From 476f8657c2562ff84d78ce899416be436f2de0b1 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Thu, 18 Sep 2025 06:44:46 +0200 Subject: [PATCH] update week 39 --- doc/src/week39/Previousversions/week39.do.txt | 2711 ++++++++++++ doc/src/week39/week39.do.txt | 3781 ++++++----------- 2 files changed, 3961 insertions(+), 2531 deletions(-) create mode 100644 doc/src/week39/Previousversions/week39.do.txt diff --git a/doc/src/week39/Previousversions/week39.do.txt b/doc/src/week39/Previousversions/week39.do.txt new file mode 100644 index 000000000..2f3e51097 --- /dev/null +++ b/doc/src/week39/Previousversions/week39.do.txt @@ -0,0 +1,2711 @@ +TITLE: Week 39: Optimization and Gradient Methods +AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo +DATE: Week 39 + +!split +===== Plan for week 39, September 22-26, 2025 ===== + + + +!split +===== Lecture Monday September 22 ===== + +!bblock Material for the lecture on Monday September 22 + * Repetition of Logistic regression equations and classification problems and discussion of Gradient methods. Examples on how to implement Logistic Regression and discussion of gradient methods + * Stochastic Gradient descent with examples and automatic differentiation (theme also for next week). +# * "Video of lecture":"https://youtu.be/ISGpTC28Vmk" +# * "Whiteboard notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember23.pdf" + * Readings and Videos: + * These lecture notes + * For a good discussion on gradient methods, we would like to recommend Goodfellow et al section 4.3-4.5 and sections 8.3-8.6. We will come back to the latter chapter in our discussion of Neural networks as well. + * Raschka et al, pages 53-76 on Logistic regression and pages 37-52 on gradient optimization + * "Video on gradient descent":"https://www.youtube.com/watch?v=sDv4f4s2SB8" + * "Video on stochastic gradient descent":"https://www.youtube.com/watch?v=vMh0zPT0tLI" +!eblock + +!split +===== Lab sessions week 39 ===== + +!bblock Material for the active learning sessions on Tuesday and Wednesday + * Discussions on how to structure your report for the first project + * Exercise for week 39 on how to write the abstract and the introduction of the report and how to include references. + * Work on project 1, in particular resampling methods like cross-validation and bootstrap. _For more discussions of project 1, chapter 5 of Goodfellow et al is a good read, in particular sections 5.1-5.5 and 5.7-5.11_. + * "Video on how to write scientific reports recorded during one of the lab sessions":"https://youtu.be/tVW1ZDmZnwM" + * A general guideline can be found at URL:"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Projects/EvaluationGrading/EvaluationForm.md". +!eblock + + + + +!split +===== Lecture Monday September 22, Optimization, the central part of any Machine Learning algortithm ===== + +The first few slides here are a repetition from last week. + +Almost every problem in machine learning and data science starts with +a dataset $X$, a model $g(\beta)$, which is a function of the +parameters $\beta$ and a cost function $C(X, g(\beta))$ that allows +us to judge how well the model $g(\beta)$ explains the observations +$X$. The model is fit by finding the values of $\beta$ that minimize +the cost function. Ideally we would be able to solve for $\beta$ +analytically, however this is not possible in general and we must use +some approximative/numerical method to compute the minimum. + + +!split +===== Revisiting our Logistic Regression case ===== + +In our discussion on Logistic Regression we studied the +case of +two classes, with $y_i$ either +$0$ or $1$. Furthermore we assumed also that we have only two +parameters $\beta$ in our fitting, that is we +defined probabilities + +!bt +\begin{align*} +p(y_i=1|x_i,\bm{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\bm{\beta}) &= 1 - p(y_i=1|x_i,\bm{\beta}), +\end{align*} +!et +where $\bm{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$. + +!split +===== The equations to solve ===== + +Our compact equations used a definition of a vector $\bm{y}$ with $n$ +elements $y_i$, an $n\times p$ matrix $\bm{X}$ which contains the +$x_i$ values and a vector $\bm{p}$ of fitted probabilities +$p(y_i\vert x_i,\bm{\beta})$. We rewrote in a more compact form +the first derivative of the cost function as + +!bt +\[ +\frac{\partial \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}} = -\bm{X}^T\left(\bm{y}-\bm{p}\right). +\] +!et + +If we in addition define a diagonal matrix $\bm{W}$ with elements +$p(y_i\vert x_i,\bm{\beta})(1-p(y_i\vert x_i,\bm{\beta})$, we can obtain a compact expression of the second derivative as + +!bt +\[ +\frac{\partial^2 \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}\partial \bm{\beta}^T} = \bm{X}^T\bm{W}\bm{X}. +\] +!et +This defines what is called the Hessian matrix. + + + + + +!split +===== Solving using Newton-Raphson's method ===== + +If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. + +Our iterative scheme is then given by + +!bt +\[ +\bm{\beta}^{\mathrm{new}} = \bm{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}\partial \bm{\beta}^T}\right)^{-1}_{\bm{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}}\right)_{\bm{\beta}^{\mathrm{old}}}, +\] +!et +or in matrix form as + +!bt +\[ +\bm{\beta}^{\mathrm{new}} = \bm{\beta}^{\mathrm{old}}-\left(\bm{X}^T\bm{W}\bm{X} \right)^{-1}\times \left(-\bm{X}^T(\bm{y}-\bm{p}) \right)_{\bm{\beta}^{\mathrm{old}}}. +\] +!et +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. + + +!split +===== Brief reminder on Newton-Raphson's method ===== + +Let us quickly 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 requires the evaluation of both the +function $f$ and its derivative $f'$ at arbitrary points. +If you can only calculate the derivative +numerically and/or your function is not of the smooth type, we +normally discourage the use of this method. + +!split +===== 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 + + +!bt +\[ + f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots. + \label{eq:taylornr} +\] +!et + +For small enough values of the function and for well-behaved +functions, the terms beyond linear are unimportant, hence we obtain + + +!bt +\[ + f(x)+(s-x)f'(x)\approx 0, +\] +!et +yielding +!bt +\[ + s\approx x-\frac{f(x)}{f'(x)}. +\] +!et + +Having in mind an iterative procedure, it is natural to start iterating with +!bt +\[ + x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. +\] +!et + +!split +===== 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 + + +!split +===== 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 +!bt +\[ + \begin{array}{cc} f_1(x_1,x_2) &=0\\ + f_2(x_1,x_2) &=0,\end{array} +\] +!et +which we Taylor expand to obtain + +!bt +\[ + \begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1 + \partial f_1/\partial x_1+h_2 + \partial f_1/\partial x_2+\dots\\ + 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1 + \partial f_2/\partial x_1+h_2 + \partial f_2/\partial x_2+\dots + \end{array}. +\] +!et +Defining the Jacobian matrix ${\bf \bm{J}}$ we have +!bt +\[ + {\bf \bm{J}}=\left( \begin{array}{cc} + \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\ + \partial f_2/\partial x_1 &\partial f_2/\partial x_2 + \end{array} \right), +\] +!et +we can rephrase Newton's method as +!bt +\[ +\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)= +\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+ +\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right), +\] +!et +where we have defined +!bt +\[ + \left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)= + -{\bf \bm{J}}^{-1} + \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right). +\] +!et +We need thus to compute the inverse of the Jacobian matrix and it +is to understand that difficulties may +arise in case ${\bf \bm{J}}$ 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. + + + +!split +===== 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 $-\nabla F(\mathbf{x})$. + +It can be shown that if +!bt +\[ +\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), +\] +!et +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. + +!split +===== More on Steepest descent ===== + +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 + +!bt +\[ +\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0. +\] +!et + +The parameter $\gamma_k$ is often referred to as the step length or +the learning rate within the context of Machine Learning. + +!split +===== The ideal ===== + +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: + +In machine learing we are often faced with non-convex high dimensional +cost functions with many local minima. Since GD is deterministic we +will get stuck in a local minimum, if the method converges, unless we +have a very good intial guess. This also implies that the scheme is +sensitive to the chosen initial condition. + +Note that the gradient is a function of $\mathbf{x} = +(x_1,\cdots,x_n)$ which makes it expensive to compute numerically. + + +!split +===== The sensitiveness of the gradient descent ===== + +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. + +Many of these shortcomings can be alleviated by introducing +randomness. One such method is that of Stochastic Gradient Descent +(SGD), see below. + + +!split +===== Convex functions ===== + +Ideally we want our cost/loss function to be convex(concave). + +First we give the definition of a convex set: A set $C$ in +$\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and +all $t \in (0,1)$ , the point $(1 − t)x + ty$ also belongs to +C. Geometrically this means that every point on the line segment +connecting $x$ and $y$ is in $C$ as discussed below. + +The convex subsets of $\mathbb{R}$ are the intervals of +$\mathbb{R}$. Examples of convex sets of $\mathbb{R}^2$ are the +regular polygons (triangles, rectangles, pentagons, etc...). + +!split +===== 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. + +!split +===== Conditions on convex functions ===== + +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":"http://stanford.edu/boyd/cvxbook/". + +!bblock First order condition +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. +!eblock + +!bblock Second order condition +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. +!eblock + +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. + +!split +===== More on convex functions ===== + +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: + +!bblock Any minimum is global for convex functions +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. +!eblock + +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. + +!split +===== Some simple problems ===== + +o 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 $\lambda \in [0,1]$ $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$. + +o Using the second order condition show that the following functions are convex on the specified domain. + * $f(x) = e^x$ is convex for $x \in \mathbb{R}$. + * $g(x) = -\ln(x)$ is convex for $x \in (0,\infty)$. +o 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. + +o A norm is any function that satisfy the following properties + * $f(\alpha x) = |\alpha| f(x)$ for all $\alpha \in \mathbb{R}$. + * $f(x+y) \leq f(x) + f(y)$ + * $f(x) \leq 0$ for all $x \in \mathbb{R}^n$ with equality if and only if $x = 0$ + +Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this). + + + + +!split +===== Standard steepest descent ===== + + +Before we proceed, we would like to discuss the approach called the +_standard Steepest descent_ (different from the above steepest descent discussion), which again leads to us having to be able +to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). + +"The success of the CG method":"https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf" +for finding solutions of non-linear problems is based on the theory +of conjugate gradients for linear systems of equations. It belongs to +the class of iterative methods for solving problems from linear +algebra of the type +!bt +\begin{equation*} +\bm{A}\bm{x} = \bm{b}. +\end{equation*} +!et + +In the iterative process we end up with a problem like + +!bt +\begin{equation*} + \bm{r}= \bm{b}-\bm{A}\bm{x}, +\end{equation*} +!et +where $\bm{r}$ is the so-called residual or error in the iterative process. + +When we have found the exact solution, $\bm{r}=0$. + +!split +===== Gradient method ===== + +The residual is zero when we reach the minimum of the quadratic equation +!bt +\begin{equation*} + P(\bm{x})=\frac{1}{2}\bm{x}^T\bm{A}\bm{x} - \bm{x}^T\bm{b}, +\end{equation*} +!et + +with the constraint that the matrix $\bm{A}$ is positive definite and +symmetric. This defines also the Hessian and we want it to be positive definite. + + +!split +===== Steepest descent method ===== + +We denote the initial guess for $\bm{x}$ as $\bm{x}_0$. +We can assume without loss of generality that +!bt +\begin{equation*} +\bm{x}_0=0, +\end{equation*} +!et +or consider the system +!bt +\begin{equation*} +\bm{A}\bm{z} = \bm{b}-\bm{A}\bm{x}_0, +\end{equation*} +!et +instead. + + +!split +===== Steepest descent method ===== +!bblock +One can show that the solution $\bm{x}$ is also the unique minimizer of the quadratic form +!bt +\begin{equation*} + f(\bm{x}) = \frac{1}{2}\bm{x}^T\bm{A}\bm{x} - \bm{x}^T \bm{x} , \quad \bm{x}\in\mathbf{R}^n. +\end{equation*} +!et +This suggests taking the first basis vector $\bm{r}_1$ (see below for definition) +to be the gradient of $f$ at $\bm{x}=\bm{x}_0$, +which equals +!bt +\begin{equation*} +\bm{A}\bm{x}_0-\bm{b}, +\end{equation*} +!et +and +$\bm{x}_0=0$ it is equal $-\bm{b}$. + +!eblock + +!split +===== Final expressions ===== +!bblock +We can compute the residual iteratively as +!bt +\begin{equation*} +\bm{r}_{k+1}=\bm{b}-\bm{A}\bm{x}_{k+1}, + \end{equation*} +!et +which equals +!bt +\begin{equation*} +\bm{b}-\bm{A}(\bm{x}_k+\alpha_k\bm{r}_k), + \end{equation*} +!et +or +!bt +\begin{equation*} +(\bm{b}-\bm{A}\bm{x}_k)-\alpha_k\bm{A}\bm{r}_k, + \end{equation*} +!et +which gives + +!bt +\[ +\alpha_k = \frac{\bm{r}_k^T\bm{r}_k}{\bm{r}_k^T\bm{A}\bm{r}_k} +\] +!et +leading to the iterative scheme +!bt +\begin{equation*} +\bm{x}_{k+1}=\bm{x}_k+\alpha_k\bm{r}_{k}, + \end{equation*} +!et +!eblock + + + +!split +===== Steepest descent example ===== + +!bc pycod +import numpy as np +import numpy.linalg as la + +import scipy.optimize as sopt + +import matplotlib.pyplot as pt +from mpl_toolkits.mplot3d import axes3d + +def f(x): + return x[0]**2 + 3.0*x[1]**2 + +def df(x): + return np.array([2*x[0], 6*x[1]]) + +fig = pt.figure() +ax = fig.add_subplot(projection = '3d') + +xmesh, ymesh = np.mgrid[-3:3:50j,-3:3:50j] +fmesh = f(np.array([xmesh, ymesh])) +ax.plot_surface(xmesh, ymesh, fmesh) +!ec +And then as countor plot +!bc pycod +pt.axis("equal") +pt.contour(xmesh, ymesh, fmesh) +guesses = [np.array([2, 2./5])] +!ec +Find guesses +!bc pycod +x = guesses[-1] +s = -df(x) +!ec +Run it! +!bc pycod +def f1d(alpha): + return f(x + alpha*s) + +alpha_opt = sopt.golden(f1d) +next_guess = x + alpha_opt * s +guesses.append(next_guess) +print(next_guess) +!ec +What happened? +!bc pycod +pt.axis("equal") +pt.contour(xmesh, ymesh, fmesh, 50) +it_array = np.array(guesses) +pt.plot(it_array.T[0], it_array.T[1], "x-") +!ec + +Note that we did only one iteration here. We can easily add more using our previous guesses. + +!split +===== Conjugate gradient method ===== +!bblock +In the CG method we define so-called conjugate directions and two vectors +$\bm{s}$ and $\bm{t}$ +are said to be +conjugate if +!bt +\begin{equation*} +\bm{s}^T\bm{A}\bm{t}= 0. +\end{equation*} +!et +The philosophy of the CG method is to perform searches in various conjugate directions +of our vectors $\bm{x}_i$ obeying the above criterion, namely +!bt +\begin{equation*} +\bm{x}_i^T\bm{A}\bm{x}_j= 0. +\end{equation*} +!et +Two vectors are conjugate if they are orthogonal with respect to +this inner product. Being conjugate is a symmetric relation: if $\bm{s}$ is conjugate to $\bm{t}$, then $\bm{t}$ is conjugate to $\bm{s}$. +!eblock + +!split +===== Conjugate gradient method ===== +!bblock +An example is given by the eigenvectors of the matrix +!bt +\begin{equation*} +\bm{v}_i^T\bm{A}\bm{v}_j= \lambda\bm{v}_i^T\bm{v}_j, +\end{equation*} +!et +which is zero unless $i=j$. +!eblock + + +!split +===== Conjugate gradient method ===== +!bblock +Assume now that we have a symmetric positive-definite matrix $\bm{A}$ of size +$n\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector +!bt +\begin{equation*} +\bm{x}_{i+1}=\bm{x}_{i}+\alpha_i\bm{p}_{i}. +\end{equation*} +!et +We assume that $\bm{p}_{i}$ is a sequence of $n$ mutually conjugate directions. +Then the $\bm{p}_{i}$ form a basis of $R^n$ and we can expand the solution +$ \bm{A}\bm{x} = \bm{b}$ in this basis, namely + +!bt +\begin{equation*} + \bm{x} = \sum^{n}_{i=1} \alpha_i \bm{p}_i. +\end{equation*} +!et +!eblock + +!split +===== Conjugate gradient method ===== +!bblock +The coefficients are given by +!bt +\begin{equation*} + \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. +\end{equation*} +!et +Multiplying with $\bm{p}_k^T$ from the left gives + +!bt +\begin{equation*} + \bm{p}_k^T \bm{A}\bm{x} = \sum^{n}_{i=1} \alpha_i\bm{p}_k^T \bm{A}\bm{p}_i= \bm{p}_k^T \bm{b}, +\end{equation*} +!et +and we can define the coefficients $\alpha_k$ as + +!bt +\begin{equation*} + \alpha_k = \frac{\bm{p}_k^T \bm{b}}{\bm{p}_k^T \bm{A} \bm{p}_k} +\end{equation*} +!et +!eblock + +!split +===== Conjugate gradient method and iterations ===== +!bblock + +If we choose the conjugate vectors $\bm{p}_k$ carefully, +then we may not need all of them to obtain a good approximation to the solution +$\bm{x}$. +We want to regard the conjugate gradient method as an iterative method. +This will us to solve systems where $n$ is so large that the direct +method would take too much time. + +We denote the initial guess for $\bm{x}$ as $\bm{x}_0$. +We can assume without loss of generality that +!bt +\begin{equation*} +\bm{x}_0=0, +\end{equation*} +!et +or consider the system +!bt +\begin{equation*} +\bm{A}\bm{z} = \bm{b}-\bm{A}\bm{x}_0, +\end{equation*} +!et +instead. +!eblock + + +!split +===== Conjugate gradient method ===== +!bblock +One can show that the solution $\bm{x}$ is also the unique minimizer of the quadratic form +!bt +\begin{equation*} + f(\bm{x}) = \frac{1}{2}\bm{x}^T\bm{A}\bm{x} - \bm{x}^T \bm{x} , \quad \bm{x}\in\mathbf{R}^n. +\end{equation*} +!et +This suggests taking the first basis vector $\bm{p}_1$ +to be the gradient of $f$ at $\bm{x}=\bm{x}_0$, +which equals +!bt +\begin{equation*} +\bm{A}\bm{x}_0-\bm{b}, +\end{equation*} +!et +and +$\bm{x}_0=0$ it is equal $-\bm{b}$. +The other vectors in the basis will be conjugate to the gradient, +hence the name conjugate gradient method. +!eblock + + +!split +===== Conjugate gradient method ===== +!bblock +Let $\bm{r}_k$ be the residual at the $k$-th step: +!bt +\begin{equation*} +\bm{r}_k=\bm{b}-\bm{A}\bm{x}_k. +\end{equation*} +!et +Note that $\bm{r}_k$ is the negative gradient of $f$ at +$\bm{x}=\bm{x}_k$, +so the gradient descent method would be to move in the direction $\bm{r}_k$. +Here, we insist that the directions $\bm{p}_k$ are conjugate to each other, +so we take the direction closest to the gradient $\bm{r}_k$ +under the conjugacy constraint. +This gives the following expression +!bt +\begin{equation*} +\bm{p}_{k+1}=\bm{r}_k-\frac{\bm{p}_k^T \bm{A}\bm{r}_k}{\bm{p}_k^T\bm{A}\bm{p}_k} \bm{p}_k. +\end{equation*} +!et +!eblock + +!split +===== Conjugate gradient method ===== +!bblock +We can also compute the residual iteratively as +!bt +\begin{equation*} +\bm{r}_{k+1}=\bm{b}-\bm{A}\bm{x}_{k+1}, + \end{equation*} +!et +which equals +!bt +\begin{equation*} +\bm{b}-\bm{A}(\bm{x}_k+\alpha_k\bm{p}_k), + \end{equation*} +!et +or +!bt +\begin{equation*} +(\bm{b}-\bm{A}\bm{x}_k)-\alpha_k\bm{A}\bm{p}_k, + \end{equation*} +!et +which gives + +!bt +\begin{equation*} +\bm{r}_{k+1}=\bm{r}_k-\bm{A}\bm{p}_{k}, + \end{equation*} +!et +!eblock + + + + + +!split +===== Revisiting our first homework ===== + +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: + +o An analytical solution (recall homework set 1). +o The gradient can be computed analytically. +o The cost function is convex which guarantees that gradient descent converges for small enough learning rates + +We revisit an example similar to what we had in the first homework set. We had a function of the type + +!bc pycod +m = 100 +x = 2*np.random.rand(m,1) +y = 4+3*x+np.random.randn(m,1) +!ec +with $x_i \in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\cal {N}(0,1)$. +The linear regression model is given by +!bt +\[ +h_\beta(x) = \bm{y} = \beta_0 + \beta_1 x, +\] +!et +such that +!bt +\[ +\bm{y}_i = \beta_0 + \beta_1 x_i. +\] +!et + +!split +===== Gradient descent example ===== + +Let $\mathbf{y} = (y_1,\cdots,y_n)^T$, $\mathbf{\bm{y}} = (\bm{y}_1,\cdots,\bm{y}_n)^T$ and $\beta = (\beta_0, \beta_1)^T$ + +It is convenient to write $\mathbf{\bm{y}} = X\beta$ where $X \in \mathbb{R}^{100 \times 2} $ is the design matrix given by (we keep the intercept here) +!bt +\[ +X \equiv \begin{bmatrix} +1 & x_1 \\ +\vdots & \vdots \\ +1 & x_{100} & \\ +\end{bmatrix}. +\] +!et +The cost/loss/risk function is given by ( +!bt +\[ +C(\beta) = \frac{1}{n}||X\beta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2\right] +\] +!et +and we want to find $\beta$ such that $C(\beta)$ is minimized. + +!split +===== 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 +!bt +\[ +\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ +\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ +\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}), +\] +!et +where $X$ is the design matrix defined above. + +!split +===== The Hessian matrix ===== +The Hessian matrix of $C(\beta)$ is given by +!bt +\[ +\bm{H} \equiv \begin{bmatrix} +\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ +\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ +\end{bmatrix} = \frac{2}{n}X^T X. +\] +!et +This result implies that $C(\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite. + + + + +!split +===== 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 +!bt +\[ +\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots +\] +!et + +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 $||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8}$. _Note that the code below does not include the latter stop criterion_. + +And finally we can compare our solution for $\beta$ with the analytic result given by +$\beta= (X^TX)^{-1} X^T \mathbf{y}$. + +!split +===== Gradient Descent Example ===== + +Here our simple example +!bc pycod + +# 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 + +# the number of datapoints +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +# Hessian matrix +H = (2.0/n)* X.T @ X +# Get the eigenvalues +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +beta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y +print(beta_linreg) +beta = np.random.randn(2,1) + +eta = 1.0/np.max(EigValues) +Niterations = 1000 + +for iter in range(Niterations): + gradient = (2.0/n)*X.T @ (X @ beta-y) + beta -= eta*gradient + +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() + +!ec + +!split +===== And a corresponding example using _scikit-learn_ ===== + +!bc pycod +# Importing various packages +from random import random, seed +import numpy as np +import matplotlib.pyplot as plt +from sklearn.linear_model import SGDRegressor + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +beta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y) +print(beta_linreg) +sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) +sgdreg.fit(x,y.ravel()) +print(sgdreg.intercept_, sgdreg.coef_) + +!ec + + + +!split +===== Gradient descent and Ridge ===== + +We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\beta$, +!bt +\[ +C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. +\] +!et + +In order to minimize $C_{\text{ridge}}(\beta)$ using GD we adjust the gradient as follows +!bt +\[ +\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ +\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ +\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (\frac{1}{n}X^T(X\beta - \mathbf{y})+\lambda \beta). +\] +!et + +We can easily extend our program to minimize $C_{\text{ridge}}(\beta)$ using gradient descent and compare with the analytical solution given by +!bt +\[ +\beta_{\text{ridge}} = \left(X^T X + n\lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}. +\] +!et + +!split +===== The Hessian matrix for Ridge Regression ===== +The Hessian matrix of Ridge Regression for our simple example is given by +!bt +\[ +\bm{H} \equiv \begin{bmatrix} +\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ +\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ +\end{bmatrix} = \frac{2}{n}X^T X+2\lambda\bm{I}. +\] +!et +This implies that the Hessian matrix is positive definite, hence the stationary point is a +minimum. +Note that the Ridge cost function is convex being a sum of two convex +functions. Therefore, the stationary point is a global +minimum of this function. + + +!split +===== Program example for gradient descent with Ridge Regression ===== +!bc pycod +from random import random, seed +import numpy as np +import matplotlib.pyplot as plt +from mpl_toolkits.mplot3d import Axes3D +from matplotlib import cm +from matplotlib.ticker import LinearLocator, FormatStrFormatter +import sys + +# the number of datapoints +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X + +#Ridge parameter lambda +lmbda = 0.001 +Id = n*lmbda* np.eye(XT_X.shape[0]) + +# Hessian matrix +H = (2.0/n)* XT_X+2*lmbda* np.eye(XT_X.shape[0]) +# Get the eigenvalues +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + + +beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y +print(beta_linreg) +# Start plain gradient descent +beta = np.random.randn(2,1) + +eta = 1.0/np.max(EigValues) +Niterations = 100 + +for iter in range(Niterations): + gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta + beta -= eta*gradients + +print(beta) +ypredict = X @ beta +ypredict2 = X @ beta_linreg +plt.plot(x, ypredict, "r-") +plt.plot(x, ypredict2, "b-") +plt.plot(x, y ,'ro') +plt.axis([0,2.0,0, 15.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Gradient descent example for Ridge') +plt.show() + + +!ec + +!split +===== Using gradient descent methods, limitations ===== + +* _Gradient descent (GD) finds local minima of our function_. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance. + +* _GD is sensitive to initial conditions_. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD. + +* _Gradients are computationally expensive to calculate for large datasets_. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called ``mini batches''. This has the added benefit of introducing stochasticity into our algorithm. + +* _GD is very sensitive to choices of learning rates_. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape. + +* _GD treats all directions in parameter space uniformly._ Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. + +* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points. + +!split +===== Improving gradient descent with momentum ===== + +We discuss here some simple examples where we introduce what is called 'memory'about previous steps, or what is normally called momentum gradient descent. The mathematics is explained below in connection with Stochastic gradient descent. + +!bc pycod +from numpy import asarray +from numpy import arange +from numpy.random import rand +from numpy.random import seed +from matplotlib import pyplot + +# objective function +def objective(x): + return x**2.0 + +# derivative of objective function +def derivative(x): + return x * 2.0 + +# gradient descent algorithm +def gradient_descent(objective, derivative, bounds, n_iter, step_size): + # track all solutions + solutions, scores = list(), list() + # generate an initial point + solution = bounds[:, 0] + rand(len(bounds)) * (bounds[:, 1] - bounds[:, 0]) + # run the gradient descent + for i in range(n_iter): + # calculate gradient + gradient = derivative(solution) + # take a step + solution = solution - step_size * gradient + # evaluate candidate point + solution_eval = objective(solution) + # store solution + solutions.append(solution) + scores.append(solution_eval) + # report progress + print('>%d f(%s) = %.5f' % (i, solution, solution_eval)) + return [solutions, scores] + +# seed the pseudo random number generator +seed(4) +# define range for input +bounds = asarray([[-1.0, 1.0]]) +# define the total iterations +n_iter = 30 +# define the step size +step_size = 0.1 +# perform the gradient descent search +solutions, scores = gradient_descent(objective, derivative, bounds, n_iter, step_size) +# sample input range uniformly at 0.1 increments +inputs = arange(bounds[0,0], bounds[0,1]+0.1, 0.1) +# compute targets +results = objective(inputs) +# create a line plot of input vs result +pyplot.plot(inputs, results) +# plot the solutions found +pyplot.plot(solutions, scores, '.-', color='red') +# show the plot +pyplot.show() + +!ec + + +!split +===== Same code but now with momentum gradient descent ===== + +!bc pycod +from numpy import asarray +from numpy import arange +from numpy.random import rand +from numpy.random import seed +from matplotlib import pyplot + +# objective function +def objective(x): + return x**2.0 + +# derivative of objective function +def derivative(x): + return x * 2.0 + +# gradient descent algorithm +def gradient_descent(objective, derivative, bounds, n_iter, step_size, momentum): + # track all solutions + solutions, scores = list(), list() + # generate an initial point + solution = bounds[:, 0] + rand(len(bounds)) * (bounds[:, 1] - bounds[:, 0]) + # keep track of the change + change = 0.0 + # run the gradient descent + for i in range(n_iter): + # calculate gradient + gradient = derivative(solution) + # calculate update + new_change = step_size * gradient + momentum * change + # take a step + solution = solution - new_change + # save the change + change = new_change + # evaluate candidate point + solution_eval = objective(solution) + # store solution + solutions.append(solution) + scores.append(solution_eval) + # report progress + print('>%d f(%s) = %.5f' % (i, solution, solution_eval)) + return [solutions, scores] + +# seed the pseudo random number generator +seed(4) +# define range for input +bounds = asarray([[-1.0, 1.0]]) +# define the total iterations +n_iter = 30 +# define the step size +step_size = 0.1 +# define momentum +momentum = 0.3 +# perform the gradient descent search with momentum +solutions, scores = gradient_descent(objective, derivative, bounds, n_iter, step_size, momentum) +# sample input range uniformly at 0.1 increments +inputs = arange(bounds[0,0], bounds[0,1]+0.1, 0.1) +# compute targets +results = objective(inputs) +# create a line plot of input vs result +pyplot.plot(inputs, results) +# plot the solutions found +pyplot.plot(solutions, scores, '.-', color='red') +# show the plot +pyplot.show() +!ec + + + +!split +===== Overview video on Stochastic Gradient Descent ===== + +"What is Stochastic Gradient Descent":"https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" + + +!split +===== Batches and mini-batches ===== + +In gradient descent we compute the cost function and its gradient for all data points we have. + +In large-scale applications such as the "ILSVRC challenge":"https://www.image-net.org/challenges/LSVRC/", the +training data can have on order of millions of examples. Hence, it +seems wasteful to compute the full cost function over the entire +training set in order to perform only a single parameter update. A +very common approach to addressing this challenge is to compute the +gradient over batches of the training data. For example, a typical batch could contain some thousand examples from +an entire training set of several millions. This batch is then used to +perform a parameter update. + +!split +===== Stochastic Gradient Descent (SGD) ===== + +In stochastic gradient descent, the extreme case is the case where we +have only one batch, that is we include the whole data set. + +This process is called Stochastic Gradient +Descent (SGD) (or also sometimes on-line gradient descent). This is +relatively less common to see because in practice due to vectorized +code optimizations it can be computationally much more efficient to +evaluate the gradient for 100 examples, than the gradient for one +example 100 times. Even though SGD technically refers to using a +single example at a time to evaluate the gradient, you will hear +people use the term SGD even when referring to mini-batch gradient +descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD +for “Batch gradient descent” are rare to see), where it is usually +assumed that mini-batches are used. The size of the mini-batch is a +hyperparameter but it is not very common to cross-validate or bootstrap it. It is +usually based on memory constraints (if any), or set to some value, +e.g. 32, 64 or 128. We use powers of 2 in practice because many +vectorized operation implementations work faster when their inputs are +sized in powers of 2. + +In our notes with SGD we mean stochastic gradient descent with mini-batches. + + +!split +===== Stochastic Gradient Descent ===== + +Stochastic gradient descent (SGD) and variants thereof address some of +the shortcomings of the Gradient descent method discussed above. + +The underlying idea of SGD comes from the observation that the cost +function, which we want to minimize, can almost always be written as a +sum over $n$ data points $\{\mathbf{x}_i\}_{i=1}^n$, +!bt +\[ +C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, +\mathbf{\beta}). +\] +!et + +!split +===== Computation of gradients ===== + +This in turn means that the gradient can be +computed as a sum over $i$-gradients +!bt +\[ +\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}). +\] +!et + +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 +$k=1,\cdots,n/M$. + + + +!split +===== 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 +!bt +\[ +\nabla_{\beta} +C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta +c_i(\mathbf{x}_i, \mathbf{\beta}). +\] +!et + +!split +===== The gradient step ===== + +Thus a gradient descent step now looks like +!bt +\[ +\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) +\] +!et + +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. + +!split +===== Simple example code ===== + +!bc pycod +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 +!ec + +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. + +!split +===== 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. + +!split +===== Slightly different approach ===== + +Another approach is to let the step length $\gamma_j$ depend on the +number of epochs in such a way that it becomes very small after a +reasonable time such that we do not move at all. Such approaches are +also called scaling. There are many such ways to "scale the learning +rate":"https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1" +and "discussions here":"https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf". See +also +URL:"https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1" +for a discussion of different scaling functions for the learning rate. + +!split +===== Time decay rate ===== + +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. + +!bc pycod +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)) +!ec + + + + + + +!split +===== Code with a Number of Minibatches which varies ===== + +In the code here we vary the number of mini-batches. +!bc pycode +# Importing various packages +from math import exp, sqrt +from random import random, seed +import numpy as np +import matplotlib.pyplot as plt + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +theta = np.random.randn(2,1) +eta = 1.0/np.max(EigValues) +Niterations = 1000 + + +for iter in range(Niterations): + gradients = 2.0/n*X.T @ ((X @ theta)-y) + theta -= eta*gradients +print("theta from own gd") +print(theta) + +xnew = np.array([[0],[2]]) +Xnew = np.c_[np.ones((2,1)), xnew] +ypredict = Xnew.dot(theta) +ypredict2 = Xnew.dot(theta_linreg) + +n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +t0, t1 = 5, 50 + +def learning_schedule(t): + return t0/(t+t1) + +theta = np.random.randn(2,1) + +for epoch in range(n_epochs): +# Can you figure out a better way of setting up the contributions to each batch? + for i in range(m): + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi) + eta = learning_schedule(epoch*m+i) + theta = theta - eta*gradients +print("theta from own sdg") +print(theta) + +plt.plot(xnew, ypredict, "r-") +plt.plot(xnew, ypredict2, "b-") +plt.plot(x, y ,'ro') +plt.axis([0,2.0,0, 15.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Random numbers ') +plt.show() + +!ec + + + +!split +===== Replace or not ===== + +In the above code, we have use replacement in setting up the +mini-batches. The discussion +"here":"https://sebastianraschka.com/faq/docs/sgd-methods.html" may be +useful. + + +!split +===== Momentum based GD ===== + +The stochastic gradient descent (SGD) is almost always used with a +*momentum* or inertia term that serves as a memory of the direction we +are moving in parameter space. This is typically implemented as +follows + +!bt +\begin{align} +\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\ +\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}, +\end{align} +!et + +where we have introduced a momentum parameter $\gamma$, with +$0\le\gamma\le 1$, and for brevity we dropped the explicit notation to +indicate the gradient is to be taken over a different mini-batch at +each step. We call this algorithm gradient descent with momentum +(GDM). From these equations, it is clear that $\mathbf{v}_t$ is a +running average of recently encountered gradients and +$(1-\gamma)^{-1}$ sets the characteristic time scale for the memory +used in the averaging procedure. Consistent with this, when +$\gamma=0$, this just reduces down to ordinary SGD as discussed +earlier. An equivalent way of writing the updates is + +!bt +\[ +\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t), +\] +!et +where we have defined $\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}$. + +!split +===== More on momentum based approaches ===== + +Let us try to get more intuition from these equations. It is helpful +to consider a simple physical analogy with a particle of mass $m$ +moving in a viscous medium with drag coefficient $\mu$ and potential +$E(\mathbf{w})$. If we denote the particle's position by $\mathbf{w}$, +then its motion is described by + +!bt +\[ +m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}). +\] +!et + +We can discretize this equation in the usual way to get + +!bt +\[ +m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}). +\] +!et + +Rearranging this equation, we can rewrite this as + +!bt +\[ +\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t. +\] +!et + +!split +===== Momentum parameter ===== + +Notice that this equation is identical to previous one if we identify +the position of the particle, $\mathbf{w}$, with the parameters +$\boldsymbol{\theta}$. This allows us to identify the momentum +parameter and learning rate with the mass of the particle and the +viscous drag as: + +!bt +\[ +\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}. +\] +!et + +Thus, as the name suggests, the momentum parameter is proportional to +the mass of the particle and effectively provides inertia. +Furthermore, in the large viscosity/small learning rate limit, our +memory time scales as $(1-\gamma)^{-1} \approx m/(\mu \Delta t)$. + +Why is momentum useful? SGD momentum helps the gradient descent +algorithm gain speed in directions with persistent but small gradients +even in the presence of stochasticity, while suppressing oscillations +in high-curvature directions. This becomes especially important in +situations where the landscape is shallow and flat in some directions +and narrow and steep in others. It has been argued that first-order +methods (with appropriate initial conditions) can perform comparable +to more expensive second order methods, especially in the context of +complex deep learning models. + +These beneficial properties of momentum can sometimes become even more +pronounced by using a slight modification of the classical momentum +algorithm called Nesterov Accelerated Gradient (NAG). + +In the NAG algorithm, rather than calculating the gradient at the +current parameters, $\nabla_\theta E(\boldsymbol{\theta}_t)$, one +calculates the gradient at the expected value of the parameters given +our current momentum, $\nabla_\theta E(\boldsymbol{\theta}_t +\gamma +\mathbf{v}_{t-1})$. This yields the NAG update rule + +!bt +\begin{align} +\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\ +\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}. +\end{align} +!et + +One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\gamma$. + + +!split +===== Second moment of the gradient ===== + + +In stochastic gradient descent, with and without momentum, we still +have to specify a schedule for tuning the learning rates $\eta_t$ +as a function of time. As discussed in the context of Newton's +method, this presents a number of dilemmas. The learning rate is +limited by the steepest direction which can change depending on the +current position in the landscape. To circumvent this problem, ideally +our algorithm would keep track of curvature and take large steps in +shallow, flat directions and small steps in steep, narrow directions. +Second-order methods accomplish this by calculating or approximating +the Hessian and normalizing the learning rate by the +curvature. However, this is very computationally expensive for +extremely large models. Ideally, we would like to be able to +adaptively change the step size to match the landscape without paying +the steep computational price of calculating or approximating +Hessians. + +Recently, a number of methods have been introduced that accomplish +this by tracking not only the gradient, but also the second moment of +the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and +"ADAM":"https://arxiv.org/abs/1412.6980". + +!split +===== RMS prop ===== + +In RMS prop, in addition to keeping a running average of the first +moment of the gradient, we also keep track of the second moment +denoted by $\mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2]$. The update rule +for RMS prop is given by + +!bt +\begin{align} +\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\ +\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\ +\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber +\end{align} +!et + +where $\beta$ controls the averaging time of the second moment