From 20e8b8fbec9f0112c5f91855d5d28b72bd81ea99 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 27 Sep 2018 04:35:41 +0200 Subject: [PATCH] Update on steepest descent --- doc/pub/Splines/html/._Splines-bs000.html | 176 +++--- doc/pub/Splines/html/._Splines-bs001.html | 174 +++--- doc/pub/Splines/html/._Splines-bs002.html | 200 ++++--- doc/pub/Splines/html/._Splines-bs003.html | 199 ++++--- doc/pub/Splines/html/._Splines-bs004.html | 208 ++++--- doc/pub/Splines/html/._Splines-bs005.html | 198 ++++--- doc/pub/Splines/html/._Splines-bs006.html | 212 ++++--- doc/pub/Splines/html/._Splines-bs007.html | 188 +++--- doc/pub/Splines/html/._Splines-bs008.html | 257 +++++---- doc/pub/Splines/html/._Splines-bs009.html | 209 ++++--- doc/pub/Splines/html/._Splines-bs010.html | 208 ++++--- doc/pub/Splines/html/._Splines-bs011.html | 217 +++---- doc/pub/Splines/html/._Splines-bs012.html | 202 ++++--- doc/pub/Splines/html/._Splines-bs013.html | 197 ++++--- doc/pub/Splines/html/._Splines-bs014.html | 186 +++--- doc/pub/Splines/html/._Splines-bs015.html | 237 ++++---- doc/pub/Splines/html/._Splines-bs016.html | 234 ++++---- doc/pub/Splines/html/._Splines-bs017.html | 211 ++++--- doc/pub/Splines/html/._Splines-bs018.html | 239 ++++---- doc/pub/Splines/html/._Splines-bs019.html | 199 ++++--- doc/pub/Splines/html/._Splines-bs020.html | 225 +++++--- doc/pub/Splines/html/._Splines-bs021.html | 225 +++++--- doc/pub/Splines/html/._Splines-bs022.html | 210 ++++--- doc/pub/Splines/html/._Splines-bs023.html | 217 +++---- doc/pub/Splines/html/._Splines-bs024.html | 194 ++++--- doc/pub/Splines/html/._Splines-bs025.html | 226 ++++---- doc/pub/Splines/html/._Splines-bs026.html | 233 ++++---- doc/pub/Splines/html/._Splines-bs027.html | 230 +++++--- doc/pub/Splines/html/._Splines-bs028.html | 213 ++++--- doc/pub/Splines/html/._Splines-bs029.html | 244 +++++--- doc/pub/Splines/html/._Splines-bs030.html | 202 ++++--- doc/pub/Splines/html/._Splines-bs031.html | 212 ++++--- doc/pub/Splines/html/._Splines-bs032.html | 222 +++---- doc/pub/Splines/html/._Splines-bs033.html | 220 +++---- doc/pub/Splines/html/._Splines-bs034.html | 230 +++++--- doc/pub/Splines/html/._Splines-bs035.html | 219 ++++--- doc/pub/Splines/html/._Splines-bs036.html | 256 +++++---- doc/pub/Splines/html/Splines-bs.html | 176 +++--- doc/pub/Splines/html/Splines-reveal.html | 441 ++++++++++++-- doc/pub/Splines/html/Splines-solarized.html | 502 +++++++++++++--- doc/pub/Splines/html/Splines.html | 502 +++++++++++++--- doc/pub/Splines/ipynb/Splines.ipynb | 540 ++++++++++++++++-- .../Splines/ipynb/ipynb-Splines-src.tar.gz | Bin 210 -> 208 bytes doc/pub/Splines/pdf/Splines-minted.pdf | Bin 363461 -> 400749 bytes doc/src/Splines/Splines.do.txt | 310 ++++++++++ 45 files changed, 6767 insertions(+), 3633 deletions(-) diff --git a/doc/pub/Splines/html/._Splines-bs000.html b/doc/pub/Splines/html/._Splines-bs000.html index e3f254239..f8494c29a 100644 --- a/doc/pub/Splines/html/._Splines-bs000.html +++ b/doc/pub/Splines/html/._Splines-bs000.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -193,7 +225,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Sep 21, 2018

+

Sep 27, 2018


@@ -217,7 +249,7 @@ MathJax.Hub.Config({

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    Steepest descent

    +

    Revisiting our Logistic Regression case

    -The method of steepest descent 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 \( -\nabla F(\mathbf{x}) \). +In our discussion on Logistic Regression we defined we studied first 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 of the Sigmoid function, that is we +defined probabilities -

    -It can be shown that if $$ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} $$ -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. +where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).

    @@ -213,7 +243,7 @@ we are always moving towards smaller function values, i.e a minimum.

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    - + -

    More on Steepest descent

    +

    The equations to solve

    -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 +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 $$ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0. +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). $$

    -The parameter \( \gamma_k \) is often referred to as the step length or -the learning rate within the context of Machine Learning. +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 we call the Hessian.

    @@ -209,7 +248,7 @@ the learning rate within the context of Machine Learning.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs004.html b/doc/pub/Splines/html/._Splines-bs004.html index 47ec80c33..2401ae554 100644 --- a/doc/pub/Splines/html/._Splines-bs004.html +++ b/doc/pub/Splines/html/._Splines-bs004.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,30 +204,30 @@ MathJax.Hub.Config({

     

     

     

    - + -

    The ideal

    +

    Solving using Newton-Raphson's method

    -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: +If we can set up these equations, Newton-Raphson's iterative method is the nomrally the method of choice. It requires however that we setting the matrices that define the first and second derivatives.

    -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. +Our iterative scheme is then given by + +$$ +\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T}\right)^{-1}\times \left(\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}}\right)_{\hat{\beta}^{\mathrm{old}}}, +$$ + +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 \( \beta \).

    -Note that the gradient is a function of \( \mathbf{x} = -(x_1,\cdots,x_n) \) which makes it expensive to compute numerically. +If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.

    @@ -217,7 +249,7 @@ Note that the gradient is a function of \( \mathbf{x} =

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    • diff --git a/doc/pub/Splines/html/._Splines-bs005.html b/doc/pub/Splines/html/._Splines-bs005.html index a4e58bab5..ad022b42b 100644 --- a/doc/pub/Splines/html/._Splines-bs005.html +++ b/doc/pub/Splines/html/._Splines-bs005.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,23 +204,23 @@ MathJax.Hub.Config({

     

     

     

    - + -

    The sensitiveness of the gradient descent

    +

    Brief reminder on Newton-Raphson's method

    -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. +Let us quicly remind ourselves how we derive the above method.

    -Many of these shortcomings can be alleviated by introducing -randomness. One such method is that of Stochastic Gradient Descent -(SGD), see below. +Perhaps the most celebrated of all one-dimensional root-finding +routines is Newton's method, also called the Newton-Raphson +method. This method is distinguished from the previously discussed +methods by the fact that it requires the evaluation of both the +function \( f \) and its derivative \( f' \) at arbitrary points. In this +sense, it is taylored to cases with e.g., transcendental equations. +If you can only calculate the derivative +numerically and/or your function is not of the smooth type, we +discourage the use of this method.

    @@ -211,7 +243,7 @@ randomness. One such method is that of Stochastic Gradient Descent

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    • diff --git a/doc/pub/Splines/html/._Splines-bs006.html b/doc/pub/Splines/html/._Splines-bs006.html index 1ba9ffe66..6f55896c7 100644 --- a/doc/pub/Splines/html/._Splines-bs006.html +++ b/doc/pub/Splines/html/._Splines-bs006.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,24 +204,40 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Convex functions

    +

    The equations

    -Ideally we want our cost/loss function to be convex(concave). +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 + +$$ + f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots. + \tag{1} +$$

    -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. +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 +$$ + s\approx x-\frac{f(x)}{f'(x)}. +$$

    -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...). +Having in mind an iterative procedure, it is natural to start iterating with +$$ + x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. +$$

    @@ -213,7 +261,7 @@ regular polygons (triangles, rectangles, pentagons, etc...).

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    • diff --git a/doc/pub/Splines/html/._Splines-bs007.html b/doc/pub/Splines/html/._Splines-bs007.html index 57c477e77..fb48f2408 100644 --- a/doc/pub/Splines/html/._Splines-bs007.html +++ b/doc/pub/Splines/html/._Splines-bs007.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,10 +206,20 @@ MathJax.Hub.Config({ -

    Convex function

    +

    Simple geometric interpretation

    -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. +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

    @@ -202,7 +244,7 @@ MathJax.Hub.Config({

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    • diff --git a/doc/pub/Splines/html/._Splines-bs008.html b/doc/pub/Splines/html/._Splines-bs008.html index f0565b9ad..c035217c7 100644 --- a/doc/pub/Splines/html/._Splines-bs008.html +++ b/doc/pub/Splines/html/._Splines-bs008.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,46 +206,57 @@ MathJax.Hub.Config({ -

    Conditions on convex functions

    +

    Extending to more than one variable

    -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. +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.

    -

    -
    -

    -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. +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.

    @@ -239,7 +282,7 @@ This condition is particularly useful since it gives us an procedure for determi

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    • diff --git a/doc/pub/Splines/html/._Splines-bs009.html b/doc/pub/Splines/html/._Splines-bs009.html index 15e45e732..264ce754a 100644 --- a/doc/pub/Splines/html/._Splines-bs009.html +++ b/doc/pub/Splines/html/._Splines-bs009.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,33 +206,26 @@ MathJax.Hub.Config({ -

    More on convex functions

    +

    Steepest descent

    -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. +The method of steepest descent 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 \( -\nabla F(\mathbf{x}) \).

