in and out transitions for the same slide:
```html
- The train goes on …
+ The train goes on …
-
- and on …
+
+ and on …
-
+
and stops.
-
+
(Passengers entering and leaving)
@@ -690,6 +503,9 @@ You can also use different in and out transitions for the same slide:
```
+Note that this does not work with the page and cube transitions.
+
+
### Internal links
It's easy to link between slides. The first example below targets the index of another slide whereas the second targets a slide with an ID attribute (``````):
@@ -712,7 +528,7 @@ You can also add relative navigation links, similar to the built in reveal.js co
### Fragments
-Fragments are used to highlight individual elements on a slide. Every element with the class ```fragment``` will be stepped through before moving on to the next slide. Here's an example: http://revealjs.com/#/fragments
+Fragments are used to highlight individual elements on a slide. Every element with the class ```fragment``` will be stepped through before moving on to the next slide. Here's an example: http://lab.hakim.se/reveal-js/#/fragments
The default fragment style is to start out invisible and fade in. This style can be changed by appending a different class to the fragment:
@@ -721,7 +537,6 @@ The default fragment style is to start out invisible and fade in. This style can
grow
shrink
fade-out
-
fade-up (also down, left and right!)
visible only once
blue only once
highlight-red
@@ -767,41 +582,33 @@ Reveal.addEventListener( 'fragmenthidden', function( event ) {
### Code syntax highlighting
-By default, Reveal is configured with [highlight.js](https://highlightjs.org/) for code syntax highlighting. To enable syntax highlighting, you'll have to load the highlight plugin ([plugin/highlight/highlight.js](plugin/highlight/highlight.js)) and a highlight.js CSS theme (Reveal comes packaged with the zenburn theme: [lib/css/zenburn.css](lib/css/zenburn.css)).
-
-Below is an example with clojure code that will be syntax highlighted. When the `data-trim` attribute is present, surrounding whitespace is automatically removed. HTML will be escaped by default. To avoid this, for example if you are using `` to call out a line of code, add the `data-noescape` attribute to the `` element.
+By default, Reveal is configured with [highlight.js](http://softwaremaniacs.org/soft/highlight/en/) for code syntax highlighting. Below is an example with clojure code that will be syntax highlighted. When the `data-trim` attribute is present surrounding whitespace is automatically removed.
```html
-
+
(def lazy-fib
(concat
[0 1]
- ((fn rfib [a b]
+ ((fn rfib [a b]
(lazy-cons (+ a b) (rfib b (+ a b)))) 0 1)))
```
### Slide number
-If you would like to display the page number of the current slide you can do so using the ```slideNumber``` and ```showSlideNumber``` configuration values.
+If you would like to display the page number of the current slide you can do so using the ```slideNumber``` configuration value.
```javascript
// Shows the slide number using default formatting
Reveal.configure({ slideNumber: true });
// Slide number formatting can be configured using these variables:
-// "h.v": horizontal . vertical slide number (default)
-// "h/v": horizontal / vertical slide number
-// "c": flattened slide number
-// "c/t": flattened slide number / total slides
-Reveal.configure({ slideNumber: 'c/t' });
-
-// Control which views the slide number displays on using the "showSlideNumber" value:
-// "all": show on all views (default)
-// "speaker": only show slide numbers on speaker notes view
-// "print": only show slide numbers when printing to PDF
-Reveal.configure({ showSlideNumber: 'speaker' });
+// h: current slide's horizontal index
+// v: current slide's vertical index
+// c: current slide index (flattened)
+// t: total number of slides (flattened)
+Reveal.configure({ slideNumber: 'c / t' });
```
@@ -819,26 +626,20 @@ Reveal.addEventListener( 'overviewhidden', function( event ) { /* ... */ } );
Reveal.toggleOverview();
```
-
### Fullscreen mode
Just press »F« on your keyboard to show your presentation in fullscreen mode. Press the »ESC« key to exit fullscreen mode.
### Embedded media
+Embedded HTML5 `
+# reveal.js [](https://travis-ci.org/hakimel/reveal.js)
-A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://revealjs.com/).
+A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://lab.hakim.se/reveal-js/).
-reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). There's also a fully featured visual editor and platform for sharing reveal.js presentations at [slides.com](https://slides.com?ref=github).
+reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). It's best viewed in a modern browser but [fallbacks](https://github.com/hakimel/reveal.js/wiki/Browser-Support) are available to make sure your presentation can still be viewed elsewhere.
-## Table of contents
-- [Online Editor](#online-editor)
-- [Instructions](#instructions)
- - [Markup](#markup)
- - [Markdown](#markdown)
- - [Element Attributes](#element-attributes)
- - [Slide Attributes](#slide-attributes)
-- [Configuration](#configuration)
-- [Presentation Size](#presentation-size)
-- [Dependencies](#dependencies)
-- [Ready Event](#ready-event)
-- [Auto-sliding](#auto-sliding)
-- [Keyboard Bindings](#keyboard-bindings)
-- [Touch Navigation](#touch-navigation)
-- [Lazy Loading](#lazy-loading)
-- [API](#api)
- - [Slide Changed Event](#slide-changed-event)
- - [Presentation State](#presentation-state)
- - [Slide States](#slide-states)
- - [Slide Backgrounds](#slide-backgrounds)
- - [Parallax Background](#parallax-background)
- - [Slide Transitions](#slide-transitions)
- - [Internal links](#internal-links)
- - [Fragments](#fragments)
- - [Fragment events](#fragment-events)
- - [Code syntax highlighting](#code-syntax-highlighting)
- - [Slide number](#slide-number)
- - [Overview mode](#overview-mode)
- - [Fullscreen mode](#fullscreen-mode)
- - [Embedded media](#embedded-media)
- - [Stretching elements](#stretching-elements)
- - [postMessage API](#postmessage-api)
-- [PDF Export](#pdf-export)
-- [Theming](#theming)
-- [Speaker Notes](#speaker-notes)
- - [Share and Print Speaker Notes](#share-and-print-speaker-notes)
- - [Server Side Speaker Notes](#server-side-speaker-notes)
-- [Multiplexing](#multiplexing)
- - [Master presentation](#master-presentation)
- - [Client presentation](#client-presentation)
- - [Socket.io server](#socketio-server)
-- [MathJax](#mathjax)
-- [Installation](#installation)
- - [Basic setup](#basic-setup)
- - [Full setup](#full-setup)
- - [Folder Structure](#folder-structure)
-- [License](#license)
-#### More reading
+#### More reading:
+- [Installation](#installation): Step-by-step instructions for getting reveal.js running on your computer.
- [Changelog](https://github.com/hakimel/reveal.js/releases): Up-to-date version history.
- [Examples](https://github.com/hakimel/reveal.js/wiki/Example-Presentations): Presentations created with reveal.js, add your own!
- [Browser Support](https://github.com/hakimel/reveal.js/wiki/Browser-Support): Explanation of browser support and fallbacks.
@@ -60,36 +14,14 @@ reveal.js comes with a broad range of features including [nested slides](https:/
## Online Editor
-Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [https://slides.com](https://slides.com?ref=github).
+Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [http://slides.com](http://slides.com).
## Instructions
### Markup
-Here's a barebones example of a fully working reveal.js presentation:
-```html
-
-
-
-
-
-
-