and is +typically taken to be about $\beta=0.9$, $\eta_t$ is a learning rate +typically chosen to be $10^{-3}$, and $\epsilon\sim 10^{-8} $ is a +small regularization constant to prevent divergences. Multiplication +and division by vectors is understood as an element-wise operation. It +is clear from this formula that the learning rate is reduced in +directions where the norm of the gradient is consistently large. This +greatly speeds up the convergence by allowing us to use a larger +learning rate for flat directions. + + +!split +===== "ADAM optimizer":"https://arxiv.org/abs/1412.6980" ===== + +A related algorithm is the ADAM optimizer. In +"ADAM":"https://arxiv.org/abs/1412.6980", we keep a running average of +both the first and second moment of the gradient and use this +information to adaptively change the learning rate for different +parameters. The method isefficient when working with large +problems involving lots data and/or parameters. It is a combination of the +gradient descent with momentum algorithm and the RMSprop algorithm +discussed above. + +In addition to keeping a running average of the first and +second moments of the gradient +(i.e. $\mathbf{m}_t=\mathbb{E}[\mathbf{g}_t]$ and +$\mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t]$, respectively), ADAM +performs an additional bias correction to account for the fact that we +are estimating the first two moments of the gradient using a running +average (denoted by the hats in the update rule below). The update +rule for ADAM is given by (where multiplication and division are once +again understood to be element-wise operations below) + +!bt +\begin{align} +\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\ +\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\ +\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\ +\bm{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\ +\bm{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\ +\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \bm{\mathbf{m}}_t \over \sqrt{\bm{\mathbf{s}}_t} +\epsilon}, \nonumber \\ +\end{align} +!et + +where $\beta_1$ and $\beta_2$ set the memory lifetime of the first and +second moment and are typically taken to be $0.9$ and $0.99$ +respectively, and $\eta$ and $\epsilon$ are identical to RMSprop. + +Like in RMSprop, the effective step size of a parameter depends on the +magnitude of its gradient squared. To understand this better, let us +rewrite this expression in terms of the variance +$\boldsymbol{\sigma}_t^2 = \bm{\mathbf{s}}_t - +(\bm{\mathbf{m}}_t)^2$. Consider a single parameter $\theta_t$. The +update rule for this parameter is given by + +!bt +\[ +\Delta \theta_{t+1}= -\eta_t { \bm{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}. +\] +!et + +!split +===== Algorithms and codes for Adagrad, RMSprop and Adam ===== + +The algorithms we have implemented are well described in the text by "Goodfellow, Bengio and Courville, chapter 8":"https://www.deeplearningbook.org/contents/optimization.html". + +The codes which implement these algorithms are discussed after our presentation of automatic differentiation. + + +!split +===== Practical tips ===== + +* _Randomize the data when making mini-batches_. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented. + +* _Transform your inputs_. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case. + +* _Monitor the out-of-sample performance._ Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings. + +* _Adaptive optimization methods don't always have good generalization._ Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications. + +Geron's text, see chapter 11, has several interesting discussions. + + + +!split +===== Automatic differentiation ===== + +"Automatic differentiation (AD)":"https://en.wikipedia.org/wiki/Automatic_differentiation", +also called algorithmic +differentiation or computational differentiation,is a set of +techniques to numerically evaluate the derivative of a function +specified by a computer program. AD exploits the fact that every +computer program, no matter how complicated, executes a sequence of +elementary arithmetic operations (addition, subtraction, +multiplication, division, etc.) and elementary functions (exp, log, +sin, cos, etc.). By applying the chain rule repeatedly to these +operations, derivatives of arbitrary order can be computed +automatically, accurately to working precision, and using at most a +small constant factor more arithmetic operations than the original +program. + +Automatic differentiation is neither: + +* Symbolic differentiation, nor +* Numerical differentiation (the method of finite differences). + +Symbolic differentiation can lead to inefficient code and faces the +difficulty of converting a computer program into a single expression, +while numerical differentiation can introduce round-off errors in the +discretization process and cancellation + + + +Python has tools for so-called _automatic differentiation_. +Consider the following example +!bt +\[ +f(x) = \sin\left(2\pi x + x^2\right) +\] +!et +which has the following derivative +!bt +\[ +f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) +\] +!et +Using _autograd_ we have + +!bc pycod +import autograd.numpy as np + +# To do elementwise differentiation: +from autograd import elementwise_grad as egrad + +# To plot: +import matplotlib.pyplot as plt + + +def f(x): + return np.sin(2*np.pi*x + x**2) + +def f_grad_analytic(x): + return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x) + +# Do the comparison: +x = np.linspace(0,1,1000) + +f_grad = egrad(f) + +computed = f_grad(x) +analytic = f_grad_analytic(x) + +plt.title('Derivative computed from Autograd compared with the analytical derivative') +plt.plot(x,computed,label='autograd') +plt.plot(x,analytic,label='analytic') + +plt.xlabel('x') +plt.ylabel('y') +plt.legend() + +plt.show() + +print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic)))) +!ec + +!split +===== Using autograd ===== + +Here we +experiment with what kind of functions Autograd is capable +of finding the gradient of. The following Python functions are just +meant to illustrate what Autograd can do, but please feel free to +experiment with other, possibly more complicated, functions as well. + +!bc pycod +import autograd.numpy as np +from autograd import grad + +def f1(x): + return x**3 + 1 + +f1_grad = grad(f1) + +# Remember to send in float as argument to the computed gradient from Autograd! +a = 1.0 + +# See the evaluated gradient at a using autograd: +print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) + +# Compare with the analytical derivative, that is f1'(x) = 3*x**2 +grad_analytical = 3*a**2 +print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical)) +!ec + + +!split +===== Autograd with more complicated functions ===== + +To differentiate with respect to two (or more) arguments of a Python +function, Autograd need to know at which variable the function if +being differentiated with respect to. + +!bc pycod +import autograd.numpy as np +from autograd import grad +def f2(x1,x2): + return 3*x1**3 + x2*(x1 - 5) + 1 + +# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1 +f2_grad_x1 = grad(f2,0) + +# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad +f2_grad_x2 = grad(f2,1) + +x1 = 1.0 +x2 = 3.0 + +print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) +print("-"*30) + +# Compare with the analytical derivatives: + +# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2: +f2_grad_x1_analytical = 9*x1**2 + x2 + +# Derivative of f2 w.r.t x2 is: x1 - 5: +f2_grad_x2_analytical = x1 - 5 + +# See the evaluated derivations: +print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) +print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) + +print() + +print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) +!ec + +Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. + + +!split +===== More complicated functions using the elements of their arguments directly ===== + +!bc pycod +import autograd.numpy as np +from autograd import grad +def f3(x): # Assumes x is an array of length 5 or higher + return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2 + +f3_grad = grad(f3) + +x = np.linspace(0,4,5) + +# Print the computed gradient: +print("The computed gradient of f3 is: ", f3_grad(x)) + +# The analytical gradient is: (2, 3, 5, 7, 22*x[4]) +f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]]) + +# Print the analytical gradient: +print("The analytical gradient of f3 is: ", f3_grad_analytical) +!ec + +Note that in this case, when sending an array as input argument, the +output from Autograd is another array. This is the true gradient of +the function, as opposed to the function in the previous example. By +using arrays to represent the variables, the output from Autograd +might be easier to work with, as the output is closer to what one +could expect form a gradient-evaluting function. + +!split +===== Functions using mathematical functions from Numpy ===== + +!bc pycod +import autograd.numpy as np +from autograd import grad +def f4(x): + return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x) + +f4_grad = grad(f4) + +x = 2.7 + +# Print the computed derivative: +print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) + +# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi +f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi + +# Print the analytical gradient: +print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical)) +!ec + + +!split +===== More autograd ===== + +!bc pycod +import autograd.numpy as np +from autograd import grad +def f5(x): + if x >= 0: + return x**2 + else: + return -3*x + 1 + +f5_grad = grad(f5) + +x = 2.7 + +# Print the computed derivative: +print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) +!ec + + +!split +===== And with loops ===== + +!bc pycod +import autograd.numpy as np +from autograd import grad +def f6_for(x): + val = 0 + for i in range(10): + val = val + x**i + return val + +def f6_while(x): + val = 0 + i = 0 + while i < 10: + val = val + x**i + i = i + 1 + return val + +f6_for_grad = grad(f6_for) +f6_while_grad = grad(f6_while) + +x = 0.5 + +# Print the computed derivaties of f6_for and f6_while +print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) +print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x))) +!ec +!bc pycod +import autograd.numpy as np +from autograd import grad +# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9 +# The analytical derivative is: sum(i*x**(i-1)) +f6_grad_analytical = 0 +for i in range(10): + f6_grad_analytical += i*x**(i-1) + +print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) +!ec + +!split +===== Using recursion ===== +!bc pycod +import autograd.numpy as np +from autograd import grad + +def f7(n): # Assume that n is an integer + if n == 1 or n == 0: + return 1 + else: + return n*f7(n-1) + +f7_grad = grad(f7) + +n = 2.0 + +print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) + +# The function f7 is an implementation of the factorial of n. +# By using the product rule, one can find that the derivative is: + +f7_grad_analytical = 0 +for i in range(int(n)-1): + tmp = 1 + for k in range(int(n)-1): + if k != i: + tmp *= (n - k) + f7_grad_analytical += tmp + +print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical)) + +!ec +Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. + +!split +===== Unsupported functions ===== +Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd. + +Assigning a value to the variable being differentiated with respect to +!bc pycod +import autograd.numpy as np +from autograd import grad +def f8(x): # Assume x is an array + x[2] = 3 + return x*2 + +#f8_grad = grad(f8) + +#x = 8.4 + +#print("The derivative of f8 is:",f8_grad(x)) +!ec +Here, running this code, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. + +!split +===== The syntax a.dot(b) when finding the dot product ===== +!bc pycod +import autograd.numpy as np +from autograd import grad +def f9(a): # Assume a is an array with 2 elements + b = np.array([1.0,2.0]) + return a.dot(b) + +#f9_grad = grad(f9) + +#x = np.array([1.0,0.0]) + +#print("The derivative of f9 is:",f9_grad(x)) +!ec + +Here we are told that the 'dot' function does not belong to Autograd's +version of a Numpy array. To overcome this, an alternative syntax +which also computed the dot product can be used: + +!bc pycod +import autograd.numpy as np +from autograd import grad +def f9_alternative(x): # Assume a is an array with 2 elements + b = np.array([1.0,2.0]) + return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2 + +f9_alternative_grad = grad(f9_alternative) + +x = np.array([3.0,0.0]) + +print("The gradient of f9 is:",f9_alternative_grad(x)) + +# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively +# w.r.t x is (b_1, b_2). +!ec + + +!split +===== Using Autograd with OLS ===== + +We conclude the part on optmization by showing how we can make codes +for linear regression and logistic regression using _autograd_. The +first example shows results with ordinary leats squares. + +!bc pycod +# Using Autograd to calculate gradients for OLS +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +def CostOLS(beta): + return (1.0/n)*np.sum((y-X @ beta)**2) + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +theta = np.random.randn(2,1) +eta = 1.0/np.max(EigValues) +Niterations = 1000 +# define the gradient +training_gradient = grad(CostOLS) + +for iter in range(Niterations): + gradients = training_gradient(theta) + theta -= eta*gradients +print("theta from own gd") +print(theta) + +xnew = np.array([[0],[2]]) +Xnew = np.c_[np.ones((2,1)), xnew] +ypredict = Xnew.dot(theta) +ypredict2 = Xnew.dot(theta_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'Random numbers ') +plt.show() + +!ec + + +!split +===== Same code but now with momentum gradient descent ===== +!bc pycod +# Using Autograd to calculate gradients for OLS +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +def CostOLS(beta): + return (1.0/n)*np.sum((y-X @ beta)**2) + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x#+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +theta = np.random.randn(2,1) +eta = 1.0/np.max(EigValues) +Niterations = 30 + +# define the gradient +training_gradient = grad(CostOLS) + +for iter in range(Niterations): + gradients = training_gradient(theta) + theta -= eta*gradients + print(iter,gradients[0],gradients[1]) +print("theta from own gd") +print(theta) + +# Now improve with momentum gradient descent +change = 0.0 +delta_momentum = 0.3 +for iter in range(Niterations): + # calculate gradient + gradients = training_gradient(theta) + # calculate update + new_change = eta*gradients+delta_momentum*change + # take a step + theta -= new_change + # save the change + change = new_change + print(iter,gradients[0],gradients[1]) +print("theta from own gd wth momentum") +print(theta) + +!ec + +!split +===== But none of these can compete with Newton's method ===== + +!bc pycod +# Using Newton's method +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +def CostOLS(beta): + return (1.0/n)*np.sum((y-X @ beta)**2) + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +beta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(beta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +# Note that here the Hessian does not depend on the parameters beta +invH = np.linalg.pinv(H) +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +beta = np.random.randn(2,1) +Niterations = 5 + +# define the gradient +training_gradient = grad(CostOLS) + +for iter in range(Niterations): + gradients = training_gradient(beta) + beta -= invH @ gradients + print(iter,gradients[0],gradients[1]) +print("beta from own Newton code") +print(beta) +!ec + + +!split +===== Including Stochastic Gradient Descent with Autograd ===== +In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using _autograd_. + +!bc pycod +# Using Autograd to calculate gradients using SGD +# OLS example +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +# Note change from previous example +def CostOLS(y,X,theta): + return np.sum((y-X @ theta)**2) + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +theta = np.random.randn(2,1) +eta = 1.0/np.max(EigValues) +Niterations = 1000 + +# Note that we request the derivative wrt third argument (theta, 2 here) +training_gradient = grad(CostOLS,2) + +for iter in range(Niterations): + gradients = (1.0/n)*training_gradient(y, X, theta) + theta -= eta*gradients +print("theta from own gd") +print(theta) + +xnew = np.array([[0],[2]]) +Xnew = np.c_[np.ones((2,1)), xnew] +ypredict = Xnew.dot(theta) +ypredict2 = Xnew.dot(theta_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'Random numbers ') +plt.show() + +n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +t0, t1 = 5, 50 +def learning_schedule(t): + return t0/(t+t1) + +theta = np.random.randn(2,1) + +for epoch in range(n_epochs): +# Can you figure out a better way of setting up the contributions to each batch? + for i in range(m): + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (1.0/M)*training_gradient(yi, xi, theta) + eta = learning_schedule(epoch*m+i) + theta = theta - eta*gradients +print("theta from own sdg") +print(theta) + + +!ec + + +!split +===== Same code but now with momentum gradient descent ===== +!bc pycod +# Using Autograd to calculate gradients using SGD +# OLS example +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +# Note change from previous example +def CostOLS(y,X,theta): + return np.sum((y-X @ theta)**2) + +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) +# Hessian matrix +H = (2.0/n)* XT_X +EigValues, EigVectors = np.linalg.eig(H) +print(f"Eigenvalues of Hessian Matrix:{EigValues}") + +theta = np.random.randn(2,1) +eta = 1.0/np.max(EigValues) +Niterations = 100 + +# Note that we request the derivative wrt third argument (theta, 2 here) +training_gradient = grad(CostOLS,2) + +for iter in range(Niterations): + gradients = (1.0/n)*training_gradient(y, X, theta) + theta -= eta*gradients +print("theta from own gd") +print(theta) + + +n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +t0, t1 = 5, 50 +def learning_schedule(t): + return t0/(t+t1) + +theta = np.random.randn(2,1) + +change = 0.0 +delta_momentum = 0.3 + +for epoch in range(n_epochs): + for i in range(m): + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (1.0/M)*training_gradient(yi, xi, theta) + eta = learning_schedule(epoch*m+i) + # calculate update + new_change = eta*gradients+delta_momentum*change + # take a step + theta -= new_change + # save the change + change = new_change +print("theta from own sdg with momentum") +print(theta) +!ec + + +!split +===== Similar (second order function now) problem but now with AdaGrad ===== +!bc pycod +# Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent +# OLS example +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +# Note change from previous example +def CostOLS(y,X,theta): + return np.sum((y-X @ theta)**2) + +n = 1000 +x = np.random.rand(n,1) +y = 2.0+3*x +4*x*x + +X = np.c_[np.ones((n,1)), x, x*x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) + + +# Note that we request the derivative wrt third argument (theta, 2 here) +training_gradient = grad(CostOLS,2) +# Define parameters for Stochastic Gradient Descent +n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +# Guess for unknown parameters theta +theta = np.random.randn(3,1) + +# Value for learning rate +eta = 0.01 +# Including AdaGrad parameter to avoid possible division by zero +delta = 1e-8 +for epoch in range(n_epochs): + Giter = 0.0 + for i in range(m): + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (1.0/M)*training_gradient(yi, xi, theta) + Giter += gradients*gradients + update = gradients*eta/(delta+np.sqrt(Giter)) + theta -= update +print("theta from own AdaGrad") +print(theta) + + +!ec + +Running this code we note an almost perfect agreement with the results from matrix inversion. + +!split +===== RMSprop for adaptive learning rate with Stochastic Gradient Descent ===== +!bc pycod +# Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent +# OLS example +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +# Note change from previous example +def CostOLS(y,X,theta): + return np.sum((y-X @ theta)**2) + +n = 1000 +x = np.random.rand(n,1) +y = 2.0+3*x +4*x*x# +np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x, x*x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) + + +# Note that we request the derivative wrt third argument (theta, 2 here) +training_gradient = grad(CostOLS,2) +# Define parameters for Stochastic Gradient Descent +n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +# Guess for unknown parameters theta +theta = np.random.randn(3,1) + +# Value for learning rate +eta = 0.01 +# Value for parameter rho +rho = 0.99 +# Including AdaGrad parameter to avoid possible division by zero +delta = 1e-8 +for epoch in range(n_epochs): + Giter = 0.0 + for i in range(m): + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (1.0/M)*training_gradient(yi, xi, theta) + # Accumulated gradient + # Scaling with rho the new and the previous results + Giter = (rho*Giter+(1-rho)*gradients*gradients) + # Taking the diagonal only and inverting + update = gradients*eta/(delta+np.sqrt(Giter)) + # Hadamard product + theta -= update +print("theta from own RMSprop") +print(theta) +!ec + +!split +===== And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf" ===== + +!bc