    -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: +It can be shown that if +$$ +\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), +$$ + +with \( \gamma_k > 0 \).

    -

    -
    -

    -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. -

    -
    - - -

    -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. +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.

    @@ -227,7 +252,7 @@ This result means that if we know that the cost/loss function is convex and we a

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    • diff --git a/doc/pub/Splines/html/._Splines-bs010.html b/doc/pub/Splines/html/._Splines-bs010.html index 91bd8be2f..18fbf8251 100644 --- a/doc/pub/Splines/html/._Splines-bs010.html +++ b/doc/pub/Splines/html/._Splines-bs010.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,31 +204,23 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Some simple problems

    +

    More on Steepest descent

    -
      -
    1. 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 \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
      2. -
    2. Using the second order condition show that the following functions are convex on the specified domain.
      3. +

      +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 -

      +$$ +\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0. +$$ -
    3. 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.
      4. -
    4. A norm is any function that satisfy the following properties
      5. - - - -
    - -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). +

    +The parameter \( \gamma_k \) is often referred to as the step length or +the learning rate within the context of Machine Learning.

    @@ -224,7 +248,7 @@ Using the definition of convexity, try to show that a function satisfying the pr

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    • diff --git a/doc/pub/Splines/html/._Splines-bs011.html b/doc/pub/Splines/html/._Splines-bs011.html index fc3bba3df..c96d82bbe 100644 --- a/doc/pub/Splines/html/._Splines-bs011.html +++ b/doc/pub/Splines/html/._Splines-bs011.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,35 +206,28 @@ MathJax.Hub.Config({ -

    Revisiting our first homework

    +

    The ideal

    -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: +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: -

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

    +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. -We revisit the example from homework set 1 where we had -$$ -y_i = 5x_i^2 + 0.1\xi_i, \ i=1,\cdots,100 -$$ - -with \( x_i \in [0,1] \) chosen randomly with a uniform distribution. Additionally \( \xi_i \) represents stochastic noise chosen according to a normal distribution \( \cal {N}(0,1) \). -The linear regression model is given by -$$ -h_\beta(x) = \hat{y} = \beta_0 + \beta_1 x, -$$ - -such that -$$ -\hat{y}_i = \beta_0 + \beta_1 x_i. -$$ +

    +Note that the gradient is a function of \( \mathbf{x} = +(x_1,\cdots,x_n) \) which makes it expensive to compute numerically.

    @@ -230,7 +255,7 @@ $$

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    • diff --git a/doc/pub/Splines/html/._Splines-bs012.html b/doc/pub/Splines/html/._Splines-bs012.html index 27137f59b..0b9edbd30 100644 --- a/doc/pub/Splines/html/._Splines-bs012.html +++ b/doc/pub/Splines/html/._Splines-bs012.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,27 +206,21 @@ MathJax.Hub.Config({ -

    Gradient descent example

    +

    The sensitiveness of the gradient descent

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

    -It is convenient to write \( \mathbf{\hat{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by -$$ -X \equiv \begin{bmatrix} -1 & x_1 \\ -\vdots & \vdots \\ -1 & x_{100} & \\ -\end{bmatrix}. -$$ - -The loss function is given by -$$ -C(\beta) = ||X\beta-\mathbf{y}||^2 = ||X\beta||^2 - 2 \mathbf{y}^T X\beta + ||\mathbf{y}||^2 = \sum_{i=1}^{100} (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2 -$$ - -and we want to find \( \beta \) such that \( C(\beta) \) is minimized. +Many of these shortcomings can be alleviated by introducing +randomness. One such method is that of Stochastic Gradient Descent +(SGD), see below.

    @@ -222,7 +248,7 @@ and we want to find \( \beta \) such that \( C(\beta) \) is minimized.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs013.html b/doc/pub/Splines/html/._Splines-bs013.html index 372867cf3..b211a4e22 100644 --- a/doc/pub/Splines/html/._Splines-bs013.html +++ b/doc/pub/Splines/html/._Splines-bs013.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,19 +204,24 @@ MathJax.Hub.Config({

     

     

     

    - + -

    The derivative of the cost/loss function

    +

    Convex functions

    -Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as -$$ -\nabla_{\beta} C(\beta) = (\partial C(\beta) / \partial \beta_0, \partial C(\beta) / \partial \beta_1)^T = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} = 2X^T(X\beta - \mathbf{y}), -$$ +Ideally we want our cost/loss function to be convex(concave). -where \( X \) is the design matrix defined above. +

    +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...).

    @@ -212,7 +249,7 @@ where \( X \) is the design matrix defined above.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs014.html b/doc/pub/Splines/html/._Splines-bs014.html index 9374c1ab0..bca2a9dfa 100644 --- a/doc/pub/Splines/html/._Splines-bs014.html +++ b/doc/pub/Splines/html/._Splines-bs014.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,16 +206,10 @@ MathJax.Hub.Config({ -

    The Hessian matrix

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

    Convex function

    -This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite. +

    +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.

    @@ -211,7 +237,7 @@ This result implies that \( C(\beta) \) is a convex function since the matrix \(

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    • diff --git a/doc/pub/Splines/html/._Splines-bs015.html b/doc/pub/Splines/html/._Splines-bs015.html index 36d1bdfcf..e7589548b 100644 --- a/doc/pub/Splines/html/._Splines-bs015.html +++ b/doc/pub/Splines/html/._Splines-bs015.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,44 +206,47 @@ MathJax.Hub.Config({ -

    Simple program

    +

    Conditions on convex functions

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

    -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 \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \). +

    +
    +

    +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. +

    +
    +

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

    +
    +

    +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. - -

    import numpy as np
-
-"""
-The following setup is just a suggestion, feel free to write it the way you like.
-"""
-
-#Setup problem described in the exercise
-N  = 100 #Nr of datapoints
-M  = 2 #Nr of features
-x  = np.random.rand(N) #Uniformly generated x-values in [0,1]
-y  = 5*x**2 + 0.1*np.random.randn(N)
-X  = np.c_[np.ones(N),x] #Construct design matrix
-
-#Compute beta according to normal equations to compare with GD solution
-Xt_X_inv = np.linalg.inv(np.dot(X.T,X))
-Xt_y     = np.dot(X.transpose(),y)
-beta_NE = np.dot(Xt_X_inv,Xt_y)
-print(beta_NE)
-

    @@ -238,7 +273,7 @@ beta_NE = np.24

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    • diff --git a/doc/pub/Splines/html/._Splines-bs016.html b/doc/pub/Splines/html/._Splines-bs016.html index 8f6c8a419..f68c92deb 100644 --- a/doc/pub/Splines/html/._Splines-bs016.html +++ b/doc/pub/Splines/html/._Splines-bs016.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,52 +206,34 @@ MathJax.Hub.Config({ -

    Gradient Descent Example

    +

    More on convex functions

    -Another simple example is here +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: - -

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

    @@ -246,7 +260,7 @@ plt.show()

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    • diff --git a/doc/pub/Splines/html/._Splines-bs017.html b/doc/pub/Splines/html/._Splines-bs017.html index b6e338b6b..0497c1c2e 100644 --- a/doc/pub/Splines/html/._Splines-bs017.html +++ b/doc/pub/Splines/html/._Splines-bs017.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,27 +206,30 @@ MathJax.Hub.Config({ -

    And a corresponding example using scikit-learn

    +

    Some simple problems

    -

    +

      +
    1. 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 \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
      2. +
    2. Using the second order condition show that the following functions are convex on the specified domain.
      3. - -
      # Importing various packages
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
+
        
+ 
      •  \( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).
        • 
+ 
      •  \( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).
        • 
+
      
 
-x = 2*np.random.rand(100,1)
-y = 4+3*x+np.random.randn(100,1)
+
    3.  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.
      4. 
+
    4.  A norm is any function that satisfy the following properties
      5. 
+
+
        
+ 
      •  \( f(\alpha x) = |\alpha| f(x) \) 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). -xb = np.c_[np.ones((100,1)), x] -beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y) -print(beta_linreg) -sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1) -sgdreg.fit(x,y.ravel()) -print(sgdreg.intercept_, sgdreg.coef_) -

    @@ -221,7 +256,7 @@ sgdreg.fit(x,y.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs018.html b/doc/pub/Splines/html/._Splines-bs018.html index 507b4a0f3..cefc870b1 100644 --- a/doc/pub/Splines/html/._Splines-bs018.html +++ b/doc/pub/Splines/html/._Splines-bs018.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,62 +204,39 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Gradient descent and Ridge

    +

    Standard steepest descent

    -We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \), +Before we proceed, we would like to mention the approach called the standard Steepest descent, which again leads to us having to be able to compute a matrix. + +

    +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 $$ -C_{\text{ridge}}(\beta) = ||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. +\begin{equation*} +\hat{A}\hat{x} = \hat{b}. +\end{equation*} $$

    -In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows +In the iterative process we end up with a problem like + $$ -\nabla_\beta C_{\text{ridge}}(\beta) = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta). +\begin{equation*} + \hat{r}= \hat{b}-\hat{A}\hat{x}, +\end{equation*} $$ +where \( \hat{r} \) is the so-called residual or error in the iterative process. +

    -We can now extend our program to minimize \( C_{\text{ridge}}(\beta) \) using gradient descent and compare with the analytical solution given by -$$ -\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}, -$$ +When we have found the exact solution, \( \hat{r}=0 \). -for \( \lambda = {0,1,10,50,100} \) (\( \lambda = 0 \) corresponds to ordinary least squares). -We can then compute \( ||\beta_{\text{ridge}}|| \) for each \( \lambda \). - -

    - - -

    import numpy as np
-
-"""
-The following setup is just a suggestion, feel free to write it the way you like.
-"""
-
-#Setup problem described in the exercise
-N  = 100 #Nr of datapoints
-M  = 2   #Nr of features
-x  = np.random.rand(N)
-y  = 5*x**2 + 0.1*np.random.randn(N)
-
-
-#Compute analytic beta for Ridge regression 
-X    = np.c_[np.ones(N),x]
-XT_X = np.dot(X.T,X)
-
-l  = 0.1 #Ridge parameter lambda
-Id = np.eye(XT_X.shape[0])
-
-Z = np.linalg.inv(XT_X+l*Id)
-beta_ridge = np.dot(Z,np.dot(X.T,y))
-
-print(beta_ridge)
-print(np.linalg.norm(beta_ridge)) #||beta||
-

    @@ -254,7 +263,7 @@ beta_ridge = np

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    • diff --git a/doc/pub/Splines/html/._Splines-bs019.html b/doc/pub/Splines/html/._Splines-bs019.html index dda06818b..47978e50c 100644 --- a/doc/pub/Splines/html/._Splines-bs019.html +++ b/doc/pub/Splines/html/._Splines-bs019.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,20 +206,25 @@ MathJax.Hub.Config({ -

    Stochastic Gradient Descent

    +

    Conjugate gradient method

    -Stochastic gradient descent (SGD) and variants thereof address some of -the shortcomings of the Gradient descent method discussed above. +The residual is zero when we reach the minimum of the quadratic equation +$$ +\begin{equation*} + P(\hat{x})=\frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T\hat{b}, +\end{equation*} +$$

    -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 \( \{\mathbf{x}_i\}_{i=1}^n \), -$$ -C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, -\mathbf{\beta}). -$$ +with the constraint that the matrix \( \hat{A} \) is positive definite and +symmetric. If we search for a minimum of the quantum mechanical +variance, then the matrix \( \hat{A} \), which is called the Hessian, is +given by the second-derivative of the function we want to minimize. +This quantity is always positive definite. + +

    +More details will be added here soon.

    @@ -215,7 +252,7 @@ $$

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    • diff --git a/doc/pub/Splines/html/._Splines-bs020.html b/doc/pub/Splines/html/._Splines-bs020.html index 2b13ebb75..8226361b5 100644 --- a/doc/pub/Splines/html/._Splines-bs020.html +++ b/doc/pub/Splines/html/._Splines-bs020.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,22 +206,47 @@ MathJax.Hub.Config({ -

    Computation of gradients

    - +

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

    +
    +
    +

    -This in turn means that the gradient can be -computed as a sum over \( i \)-gradients -$$ -\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}). -$$ + +

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

    -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 -\( k=1,\cdots,n/M \). +

    +
    +

    @@ -217,7 +274,7 @@ minibatches. We denote these minibatches by \( B_k \) where

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    • diff --git a/doc/pub/Splines/html/._Splines-bs021.html b/doc/pub/Splines/html/._Splines-bs021.html index 6525dab00..18036c43b 100644 --- a/doc/pub/Splines/html/._Splines-bs021.html +++ b/doc/pub/Splines/html/._Splines-bs021.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,26 +206,39 @@ MathJax.Hub.Config({ -

    SGD example

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

    The routine for the steepest descent method

    +
    +
    +

    -The idea is now to approximate the gradient by replacing the sum over -all data points with a sum over the data points in one the minibatches -picked at random in each gradient descent step -$$ -\nabla_{\beta} -C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta -c_i(\mathbf{x}_i, \mathbf{\beta}). -$$ + + +

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

    +

    +
    +

    @@ -221,7 +266,7 @@ $$

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    • diff --git a/doc/pub/Splines/html/._Splines-bs022.html b/doc/pub/Splines/html/._Splines-bs022.html index c7901fea0..768900c4b 100644 --- a/doc/pub/Splines/html/._Splines-bs022.html +++ b/doc/pub/Splines/html/._Splines-bs022.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,23 +204,37 @@ MathJax.Hub.Config({

     

     

     

    - + -

    The gradient step

    +

    Revisiting our first homework

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

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

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

    @@ -216,7 +262,7 @@ the number of minibatches, as exemplified in the code below.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs023.html b/doc/pub/Splines/html/._Splines-bs023.html index ccc0d0fab..05ed134f0 100644 --- a/doc/pub/Splines/html/._Splines-bs023.html +++ b/doc/pub/Splines/html/._Splines-bs023.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,36 +204,29 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Simple example code

    +

    Gradient descent example

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

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

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

    @@ -229,7 +254,7 @@ all \( n \) datapoints.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs024.html b/doc/pub/Splines/html/._Splines-bs024.html index 084a9a24f..f44e7b0cf 100644 --- a/doc/pub/Splines/html/._Splines-bs024.html +++ b/doc/pub/Splines/html/._Splines-bs024.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,19 +206,17 @@ MathJax.Hub.Config({ -

    When do we stop?

    +

    The derivative of the cost/loss function

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

    @@ -214,7 +244,7 @@ gave the lowest value.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs025.html b/doc/pub/Splines/html/._Splines-bs025.html index 2568a5049..4ef788afe 100644 --- a/doc/pub/Splines/html/._Splines-bs025.html +++ b/doc/pub/Splines/html/._Splines-bs025.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,51 +206,17 @@ MathJax.Hub.Config({ -

    Slightly different approach

    +

    The Hessian matrix

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

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

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

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

    - - -

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

    @@ -245,7 +243,7 @@ j = 0

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    • diff --git a/doc/pub/Splines/html/._Splines-bs026.html b/doc/pub/Splines/html/._Splines-bs026.html index 069443c14..cdd5ad290 100644 --- a/doc/pub/Splines/html/._Splines-bs026.html +++ b/doc/pub/Splines/html/._Splines-bs026.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,35 +206,44 @@ MathJax.Hub.Config({ -

    Conjugate gradient (CG) method

    -
    -
    -

    -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. +

    Simple program

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

    -
    +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 +\( \beta= (X^TX)^{-1} X^T \mathbf{y} \). +

    + + +

    import numpy as np
+
+"""
+The following setup is just a suggestion, feel free to write it the way you like.
+"""
+
+#Setup problem described in the exercise
+N  = 100 #Nr of datapoints
+M  = 2 #Nr of features
+x  = np.random.rand(N) #Uniformly generated x-values in [0,1]
+y  = 5*x**2 + 0.1*np.random.randn(N)
+X  = np.c_[np.ones(N),x] #Construct design matrix
+
+#Compute beta according to normal equations to compare with GD solution
+Xt_X_inv = np.linalg.inv(np.dot(X.T,X))
+Xt_y     = np.dot(X.transpose(),y)
+beta_NE = np.dot(Xt_X_inv,Xt_y)
+print(beta_NE)
+

    @@ -229,7 +270,7 @@ When we have found the exact solution, \( \hat{r}=0 \).

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    • diff --git a/doc/pub/Splines/html/._Splines-bs027.html b/doc/pub/Splines/html/._Splines-bs027.html index faf2c87d5..aa168e923 100644 --- a/doc/pub/Splines/html/._Splines-bs027.html +++ b/doc/pub/Splines/html/._Splines-bs027.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,26 +206,52 @@ MathJax.Hub.Config({ -

    Conjugate gradient method

    -
    -
    -

    +

    Gradient Descent Example

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

    -with the constraint that the matrix \( \hat{A} \) is positive definite and symmetric. -If we search for a minimum of the quantum mechanical variance, then the matrix -\( \hat{A} \), which is called the Hessian, is given by the second-derivative of the function we want to minimize. This quantity is always positive definite. In our case this corresponds normally to the second derivative of the energy. -

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

    @@ -219,6 +277,8 @@ If we search for a minimum of the quantum mechanical variance, then the matrix

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    • diff --git a/doc/pub/Splines/html/._Splines-bs028.html b/doc/pub/Splines/html/._Splines-bs028.html index b6e89d1b7..841560ab0 100644 --- a/doc/pub/Splines/html/._Splines-bs028.html +++ b/doc/pub/Splines/html/._Splines-bs028.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,29 +206,27 @@ MathJax.Hub.Config({ -

    Conjugate gradient method, Newton's method first

    -
    -
    -

    -We seek the minimum of the energy or the variance as function of various variational parameters. -In our case we have thus a function \( f \) whose minimum we are seeking. -In Newton's method we set \( \nabla f = 0 \) and we can thus compute the next iteration point -$$ -\begin{equation*} -\hat{x}-\hat{x}_i=\hat{A}^{-1}\nabla f(\hat{x}_i). -\end{equation*} -$$ +

    And a corresponding example using scikit-learn

    -Subtracting this equation from that of \( \hat{x}_{i+1} \) we have -$$ -\begin{equation*} -\hat{x}_{i+1}-\hat{x}_i=\hat{A}^{-1}(\nabla f(\hat{x}_{i+1})-\nabla f(\hat{x}_i)). -\end{equation*} -$$ -
    -
    +

    + +

    # Importing various packages
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import SGDRegressor
 
+x = 2*np.random.rand(100,1)
+y = 4+3*x+np.random.randn(100,1)
+
+xb = np.c_[np.ones((100,1)), x]
+beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)
+print(beta_linreg)
+sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1)
+sgdreg.fit(x,y.ravel())
+print(sgdreg.intercept_, sgdreg.coef_)
+