pycod +# Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent +# OLS example +from random import random, seed +import numpy as np +import autograd.numpy as np +import matplotlib.pyplot as plt +from autograd import grad + +# Note change from previous example +def CostOLS(y,X,theta): + return np.sum((y-X @ theta)**2) + +n = 1000 +x = np.random.rand(n,1) +y = 2.0+3*x +4*x*x# +np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x, x*x] +XT_X = X.T @ X +theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) +print("Own inversion") +print(theta_linreg) + + +# Note that we request the derivative wrt third argument (theta, 2 here) +training_gradient = grad(CostOLS,2) +# Define parameters for Stochastic Gradient Descent +n_epochs = 50 +M = 5 #size of each minibatch +m = int(n/M) #number of minibatches +# Guess for unknown parameters theta +theta = np.random.randn(3,1) + +# Value for learning rate +eta = 0.01 +# Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980 +beta1 = 0.9 +beta2 = 0.999 +# Including AdaGrad parameter to avoid possible division by zero +delta = 1e-7 +iter = 0 +for epoch in range(n_epochs): + first_moment = 0.0 + second_moment = 0.0 + iter += 1 + for i in range(m): + random_index = M*np.random.randint(m) + xi = X[random_index:random_index+M] + yi = y[random_index:random_index+M] + gradients = (1.0/M)*training_gradient(yi, xi, theta) + # Computing moments first + first_moment = beta1*first_moment + (1-beta1)*gradients + second_moment = beta2*second_moment+(1-beta2)*gradients*gradients + first_term = first_moment/(1.0-beta1**iter) + second_term = second_moment/(1.0-beta2**iter) + # Scaling with rho the new and the previous results + update = eta*first_term/(np.sqrt(second_term)+delta) + theta -= update +print("theta from own ADAM") +print(theta) +!ec + +!split +===== And Logistic Regression ===== + +!bc pycod +import autograd.numpy as np +from autograd import grad + +def sigmoid(x): + return 0.5 * (np.tanh(x / 2.) + 1) + +def logistic_predictions(weights, inputs): + # Outputs probability of a label being true according to logistic model. + return sigmoid(np.dot(inputs, weights)) + +def training_loss(weights): + # Training loss is the negative log-likelihood of the training labels. + preds = logistic_predictions(weights, inputs) + label_probabilities = preds * targets + (1 - preds) * (1 - targets) + return -np.sum(np.log(label_probabilities)) + +# Build a toy dataset. +inputs = np.array([[0.52, 1.12, 0.77], + [0.88, -1.08, 0.15], + [0.52, 0.06, -1.30], + [0.74, -2.49, 1.39]]) +targets = np.array([True, True, False, True]) + +# Define a function that returns gradients of training loss using Autograd. +training_gradient_fun = grad(training_loss) + +# Optimize weights using gradient descent. +weights = np.array([0.0, 0.0, 0.0]) +print("Initial loss:", training_loss(weights)) +for i in range(100): + weights -= training_gradient_fun(weights) * 0.01 + +print("Trained loss:", training_loss(weights)) +!ec + + +!split +===== Introducing "JAX":"https://jax.readthedocs.io/en/latest/" ===== + +Presently, instead of using _autograd_, we recommend using "JAX":"https://jax.readthedocs.io/en/latest/" + +_JAX_ is Autograd and "XLA (Accelerated Linear Algebra))":"https://www.tensorflow.org/xla", +brought together for high-performance numerical computing and machine learning research. +It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more. + +Here's a simple example on how you can use _JAX_ to compute the derivate of the logistic function. + +!bc pycod +import jax.numpy as jnp +from jax import grad, jit, vmap + +def sum_logistic(x): + return jnp.sum(1.0 / (1.0 + jnp.exp(-x))) + +x_small = jnp.arange(3.) +derivative_fn = grad(sum_logistic) +print(derivative_fn(x_small)) + +!ec + + + + + diff --git a/doc/src/week39/week39.do.txt b/doc/src/week39/week39.do.txt index 2f3e51097..7b190e701 100644 --- a/doc/src/week39/week39.do.txt +++ b/doc/src/week39/week39.do.txt @@ -38,2674 +38,1393 @@ DATE: Week 39 !split -===== Lecture Monday September 22, Optimization, the central part of any Machine Learning algortithm ===== +===== Lecture Monday September 22, Resampling techniques and Logistic regression ===== -The first few slides here are a repetition from last week. +TITLE: Week 38: Statistical analysis, bias-variance tradeoff and resampling methods +AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics and Center for Computing in Science Education, University of Oslo, Norway +DATE: September 15-19, 2025 -Almost every problem in machine learning and data science starts with -a dataset $X$, a model $g(\beta)$, which is a function of the -parameters $\beta$ and a cost function $C(X, g(\beta))$ that allows -us to judge how well the model $g(\beta)$ explains the observations -$X$. The model is fit by finding the values of $\beta$ that minimize -the cost function. Ideally we would be able to solve for $\beta$ -analytically, however this is not possible in general and we must use -some approximative/numerical method to compute the minimum. !split -===== Revisiting our Logistic Regression case ===== +===== Plans for week 38, lecture Monday September 15 ===== -In our discussion on Logistic Regression we studied the -case of -two classes, with $y_i$ either -$0$ or $1$. Furthermore we assumed also that we have only two -parameters $\beta$ in our fitting, that is we -defined probabilities +!bblock Material for the lecture on Monday September 15 +o Statistical interpretation of OLS and various expectation values +o Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff +!eblock + + + + + + + +!split +===== Readings and Videos ===== +!bblock +o Raschka et al, pages 175-192 +o Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See URL:"https://link.springer.com/book/10.1007/978-0-387-84858-7". +o "Video on bias-variance tradeoff":"https://www.youtube.com/watch?v=EuBBz3bI-aA" +o "Video on Bootstrapping":"https://www.youtube.com/watch?v=Xz0x-8-cgaQ" +o "Video on cross validation":"https://www.youtube.com/watch?v=fSytzGwwBVw" +For the lab session, the following video on cross validation (from 2024), could be helpful, see URL:"https://www.youtube.com/watch?v=T9jjWsmsd1o" +!eblock + + +!split +===== Linking the regression analysis with a statistical interpretation ===== + +We will now couple the discussions of ordinary least squares, Ridge +and Lasso regression with a statistical interpretation, that is we +move from a linear algebra analysis to a statistical analysis. In +particular, we will focus on what the regularization terms can result +in. We will amongst other things show that the regularization +parameter can reduce considerably the variance of the parameters +$\theta$. + + +On of the advantages of doing linear regression is that we actually end up with +analytical expressions for several statistical quantities. +Standard least squares and Ridge regression allow us to +derive quantities like the variance and other expectation values in a +rather straightforward way. + + +It is assumed that $\varepsilon_i +\sim \mathcal{N}(0, \sigma^2)$ and the $\varepsilon_{i}$ are +independent, i.e.: !bt -\begin{align*} -p(y_i=1|x_i,\bm{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ -p(y_i=0|x_i,\bm{\beta}) &= 1 - p(y_i=1|x_i,\bm{\beta}), +\begin{align*} +\mbox{Cov}(\varepsilon_{i_1}, +\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if} +& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right. +\end{align*} +!et +The randomness of $\varepsilon_i$ implies that +$\mathbf{y}_i$ is also a random variable. In particular, +$\mathbf{y}_i$ is normally distributed, because $\varepsilon_i \sim +\mathcal{N}(0, \sigma^2)$ and $\mathbf{X}_{i,\ast} \, \bm{\theta}$ is a +non-random scalar. To specify the parameters of the distribution of +$\mathbf{y}_i$ we need to calculate its first two moments. + +Recall that $\bm{X}$ is a matrix of dimensionality $n\times p$. The +notation above $\mathbf{X}_{i,\ast}$ means that we are looking at the +row number $i$ and perform a sum over all values $p$. + + +!split +===== Assumptions made ===== + +The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off) +that there exists a function $f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim \mathcal{N}(0, \sigma^2)$ +which describe our data +!bt +\[ +\bm{y} = f(\bm{x})+\bm{\varepsilon} +\] +!et + +We approximate this function with our model from the solution of the linear regression equations, that is our +function $f$ is approximated by $\bm{\tilde{y}}$ where we want to minimize $(\bm{y}-\bm{\tilde{y}})^2$, our MSE, with +!bt +\[ +\bm{\tilde{y}} = \bm{X}\bm{\theta}. +\] +!et + +!split +===== Expectation value and variance ===== + +We can calculate the expectation value of $\bm{y}$ for a given element $i$ +!bt +\begin{align*} +\mathbb{E}(y_i) & = +\mathbb{E}(\mathbf{X}_{i, \ast} \, \bm{\theta}) + \mathbb{E}(\varepsilon_i) +\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \theta, +\end{align*} +!et +while +its variance is +!bt +\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i +- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - +[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, +\theta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \bm{\theta})^2 \\ & += \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \bm{\theta})^2 + 2 \varepsilon_i +\mathbf{X}_{i, \ast} \, \bm{\theta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, +\ast} \, \theta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \bm{\theta})^2 + 2 +\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \bm{\theta} + +\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \bm{\theta})^2 +\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, +\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. \end{align*} !et -where $\bm{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$. +Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \bm{\theta}, \sigma^2)$, that is $\bm{y}$ follows a normal distribution with +mean value $\bm{X}\bm{\theta}$ and variance $\sigma^2$ (not be confused with the singular values of the SVD). !split -===== The equations to solve ===== +===== Expectation value and variance for $\bm{\theta}$ ===== -Our compact equations used a definition of a vector $\bm{y}$ with $n$ -elements $y_i$, an $n\times p$ matrix $\bm{X}$ which contains the -$x_i$ values and a vector $\bm{p}$ of fitted probabilities -$p(y_i\vert x_i,\bm{\beta})$. We rewrote in a more compact form -the first derivative of the cost function as +With the OLS expressions for the optimal parameters $\bm{\hat{\theta}}$ we can evaluate the expectation value +!bt +\[ +\mathbb{E}(\bm{\hat{\theta}}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\bm{\theta}=\bm{\theta}. +\] +!et +This means that the estimator of the regression parameters is unbiased. + +We can also calculate the variance + +The variance of the optimal value $\bm{\hat{\theta}}$ is +!bt +\begin{eqnarray*} +\mbox{Var}(\bm{\hat{\theta}}) & = & \mathbb{E} \{ [\bm{\theta} - \mathbb{E}(\bm{\theta})] [\bm{\theta} - \mathbb{E}(\bm{\theta})]^{T} \} +\\ +& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\theta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\theta}]^{T} \} +\\ +% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \bm{\theta} \, \bm{\theta}^{T} +% \\ +% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \bm{\theta} \, \bm{\theta}^{T} +% \\ +& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\theta} \, \bm{\theta}^{T} +\\ +& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \bm{\theta} \, \bm{\theta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\theta} \, \bm{\theta}^{T} +% \\ +% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \bm{\theta} \, \bm{\theta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} +% \\ +% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \bm{\theta} \bm{\theta}^T +\\ +& = & \bm{\theta} \, \bm{\theta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\theta} \, \bm{\theta}^{T} +\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, +\end{eqnarray*} +!et + +where we have used that $\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = +\mathbf{X} \, \bm{\theta} \, \bm{\theta}^{T} \, \mathbf{X}^{T} + +\sigma^2 \, \mathbf{I}_{nn}$. From $\mbox{Var}(\bm{\theta}) = \sigma^2 +\, (\mathbf{X}^{T} \mathbf{X})^{-1}$, one obtains an estimate of the +variance of the estimate of the $j$-th regression coefficient: +$\bm{\sigma}^2 (\bm{\theta}_j ) = \bm{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} $. This may be used to +construct a confidence interval for the estimates. + + +In a similar way, we can obtain analytical expressions for say the +expectation values of the parameters $\bm{\theta}$ and their variance +when we employ Ridge regression, allowing us again to define a confidence interval. + +It is rather straightforward to show that +!bt +\[ +\mathbb{E} \big[ \bm{\theta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\bm{\theta}^{\mathrm{OLS}}. +\] +!et +We see clearly that +$\mathbb{E} \big[ \bm{\theta}^{\mathrm{Ridge}} \big] \not= \bm{\theta}^{\mathrm{OLS}}$ for any $\lambda > 0$. We say then that the ridge estimator is biased. + +We can also compute the variance as !bt \[ -\frac{\partial \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}} = -\bm{X}^T\left(\bm{y}-\bm{p}\right). +\mbox{Var}[\bm{\theta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, \] !et +and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of Ridge parameters $\bm{\theta}$ goes to zero. -If we in addition define a diagonal matrix $\bm{W}$ with elements -$p(y_i\vert x_i,\bm{\beta})(1-p(y_i\vert x_i,\bm{\beta})$, we can obtain a compact expression of the second derivative as +With this, we can compute the difference !bt \[ -\frac{\partial^2 \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}\partial \bm{\beta}^T} = \bm{X}^T\bm{W}\bm{X}. +\mbox{Var}[\bm{\theta}^{\mathrm{OLS}}]-\mbox{Var}(\bm{\theta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. \] !et -This defines what is called the Hessian matrix. - - - +The difference is non-negative definite since each component of the +matrix product is non-negative definite. +This means the variance we obtain with the standard OLS will always for $\lambda > 0$ be larger than the variance of $\bm{\theta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. !split -===== Solving using Newton-Raphson's method ===== +===== Deriving OLS from a probability distribution ===== -If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. +Our basic assumption when we derived the OLS equations was to assume +that our output is determined by a given continuous function +$f(\bm{x})$ and a random noise $\bm{\epsilon}$ given by the normal +distribution with zero mean value and an undetermined variance +$\sigma^2$. -Our iterative scheme is then given by +We found above that the outputs $\bm{y}$ have a mean value given by +$\bm{X}\hat{\bm{\theta}}$ and variance $\sigma^2$. Since the entries to +the design matrix are not stochastic variables, we can assume that the +probability distribution of our targets is also a normal distribution +but now with mean value $\bm{X}\hat{\bm{\theta}}$. This means that a +single output $y_i$ is given by the Gaussian distribution !bt \[ -\bm{\beta}^{\mathrm{new}} = \bm{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}\partial \bm{\beta}^T}\right)^{-1}_{\bm{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\bm{\beta})}{\partial \bm{\beta}}\right)_{\bm{\beta}^{\mathrm{old}}}, -\] -!et -or in matrix form as - -!bt -\[ -\bm{\beta}^{\mathrm{new}} = \bm{\beta}^{\mathrm{old}}-\left(\bm{X}^T\bm{W}\bm{X} \right)^{-1}\times \left(-\bm{X}^T(\bm{y}-\bm{p}) \right)_{\bm{\beta}^{\mathrm{old}}}. -\] -!et -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. - - -!split -===== Brief reminder on Newton-Raphson's method ===== - -Let us quickly 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 requires the evaluation of both the -function $f$ and its derivative $f'$ at arbitrary points. -If you can only calculate the derivative -numerically and/or your function is not of the smooth type, we -normally discourage the use of this method. - -!split -===== 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 - - -!bt -\[ - f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots. - \label{eq:taylornr} -\] -!et - -For small enough values of the function and for well-behaved -functions, the terms beyond linear are unimportant, hence we obtain - - -!bt -\[ - f(x)+(s-x)f'(x)\approx 0, -\] -!et -yielding -!bt -\[ - s\approx x-\frac{f(x)}{f'(x)}. -\] -!et - -Having in mind an iterative procedure, it is natural to start iterating with -!bt -\[ - x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. +y_i\sim \mathcal{N}(\bm{X}_{i,*}\bm{\theta}, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\bm{X}_{i,*}\bm{\theta})^2}{2\sigma^2}\right]}. \] !et !split -===== Simple geometric interpretation ===== +===== Independent and Identically Distributed (iid) ===== -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 - - -!split -===== 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 +We assume now that the various $y_i$ values are stochastically distributed according to the above Gaussian distribution. +We define this distribution as !bt \[ - \begin{array}{cc} f_1(x_1,x_2) &=0\\ - f_2(x_1,x_2) &=0,\end{array} +p(y_i, \bm{X}\vert\bm{\theta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\bm{X}_{i,*}\bm{\theta})^2}{2\sigma^2}\right]}, \] !et -which we Taylor expand to obtain +which reads as finding the likelihood of an event $y_i$ with the input variables $\bm{X}$ given the parameters (to be determined) $\bm{\theta}$. + +Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event $\bm{y}$ as the product of the single events, that is we have !bt \[ - \begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1 - \partial f_1/\partial x_1+h_2 - \partial f_1/\partial x_2+\dots\\ - 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1 - \partial f_2/\partial x_1+h_2 - \partial f_2/\partial x_2+\dots - \end{array}. -\] -!et -Defining the Jacobian matrix ${\bf \bm{J}}$ we have -!bt -\[ - {\bf \bm{J}}=\left( \begin{array}{cc} - \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\ - \partial f_2/\partial x_1 &\partial f_2/\partial x_2 - \end{array} \right), -\] -!et -we can rephrase Newton's method as -!bt -\[ -\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)= -\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+ -\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right), -\] -!et -where we have defined -!bt -\[ - \left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)= - -{\bf \bm{J}}^{-1} - \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right). -\] -!et -We need thus to compute the inverse of the Jacobian matrix and it -is to understand that difficulties may -arise in case ${\bf \bm{J}}$ 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. - - - -!split -===== 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 $-\nabla F(\mathbf{x})$. - -It can be shown that if -!bt -\[ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), -\] -!et -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. - -!split -===== More on Steepest descent ===== - -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 - -!bt -\[ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0. +p(\bm{y},\bm{X}\vert\bm{\theta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\bm{X}_{i,*}\bm{\theta})^2}{2\sigma^2}\right]}=\prod_{i=0}^{n-1}p(y_i,\bm{X}\vert\bm{\theta}). \] !et -The parameter $\gamma_k$ is often referred to as the step length or -the learning rate within the context of Machine Learning. - -!split -===== The ideal ===== - -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: - -In machine learing we are often faced with non-convex high dimensional -cost functions with many local minima. Since GD is deterministic we -will get stuck in a local minimum, if the method converges, unless we -have a very good intial guess. This also implies that the scheme is -sensitive to the chosen initial condition. - -Note that the gradient is a function of $\mathbf{x} = -(x_1,\cdots,x_n)$ which makes it expensive to compute numerically. - - -!split -===== The sensitiveness of the gradient descent ===== - -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. - -Many of these shortcomings can be alleviated by introducing -randomness. One such method is that of Stochastic Gradient Descent -(SGD), see below. - - -!split -===== Convex functions ===== - -Ideally we want our cost/loss function to be convex(concave). - -First we give the definition of a convex set: A set $C$ in -$\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and -all $t \in (0,1)$ , the point $(1 − t)x + ty$ also belongs to -C. Geometrically this means that every point on the line segment -connecting $x$ and $y$ is in $C$ as discussed below. - -The convex subsets of $\mathbb{R}$ are the intervals of -$\mathbb{R}$. Examples of convex sets of $\mathbb{R}^2$ are the -regular polygons (triangles, rectangles, pentagons, etc...). - -!split -===== 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. - -!split -===== Conditions on convex functions ===== - -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":"http://stanford.edu/boyd/cvxbook/". - -!bblock First order condition -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. -!eblock - -!bblock Second order condition -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. -!eblock - -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. - -!split -===== More on convex functions ===== - -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: - -!bblock Any minimum is global for convex functions -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. -!eblock - -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. - -!split -===== Some simple problems ===== - -o 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 $\lambda \in [0,1]$ $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$. - -o Using the second order condition show that the following functions are convex on the specified domain. - * $f(x) = e^x$ is convex for $x \in \mathbb{R}$. - * $g(x) = -\ln(x)$ is convex for $x \in (0,\infty)$. -o 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. - -o A norm is any function that satisfy the following properties - * $f(\alpha x) = |\alpha| f(x)$ for all $\alpha \in \mathbb{R}$. - * $f(x+y) \leq f(x) + f(y)$ - * $f(x) \leq 0$ for all $x \in \mathbb{R}^n$ with equality if and only if $x = 0$ - -Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this). - - - - -!split -===== Standard steepest descent ===== - - -Before we proceed, we would like to discuss the approach called the -_standard Steepest descent_ (different from the above steepest descent discussion), which again leads to us having to be able -to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). - -"The success of the CG method":"https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf" -for finding solutions of non-linear problems is based on the theory -of conjugate gradients for linear systems of equations. It belongs to -the class of iterative methods for solving problems from linear -algebra of the type -!bt -\begin{equation*} -\bm{A}\bm{x} = \bm{b}. -\end{equation*} -!et - -In the iterative process we end up with a problem like - -!bt -\begin{equation*} - \bm{r}= \bm{b}-\bm{A}\bm{x}, -\end{equation*} -!et -where $\bm{r}$ is the so-called residual or error in the iterative process. - -When we have found the exact solution, $\bm{r}=0$. - -!split -===== Gradient method ===== - -The residual is zero when we reach the minimum of the quadratic equation -!bt -\begin{equation*} - P(\bm{x})=\frac{1}{2}\bm{x}^T\bm{A}\bm{x} - \bm{x}^T\bm{b}, -\end{equation*} -!et - -with the constraint that the matrix $\bm{A}$ is positive definite and -symmetric. This defines also the Hessian and we want it to be positive definite. - - -!split -===== Steepest descent method ===== - -We denote the initial guess for $\bm{x}$ as $\bm{x}_0$. -We can assume without loss of generality that -!bt -\begin{equation*} -\bm{x}_0=0, -\end{equation*} -!et -or consider the system -!bt -\begin{equation*} -\bm{A}\bm{z} = \bm{b}-\bm{A}\bm{x}_0, -\end{equation*} -!et -instead. - - -!split -===== Steepest descent method ===== -!bblock -One can show that the solution $\bm{x}$ is also the unique minimizer of the quadratic form -!bt -\begin{equation*} - f(\bm{x}) = \frac{1}{2}\bm{x}^T\bm{A}\bm{x} - \bm{x}^T \bm{x} , \quad \bm{x}\in\mathbf{R}^n. -\end{equation*} -!et -This suggests taking the first basis vector $\bm{r}_1$ (see below for definition) -to be the gradient of $f$ at $\bm{x}=\bm{x}_0$, -which equals -!bt -\begin{equation*} -\bm{A}\bm{x}_0-\bm{b}, -\end{equation*} -!et -and -$\bm{x}_0=0$ it is equal $-\bm{b}$. - -!eblock - -!split -===== Final expressions ===== -!bblock -We can compute the residual iteratively as -!bt -\begin{equation*} -\bm{r}_{k+1}=\bm{b}-\bm{A}\bm{x}_{k+1}, - \end{equation*} -!et -which equals -!bt -\begin{equation*} -\bm{b}-\bm{A}(\bm{x}_k+\alpha_k\bm{r}_k), - \end{equation*} -!et -or -!bt -\begin{equation*} -(\bm{b}-\bm{A}\bm{x}_k)-\alpha_k\bm{A}\bm{r}_k, - \end{equation*} -!et -which gives - +We will write this in a more compact form reserving $\bm{D}$ for the domain of events, including the ouputs (targets) and the inputs. That is +in case we have a simple one-dimensional input and output case !bt \[ -\alpha_k = \frac{\bm{r}_k^T\bm{r}_k}{\bm{r}_k^T\bm{A}\bm{r}_k} +\bm{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})]. \] !et -leading to the iterative scheme -!bt -\begin{equation*} -\bm{x}_{k+1}=\bm{x}_k+\alpha_k\bm{r}_{k}, - \end{equation*} -!et -!eblock - - - -!split -===== Steepest descent example ===== - -!bc pycod -import numpy as np -import numpy.linalg as la - -import scipy.optimize as sopt - -import matplotlib.pyplot as pt -from mpl_toolkits.mplot3d import axes3d - -def f(x): - return x[0]**2 + 3.0*x[1]**2 - -def df(x): - return np.array([2*x[0], 6*x[1]]) - -fig = pt.figure() -ax = fig.add_subplot(projection = '3d') - -xmesh, ymesh = np.mgrid[-3:3:50j,-3:3:50j] -fmesh = f(np.array([xmesh, ymesh])) -ax.plot_surface(xmesh, ymesh, fmesh) -!ec -And then as countor plot -!bc pycod -pt.axis("equal") -pt.contour(xmesh, ymesh, fmesh) -guesses = [np.array([2, 2./5])] -!ec -Find guesses -!bc pycod -x = guesses[-1] -s = -df(x) -!ec -Run it! -!bc pycod -def f1d(alpha): - return f(x + alpha*s) - -alpha_opt = sopt.golden(f1d) -next_guess = x + alpha_opt * s -guesses.append(next_guess) -print(next_guess) -!ec -What happened? -!bc pycod -pt.axis("equal") -pt.contour(xmesh, ymesh, fmesh, 50) -it_array = np.array(guesses) -pt.plot(it_array.T[0], it_array.T[1], "x-") -!ec - -Note that we did only one iteration here. We can easily add more using our previous guesses. - -!split -===== Conjugate gradient method ===== -!bblock -In the CG method we define so-called conjugate directions and two vectors -$\bm{s}$ and $\bm{t}$ -are said to be -conjugate if -!bt -\begin{equation*} -\bm{s}^T\bm{A}\bm{t}= 0. -\end{equation*} -!et -The philosophy of the CG method is to perform searches in various conjugate directions -of our vectors $\bm{x}_i$ obeying the above criterion, namely -!bt -\begin{equation*} -\bm{x}_i^T\bm{A}\bm{x}_j= 0. -\end{equation*} -!et -Two vectors are conjugate if they are orthogonal with respect to -this inner product. Being conjugate is a symmetric relation: if $\bm{s}$ is conjugate to $\bm{t}$, then $\bm{t}$ is conjugate to $\bm{s}$. -!eblock - -!split -===== Conjugate gradient method ===== -!bblock -An example is given by the eigenvectors of the matrix -!bt -\begin{equation*} -\bm{v}_i^T\bm{A}\bm{v}_j= \lambda\bm{v}_i^T\bm{v}_j, -\end{equation*} -!et -which is zero unless $i=j$. -!eblock - - -!split -===== Conjugate gradient method ===== -!bblock -Assume now that we have a symmetric positive-definite matrix $\bm{A}$ of size -$n\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector -!bt -\begin{equation*} -\bm{x}_{i+1}=\bm{x}_{i}+\alpha_i\bm{p}_{i}. -\end{equation*} -!et -We assume that $\bm{p}_{i}$ is a sequence of $n$ mutually conjugate directions. -Then the $\bm{p}_{i}$ form a basis of $R^n$ and we can expand the solution -$ \bm{A}\bm{x} = \bm{b}$ in this basis, namely - -!bt -\begin{equation*} - \bm{x} = \sum^{n}_{i=1} \alpha_i \bm{p}_i. -\end{equation*} -!et -!eblock - -!split -===== Conjugate gradient method ===== -!bblock -The coefficients are given by -!bt -\begin{equation*} - \mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. -\end{equation*} -!et -Multiplying with $\bm{p}_k^T$ from the left gives - -!bt -\begin{equation*} - \bm{p}_k^T \bm{A}\bm{x} = \sum^{n}_{i=1} \alpha_i\bm{p}_k^T \bm{A}\bm{p}_i= \bm{p}_k^T \bm{b}, -\end{equation*} -!et -and we can define the coefficients $\alpha_k$ as - -!bt -\begin{equation*} - \alpha_k = \frac{\bm{p}_k^T \bm{b}}{\bm{p}_k^T \bm{A} \bm{p}_k} -\end{equation*} -!et -!eblock - -!split -===== Conjugate gradient method and iterations ===== -!bblock - -If we choose the conjugate vectors $\bm{p}_k$ carefully, -then we may not need all of them to obtain a good approximation to the solution -$\bm{x}$. -We want to regard the conjugate gradient method as an iterative method. -This will us to solve systems where $n$ is so large that the direct -method would take too much time. - -We denote the initial guess for $\bm{x}$ as $\bm{x}_0$. -We can assume without loss of generality that -!bt -\begin{equation*} -\bm{x}_0=0, -\end{equation*} -!et -or consider the system -!bt -\begin{equation*} -\bm{A}\bm{z} = \bm{b}-\bm{A}\bm{x}_0, -\end{equation*} -!et -instead. -!eblock - - -!split -===== Conjugate gradient method ===== -!bblock -One can show that the solution $\bm{x}$ is also the unique minimizer of the quadratic form -!bt -\begin{equation*} - f(\bm{x}) = \frac{1}{2}\bm{x}^T\bm{A}\bm{x} - \bm{x}^T \bm{x} , \quad \bm{x}\in\mathbf{R}^n. -\end{equation*} -!et -This suggests taking the first basis vector $\bm{p}_1$ -to be the gradient of $f$ at $\bm{x}=\bm{x}_0$, -which equals -!bt -\begin{equation*} -\bm{A}\bm{x}_0-\bm{b}, -\end{equation*} -!et -and -$\bm{x}_0=0$ it is equal $-\bm{b}$. -The other vectors in the basis will be conjugate to the gradient, -hence the name conjugate gradient method. -!eblock - - -!split -===== Conjugate gradient method ===== -!bblock -Let $\bm{r}_k$ be the residual at the $k$-th step: -!bt -\begin{equation*} -\bm{r}_k=\bm{b}-\bm{A}\bm{x}_k. -\end{equation*} -!et -Note that $\bm{r}_k$ is the negative gradient of $f$ at -$\bm{x}=\bm{x}_k$, -so the gradient descent method would be to move in the direction $\bm{r}_k$. -Here, we insist that the directions $\bm{p}_k$ are conjugate to each other, -so we take the direction closest to the gradient $\bm{r}_k$ -under the conjugacy constraint. -This gives the following expression -!bt -\begin{equation*} -\bm{p}_{k+1}=\bm{r}_k-\frac{\bm{p}_k^T \bm{A}\bm{r}_k}{\bm{p}_k^T\bm{A}\bm{p}_k} \bm{p}_k. -\end{equation*} -!et -!eblock - -!split -===== Conjugate gradient method ===== -!bblock -We can also compute the residual iteratively as -!bt -\begin{equation*} -\bm{r}_{k+1}=\bm{b}-\bm{A}\bm{x}_{k+1}, - \end{equation*} -!et -which equals -!bt -\begin{equation*} -\bm{b}-\bm{A}(\bm{x}_k+\alpha_k\bm{p}_k), - \end{equation*} -!et -or -!bt -\begin{equation*} -(\bm{b}-\bm{A}\bm{x}_k)-\alpha_k\bm{A}\bm{p}_k, - \end{equation*} -!et -which gives - -!bt -\begin{equation*} -\bm{r}_{k+1}=\bm{r}_k-\bm{A}\bm{p}_{k}, - \end{equation*} -!et -!eblock - - - - - -!split -===== Revisiting our first homework ===== - -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: - -o An analytical solution (recall homework set 1). -o The gradient can be computed analytically. -o The cost function is convex which guarantees that gradient descent converges for small enough learning rates - -We revisit an example similar to what we had in the first homework set. We had a function of the type - -!bc pycod -m = 100 -x = 2*np.random.rand(m,1) -y = 4+3*x+np.random.randn(m,1) -!ec -with $x_i \in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\cal {N}(0,1)$. -The linear regression model is given by +In the more general case the various inputs should be replaced by the possible features represented by the input data set $\bm{X}$. +We can now rewrite the above probability as !bt \[ -h_\beta(x) = \bm{y} = \beta_0 + \beta_1 x, -\] -!et -such that -!bt -\[ -\bm{y}_i = \beta_0 + \beta_1 x_i. +p(\bm{D}\vert\bm{\theta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\bm{X}_{i,*}\bm{\theta})^2}{2\sigma^2}\right]}. \] !et -!split -===== Gradient descent example ===== - -Let $\mathbf{y} = (y_1,\cdots,y_n)^T$, $\mathbf{\bm{y}} = (\bm{y}_1,\cdots,\bm{y}_n)^T$ and $\beta = (\beta_0, \beta_1)^T$ - -It is convenient to write $\mathbf{\bm{y}} = X\beta$ where $X \in \mathbb{R}^{100 \times 2} $ is the design matrix given by (we keep the intercept here) -!bt -\[ -X \equiv \begin{bmatrix} -1 & x_1 \\ -\vdots & \vdots \\ -1 & x_{100} & \\ -\end{bmatrix}. -\] -!et -The cost/loss/risk function is given by ( -!bt -\[ -C(\beta) = \frac{1}{n}||X\beta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2\right] -\] -!et -and we want to find $\beta$ such that $C(\beta)$ is minimized. +It is a conditional probability (see below) and reads as the likelihood of a domain of events $\bm{D}$ given a set of parameters $\bm{\theta}$. !split -===== The derivative of the cost/loss function ===== +===== Maximum Likelihood Estimation (MLE) ===== -Computing $\partial C(\beta) / \partial \beta_0$ and $\partial C(\beta) / \partial \beta_1$ we can show that the gradient can be written as -!bt -\[ -\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}), -\] -!et -where $X$ is the design matrix defined above. +In statistics, maximum likelihood estimation (MLE) is a method of +estimating the parameters of an assumed probability distribution, +given some observed data. This is achieved by maximizing a likelihood +function so that, under the assumed statistical model, the observed +data is the most probable. -!split -===== The Hessian matrix ===== -The Hessian matrix of $C(\beta)$ is given by -!bt -\[ -\bm{H} \equiv \begin{bmatrix} -\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ -\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ -\end{bmatrix} = \frac{2}{n}X^T X. -\] -!et -This result implies that $C(\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite. +We will assume here that our events are given by the above Gaussian +distribution and we will determine the optimal parameters $\theta$ by +maximizing the above PDF. However, computing the derivatives of a +product function is cumbersome and can easily lead to overflow and/or +underflowproblems, with potentials for loss of numerical precision. - -!split -===== 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 -!bt -\[ -\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots -\] -!et - -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 $||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8}$. _Note that the code below does not include the latter stop criterion_. - -And finally we can compare our solution for $\beta$ with the analytic result given by -$\beta= (X^TX)^{-1} X^T \mathbf{y}$. - -!split -===== Gradient Descent Example ===== - -Here our simple example -!bc pycod - -# 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 - -# the number of datapoints -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -# Hessian matrix -H = (2.0/n)* X.T @ X -# Get the eigenvalues -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") - -beta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y -print(beta_linreg) -beta = np.random.randn(2,1) - -eta = 1.0/np.max(EigValues) -Niterations = 1000 - -for iter in range(Niterations): - gradient = (2.0/n)*X.T @ (X @ beta-y) - beta -= eta*gradient - -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() - -!ec - -!split -===== And a corresponding example using _scikit-learn_ ===== - -!bc pycod -# Importing various packages -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import SGDRegressor - -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -beta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y) -print(beta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) -sgdreg.fit(x,y.ravel()) -print(sgdreg.intercept_, sgdreg.coef_) - -!ec - - - -!split -===== Gradient descent and Ridge ===== - -We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\beta$, -!bt -\[ -C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. -\] -!et - -In order to minimize $C_{\text{ridge}}(\beta)$ using GD we adjust the gradient as follows -!bt -\[ -\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (\frac{1}{n}X^T(X\beta - \mathbf{y})+\lambda \beta). -\] -!et - -We can easily extend our program to minimize $C_{\text{ridge}}(\beta)$ using gradient descent and compare with the analytical solution given by -!bt -\[ -\beta_{\text{ridge}} = \left(X^T X + n\lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}. -\] -!et - -!split -===== The Hessian matrix for Ridge Regression ===== -The Hessian matrix of Ridge Regression for our simple example is given by -!bt -\[ -\bm{H} \equiv \begin{bmatrix} -\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ -\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ -\end{bmatrix} = \frac{2}{n}X^T X+2\lambda\bm{I}. -\] -!et -This implies that the Hessian matrix is positive definite, hence the stationary point is a -minimum. -Note that the Ridge cost function is convex being a sum of two convex -functions. Therefore, the stationary point is a global -minimum of this function. - - -!split -===== Program example for gradient descent with Ridge Regression ===== -!bc pycod -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt -from mpl_toolkits.mplot3d import Axes3D -from matplotlib import cm -from matplotlib.ticker import LinearLocator, FormatStrFormatter -import sys - -# the number of datapoints -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X - -#Ridge parameter lambda -lmbda = 0.001 -Id = n*lmbda* np.eye(XT_X.shape[0]) - -# Hessian matrix -H = (2.0/n)* XT_X+2*lmbda* np.eye(XT_X.shape[0]) -# Get the eigenvalues -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") - - -beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y -print(beta_linreg) -# Start plain gradient descent -beta = np.random.randn(2,1) - -eta = 1.0/np.max(EigValues) -Niterations = 100 - -for iter in range(Niterations): - gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta - beta -= eta*gradients - -print(beta) -ypredict = X @ beta -ypredict2 = X @ beta_linreg -plt.plot(x, ypredict, "r-") -plt.plot(x, ypredict2, "b-") -plt.plot(x, y ,'ro') -plt.axis([0,2.0,0, 15.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Gradient descent example for Ridge') -plt.show() - - -!ec - -!split -===== Using gradient descent methods, limitations ===== - -* _Gradient descent (GD) finds local minima of our function_. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance. - -* _GD is sensitive to initial conditions_. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD. - -* _Gradients are computationally expensive to calculate for large datasets_. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called ``mini batches''. This has the added benefit of introducing stochasticity into our algorithm. - -* _GD is very sensitive to choices of learning rates_. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape. - -* _GD treats all directions in parameter space uniformly._ Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. - -* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points. - -!split -===== Improving gradient descent with momentum ===== - -We discuss here some simple examples where we introduce what is called 'memory'about previous steps, or what is normally called momentum gradient descent. The mathematics is explained below in connection with Stochastic gradient descent. - -!bc pycod -from numpy import asarray -from numpy import arange -from numpy.random import rand -from numpy.random import seed -from matplotlib import pyplot - -# objective function -def objective(x): - return x**2.0 - -# derivative of objective function -def derivative(x): - return x * 2.0 - -# gradient descent algorithm -def gradient_descent(objective, derivative, bounds, n_iter, step_size): - # track all solutions - solutions, scores = list(), list() - # generate an initial point - solution = bounds[:, 0] + rand(len(bounds)) * (bounds[:, 1] - bounds[:, 0]) - # run the gradient descent - for i in range(n_iter): - # calculate gradient - gradient = derivative(solution) - # take a step - solution = solution - step_size * gradient - # evaluate candidate point - solution_eval = objective(solution) - # store solution - solutions.append(solution) - scores.append(solution_eval) - # report progress - print('>%d f(%s) = %.5f' % (i, solution, solution_eval)) - return [solutions, scores] - -# seed the pseudo random number generator -seed(4) -# define range for input -bounds = asarray([[-1.0, 1.0]]) -# define the total iterations -n_iter = 30 -# define the step size -step_size = 0.1 -# perform the gradient descent search -solutions, scores = gradient_descent(objective, derivative, bounds, n_iter, step_size) -# sample input range uniformly at 0.1 increments -inputs = arange(bounds[0,0], bounds[0,1]+0.1, 0.1) -# compute targets -results = objective(inputs) -# create a line plot of input vs result -pyplot.plot(inputs, results) -# plot the solutions found -pyplot.plot(solutions, scores, '.-', color='red') -# show the plot -pyplot.show() - -!ec - - -!split -===== Same code but now with momentum gradient descent ===== - -!bc pycod -from numpy import asarray -from numpy import arange -from numpy.random import rand -from numpy.random import seed -from matplotlib import pyplot - -# objective function -def objective(x): - return x**2.0 - -# derivative of objective function -def derivative(x): - return x * 2.0 - -# gradient descent algorithm -def gradient_descent(objective, derivative, bounds, n_iter, step_size, momentum): - # track all solutions - solutions, scores = list(), list() - # generate an initial point - solution = bounds[:, 0] + rand(len(bounds)) * (bounds[:, 1] - bounds[:, 0]) - # keep track of the change - change = 0.0 - # run the gradient descent - for i in range(n_iter): - # calculate gradient - gradient = derivative(solution) - # calculate update - new_change = step_size * gradient + momentum * change - # take a step - solution = solution - new_change - # save the change - change = new_change - # evaluate candidate point - solution_eval = objective(solution) - # store solution - solutions.append(solution) - scores.append(solution_eval) - # report progress - print('>%d f(%s) = %.5f' % (i, solution, solution_eval)) - return [solutions, scores] - -# seed the pseudo random number generator -seed(4) -# define range for input -bounds = asarray([[-1.0, 1.0]]) -# define the total iterations -n_iter = 30 -# define the step size -step_size = 0.1 -# define momentum -momentum = 0.3 -# perform the gradient descent search with momentum -solutions, scores = gradient_descent(objective, derivative, bounds, n_iter, step_size, momentum) -# sample input range uniformly at 0.1 increments -inputs = arange(bounds[0,0], bounds[0,1]+0.1, 0.1) -# compute targets -results = objective(inputs) -# create a line plot of input vs result -pyplot.plot(inputs, results) -# plot the solutions found -pyplot.plot(solutions, scores, '.