    @@ -221,6 +251,9 @@ $$

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    • diff --git a/doc/pub/Splines/html/._Splines-bs029.html b/doc/pub/Splines/html/._Splines-bs029.html index 7ea3b65e7..b7c1a94cf 100644 --- a/doc/pub/Splines/html/._Splines-bs029.html +++ b/doc/pub/Splines/html/._Splines-bs029.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -172,36 +204,62 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Conjugate gradient method

    -
    -
    -

    -In the CG method we define so-called conjugate directions and two vectors -\( \hat{s} \) and \( \hat{t} \) -are said to be -conjugate if +

    Gradient descent and Ridge

    + +

    +We have also discussed Ridge regression where the loss function contains a regularized given by the \( L_2 \) norm of \( \beta \), $$ -\begin{equation*} -\hat{s}^T\hat{A}\hat{t}= 0. -\end{equation*} +C_{\text{ridge}}(\beta) = ||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. $$ -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 +

    +In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows $$ -\begin{equation*} -\hat{x}_i^T\hat{A}\hat{x}_j= 0. -\end{equation*} +\nabla_\beta C_{\text{ridge}}(\beta) = 2\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ +\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ +\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta). $$ -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 \( \hat{s} \). -

    -
    +

    +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 \( \lambda \). + +

    + + +

    import numpy as np
+
+"""
+The following setup is just a suggestion, feel free to write it the way you like.
+"""
+
+#Setup problem described in the exercise
+N  = 100 #Nr of datapoints
+M  = 2   #Nr of features
+x  = np.random.rand(N)
+y  = 5*x**2 + 0.1*np.random.randn(N)
 
 
+#Compute analytic beta for Ridge regression 
+X    = np.c_[np.ones(N),x]
+XT_X = np.dot(X.T,X)
+
+l  = 0.1 #Ridge parameter lambda
+Id = np.eye(XT_X.shape[0])
+
+Z = np.linalg.inv(XT_X+l*Id)
+beta_ridge = np.dot(Z,np.dot(X.T,y))
+
+print(beta_ridge)
+print(np.linalg.norm(beta_ridge)) #||beta||
+

    @@ -225,6 +283,10 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is

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    • diff --git a/doc/pub/Splines/html/._Splines-bs030.html b/doc/pub/Splines/html/._Splines-bs030.html index 03448fbfc..9273bfbd6 100644 --- a/doc/pub/Splines/html/._Splines-bs030.html +++ b/doc/pub/Splines/html/._Splines-bs030.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,21 +206,20 @@ MathJax.Hub.Config({ -

    Conjugate gradient method

    -
    -
    -

    -An example is given by the eigenvectors of the matrix -$$ -\begin{equation*} -\hat{v}_i^T\hat{A}\hat{v}_j= \lambda\hat{v}_i^T\hat{v}_j, -\end{equation*} -$$ +

    Stochastic Gradient Descent

    -which is zero unless \( i=j \). -
    -
    +

    +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 \( \{\mathbf{x}_i\}_{i=1}^n \), +$$ +C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, +\mathbf{\beta}). +$$

    @@ -212,6 +243,11 @@ which is zero unless \( i=j \).

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    • diff --git a/doc/pub/Splines/html/._Splines-bs031.html b/doc/pub/Splines/html/._Splines-bs031.html index 0237a96c7..66828afea 100644 --- a/doc/pub/Splines/html/._Splines-bs031.html +++ b/doc/pub/Splines/html/._Splines-bs031.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,30 +206,22 @@ MathJax.Hub.Config({ -

    Conjugate gradient method

    -
    -
    -

    -Assume now that we have a symmetric positive-definite matrix \( \hat{A} \) of size -\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector +

    Computation of gradients

    + +

    +This in turn means that the gradient can be +computed as a sum over \( i \)-gradients $$ -\begin{equation*} -\hat{x}_{i+1}=\hat{x}_{i}+\alpha_i\hat{p}_{i}. -\end{equation*} +\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}). $$ -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 - -$$ -\begin{equation*} - \hat{x} = \sum^{n}_{i=1} \alpha_i \hat{p}_i. -\end{equation*} -$$ -

    -
    - +

    +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 +\( k=1,\cdots,n/M \).

    @@ -220,6 +244,12 @@ $$

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    • diff --git a/doc/pub/Splines/html/._Splines-bs032.html b/doc/pub/Splines/html/._Splines-bs032.html index 0c101dce5..e93b4f710 100644 --- a/doc/pub/Splines/html/._Splines-bs032.html +++ b/doc/pub/Splines/html/._Splines-bs032.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,36 +206,27 @@ MathJax.Hub.Config({ -

    Conjugate gradient method

    -
    -
    -

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

    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 $$ -\begin{equation*} - \hat{p}_k^T \hat{A}\hat{x} = \sum^{n}_{i=1} \alpha_i\hat{p}_k^T \hat{A}\hat{p}_i= \hat{p}_k^T \hat{b}, -\end{equation*} +\nabla_{\beta} +C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta +c_i(\mathbf{x}_i, \mathbf{\beta}). $$ -and we can define the coefficients \( \alpha_k \) as - -$$ -\begin{equation*} - \alpha_k = \frac{\hat{p}_k^T \hat{b}}{\hat{p}_k^T \hat{A} \hat{p}_k} -\end{equation*} -$$ -

    -
    - -

    @@ -224,6 +247,13 @@ $$

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    • diff --git a/doc/pub/Splines/html/._Splines-bs033.html b/doc/pub/Splines/html/._Splines-bs033.html index ea7e4344b..88c682dbd 100644 --- a/doc/pub/Splines/html/._Splines-bs033.html +++ b/doc/pub/Splines/html/._Splines-bs033.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,39 +206,21 @@ MathJax.Hub.Config({ -

    Conjugate gradient method and iterations

    -
    -
    -

    +

    The gradient step

    -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. +Thus a gradient descent step now looks like +$$ +\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) +$$

    -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. -

    -
    - +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.

    @@ -227,6 +241,14 @@ instead.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs034.html b/doc/pub/Splines/html/._Splines-bs034.html index f96293d33..9ea34b39c 100644 --- a/doc/pub/Splines/html/._Splines-bs034.html +++ b/doc/pub/Splines/html/._Splines-bs034.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,33 +206,34 @@ MathJax.Hub.Config({ -

    Conjugate gradient method

    -
    -
    -

    -One can show that the solution \( \hat{x} \) is also the unique minimizer of the quadratic form -$$ -\begin{equation*} - f(\hat{x}) = \frac{1}{2}\hat{x}^T\hat{A}\hat{x} - \hat{x}^T \hat{x} , \quad \hat{x}\in\mathbf{R}^n. -\end{equation*} -$$ +

    Simple example code

    -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. -

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

    +Taking the gradient only on a subset of the data has two important +benefits. First, it introduces randomness which decreases the chance +that our opmization scheme gets stuck in a local minima. Second, if +the size of the minibatches are small relative to the number of +datapoints (\( M < n \)), the computation of the gradient is much +cheaper since we sum over the datapoints in the \( k-th \) minibatch and not +all \( n \) datapoints.

    @@ -220,6 +253,15 @@ hence the name conjugate gradient method.

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    • diff --git a/doc/pub/Splines/html/._Splines-bs035.html b/doc/pub/Splines/html/._Splines-bs035.html index 63617ce8f..3728b6100 100644 --- a/doc/pub/Splines/html/._Splines-bs035.html +++ b/doc/pub/Splines/html/._Splines-bs035.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,32 +206,19 @@ MathJax.Hub.Config({ -

    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 -$$ -\begin{equation*} -\hat{p}_{k+1}=\hat{r}_k-\frac{\hat{p}_k^T \hat{A}\hat{r}_k}{\hat{p}_k^T\hat{A}\hat{p}_k} \hat{p}_k. -\end{equation*} -$$ -

    -
    +

    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.

    @@ -218,6 +237,16 @@ $$

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    • diff --git a/doc/pub/Splines/html/._Splines-bs036.html b/doc/pub/Splines/html/._Splines-bs036.html index 93e47979b..1b3b69564 100644 --- a/doc/pub/Splines/html/._Splines-bs036.html +++ b/doc/pub/Splines/html/._Splines-bs036.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -174,44 +206,52 @@ MathJax.Hub.Config({ -

    Conjugate gradient method

    -
    -
    -

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

    Slightly different approach

    -which equals -$$ -\begin{equation*} -\hat{b}-\hat{A}(\hat{x}_k+\alpha_k\hat{p}_k), - \end{equation*} -$$ +

    +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. -or -$$ -\begin{equation*} -(\hat{b}-\hat{A}\hat{x}_k)-\alpha_k\hat{A}\hat{p}_k, - \end{equation*} -$$ - -which gives - -$$ -\begin{equation*} -\hat{r}_{k+1}=\hat{r}_k-\hat{A}\hat{p}_{k}, - \end{equation*} -$$ -

    -
    +

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

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

    + +

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

    diff --git a/doc/pub/Splines/html/Splines-bs.html b/doc/pub/Splines/html/Splines-bs.html index e3f254239..f8494c29a 100644 --- a/doc/pub/Splines/html/Splines-bs.html +++ b/doc/pub/Splines/html/Splines-bs.html @@ -45,47 +45,68 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -124,41 +145,52 @@ MathJax.Hub.Config({ Contents @@ -193,7 +225,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 21, 2018

    +

    Sep 27, 2018


    @@ -217,7 +249,7 @@ MathJax.Hub.Config({

  • 9
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    • -
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    • +
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    • diff --git a/doc/pub/Splines/html/Splines-reveal.html b/doc/pub/Splines/html/Splines-reveal.html index 53afa7c72..b63f29483 100644 --- a/doc/pub/Splines/html/Splines-reveal.html +++ b/doc/pub/Splines/html/Splines-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Sep 21, 2018

    +

    Sep 27, 2018


    @@ -174,7 +174,237 @@ some approximative/numerical method to compute the minimum.

    -

    Steepest descent

    +

    Revisiting our Logistic Regression case

    + +

    +In our discussion on Logistic Regression we defined we studied first 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 of the Sigmoid function, 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 \( \beta_1 \). +

    + + +
    +

    The equations to solve

    + +

    +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 + +

     
    +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ +

     
    + +

    +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 we call the Hessian. +

    + + +
    +

    Solving using Newton-Raphson's method

    + +

    +If we can set up these equations, Newton-Raphson's iterative method is the nomrally the method of choice. It requires however that we setting the matrices that define the first and second derivatives. + +

    +Our iterative scheme is then given by + +

     
    +$$ +\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T}\right)^{-1}\times \left(\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}}\right)_{\hat{\beta}^{\mathrm{old}}}, +$$ +

     
    + +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. +

    + + +
    +

    Brief reminder on Newton-Raphson's method

    + +

    +Let us quicly remind ourselves how we derive the above method. + +

    +Perhaps the most celebrated of all one-dimensional root-finding +routines is Newton's method, also called the Newton-Raphson +method. This method is distinguished from the previously discussed +methods by the fact that it requires the evaluation of both the +function \( f \) and its derivative \( f' \) at arbitrary points. In this +sense, it is taylored to cases with e.g., transcendental equations. +If you can only calculate the derivative +numerically and/or your function is not of the smooth type, we +discourage the use of this method. +

    + + +
    +

    The equations

    + +

    +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 +

     
    +$$ + x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. +$$ +

     
    +

    + + +
    +

    Simple geometric interpretation

    + +

    +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 +

    + + +
    +

    Extending to more than one variable

    + +

    +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. +

    + + +
    +

    Steepest descent

    The method of steepest descent The basic idea of gradient descent is @@ -200,7 +430,7 @@ 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 @@ -221,7 +451,7 @@ 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 @@ -247,7 +477,7 @@ 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 @@ -266,7 +496,7 @@ 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). @@ -286,7 +516,7 @@ 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

     
    @@ -296,7 +526,7 @@ $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$

    -

    Conditions on convex functions

    +

    Conditions on convex functions

    In the following we state first and second-order conditions which @@ -338,7 +568,7 @@ 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 @@ -367,7 +597,7 @@ This result means that if we know that the cost/loss function is convex and we a

    -

    Some simple problems

    +

    Some simple problems

    1. 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 \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
      2. @@ -394,7 +624,146 @@ Using the definition of convexity, try to show that a function satisfying the pr
      -

      Revisiting our first homework

      +

      Standard steepest descent

      + +

      +Before we proceed, we would like to mention the approach called the standard Steepest descent, which again leads to us having to be able to compute a matrix. + +

      +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, \( \hat{r}=0 \). +

      + + +
      +

      Conjugate 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. If we search for a minimum of the quantum mechanical +variance, then the matrix \( \hat{A} \), which is called the Hessian, is +given by the second-derivative of the function we want to minimize. +This quantity is always positive definite. + +

      +More details will be added here soon. +

      + + +
      +

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

      +
      + +

      + + +

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

      The routine for the steepest descent method

      +
      + +

      + + +

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

      Revisiting our first homework

      We will use linear regression as a case study for the gradient descent @@ -434,7 +803,7 @@ $$

      -

      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 \( \beta = (\beta_0, \beta_1)^T \) @@ -463,7 +832,7 @@ 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 @@ -480,7 +849,7 @@ where \( X \) is the design matrix defined above.

      -

      The Hessian matrix

      +

      The Hessian matrix

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

       
      $$ @@ -496,7 +865,7 @@ 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 @@ -540,7 +909,7 @@ beta_NE = np.dot(Xt_X_inv,Xt_y)

      -

      Gradient Descent Example

      +

      Gradient Descent Example

      Another simple example is here @@ -590,7 +959,7 @@ plt.show()

      -

      And a corresponding example using scikit-learn

      +

      And a corresponding example using scikit-learn

      @@ -615,7 +984,7 @@ 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 \( \beta \), @@ -679,7 +1048,7 @@ beta_ridge = np.dot(Z,np.dot(X.T,y))

      -

      Stochastic Gradient Descent

      +

      Stochastic Gradient Descent

      Stochastic gradient descent (SGD) and variants thereof address some of @@ -699,7 +1068,7 @@ $$

      -

      Computation of gradients

      +

      Computation of gradients

      This in turn means that the gradient can be @@ -721,7 +1090,7 @@ 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 @@ -747,7 +1116,7 @@ $$
      -

      The gradient step

      +

      The gradient step

      Thus a gradient descent step now looks like @@ -768,7 +1137,7 @@ the number of minibatches, as exemplified in the code below.

      -

      Simple example code

      +

      Simple example code

      @@ -800,7 +1169,7 @@ 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? @@ -817,7 +1186,7 @@ gave the lowest value.

      -

      Slightly different approach

      +

      Slightly different approach

      Another approach is to let the step length \( \gamma_j \) depend on the @@ -868,7 +1237,7 @@ j = 0

      -

      Conjugate gradient (CG) method

      +

      Conjugate gradient (CG) method

      @@ -902,7 +1271,7 @@ When we have found the exact solution, \( \hat{r}=0 \).

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      @@ -923,7 +1292,7 @@ If we search for a minimum of the quantum mechanical variance, then the matrix

      -

      Conjugate gradient method, Newton's method first

      +

      Conjugate gradient method, Newton's method first

      @@ -951,7 +1320,7 @@ $$

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      @@ -984,7 +1353,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      @@ -1003,7 +1372,7 @@ which is zero unless \( i=j \).

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      @@ -1033,7 +1402,7 @@ $$

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      @@ -1070,7 +1439,7 @@ $$

      -

      Conjugate gradient method and iterations

      +

      Conjugate gradient method and iterations

      @@ -1107,7 +1476,7 @@ instead.

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      @@ -1140,7 +1509,7 @@ hence the name conjugate gradient method.

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      @@ -1172,7 +1541,7 @@ $$

      -

      Conjugate gradient method

      +

      Conjugate gradient method

      diff --git a/doc/pub/Splines/html/Splines-solarized.html b/doc/pub/Splines/html/Splines-solarized.html index 752bc03f2..b342a25ff 100644 --- a/doc/pub/Splines/html/Splines-solarized.html +++ b/doc/pub/Splines/html/Splines-solarized.html @@ -65,47 +65,68 @@ div { text-align: justify; text-justify: inter-word; } 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -147,7 +168,7 @@ MathJax.Hub.Config({

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

      -

      Sep 21, 2018

      +

      Sep 27, 2018












      @@ -167,7 +188,209 @@ some approximative/numerical method to compute the minimum.











      -

      Steepest descent

      +

      Revisiting our Logistic Regression case

      + +

      +In our discussion on Logistic Regression we defined we studied first 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 of the Sigmoid function, 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 \( \beta_1 \). + +

      +









      + +

      The equations to solve

      + +

      +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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ + +

      +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 we call the Hessian. + +

      +









      + +

      Solving using Newton-Raphson's method

      + +

      +If we can set up these equations, Newton-Raphson's iterative method is the nomrally the method of choice. It requires however that we setting the matrices that define the first and second derivatives. + +

      +Our iterative scheme is then given by + +$$ +\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T}\right)^{-1}\times \left(\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}}\right)_{\hat{\beta}^{\mathrm{old}}}, +$$ + +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. + +

      +









      + +

      Brief reminder on Newton-Raphson's method

      + +

      +Let us quicly remind ourselves how we derive the above method. + +

      +Perhaps the most celebrated of all one-dimensional root-finding +routines is Newton's method, also called the Newton-Raphson +method. This method is distinguished from the previously discussed +methods by the fact that it requires the evaluation of both the +function \( f \) and its derivative \( f' \) at arbitrary points. In this +sense, it is taylored to cases with e.g., transcendental equations. +If you can only calculate the derivative +numerically and/or your function is not of the smooth type, we +discourage the use of this method. + +

      +









      + +

      The equations

      + +

      +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 + +$$ + f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots. + \label{eq:taylornr} +$$ + +

      +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 +$$ + x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. +$$ + +

      +









      + +

      Simple geometric interpretation

      + +

      +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 + +

      +









      + +

      Extending to more than one variable

      + +

      +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. + +

      +









      + +

      Steepest descent

      The method of steepest descent The basic idea of gradient descent is @@ -191,7 +414,7 @@ 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 @@ -210,7 +433,7 @@ 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 @@ -236,7 +459,7 @@ 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 @@ -255,7 +478,7 @@ 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). @@ -275,7 +498,7 @@ 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. @@ -283,7 +506,7 @@ 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 @@ -325,7 +548,7 @@ 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 @@ -355,7 +578,7 @@ This result means that if we know that the cost/loss function is convex and we a