-', color='red') -# show the plot -pyplot.show() -!ec - - - -!split -===== Overview video on Stochastic Gradient Descent ===== - -"What is Stochastic Gradient Descent":"https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" - - -!split -===== Batches and mini-batches ===== - -In gradient descent we compute the cost function and its gradient for all data points we have. - -In large-scale applications such as the "ILSVRC challenge":"https://www.image-net.org/challenges/LSVRC/", the -training data can have on order of millions of examples. Hence, it -seems wasteful to compute the full cost function over the entire -training set in order to perform only a single parameter update. A -very common approach to addressing this challenge is to compute the -gradient over batches of the training data. For example, a typical batch could contain some thousand examples from -an entire training set of several millions. This batch is then used to -perform a parameter update. - -!split -===== Stochastic Gradient Descent (SGD) ===== - -In stochastic gradient descent, the extreme case is the case where we -have only one batch, that is we include the whole data set. - -This process is called Stochastic Gradient -Descent (SGD) (or also sometimes on-line gradient descent). This is -relatively less common to see because in practice due to vectorized -code optimizations it can be computationally much more efficient to -evaluate the gradient for 100 examples, than the gradient for one -example 100 times. Even though SGD technically refers to using a -single example at a time to evaluate the gradient, you will hear -people use the term SGD even when referring to mini-batch gradient -descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD -for “Batch gradient descent” are rare to see), where it is usually -assumed that mini-batches are used. The size of the mini-batch is a -hyperparameter but it is not very common to cross-validate or bootstrap it. It is -usually based on memory constraints (if any), or set to some value, -e.g. 32, 64 or 128. We use powers of 2 in practice because many -vectorized operation implementations work faster when their inputs are -sized in powers of 2. - -In our notes with SGD we mean stochastic gradient descent with mini-batches. - - -!split -===== Stochastic Gradient Descent ===== - -Stochastic gradient descent (SGD) and variants thereof address some of -the shortcomings of the Gradient descent method discussed above. - -The underlying idea of SGD comes from the observation that the cost -function, which we want to minimize, can almost always be written as a -sum over $n$ data points $\{\mathbf{x}_i\}_{i=1}^n$, -!bt -\[ -C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, -\mathbf{\beta}). -\] -!et - -!split -===== Computation of gradients ===== - -This in turn means that the gradient can be -computed as a sum over $i$-gradients -!bt -\[ -\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}). -\] -!et - -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 -$k=1,\cdots,n/M$. - - - -!split -===== 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 -!bt -\[ -\nabla_{\beta} -C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta -c_i(\mathbf{x}_i, \mathbf{\beta}). -\] -!et - -!split -===== The gradient step ===== - -Thus a gradient descent step now looks like -!bt -\[ -\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) -\] -!et - -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. - -!split -===== Simple example code ===== - -!bc pycod -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 -!ec - -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. - -!split -===== 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. - -!split -===== Slightly different approach ===== - -Another approach is to let the step length $\gamma_j$ depend on the -number of epochs in such a way that it becomes very small after a -reasonable time such that we do not move at all. Such approaches are -also called scaling. There are many such ways to "scale the learning -rate":"https://towardsdatascience.com/gradient-descent-the-learning-rate-and-the-importance-of-feature-scaling-6c0b416596e1" -and "discussions here":"https://www.jmlr.org/papers/volume23/20-1258/20-1258.pdf". See -also -URL:"https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1" -for a discussion of different scaling functions for the learning rate. - -!split -===== Time decay rate ===== - -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 +In practice, it is more convenient to maximize the logarithm of the +PDF because it is a monotonically increasing function of the argument. +Alternatively, and this will be our option, we will minimize the +negative of the logarithm since this is a monotonically decreasing function. -!bc pycod -import numpy as np +Note also that maximization/minimization of the logarithm of the PDF +is equivalent to the maximization/minimization of the function itself. -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 +!split +===== A new Cost Function ===== -print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) -!ec +We could now define a new cost function to minimize, namely the negative logarithm of the above PDF +!bt +\[ +C(\bm{\theta})=-\log{\prod_{i=0}^{n-1}p(y_i,\bm{X}\vert\bm{\theta})}=-\sum_{i=0}^{n-1}\log{p(y_i,\bm{X}\vert\bm{\theta})}, +\] +!et +which becomes +!bt +\[ +C(\bm{\theta})=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\bm{y}-\bm{X}\bm{\theta})\vert\vert_2^2}{2\sigma^2}. +\] +!et + +Taking the derivative of the *new* cost function with respect to the parameters $\theta$ we recognize our familiar OLS equation, namely + +!bt +\[ +\bm{X}^T\left(\bm{y}-\bm{X}\bm{\theta}\right) =0, +\] +!et +which leads to the well-known OLS equation for the optimal paramters $\theta$ +!bt +\[ +\hat{\bm{\theta}}^{\mathrm{OLS}}=\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}! +\] +!et + + +Next week we will make a similar analysis for Ridge and Lasso regression !split -===== Code with a Number of Minibatches which varies ===== +===== Why resampling methods ===== -In the code here we vary the number of mini-batches. -!bc pycode -# Importing various packages -from math import exp, sqrt -from random import random, seed +Before we proceed, we need to rethink what we have been doing. In our +eager to fit the data, we have omitted several important elements in +our regression analysis. In what follows we will +o look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff +o introduce resampling techniques like cross-validation, bootstrapping and jackknife and more + +and discuss how to select a given model (one of the difficult parts in machine learning). + + + + + +!split +===== Resampling methods ===== +!bblock +Resampling methods are an indispensable tool in modern +statistics. They involve repeatedly drawing samples from a training +set and refitting a model of interest on each sample in order to +obtain additional information about the fitted model. For example, in +order to estimate the variability of a linear regression fit, we can +repeatedly draw different samples from the training data, fit a linear +regression to each new sample, and then examine the extent to which +the resulting fits differ. Such an approach may allow us to obtain +information that would not be available from fitting the model only +once using the original training sample. + +Two resampling methods are often used in Machine Learning analyses, +o The _bootstrap method_ +o and _Cross-Validation_ + +In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular +cross-validation and the bootstrap method. + + +!eblock + + +!split +===== Resampling approaches can be computationally expensive ===== +!bblock + +Resampling approaches can be computationally expensive, because they +involve fitting the same statistical method multiple times using +different subsets of the training data. However, due to recent +advances in computing power, the computational requirements of +resampling methods generally are not prohibitive. In this chapter, we +discuss two of the most commonly used resampling methods, +cross-validation and the bootstrap. Both methods are important tools +in the practical application of many statistical learning +procedures. For example, cross-validation can be used to estimate the +test error associated with a given statistical learning method in +order to evaluate its performance, or to select the appropriate level +of flexibility. The process of evaluating a model’s performance is +known as model assessment, whereas the process of selecting the proper +level of flexibility for a model is known as model selection. The +bootstrap is widely used. + +!eblock + +!split +===== Why resampling methods ? ===== +!bblock Statistical analysis + +* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods which are widely used in statistical analyses. +* The results can be analysed with the same statistical tools as we would use when analysing experimental data. +* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors. + + +!eblock + +!split +===== Statistical analysis ===== +!bblock + +* As in other experiments, many numerical experiments have two classes of errors: + * Statistical errors + * Systematical errors +* Statistical errors can be estimated using standard tools from statistics +* Systematical errors are method specific and must be treated differently from case to case. +!eblock + + + + + +!split +===== Resampling methods ===== + +With all these analytical equations for both the OLS and Ridge +regression, we will now outline how to assess a given model. This will +lead to a discussion of the so-called bias-variance tradeoff (see +below) and so-called resampling methods. + +One of the quantities we have discussed as a way to measure errors is +the mean-squared error (MSE), mainly used for fitting of continuous +functions. Another choice is the absolute error. + +In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data, +we discuss the +o prediction error or simply the _test error_ $\mathrm{Err_{Test}}$, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the +o training error $\mathrm{Err_{Train}}$, which is the average loss over the training data. + +As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. +For a certain level of complexity the test error will reach minimum, before starting to increase again. The +training error reaches a saturation. + + +!split +===== Resampling methods: Bootstrap ===== +!bblock +Bootstrapping is a "non-parametric approach":"https://en.wikipedia.org/wiki/Nonparametric_statistics" to statistical inference +that substitutes computation for more traditional distributional +assumptions and asymptotic results. Bootstrapping offers a number of +advantages: +o The bootstrap is quite general, although there are some cases in which it fails. +o Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small. +o It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically. +o It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples). +!eblock + +The textbook by "Davison on the Bootstrap Methods and their Applications":"https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A" provides many more insights and proofs. In this course we will take a more practical approach and use the results and theorems provided in the literature. For those interested in reading more about the bootstrap methods, we recommend the above text and the one by "Efron and Tibshirani":"https://www.routledge.com/An-Introduction-to-the-Bootstrap/Efron-Tibshirani/p/book/9780412042317". + + +Before we proceed however, we need to remind ourselves about a central theorem in statistics, namely the so-called _central limit theorem_. + +!split +===== The Central Limit Theorem ===== + + +Suppose we have a PDF $p(x)$ from which we generate a series $N$ +of averages $\mathbb{E}[x_i]$. Each mean value $\mathbb{E}[x_i]$ +is viewed as the average of a specific measurement, e.g., throwing +dice 100 times and then taking the average value, or producing a certain +amount of random numbers. +For notational ease, we set $\mathbb{E}[x_i]=x_i$ in the discussion +which follows. We do the same for $\mathbb{E}[z]=z$. + +If we compute the mean $z$ of $m$ such mean values $x_i$ +!bt +\[ + z=\frac{x_1+x_2+\dots+x_m}{m}, +\] +!et +the question we pose is which is the PDF of the new variable $z$. + +!split +===== Finding the Limit ===== + +The probability of obtaining an average value $z$ is the product of the +probabilities of obtaining arbitrary individual mean values $x_i$, +but with the constraint that the average is $z$. We can express this through +the following expression +!bt +\[ + \tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m) + \delta(z-\frac{x_1+x_2+\dots+x_m}{m}), +\] +!et +where the $\delta$-function enbodies the constraint that the mean is $z$. +All measurements that lead to each individual $x_i$ are expected to +be independent, which in turn means that we can express $\tilde{p}$ as the +product of individual $p(x_i)$. The independence assumption is important in the derivation of the central limit theorem. + + +!split +===== Rewriting the $\delta$-function ===== + +If we use the integral expression for the $\delta$-function + +!bt +\[ + \delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty} + dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)}, +\] +!et +and inserting $e^{i\mu q-i\mu q}$ where $\mu$ is the mean value +we arrive at +!bt +\[ + \tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty} + dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty} + dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m, +\] +!et +with the integral over $x$ resulting in + +!bt +\[ + \int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}= + \int_{-\infty}^{\infty}dxp(x) + \left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right]. +\] +!et + +!split +===== Identifying Terms ===== + +The second term on the rhs disappears since this is just the mean and +employing the definition of $\sigma^2$ we have +!bt +\[ + \int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}= + 1-\frac{q^2\sigma^2}{2m^2}+\dots, +\] +!et +resulting in + +!bt +\[ + \left[\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m\approx + \left[1-\frac{q^2\sigma^2}{2m^2}+\dots \right]^m, +\] +!et +and in the limit $m\rightarrow \infty$ we obtain + +!bt +\[ + \tilde{p}(z)=\frac{1}{\sqrt{2\pi}(\sigma/\sqrt{m})} + \exp{\left(-\frac{(z-\mu)^2}{2(\sigma/\sqrt{m})^2}\right)}, +\] +!et +which is the normal distribution with variance +$\sigma^2_m=\sigma^2/m$, where $\sigma$ is the variance of the PDF $p(x)$ +and $\mu$ is also the mean of the PDF $p(x)$. + +!split +===== Wrapping it up ===== + +Thus, the central limit theorem states that the PDF $\tilde{p}(z)$ of +the average of $m$ random values corresponding to a PDF $p(x)$ +is a normal distribution whose mean is the +mean value of the PDF $p(x)$ and whose variance is the variance +of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$. + +The central limit theorem leads to the well-known expression for the +standard deviation, given by + +!bt +\[ + \sigma_m= +\frac{\sigma}{\sqrt{m}}. +\] +!et + +The latter is true only if the average value is known exactly. This is obtained in the limit +$m\rightarrow \infty$ only. Because the mean and the variance are measured quantities we obtain +the familiar expression in statistics (the so-called Bessel correction) +!bt +\[ + \sigma_m\approx +\frac{\sigma}{\sqrt{m-1}}. +\] +!et + +In many cases however the above estimate for the standard deviation, +in particular if correlations are strong, may be too simplistic. Keep +in mind that we have assumed that the variables $x$ are independent +and identically distributed. This is obviously not always the +case. For example, the random numbers (or better pseudorandom numbers) +we generate in various calculations do always exhibit some +correlations. + + + +The theorem is satisfied by a large class of PDFs. Note however that for a +finite $m$, it is not always possible to find a closed form /analytic expression for +$\tilde{p}(x)$. + + +!split +===== Confidence Intervals ===== + +Confidence intervals are used in statistics and represent a type of estimate +computed from the observed data. This gives a range of values for an +unknown parameter such as the parameters $\bm{\theta}$ from linear regression. + +With the OLS expressions for the parameters $\bm{\theta}$ we found +$\mathbb{E}(\bm{\theta}) = \bm{\theta}$, which means that the estimator of the regression parameters is unbiased. + +In the exercises this week we show that the variance of the estimate of the $j$-th regression coefficient is +$\bm{\sigma}^2 (\bm{\theta}_j ) = \bm{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} $. + +This quantity can be used to +construct a confidence interval for the estimates. + + +!split +===== Standard Approach based on the Normal Distribution ===== + +We will assume that the parameters $\theta$ follow a normal +distribution. We can then define the confidence interval. Here we will be using as +shorthands $\mu_{\theta}$ for the above mean value and $\sigma_{\theta}$ +for the standard deviation. We have then a confidence interval + +!bt +\[ +\left(\mu_{\theta}\pm \frac{z\sigma_{\theta}}{\sqrt{n}}\right), +\] +!et + +where $z$ defines the level of certainty (or confidence). For a normal +distribution typical parameters are $z=2.576$ which corresponds to a +confidence of $99\%$ while $z=1.96$ corresponds to a confidence of +$95\%$. A confidence level of $95\%$ is commonly used and it is +normally referred to as a *two-sigmas* confidence level, that is we +approximate $z\approx 2$. + +For more discussions of confidence intervals (and in particular linked with a discussion of the bootstrap method), see chapter 5 of the textbook by "Davison on the Bootstrap Methods and their Applications":"https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A" + +In this text you will also find an in-depth discussion of the +Bootstrap method, why it works and various theorems related to it. + +!split +===== Resampling methods: Bootstrap background ===== + +Since $\widehat{\theta} = \widehat{\theta}(\bm{X})$ is a function of random variables, +$\widehat{\theta}$ itself must be a random variable. Thus it has +a pdf, call this function $p(\bm{t})$. The aim of the bootstrap is to +estimate $p(\bm{t})$ by the relative frequency of +$\widehat{\theta}$. You can think of this as using a histogram +in the place of $p(\bm{t})$. If the relative frequency closely +resembles $p(\vec{t})$, then using numerics, it is straight forward to +estimate all the interesting parameters of $p(\bm{t})$ using point +estimators. + + +!split +===== Resampling methods: More Bootstrap background ===== + +In the case that $\widehat{\theta}$ has +more than one component, and the components are independent, we use the +same estimator on each component separately. If the probability +density function of $X_i$, $p(x)$, had been known, then it would have +been straightforward to do this by: +o Drawing lots of numbers from $p(x)$, suppose we call one such set of numbers $(X_1^*, X_2^*, \cdots, X_n^*)$. +o Then using these numbers, we could compute a replica of $\widehat{\theta}$ called $\widehat{\theta}^*$. + +By repeated use of the above two points, many +estimates of $\widehat{\theta}$ can be obtained. The +idea is to use the relative frequency of $\widehat{\theta}^*$ +(think of a histogram) as an estimate of $p(\bm{t})$. + +!split +===== Resampling methods: Bootstrap approach ===== + +But +unless there is enough information available about the process that +generated $X_1,X_2,\cdots,X_n$, $p(x)$ is in general +unknown. Therefore, "Efron in 1979":"https://projecteuclid.org/euclid.aos/1176344552" asked the +question: What if we replace $p(x)$ by the relative frequency +of the observation $X_i$? + +If we draw observations in accordance with +the relative frequency of the observations, will we obtain the same +result in some asymptotic sense? The answer is yes. + + + +!split +===== Resampling methods: Bootstrap steps ===== + +The independent bootstrap works like this: + +o Draw with replacement $n$ numbers for the observed variables $\bm{x} = (x_1,x_2,\cdots,x_n)$. +o Define a vector $\bm{x}^*$ containing the values which were drawn from $\bm{x}$. +o Using the vector $\bm{x}^*$ compute $\widehat{\theta}^*$ by evaluating $\widehat \theta$ under the observations $\bm{x}^*$. +o Repeat this process $k$ times. + +When you are done, you can draw a histogram of the relative frequency +of $\widehat \theta^*$. This is your estimate of the probability +distribution $p(t)$. Using this probability distribution you can +estimate any statistics thereof. In principle you never draw the +histogram of the relative frequency of $\widehat{\theta}^*$. Instead +you use