      -

      Some simple problems

      +

      Some simple problems

      1. 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 \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
        2. @@ -379,10 +602,147 @@ 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 mention the approach called the standard Steepest descent, which again leads to us having to be able to compute a matrix. + +

        +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, \( \hat{r}=0 \). + +

        +









        + +

        Conjugate 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. If we search for a minimum of the quantum mechanical +variance, then the matrix \( \hat{A} \), which is called the Hessian, is +given by the second-derivative of the function we want to minimize. +This quantity is always positive definite. + +

        +More details will be added here soon. + +

        +









        + +

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

        +
        + +

        +

        + + +

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

        +









        + +

        The routine for the steepest descent method

        +
        + +

        +

        + + +

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

        -

        Revisiting our first homework

        +

        Revisiting our first homework

        We will use linear regression as a case study for the gradient descent @@ -415,7 +775,7 @@ $$

        -

        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 \( \beta = (\beta_0, \beta_1)^T \) @@ -440,7 +800,7 @@ 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 @@ -455,7 +815,7 @@ 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} @@ -469,7 +829,7 @@ 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 @@ -510,7 +870,7 @@ beta_NE = np.dot(Xt_X_inv,Xt_y)











        -

        Gradient Descent Example

        +

        Gradient Descent Example

        Another simple example is here @@ -559,7 +919,7 @@ plt.show()











        -

        And a corresponding example using scikit-learn

        +

        And a corresponding example using scikit-learn

        @@ -583,7 +943,7 @@ 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 \( \beta \), @@ -640,7 +1000,7 @@ beta_ridge = np.dot(Z,np.dot(X.T,y))











        -

        Stochastic Gradient Descent

        +

        Stochastic Gradient Descent

        Stochastic gradient descent (SGD) and variants thereof address some of @@ -658,7 +1018,7 @@ $$











        -

        Computation of gradients

        +

        Computation of gradients

        This in turn means that the gradient can be @@ -678,7 +1038,7 @@ 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 @@ -702,7 +1062,7 @@ $$











        -

        The gradient step

        +

        The gradient step

        Thus a gradient descent step now looks like @@ -721,7 +1081,7 @@ the number of minibatches, as exemplified in the code below.











        -

        Simple example code

        +

        Simple example code

        @@ -753,7 +1113,7 @@ 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? @@ -770,7 +1130,7 @@ gave the lowest value.











        -

        Slightly different approach

        +

        Slightly different approach

        Another approach is to let the step length \( \gamma_j \) depend on the @@ -818,7 +1178,7 @@ j = 0











        -

        Conjugate gradient (CG) method

        +

        Conjugate gradient (CG) method

        @@ -849,7 +1209,7 @@ When we have found the exact solution, \( \hat{r}=0 \).











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        @@ -871,7 +1231,7 @@ If we search for a minimum of the quantum mechanical variance, then the matrix











        -

        Conjugate gradient method, Newton's method first

        +

        Conjugate gradient method, Newton's method first

        @@ -896,7 +1256,7 @@ $$











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        @@ -926,7 +1286,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        @@ -944,7 +1304,7 @@ which is zero unless \( i=j \).











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        @@ -971,7 +1331,7 @@ $$











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        @@ -1003,7 +1363,7 @@ $$











        -

        Conjugate gradient method and iterations

        +

        Conjugate gradient method and iterations

        @@ -1039,7 +1399,7 @@ instead.











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        @@ -1069,7 +1429,7 @@ hence the name conjugate gradient method.











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        @@ -1098,7 +1458,7 @@ $$











        -

        Conjugate gradient method

        +

        Conjugate gradient method

        diff --git a/doc/pub/Splines/html/Splines.html b/doc/pub/Splines/html/Splines.html index eec927569..6feb11062 100644 --- a/doc/pub/Splines/html/Splines.html +++ b/doc/pub/Splines/html/Splines.html @@ -70,47 +70,68 @@ div { text-align: justify; text-justify: inter-word; } 2, None, '___sec0'), - ('Steepest descent', 2, None, '___sec1'), - ('More on Steepest descent', 2, None, '___sec2'), - ('The ideal', 2, None, '___sec3'), - ('The sensitiveness of the gradient descent', 2, None, '___sec4'), - ('Convex functions', 2, None, '___sec5'), - ('Convex function', 2, None, '___sec6'), - ('Conditions on convex functions', 2, None, '___sec7'), - ('More on convex functions', 2, None, '___sec8'), - ('Some simple problems', 2, None, '___sec9'), - ('Revisiting our first homework', 2, None, '___sec10'), - ('Gradient descent example', 2, None, '___sec11'), - ('The derivative of the cost/loss function', 2, None, '___sec12'), - ('The Hessian matrix', 2, None, '___sec13'), - ('Simple program', 2, None, '___sec14'), - ('Gradient Descent Example', 2, None, '___sec15'), + ('Revisiting our Logistic Regression case', 2, None, '___sec1'), + ('The equations to solve', 2, None, '___sec2'), + ("Solving using Newton-Raphson's method", 2, None, '___sec3'), + ("Brief reminder on Newton-Raphson's method", 2, None, '___sec4'), + ('The equations', 2, None, '___sec5'), + ('Simple geometric interpretation', 2, None, '___sec6'), + ('Extending to more than one variable', 2, None, '___sec7'), + ('Steepest descent', 2, None, '___sec8'), + ('More on Steepest descent', 2, None, '___sec9'), + ('The ideal', 2, None, '___sec10'), + ('The sensitiveness of the gradient descent', + 2, + None, + '___sec11'), + ('Convex functions', 2, None, '___sec12'), + ('Convex function', 2, None, '___sec13'), + ('Conditions on convex functions', 2, None, '___sec14'), + ('More on convex functions', 2, None, '___sec15'), + ('Some simple problems', 2, None, '___sec16'), + ('Standard steepest descent', 2, None, '___sec17'), + ('Conjugate gradient method', 2, None, '___sec18'), + ('Simple codes for steepest descent and conjugate gradient ' + 'using a $2\\times 2$ matrix, in c++, Python code to come', + 2, + None, + '___sec19'), + ('The routine for the steepest descent method', + 2, + None, + '___sec20'), + ('Revisiting our first homework', 2, None, '___sec21'), + ('Gradient descent example', 2, None, '___sec22'), + ('The derivative of the cost/loss function', 2, None, '___sec23'), + ('The Hessian matrix', 2, None, '___sec24'), + ('Simple program', 2, None, '___sec25'), + ('Gradient Descent Example', 2, None, '___sec26'), ('And a corresponding example using _scikit-learn_', 2, None, - '___sec16'), - ('Gradient descent and Ridge', 2, None, '___sec17'), - ('Stochastic Gradient Descent', 2, None, '___sec18'), - ('Computation of gradients', 2, None, '___sec19'), - ('SGD example', 2, None, '___sec20'), - ('The gradient step', 2, None, '___sec21'), - ('Simple example code', 2, None, '___sec22'), - ('When do we stop?', 2, None, '___sec23'), - ('Slightly different approach', 2, None, '___sec24'), - ('Conjugate gradient (CG) method', 2, None, '___sec25'), - ('Conjugate gradient method', 2, None, '___sec26'), + '___sec27'), + ('Gradient descent and Ridge', 2, None, '___sec28'), + ('Stochastic Gradient Descent', 2, None, '___sec29'), + ('Computation of gradients', 2, None, '___sec30'), + ('SGD example', 2, None, '___sec31'), + ('The gradient step', 2, None, '___sec32'), + ('Simple example code', 2, None, '___sec33'), + ('When do we stop?', 2, None, '___sec34'), + ('Slightly different approach', 2, None, '___sec35'), + ('Conjugate gradient (CG) method', 2, None, '___sec36'), + ('Conjugate gradient method', 2, None, '___sec37'), ("Conjugate gradient method, Newton's method first", 2, None, - '___sec27'), - ('Conjugate gradient method', 2, None, '___sec28'), - ('Conjugate gradient method', 2, None, '___sec29'), - ('Conjugate gradient method', 2, None, '___sec30'), - ('Conjugate gradient method', 2, None, '___sec31'), - ('Conjugate gradient method and iterations', 2, None, '___sec32'), - ('Conjugate gradient method', 2, None, '___sec33'), - ('Conjugate gradient method', 2, None, '___sec34'), - ('Conjugate gradient method', 2, None, '___sec35')]} + '___sec38'), + ('Conjugate gradient method', 2, None, '___sec39'), + ('Conjugate gradient method', 2, None, '___sec40'), + ('Conjugate gradient method', 2, None, '___sec41'), + ('Conjugate gradient method', 2, None, '___sec42'), + ('Conjugate gradient method and iterations', 2, None, '___sec43'), + ('Conjugate gradient method', 2, None, '___sec44'), + ('Conjugate gradient method', 2, None, '___sec45'), + ('Conjugate gradient method', 2, None, '___sec46')]} end of tocinfo --> @@ -152,7 +173,7 @@ MathJax.Hub.Config({

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

        -

        Sep 21, 2018

        +

        Sep 27, 2018












        @@ -172,7 +193,209 @@ some approximative/numerical method to compute the minimum.











        -

        Steepest descent

        +

        Revisiting our Logistic Regression case

        + +

        +In our discussion on Logistic Regression we defined we studied first 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 of the Sigmoid function, 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 \( \beta_1 \). + +

        +









        + +

        The equations to solve

        + +

        +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 + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ + +

        +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 we call the Hessian. + +

        +









        + +

        Solving using Newton-Raphson's method

        + +

        +If we can set up these equations, Newton-Raphson's iterative method is the nomrally the method of choice. It requires however that we setting the matrices that define the first and second derivatives. + +

        +Our iterative scheme is then given by + +$$ +\hat{\beta}^{\mathrm{new}} = \hat{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T}\right)^{-1}\times \left(\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}}\right)_{\hat{\beta}^{\mathrm{old}}}, +$$ + +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. + +

        +









        + +

        Brief reminder on Newton-Raphson's method

        + +

        +Let us quicly remind ourselves how we derive the above method. + +

        +Perhaps the most celebrated of all one-dimensional root-finding +routines is Newton's method, also called the Newton-Raphson +method. This method is distinguished from the previously discussed +methods by the fact that it requires the evaluation of both the +function \( f \) and its derivative \( f' \) at arbitrary points. In this +sense, it is taylored to cases with e.g., transcendental equations. +If you can only calculate the derivative +numerically and/or your function is not of the smooth type, we +discourage the use of this method. + +

        +









        + +

        The equations

        + +

        +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 + +$$ + f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots. + \label{eq:taylornr} +$$ + +

        +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 +$$ + x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. +$$ + +

        +









        + +

        Simple geometric interpretation

        + +

        +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 + +

        +









        + +

        Extending to more than one variable

        + +

        +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. + +

        +









        + +

        Steepest descent

        The method of steepest descent The basic idea of gradient descent is @@ -196,7 +419,7 @@ 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 @@ -215,7 +438,7 @@ 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 @@ -241,7 +464,7 @@ 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 @@ -260,7 +483,7 @@ 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). @@ -280,7 +503,7 @@ 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. @@ -288,7 +511,7 @@ 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 @@ -330,7 +553,7 @@ 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 @@ -360,7 +583,7 @@ This result means that if we know that the cost/loss function is convex and we a











        -

        Some simple problems

        +

        Some simple problems

        1. 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 \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
          2. @@ -384,10 +607,147 @@ 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 mention the approach called the standard Steepest descent, which again leads to us having to be able to compute a matrix. + +

          +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, \( \hat{r}=0 \). + +

          +









          + +

          Conjugate 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. If we search for a minimum of the quantum mechanical +variance, then the matrix \( \hat{A} \), which is called the Hessian, is +given by the second-derivative of the function we want to minimize. +This quantity is always positive definite. + +

          +More details will be added here soon. + +

          +









          + +

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

          +
          + +

          +

          + + +

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

          +









          + +

          The routine for the steepest descent method

          +
          + +

          +

          + + +

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

          -

          Revisiting our first homework

          +

          Revisiting our first homework

          We will use linear regression as a case study for the gradient descent @@ -420,7 +780,7 @@ $$

          -

          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 \( \beta = (\beta_0, \beta_1)^T \) @@ -445,7 +805,7 @@ 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 @@ -460,7 +820,7 @@ 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} @@ -474,7 +834,7 @@ 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 @@ -515,7 +875,7 @@ beta_NE = np.









          -

          Gradient Descent Example

          +

          Gradient Descent Example

          Another simple example is here @@ -564,7 +924,7 @@ plt.show()











          -

          And a corresponding example using scikit-learn

          +

          And a corresponding example using scikit-learn

          @@ -588,7 +948,7 @@ 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 \( \beta \), @@ -645,7 +1005,7 @@ beta_ridge = np











          -

          Stochastic Gradient Descent

          +

          Stochastic Gradient Descent

          Stochastic gradient descent (SGD) and variants thereof address some of @@ -663,7 +1023,7 @@ $$











          -

          Computation of gradients

          +

          Computation of gradients

          This in turn means that the gradient can be @@ -683,7 +1043,7 @@ 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 @@ -707,7 +1067,7 @@ $$











          -

          The gradient step

          +

          The gradient step

          Thus a gradient descent step now looks like @@ -726,7 +1086,7 @@ the number of minibatches, as exemplified in the code below.











          -

          Simple example code

          +

          Simple example code

          @@ -758,7 +1118,7 @@ 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? @@ -775,7 +1135,7 @@ gave the lowest value.











          -

          Slightly different approach

          +

          Slightly different approach

          Another approach is to let the step length \( \gamma_j \) depend on the @@ -823,7 +1183,7 @@ j = 0











          -

          Conjugate gradient (CG) method

          +

          Conjugate gradient (CG) method

          @@ -854,7 +1214,7 @@ When we have found the exact solution, \( \hat{r}=0 \).











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          @@ -876,7 +1236,7 @@ If we search for a minimum of the quantum mechanical variance, then the matrix











          -

          Conjugate gradient method, Newton's method first

          +

          Conjugate gradient method, Newton's method first

          @@ -901,7 +1261,7 @@ $$











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          @@ -931,7 +1291,7 @@ this inner product. Being conjugate is a symmetric relation: if \( \hat{s} \) is











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          @@ -949,7 +1309,7 @@ which is zero unless \( i=j \).











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          @@ -976,7 +1336,7 @@ $$











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          @@ -1008,7 +1368,7 @@ $$











          -

          Conjugate gradient method and iterations

          +

          Conjugate gradient method and iterations

          @@ -1044,7 +1404,7 @@ instead.











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          @@ -1074,7 +1434,7 @@ hence the name conjugate gradient method.











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          @@ -1103,7 +1463,7 @@ $$











          -

          Conjugate gradient method

          +

          Conjugate gradient method

          diff --git a/doc/pub/Splines/ipynb/Splines.ipynb b/doc/pub/Splines/ipynb/Splines.ipynb index 688508aaa..a46ac7740 100644 --- a/doc/pub/Splines/ipynb/Splines.ipynb +++ b/doc/pub/Splines/ipynb/Splines.ipynb @@ -10,7 +10,7 @@ " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Sep 21, 2018**\n", + "Date: **Sep 27, 2018**\n", "\n", "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -28,6 +28,327 @@ "analytically, however this is not possible in general and we must use\n", "some approximative/numerical method to compute the minimum.\n", "\n", + "\n", + "## Revisiting our Logistic Regression case\n", + "\n", + "In our discussion on Logistic Regression we defined we studied first the \n", + "case of\n", + "two classes, with $y_i$ either\n", + "$0$ or $1$. Furthermore we assumed also that we have only two\n", + "parameters $\\beta$ in our fitting of the Sigmoid function, that is we\n", + "defined probabilities" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", + "p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", + "\n", + "## The equations to solve\n", + "\n", + "Our compact equations used a definition of a vector $\\hat{y}$ with $n$\n", + "elements $y_i$, an $n\\times p$ matrix $\\hat{X}$ which contains the\n", + "$x_i$ values and a vector $\\hat{p}$ of fitted probabilities\n", + "$p(y_i\\vert x_i,\\hat{\\beta})$. We rewrote in a more compact form\n", + "the first derivative of the cost function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}} = -\\hat{X}^T\\left(\\hat{y}-\\hat{p}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we in addition define a diagonal matrix $\\hat{W}$ with elements \n", + "$p(y_i\\vert x_i,\\hat{\\beta})(1-p(y_i\\vert x_i,\\hat{\\beta})$, we can obtain a compact expression of the second derivative as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T} = \\hat{X}^T\\hat{W}\\hat{X}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This defines what we call the Hessian.\n", + "\n", + "## Solving using Newton-Raphson's method\n", + "\n", + "If we can set up these equations, Newton-Raphson's iterative method is the nomrally the method of choice. It requires however that we setting the matrices that define the first and second derivatives. \n", + "\n", + "Our iterative scheme is then given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{\\beta}^{\\mathrm{new}} = \\hat{\\beta}^{\\mathrm{old}}-\\left(\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T}\\right)^{-1}\\times \\left(\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}}\\right)_{\\hat{\\beta}^{\\mathrm{old}}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or in matrix form as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{\\beta}^{\\mathrm{new}} = \\hat{\\beta}^{\\mathrm{old}}-\\left(\\hat{X}^T\\hat{W}\\hat{X} \\right)^{-1}\\times \\left(-\\hat{X}^T(\\hat{y}-\\hat{p}) \\right)_{\\hat{\\beta}^{\\mathrm{old}}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The right-hand side is computed with the old values of $\\beta$. \n", + "\n", + "If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. \n", + "\n", + "\n", + "## Brief reminder on Newton-Raphson's method\n", + "\n", + "Let us quicly remind ourselves how we derive the above method.\n", + "\n", + "Perhaps the most celebrated of all one-dimensional root-finding\n", + "routines is Newton's method, also called the Newton-Raphson\n", + "method. This method is distinguished from the previously discussed\n", + "methods by the fact that it requires the evaluation of both the\n", + "function $f$ and its derivative $f'$ at arbitrary points. In this\n", + "sense, it is taylored to cases with e.g., transcendental equations.\n", + "If you can only calculate the derivative\n", + "numerically and/or your function is not of the smooth type, we\n", + "discourage the use of this method.\n", + "\n", + "## The equations\n", + "\n", + "The Newton-Raphson formula consists geometrically of extending the\n", + "tangent line at a current point until it crosses zero, then setting\n", + "the next guess to the abscissa of that zero-crossing. The mathematics\n", + "behind this method is rather simple. Employing a Taylor expansion for\n", + "$x$ sufficiently close to the solution $s$, we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "

          \n", + "\n", + "$$\n", + "f(s)=0=f(x)+(s-x)f'(x)+\\frac{(s-x)^2}{2}f''(x) +\\dots.\n", + " \\label{eq:taylornr} \\tag{1}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For small enough values of the function and for well-behaved\n", + "functions, the terms beyond linear are unimportant, hence we obtain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f(x)+(s-x)f'(x)\\approx 0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "yielding" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "s\\approx x-\\frac{f(x)}{f'(x)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Having in mind an iterative procedure, it is natural to start iterating with" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Simple geometric interpretation\n", + "\n", + "The above is Newton-Raphson's method. It has a simple geometric\n", + "interpretation, namely $x_{n+1}$ is the point where the tangent from\n", + "$(x_n,f(x_n))$ crosses the $x-$axis. Close to the solution,\n", + "Newton-Raphson converges fast to the desired result. However, if we\n", + "are far from a root, where the higher-order terms in the series are\n", + "important, the Newton-Raphson formula can give grossly inaccurate\n", + "results. For instance, the initial guess for the root might be so far\n", + "from the true root as to let the search interval include a local\n", + "maximum or minimum of the function. If an iteration places a trial\n", + "guess near such a local extremum, so that the first derivative nearly\n", + "vanishes, then Newton-Raphson may fail totally\n", + "\n", + "\n", + "## Extending to more than one variable\n", + "\n", + "Newton's method can be generalized to systems of several non-linear equations\n", + "and variables. Consider the case with two equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{array}{cc} f_1(x_1,x_2) &=0\\\\\n", + " f_2(x_1,x_2) &=0\\end{array},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which we Taylor expand to obtain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1\n", + " \\partial f_1/\\partial x_1+h_2\n", + " \\partial f_1/\\partial x_2+\\dots\\\\\n", + " 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1\n", + " \\partial f_2/\\partial x_1+h_2\n", + " \\partial f_2/\\partial x_2+\\dots\n", + " \\end{array}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Defining the Jacobian matrix ${\\bf \\hat{J}}$ we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "{\\bf \\hat{J}}=\\left( \\begin{array}{cc}\n", + " \\partial f_1/\\partial x_1 & \\partial f_1/\\partial x_2 \\\\\n", + " \\partial f_2/\\partial x_1 &\\partial f_2/\\partial x_2\n", + " \\end{array} \\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we can rephrase Newton's method as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\left(\\begin{array}{c} x_1^{n+1} \\\\ x_2^{n+1} \\end{array} \\right)=\n", + "\\left(\\begin{array}{c} x_1^{n} \\\\ x_2^{n} \\end{array} \\right)+\n", + "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have defined" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right)=\n", + " -{\\bf \\hat{J}}^{-1}\n", + " \\left(\\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\\\ f_2(x_1^{n},x_2^{n}) \\end{array} \\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We need thus to compute the inverse of the Jacobian matrix and it\n", + "is to understand that difficulties may\n", + "arise in case ${\\bf \\hat{J}}$ is nearly singular.\n", + "\n", + "It is rather straightforward to extend the above scheme to systems of\n", + "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. \n", + "\n", + "\n", + "\n", "## Steepest descent\n", "\n", "The method of steepest descent The basic idea of gradient descent is\n", @@ -218,6 +539,154 @@ "\n", "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).\n", "\n", + "\n", + "## Standard steepest descent\n", + "\n", + "\n", + "Before we proceed, we would like to mention the approach called the **standard Steepest descent**, which again leads to us having to be able to compute a matrix.\n", + "\n", + "[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)\n", + "for finding solutions of non-linear problems is based on the theory\n", + "of conjugate gradients for linear systems of equations. It belongs to\n", + "the class of iterative methods for solving problems from linear\n", + "algebra of the type" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{A}\\hat{x} = \\hat{b}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the iterative process we end up with a problem like" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{r}= \\hat{b}-\\hat{A}\\hat{x},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\hat{r}$ is the so-called residual or error in the iterative process.\n", + "\n", + "When we have found the exact solution, $\\hat{r}=0$.\n", + "\n", + "## Conjugate gradient method\n", + "\n", + "The residual is zero when we reach the minimum of the quadratic equation" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "P(\\hat{x})=\\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T\\hat{b},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with the constraint that the matrix $\\hat{A}$ is positive definite and\n", + "symmetric. If we search for a minimum of the quantum mechanical\n", + "variance, then the matrix $\\hat{A}$, which is called the Hessian, is\n", + "given by the second-derivative of the function we want to minimize.\n", + "This quantity is always positive definite. \n", + "\n", + "More details will be added here soon.\n", + "\n", + "## Simple codes for steepest descent and conjugate gradient using a $2\\times 2$ matrix, in c++, Python code to come" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " #include \n", + " #include \n", + " #include \n", + " #include \n", + " #include \"vectormatrixclass.h\"\n", + " using namespace std;\n", + " // Main function begins here\n", + " int main(int argc, char * argv[]){\n", + " int dim = 2;\n", + " Vector x(dim),xsd(dim), b(dim),x0(dim);\n", + " Matrix A(dim,dim);\n", + " \n", + " // Set our initial guess\n", + " x0(0) = x0(1) = 0;\n", + " // Set the matrix\n", + " A(0,0) = 3; A(1,0) = 2; A(0,1) = 2; A(1,1) = 6;\n", + " b(0) = 2; b(1) = -8;\n", + " cout << \"The Matrix A that we are using: \" << endl;\n", + " A.Print();\n", + " cout << endl;\n", + " x = ConjugateGradient(A,b,x0);\n", + " xsd = SteepestDescent(A,b,x0);\n", + " cout << \"The approximate solution using Conjugate Gradient is: \" << endl;\n", + " x.Print();\n", + " cout << endl;\n", + " cout << \"The approximate solution using Steepest Descent is: \" << endl;\n", + " xsd.Print();\n", + " cout << endl;\n", + " }\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The routine for the steepest descent method" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " Vector SteepestDescent(Matrix A, Vector b, Vector x0){\n", + " int IterMax, i;\n", + " int dim = x0.Dimension();\n", + " const double tolerance = 1.0e-14;\n", + " Vector x(dim),f(dim),z(dim);\n", + " double c,alpha,d;\n", + " IterMax = 30;\n", + " x = x0;\n", + " f = A*x-b;\n", + " i = 0;\n", + " while (i <= IterMax){\n", + " z = A*f;\n", + " c = dot(f,f);\n", + " alpha = c/dot(f,z);\n", + " x = x - alpha*f;\n", + " f = A*x-b;\n", + " if(sqrt(dot(f,f)) < tolerance) break;\n", + " i++;\n", + " }\n", + " return x;\n", + " }\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "\n", "## Revisiting our first homework\n", "\n", @@ -397,7 +866,9 @@ { "cell_type": "code", "execution_count": 1, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [], "source": [ "import numpy as np\n", @@ -431,30 +902,11 @@ }, { "cell_type": "code", - "execution_count": 6, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[4.11631855]\n", - " [2.78555876]]\n", - "[[4.11631855]\n", - " [2.78555876]]\n" - ] - }, - { - "data": { - "image/png": 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z#c_Gf^+hlFdlsy`>oP}kRCCf~s&wjiZqL>(v-8~yY1qkQBW__jCj(+^okL@)W*m0_ zM9D3-rIGT`m@q_*hyRfO1yGTGkL}@aiEmwOTc~$u|GtvdX#dbVaf{)a>xYf48!G6Jh|ulA>wS?9h7jqUhS=9Y2RJR7AgL)cr4D| zIt8d(x_n$}mJYBfZzgE*-<%dIrJG~Pmaf-lB1*$d2@t9rU!Fz}NoQthTho?~Ig)=% zY~xpd&7a@3Z=K~Y)SK#j%9}qZE!8UBocjkRq@z~`72&ayN%si)|AK`A!QvsY<5)0H z8|3r~`m^>kFpn7|u^b%j^WNMZcD+IITKEManA(~+Iysmc+WgnZ&d3tM(aF%k$<4v^ TpXdzC3T9 +#include +#include +#include +#include "vectormatrixclass.h" +using namespace std; +// Main function begins here +int main(int argc, char * argv[]){ + int dim = 2; + Vector x(dim),xsd(dim), b(dim),x0(dim); + Matrix A(dim,dim); + + // Set our initial guess + x0(0) = x0(1) = 0; + // Set the matrix + A(0,0) = 3; A(1,0) = 2; A(0,1) = 2; A(1,1) = 6; + b(0) = 2; b(1) = -8; + cout << "The Matrix A that we are using: " << endl; + A.Print(); + cout << endl; + x = ConjugateGradient(A,b,x0); + xsd = SteepestDescent(A,b,x0); + cout << "The approximate solution using Conjugate Gradient is: " << endl; + x.Print(); + cout << endl; + cout << "The approximate solution using Steepest Descent is: " << endl; + xsd.Print(); + cout << endl; +} +!ec +!eblock + +!split +===== The routine for the steepest descent method ===== +!bblock +!bc cppcod +Vector SteepestDescent(Matrix A, Vector b, Vector x0){ + int IterMax, i; + int dim = x0.Dimension(); + const double tolerance = 1.0e-14; + Vector x(dim),f(dim),z(dim); + double c,alpha,d; + IterMax = 30; + x = x0; + f = A*x-b; + i = 0; + while (i <= IterMax){ + z = A*f; + c = dot(f,f); + alpha = c/dot(f,z); + x = x - alpha*f; + f = A*x-b; + if(sqrt(dot(f,f)) < tolerance) break; + i++; + } + return x; +} +!ec +!eblock + + !split ===== Revisiting our first homework ===== @@ -839,3 +1148,4 @@ which gives +