the estimators corresponding to the statistic of interest. For +example, if you are interested in estimating the variance of $\widehat +\theta$, apply the etsimator $\widehat \sigma^2$ to the values +$\widehat \theta^*$. + + +!split +===== Code example for the Bootstrap method ===== + +The following code starts with a Gaussian distribution with mean value +$\mu =100$ and variance $\sigma=15$. We use this to generate the data +used in the bootstrap analysis. The bootstrap analysis returns a data +set after a given number of bootstrap operations (as many as we have +data points). This data set consists of estimated mean values for each +bootstrap operation. The histogram generated by the bootstrap method +shows that the distribution for these mean values is also a Gaussian, +centered around the mean value $\mu=100$ but with standard deviation +$\sigma/\sqrt{n}$, where $n$ is the number of bootstrap samples (in +this case the same as the number of original data points). The value +of the standard deviation is what we expect from the central limit +theorem. + + +!bc pycod import numpy as np +from time import time +from scipy.stats import norm import matplotlib.pyplot as plt -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) +# Returns mean of bootstrap samples +# Bootstrap algorithm +def bootstrap(data, datapoints): + t = np.zeros(datapoints) + n = len(data) + # non-parametric bootstrap + for i in range(datapoints): + t[i] = np.mean(data[np.random.randint(0,n,n)]) + # analysis + print("Bootstrap Statistics :") + print("original bias std. error") + print("%8g %8g %14g %15g" % (np.mean(data), np.std(data),np.mean(t),np.std(t))) + return t -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X -theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) -# Hessian matrix -H = (2.0/n)* XT_X -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") +# We set the mean value to 100 and the standard deviation to 15 +mu, sigma = 100, 15 +datapoints = 10000 +# We generate random numbers according to the normal distribution +x = mu + sigma*np.random.randn(datapoints) +# bootstrap returns the data sample +t = bootstrap(x, datapoints) +!ec +We see that our new variance and from that the standard deviation, agrees with the central limit theorem. -theta = np.random.randn(2,1) -eta = 1.0/np.max(EigValues) -Niterations = 1000 +!split +===== Plotting the Histogram ===== +!bc pycod +# the histogram of the bootstrapped data (normalized data if density = True) +n, binsboot, patches = plt.hist(t, 50, density=True, facecolor='red', alpha=0.75) +# add a 'best fit' line +y = norm.pdf(binsboot, np.mean(t), np.std(t)) +lt = plt.plot(binsboot, y, 'b', linewidth=1) +plt.xlabel('x') +plt.ylabel('Probability') +plt.grid(True) +plt.show() +!ec -for iter in range(Niterations): - gradients = 2.0/n*X.T @ ((X @ theta)-y) - theta -= eta*gradients -print("theta from own gd") -print(theta) -xnew = np.array([[0],[2]]) -Xnew = np.c_[np.ones((2,1)), xnew] -ypredict = Xnew.dot(theta) -ypredict2 = Xnew.dot(theta_linreg) +!split +===== The bias-variance tradeoff ===== -n_epochs = 50 -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -t0, t1 = 5, 50 -def learning_schedule(t): - return t0/(t+t1) +We will discuss the bias-variance tradeoff in the context of +continuous predictions such as regression. However, many of the +intuitions and ideas discussed here also carry over to classification +tasks. Consider a dataset $\mathcal{D}$ consisting of the data +$\mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}$. -theta = np.random.randn(2,1) +Let us assume that the true data is generated from a noisy model -for epoch in range(n_epochs): -# Can you figure out a better way of setting up the contributions to each batch? - for i in range(m): - random_index = M*np.random.randint(m) - xi = X[random_index:random_index+M] - yi = y[random_index:random_index+M] - gradients = (2.0/M)* xi.T @ ((xi @ theta)-yi) - eta = learning_schedule(epoch*m+i) - theta = theta - eta*gradients -print("theta from own sdg") -print(theta) +!bt +\[ +\bm{y}=f(\boldsymbol{x}) + \bm{\epsilon} +\] +!et -plt.plot(xnew, ypredict, "r-") -plt.plot(xnew, ypredict2, "b-") -plt.plot(x, y ,'ro') -plt.axis([0,2.0,0, 15.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Random numbers ') +where $\epsilon$ is normally distributed with mean zero and standard deviation $\sigma^2$. + +In our derivation of the ordinary least squares method we defined then +an approximation to the function $f$ in terms of the parameters +$\bm{\theta}$ and the design matrix $\bm{X}$ which embody our model, +that is $\bm{\tilde{y}}=\bm{X}\bm{\theta}$. + +Thereafter we found the parameters $\bm{\theta}$ by optimizing the means squared error via the so-called cost function +!bt +\[ +C(\bm{X},\bm{\theta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]. +\] +!et + +We can rewrite this as +!bt +\[ +\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2+\sigma^2. +\] +!et + +The three terms represent the square of the bias of the learning +method, which can be thought of as the error caused by the simplifying +assumptions built into the method. The second term represents the +variance of the chosen model and finally the last terms is variance of +the error $\bm{\epsilon}$. + +To derive this equation, we need to recall that the variance of $\bm{y}$ and $\bm{\epsilon}$ are both equal to $\sigma^2$. The mean value of $\bm{\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\bm{\tilde{y}}$. +We use a more compact notation in terms of the expectation value +!bt +\[ +\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathbb{E}\left[(\bm{f}+\bm{\epsilon}-\bm{\tilde{y}})^2\right], +\] +!et +and adding and subtracting $\mathbb{E}\left[\bm{\tilde{y}}\right]$ we get +!bt +\[ +\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathbb{E}\left[(\bm{f}+\bm{\epsilon}-\bm{\tilde{y}}+\mathbb{E}\left[\bm{\tilde{y}}\right]-\mathbb{E}\left[\bm{\tilde{y}}\right])^2\right], +\] +!et +which, using the abovementioned expectation values can be rewritten as +!bt +\[ +\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathbb{E}\left[(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\bm{\tilde{y}}\right]+\sigma^2, +\] +!et +that is the rewriting in terms of the so-called bias, the variance of the model $\bm{\tilde{y}}$ and the variance of $\bm{\epsilon}$. + + +!split +===== A way to Read the Bias-Variance Tradeoff ===== + +FIGURE: [figures/BiasVariance.png, width=600 frac=0.9] + + +!split +===== Example code for Bias-Variance tradeoff ===== +!bc pycod +import matplotlib.pyplot as plt +import numpy as np +from sklearn.linear_model import LinearRegression, Ridge, Lasso +from sklearn.preprocessing import PolynomialFeatures +from sklearn.model_selection import train_test_split +from sklearn.pipeline import make_pipeline +from sklearn.utils import resample + +np.random.seed(2018) + +n = 500 +n_boostraps = 100 +degree = 18 # A quite high value, just to show. +noise = 0.1 + +# Make data set. +x = np.linspace(-1, 3, n).reshape(-1, 1) +y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape) + +# Hold out some test data that is never used in training. +x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2) + +# Combine x transformation and model into one operation. +# Not neccesary, but convenient. +model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False)) + +# The following (m x n_bootstraps) matrix holds the column vectors y_pred +# for each bootstrap iteration. +y_pred = np.empty((y_test.shape[0], n_boostraps)) +for i in range(n_boostraps): + x_, y_ = resample(x_train, y_train) + + # Evaluate the new model on the same test data each time. + y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel() + +# Note: Expectations and variances taken w.r.t. different training +# data sets, hence the axis=1. Subsequent means are taken across the test data +# set in order to obtain a total value, but before this we have error/bias/variance +# calculated per data point in the test set. +# Note 2: The use of keepdims=True is important in the calculation of bias as this +# maintains the column vector form. Dropping this yields very unexpected results. +error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) ) +bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 ) +variance = np.mean( np.var(y_pred, axis=1, keepdims=True) ) +print('Error:', error) +print('Bias^2:', bias) +print('Var:', variance) +print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance)) + +plt.plot(x[::5, :], y[::5, :], label='f(x)') +plt.scatter(x_test, y_test, label='Data points') +plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred') +plt.legend() plt.show() !ec +!split +===== Understanding what happens ===== +!bc pycod +import matplotlib.pyplot as plt +import numpy as np +from sklearn.linear_model import LinearRegression, Ridge, Lasso +from sklearn.preprocessing import PolynomialFeatures +from sklearn.model_selection import train_test_split +from sklearn.pipeline import make_pipeline +from sklearn.utils import resample + +np.random.seed(2018) + +n = 40 +n_boostraps = 100 +maxdegree = 14 + + +# Make data set. +x = np.linspace(-3, 3, n).reshape(-1, 1) +y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape) +error = np.zeros(maxdegree) +bias = np.zeros(maxdegree) +variance = np.zeros(maxdegree) +polydegree = np.zeros(maxdegree) +x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2) + +for degree in range(maxdegree): + model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False)) + y_pred = np.empty((y_test.shape[0], n_boostraps)) + for i in range(n_boostraps): + x_, y_ = resample(x_train, y_train) + y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel() + + polydegree[degree] = degree + error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) ) + bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 ) + variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) ) + print('Polynomial degree:', degree) + print('Error:', error[degree]) + print('Bias^2:', bias[degree]) + print('Var:', variance[degree]) + print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree])) + +plt.plot(polydegree, error, label='Error') +plt.plot(polydegree, bias, label='bias') +plt.plot(polydegree, variance, label='Variance') +plt.legend() +plt.show() + + + + +!ec + +!split +===== Summing up ===== + + + + +The bias-variance tradeoff summarizes the fundamental tension in +machine learning, particularly supervised learning, between the +complexity of a model and the amount of training data needed to train +it. Since data is often limited, in practice it is often useful to +use a less-complex model with higher bias, that is a model whose asymptotic +performance is worse than another model because it is easier to +train and less sensitive to sampling noise arising from having a +finite-sized training dataset (smaller variance). + + + +The above equations tell us that in +order to minimize the expected test error, we need to select a +statistical learning method that simultaneously achieves low variance +and low bias. Note that variance is inherently a nonnegative quantity, +and squared bias is also nonnegative. Hence, we see that the expected +test MSE can never lie below $Var(\epsilon)$, the irreducible error. + + +What do we mean by the variance and bias of a statistical learning +method? The variance refers to the amount by which our model would change if we +estimated it using a different training data set. Since the training +data are used to fit the statistical learning method, different +training data sets will result in a different estimate. But ideally the +estimate for our model should not vary too much between training +sets. However, if a method has high variance then small changes in +the training data can result in large changes in the model. In general, more +flexible statistical methods have higher variance. + + +You may also find this recent "article":"https://www.pnas.org/content/116/32/15849" of interest. !split -===== Replace or not ===== +===== Another Example from Scikit-Learn's Repository ===== -In the above code, we have use replacement in setting up the -mini-batches. The discussion -"here":"https://sebastianraschka.com/faq/docs/sgd-methods.html" may be -useful. - - -!split -===== Momentum based GD ===== - -The stochastic gradient descent (SGD) is almost always used with a -*momentum* or inertia term that serves as a memory of the direction we -are moving in parameter space. This is typically implemented as -follows - -!bt -\begin{align} -\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\ -\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}, -\end{align} -!et - -where we have introduced a momentum parameter $\gamma$, with -$0\le\gamma\le 1$, and for brevity we dropped the explicit notation to -indicate the gradient is to be taken over a different mini-batch at -each step. We call this algorithm gradient descent with momentum -(GDM). From these equations, it is clear that $\mathbf{v}_t$ is a -running average of recently encountered gradients and -$(1-\gamma)^{-1}$ sets the characteristic time scale for the memory -used in the averaging procedure. Consistent with this, when -$\gamma=0$, this just reduces down to ordinary SGD as discussed -earlier. An equivalent way of writing the updates is - -!bt -\[ -\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t), -\] -!et -where we have defined $\Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1}$. - -!split -===== More on momentum based approaches ===== - -Let us try to get more intuition from these equations. It is helpful -to consider a simple physical analogy with a particle of mass $m$ -moving in a viscous medium with drag coefficient $\mu$ and potential -$E(\mathbf{w})$. If we denote the particle's position by $\mathbf{w}$, -then its motion is described by - -!bt -\[ -m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}). -\] -!et - -We can discretize this equation in the usual way to get - -!bt -\[ -m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}). -\] -!et - -Rearranging this equation, we can rewrite this as - -!bt -\[ -\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t. -\] -!et - -!split -===== Momentum parameter ===== - -Notice that this equation is identical to previous one if we identify -the position of the particle, $\mathbf{w}$, with the parameters -$\boldsymbol{\theta}$. This allows us to identify the momentum -parameter and learning rate with the mass of the particle and the -viscous drag as: - -!bt -\[ -\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}. -\] -!et - -Thus, as the name suggests, the momentum parameter is proportional to -the mass of the particle and effectively provides inertia. -Furthermore, in the large viscosity/small learning rate limit, our -memory time scales as $(1-\gamma)^{-1} \approx m/(\mu \Delta t)$. - -Why is momentum useful? SGD momentum helps the gradient descent -algorithm gain speed in directions with persistent but small gradients -even in the presence of stochasticity, while suppressing oscillations -in high-curvature directions. This becomes especially important in -situations where the landscape is shallow and flat in some directions -and narrow and steep in others. It has been argued that first-order -methods (with appropriate initial conditions) can perform comparable -to more expensive second order methods, especially in the context of -complex deep learning models. - -These beneficial properties of momentum can sometimes become even more -pronounced by using a slight modification of the classical momentum -algorithm called Nesterov Accelerated Gradient (NAG). - -In the NAG algorithm, rather than calculating the gradient at the -current parameters, $\nabla_\theta E(\boldsymbol{\theta}_t)$, one -calculates the gradient at the expected value of the parameters given -our current momentum, $\nabla_\theta E(\boldsymbol{\theta}_t +\gamma -\mathbf{v}_{t-1})$. This yields the NAG update rule - -!bt -\begin{align} -\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\ -\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}. -\end{align} -!et - -One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\gamma$. - - -!split -===== Second moment of the gradient ===== - - -In stochastic gradient descent, with and without momentum, we still -have to specify a schedule for tuning the learning rates $\eta_t$ -as a function of time. As discussed in the context of Newton's -method, this presents a number of dilemmas. The learning rate is -limited by the steepest direction which can change depending on the -current position in the landscape. To circumvent this problem, ideally -our algorithm would keep track of curvature and take large steps in -shallow, flat directions and small steps in steep, narrow directions. -Second-order methods accomplish this by calculating or approximating -the Hessian and normalizing the learning rate by the -curvature. However, this is very computationally expensive for -extremely large models. Ideally, we would like to be able to -adaptively change the step size to match the landscape without paying -the steep computational price of calculating or approximating -Hessians. - -Recently, a number of methods have been introduced that accomplish -this by tracking not only the gradient, but also the second moment of -the gradient. These methods include AdaGrad, AdaDelta, Root Mean Squared Propagation (RMS-Prop), and -"ADAM":"https://arxiv.org/abs/1412.6980". - -!split -===== RMS prop ===== - -In RMS prop, in addition to keeping a running average of the first -moment of the gradient, we also keep track of the second moment -denoted by $\mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2]$. The update rule -for RMS prop is given by - -!bt -\begin{align} -\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\ -\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\ -\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber -\end{align} -!et - -where $\beta$ controls the averaging time of the second moment and is -typically taken to be about $\beta=0.9$, $\eta_t$ is a learning rate -typically chosen to be $10^{-3}$, and $\epsilon\sim 10^{-8} $ is a -small regularization constant to prevent divergences. Multiplication -and division by vectors is understood as an element-wise operation. It -is clear from this formula that the learning rate is reduced in -directions where the norm of the gradient is consistently large. This -greatly speeds up the convergence by allowing us to use a larger -learning rate for flat directions. - - -!split -===== "ADAM optimizer":"https://arxiv.org/abs/1412.6980" ===== - -A related algorithm is the ADAM optimizer. In -"ADAM":"https://arxiv.org/abs/1412.6980", we keep a running average of -both the first and second moment of the gradient and use this -information to adaptively change the learning rate for different -parameters. The method isefficient when working with large -problems involving lots data and/or parameters. It is a combination of the -gradient descent with momentum algorithm and the RMSprop algorithm -discussed above. - -In addition to keeping a running average of the first and -second moments of the gradient -(i.e. $\mathbf{m}_t=\mathbb{E}[\mathbf{g}_t]$ and -$\mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t]$, respectively), ADAM -performs an additional bias correction to account for the fact that we -are estimating the first two moments of the gradient using a running -average (denoted by the hats in the update rule below). The update -rule for ADAM is given by (where multiplication and division are once -again understood to be element-wise operations below) - -!bt -\begin{align} -\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) \\ -\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\ -\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\ -\bm{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\ -\bm{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\ -\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \bm{\mathbf{m}}_t \over \sqrt{\bm{\mathbf{s}}_t} +\epsilon}, \nonumber \\ -\end{align} -!et - -where $\beta_1$ and $\beta_2$ set the memory lifetime of the first and -second moment and are typically taken to be $0.9$ and $0.99$ -respectively, and $\eta$ and $\epsilon$ are identical to RMSprop. - -Like in RMSprop, the effective step size of a parameter depends on the -magnitude of its gradient squared. To understand this better, let us -rewrite this expression in terms of the variance -$\boldsymbol{\sigma}_t^2 = \bm{\mathbf{s}}_t - -(\bm{\mathbf{m}}_t)^2$. Consider a single parameter $\theta_t$. The -update rule for this parameter is given by - -!bt -\[ -\Delta \theta_{t+1}= -\eta_t { \bm{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}. -\] -!et - -!split -===== Algorithms and codes for Adagrad, RMSprop and Adam ===== - -The algorithms we have implemented are well described in the text by "Goodfellow, Bengio and Courville, chapter 8":"https://www.deeplearningbook.org/contents/optimization.html". - -The codes which implement these algorithms are discussed after our presentation of automatic differentiation. - - -!split -===== Practical tips ===== - -* _Randomize the data when making mini-batches_. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented. - -* _Transform your inputs_. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case. - -* _Monitor the out-of-sample performance._ Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings. - -* _Adaptive optimization methods don't always have good generalization._ Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications. - -Geron's text, see chapter 11, has several interesting discussions. - - - -!split -===== Automatic differentiation ===== - -"Automatic differentiation (AD)":"https://en.wikipedia.org/wiki/Automatic_differentiation", -also called algorithmic -differentiation or computational differentiation,is a set of -techniques to numerically evaluate the derivative of a function -specified by a computer program. AD exploits the fact that every -computer program, no matter how complicated, executes a sequence of -elementary arithmetic operations (addition, subtraction, -multiplication, division, etc.) and elementary functions (exp, log, -sin, cos, etc.). By applying the chain rule repeatedly to these -operations, derivatives of arbitrary order can be computed -automatically, accurately to working precision, and using at most a -small constant factor more arithmetic operations than the original -program. - -Automatic differentiation is neither: - -* Symbolic differentiation, nor -* Numerical differentiation (the method of finite differences). - -Symbolic differentiation can lead to inefficient code and faces the -difficulty of converting a computer program into a single expression, -while numerical differentiation can introduce round-off errors in the -discretization process and cancellation - - - -Python has tools for so-called _automatic differentiation_. -Consider the following example -!bt -\[ -f(x) = \sin\left(2\pi x + x^2\right) -\] -!et -which has the following derivative -!bt -\[ -f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) -\] -!et -Using _autograd_ we have +This example demonstrates the problems of underfitting and overfitting and +how we can use linear regression with polynomial features to approximate +nonlinear functions. The plot shows the function that we want to approximate, +which is a part of the cosine function. In addition, the samples from the +real function and the approximations of different models are displayed. The +models have polynomial features of different degrees. We can see that a +linear function (polynomial with degree 1) is not sufficient to fit the +training samples. This is called _underfitting_. A polynomial of degree 4 +approximates the true function almost perfectly. However, for higher degrees +the model will _overfit_ the training data, i.e. it learns the noise of the +training data. +We evaluate quantitatively overfitting and underfitting by using +cross-validation. We calculate the mean squared error (MSE) on the validation +set, the higher, the less likely the model generalizes correctly from the +training data. !bc pycod -import autograd.numpy as np - -# To do elementwise differentiation: -from autograd import elementwise_grad as egrad - -# To plot: -import matplotlib.pyplot as plt -def f(x): - return np.sin(2*np.pi*x + x**2) +#print(__doc__) -def f_grad_analytic(x): - return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x) +import numpy as np +import matplotlib.pyplot as plt +from sklearn.pipeline import Pipeline +from sklearn.preprocessing import PolynomialFeatures +from sklearn.linear_model import LinearRegression +from sklearn.model_selection import cross_val_score -# Do the comparison: -x = np.linspace(0,1,1000) -f_grad = egrad(f) +def true_fun(X): + return np.cos(1.5 * np.pi * X) -computed = f_grad(x) -analytic = f_grad_analytic(x) +np.random.seed(0) -plt.title('Derivative computed from Autograd compared with the analytical derivative') -plt.plot(x,computed,label='autograd') -plt.plot(x,analytic,label='analytic') +n_samples = 30 +degrees = [1, 4, 15] + +X = np.sort(np.random.rand(n_samples)) +y = true_fun(X) + np.random.randn(n_samples) * 0.1 + +plt.figure(figsize=(14, 5)) +for i in range(len(degrees)): + ax = plt.subplot(1, len(degrees), i + 1) + plt.setp(ax, xticks=(), yticks=()) + + polynomial_features = PolynomialFeatures(degree=degrees[i], + include_bias=False) + linear_regression = LinearRegression() + pipeline = Pipeline([("polynomial_features", polynomial_features), + ("linear_regression", linear_regression)]) + pipeline.fit(X[:, np.newaxis], y) + + # Evaluate the models using crossvalidation + scores = cross_val_score(pipeline, X[:, np.newaxis], y, + scoring="neg_mean_squared_error", cv=10) + + X_test = np.linspace(0, 1, 100) + plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model") + plt.plot(X_test, true_fun(X_test), label="True function") + plt.scatter(X, y, edgecolor='b', s=20, label="Samples") + plt.xlabel("x") + plt.ylabel("y") + plt.xlim((0, 1)) + plt.ylim((-2, 2)) + plt.legend(loc="best") + plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format( + degrees[i], -scores.mean(), scores.std())) +plt.show() +!ec + + + + +!split +===== Various steps in cross-validation ===== + +When the repetitive splitting of the data set is done randomly, +samples may accidently end up in a fast majority of the splits in +either training or test set. Such samples may have an unbalanced +influence on either model building or prediction evaluation. To avoid +this $k$-fold cross-validation structures the data splitting. The +samples are divided into $k$ more or less equally sized exhaustive and +mutually exclusive subsets. In turn (at each split) one of these +subsets plays the role of the test set while the union of the +remaining subsets constitutes the training set. Such a splitting +warrants a balanced representation of each sample in both training and +test set over the splits. Still the division into the $k$ subsets +involves a degree of randomness. This may be fully excluded when +choosing $k=n$. This particular case is referred to as leave-one-out +cross-validation (LOOCV). + + +!split +===== Cross-validation in brief ===== + +For the various values of $k$ + +o shuffle the dataset randomly. +o Split the dataset into $k$ groups. +o For each unique group: + o Decide which group to use as set for test data + o Take the remaining groups as a training data set + o Fit a model on the training set and evaluate it on the test set + o Retain the evaluation score and discard the model +o Summarize the model using the sample of model evaluation scores + + + +!split +===== Code Example for Cross-validation and $k$-fold Cross-validation ===== + +The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial. +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from sklearn.model_selection import KFold +from sklearn.linear_model import Ridge +from sklearn.model_selection import cross_val_score +from sklearn.preprocessing import PolynomialFeatures + +# A seed just to ensure that the random numbers are the same for every run. +# Useful for eventual debugging. +np.random.seed(3155) + +# Generate the data. +nsamples = 100 +x = np.random.randn(nsamples) +y = 3*x**2 + np.random.randn(nsamples) + +## Cross-validation on Ridge regression using KFold only + +# Decide degree on polynomial to fit +poly = PolynomialFeatures(degree = 6) + +# Decide which values of lambda to use +nlambdas = 500 +lambdas = np.logspace(-3, 5, nlambdas) + +# Initialize a KFold instance +k = 5 +kfold = KFold(n_splits = k) + +# Perform the cross-validation to estimate MSE +scores_KFold = np.zeros((nlambdas, k)) + +i = 0 +for lmb in lambdas: + ridge = Ridge(alpha = lmb) + j = 0 + for train_inds, test_inds in kfold.split(x): + xtrain = x[train_inds] + ytrain = y[train_inds] + + xtest = x[test_inds] + ytest = y[test_inds] + + Xtrain = poly.fit_transform(xtrain[:, np.newaxis]) + ridge.fit(Xtrain, ytrain[:, np.newaxis]) + + Xtest = poly.fit_transform(xtest[:, np.newaxis]) + ypred = ridge.predict(Xtest) + + scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred) + + j += 1 + i += 1 + + +estimated_mse_KFold = np.mean(scores_KFold, axis = 1) + +## Cross-validation using cross_val_score from sklearn along with KFold + +# kfold is an instance initialized above as: +# kfold = KFold(n_splits = k) + +estimated_mse_sklearn = np.zeros(nlambdas) +i = 0 +for lmb in lambdas: + ridge = Ridge(alpha = lmb) + + X = poly.fit_transform(x[:, np.newaxis]) + estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold) + + # cross_val_score return an array containing the estimated negative mse for every fold. + # we have to the the mean of every array in order to get an estimate of the mse of the model + estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds) + + i += 1 + +## Plot and compare the slightly different ways to perform cross-validation + +plt.figure() + +plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score') +plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold') + +plt.xlabel('log10(lambda)') +plt.ylabel('mse') -plt.xlabel('x') -plt.ylabel('y') plt.legend() plt.show() -print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic)))) !ec -!split -===== Using autograd ===== - -Here we -experiment with what kind of functions Autograd is capable -of finding the gradient of. The following Python functions are just -meant to illustrate what Autograd can do, but please feel free to -experiment with other, possibly more complicated, functions as well. - -!bc pycod -import autograd.numpy as np -from autograd import grad - -def f1(x): - return x**3 + 1 - -f1_grad = grad(f1) - -# Remember to send in float as argument to the computed gradient from Autograd! -a = 1.0 - -# See the evaluated gradient at a using autograd: -print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a))) - -# Compare with the analytical derivative, that is f1'(x) = 3*x**2 -grad_analytical = 3*a**2 -print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical)) -!ec !split -===== Autograd with more complicated functions ===== - -To differentiate with respect to two (or more) arguments of a Python -function, Autograd need to know at which variable the function if -being differentiated with respect to. +===== More examples on bootstrap and cross-validation and errors ===== !bc pycod -import autograd.numpy as np -from autograd import grad -def f2(x1,x2): - return 3*x1**3 + x2*(x1 - 5) + 1 - -# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1 -f2_grad_x1 = grad(f2,0) - -# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad -f2_grad_x2 = grad(f2,1) - -x1 = 1.0 -x2 = 3.0 - -print("Evaluating at x1 = %g, x2 = %g"%(x1,x2)) -print("-"*30) - -# Compare with the analytical derivatives: - -# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2: -f2_grad_x1_analytical = 9*x1**2 + x2 - -# Derivative of f2 w.r.t x2 is: x1 - 5: -f2_grad_x2_analytical = x1 - 5 - -# See the evaluated derivations: -print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) -print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) )) - -print() - -print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) -print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) )) -!ec - -Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. - - -!split -===== More complicated functions using the elements of their arguments directly ===== - -!bc pycod -import autograd.numpy as np -from autograd import grad -def f3(x): # Assumes x is an array of length 5 or higher - return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2 - -f3_grad = grad(f3) - -x = np.linspace(0,4,5) - -# Print the computed gradient: -print("The computed gradient of f3 is: ", f3_grad(x)) - -# The analytical gradient is: (2, 3, 5, 7, 22*x[4]) -f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]]) - -# Print the analytical gradient: -print("The analytical gradient of f3 is: ", f3_grad_analytical) -!ec - -Note that in this case, when sending an array as input argument, the -output from Autograd is another array. This is the true gradient of -the function, as opposed to the function in the previous example. By -using arrays to represent the variables, the output from Autograd -might be easier to work with, as the output is closer to what one -could expect form a gradient-evaluting function. - -!split -===== Functions using mathematical functions from Numpy ===== - -!bc pycod -import autograd.numpy as np -from autograd import grad -def f4(x): - return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x) - -f4_grad = grad(f4) - -x = 2.7 - -# Print the computed derivative: -print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x))) - -# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi -f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi - -# Print the analytical gradient: -print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical)) -!ec - - -!split -===== More autograd ===== - -!bc pycod -import autograd.numpy as np -from autograd import grad -def f5(x): - if x >= 0: - return x**2 - else: - return -3*x + 1 - -f5_grad = grad(f5) - -x = 2.7 - -# Print the computed derivative: -print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x))) -!ec - - -!split -===== And with loops ===== - -!bc pycod -import autograd.numpy as np -from autograd import grad -def f6_for(x): - val = 0 - for i in range(10): - val = val + x**i - return val - -def f6_while(x): - val = 0 - i = 0 - while i < 10: - val = val + x**i - i = i + 1 - return val - -f6_for_grad = grad(f6_for) -f6_while_grad = grad(f6_while) - -x = 0.5 - -# Print the computed derivaties of f6_for and f6_while -print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x))) -print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x))) -!ec -!bc pycod -import autograd.numpy as np -from autograd import grad -# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9 -# The analytical derivative is: sum(i*x**(i-1)) -f6_grad_analytical = 0 -for i in range(10): - f6_grad_analytical += i*x**(i-1) - -print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical)) -!ec - -!split -===== Using recursion ===== -!bc pycod -import autograd.numpy as np -from autograd import grad - -def f7(n): # Assume that n is an integer - if n == 1 or n == 0: - return 1 - else: - return n*f7(n-1) - -f7_grad = grad(f7) - -n = 2.0 - -print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n))) - -# The function f7 is an implementation of the factorial of n. -# By using the product rule, one can find that the derivative is: - -f7_grad_analytical = 0 -for i in range(int(n)-1): - tmp = 1 - for k in range(int(n)-1): - if k != i: - tmp *= (n - k) - f7_grad_analytical += tmp - -print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical)) - -!ec -Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. - -!split -===== Unsupported functions ===== -Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd. - -Assigning a value to the variable being differentiated with respect to -!bc pycod -import autograd.numpy as np -from autograd import grad -def f8(x): # Assume x is an array - x[2] = 3 - return x*2 - -#f8_grad = grad(f8) - -#x = 8.4 - -#print("The derivative of f8 is:",f8_grad(x)) -!ec -Here, running this code, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. - -!split -===== The syntax a.dot(b) when finding the dot product ===== -!bc pycod -import autograd.numpy as np -from autograd import grad -def f9(a): # Assume a is an array with 2 elements - b = np.array([1.0,2.0]) - return a.dot(b) - -#f9_grad = grad(f9) - -#x = np.array([1.0,0.0]) - -#print("The derivative of f9 is:",f9_grad(x)) -!ec - -Here we are told that the 'dot' function does not belong to Autograd's -version of a Numpy array. To overcome this, an alternative syntax -which also computed the dot product can be used: - -!bc pycod -import autograd.numpy as np -from autograd import grad -def f9_alternative(x): # Assume a is an array with 2 elements - b = np.array([1.0,2.0]) - return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2 - -f9_alternative_grad = grad(f9_alternative) - -x = np.array([3.0,0.0]) - -print("The gradient of f9 is:",f9_alternative_grad(x)) - -# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively -# w.r.t x is (b_1, b_2). -!ec - - -!split -===== Using Autograd with OLS ===== - -We conclude the part on optmization by showing how we can make codes -for linear regression and logistic regression using _autograd_. The -first example shows results with ordinary leats squares. - -!bc pycod -# Using Autograd to calculate gradients for OLS -from random import random, seed +# Common imports +import os import numpy as np -import autograd.numpy as np +import pandas as pd import matplotlib.pyplot as plt -from autograd import grad +from sklearn.linear_model import LinearRegression, Ridge, Lasso +from sklearn.model_selection import train_test_split +from sklearn.utils import resample +from sklearn.metrics import mean_squared_error +# Where to save the figures and data files +PROJECT_ROOT_DIR = "Results" +FIGURE_ID = "Results/FigureFiles" +DATA_ID = "DataFiles/" -def CostOLS(beta): - return (1.0/n)*np.sum((y-X @ beta)**2) +if not os.path.exists(PROJECT_ROOT_DIR): + os.mkdir(PROJECT_ROOT_DIR) -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) +if not os.path.exists(FIGURE_ID): + os.makedirs(FIGURE_ID) -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X -theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) -# Hessian matrix -H = (2.0/n)* XT_X -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") +if not os.path.exists(DATA_ID): + os.makedirs(DATA_ID) -theta = np.random.randn(2,1) -eta = 1.0/np.max(EigValues) -Niterations = 1000 -# define the gradient -training_gradient = grad(CostOLS) +def image_path(fig_id): + return os.path.join(FIGURE_ID, fig_id) -for iter in range(Niterations): - gradients = training_gradient(theta) - theta -= eta*gradients -print("theta from own gd") -print(theta) +def data_path(dat_id): + return os.path.join(DATA_ID, dat_id) -xnew = np.array([[0],[2]]) -Xnew = np.c_[np.ones((2,1)), xnew] -ypredict = Xnew.dot(theta) -ypredict2 = Xnew.dot(theta_linreg) +def save_fig(fig_id): + plt.savefig(image_path(fig_id) + ".png", format='png') -plt.plot(xnew, ypredict, "r-") -plt.plot(xnew, ypredict2, "b-") -plt.plot(x, y ,'ro') -plt.axis([0,2.0,0, 15.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Random numbers ') +infile = open(data_path("EoS.csv"),'r') + +# Read the EoS data as csv file and organize the data into two arrays with density and energies +EoS = pd.read_csv(infile, names=('Density', 'Energy')) +EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') +EoS = EoS.dropna() +Energies = EoS['Energy'] +Density = EoS['Density'] +# The design matrix now as function of various polytrops + +Maxpolydegree = 30 +X = np.zeros((len(Density),Maxpolydegree)) +X[:,0] = 1.0 +testerror = np.zeros(Maxpolydegree) +trainingerror = np.zeros(Maxpolydegree) +polynomial = np.zeros(Maxpolydegree) + +trials = 100 +for polydegree in range(1, Maxpolydegree): + polynomial[polydegree] = polydegree + for degree in range(polydegree): + X[:,degree] = Density**(degree/3.0) + +# loop over trials in order to estimate the expectation value of the MSE + testerror[polydegree] = 0.0 + trainingerror[polydegree] = 0.0 + for samples in range(trials): + x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2) + model = LinearRegression(fit_intercept=False).fit(x_train, y_train) + ypred = model.predict(x_train) + ytilde = model.predict(x_test) + testerror[polydegree] += mean_squared_error(y_test, ytilde) + trainingerror[polydegree] += mean_squared_error(y_train, ypred) + + testerror[polydegree] /= trials + trainingerror[polydegree] /= trials + print("Degree of polynomial: %3d"% polynomial[polydegree]) + print("Mean squared error on training data: %.8f" % trainingerror[polydegree]) + print("Mean squared error on test data: %.8f" % testerror[polydegree]) + +plt.plot(polynomial, np.log10(trainingerror), label='Training Error') +plt.plot(polynomial, np.log10(testerror), label='Test Error') +plt.xlabel('Polynomial degree') +plt.ylabel('log10[MSE]') +plt.legend() +plt.show() + +!ec + +Note that we kept the intercept column in the fitting here. This means that we need to set the _intercept_ in the call to the _Scikit-Learn_ function as _False_. Alternatively, we could have set up the design matrix $X$ without the first column of ones. + +!split +===== The same example but now with cross-validation ===== + +In this example we keep the intercept column again but add cross-validation in order to estimate the best possible value of the means squared error. +!bc pycod +# Common imports +import os +import numpy as np +import pandas as pd +import matplotlib.pyplot as plt +from sklearn.linear_model import LinearRegression, Ridge, Lasso +from sklearn.metrics import mean_squared_error +from sklearn.model_selection import KFold +from sklearn.model_selection import cross_val_score + + +# Where to save the figures and data files +PROJECT_ROOT_DIR = "Results" +FIGURE_ID = "Results/FigureFiles" +DATA_ID = "DataFiles/" + +if not os.path.exists(PROJECT_ROOT_DIR): + os.mkdir(PROJECT_ROOT_DIR) + +if not os.path.exists(FIGURE_ID): + os.makedirs(FIGURE_ID) + +if not os.path.exists(DATA_ID): + os.makedirs(DATA_ID) + +def image_path(fig_id): + return os.path.join(FIGURE_ID, fig_id) + +def data_path(dat_id): + return os.path.join(DATA_ID, dat_id) + +def save_fig(fig_id): + plt.savefig(image_path(fig_id) + ".png", format='png') + +infile = open(data_path("EoS.csv"),'r') + +# Read the EoS data as csv file and organize the data into two arrays with density and energies +EoS = pd.read_csv(infile, names=('Density', 'Energy')) +EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') +EoS = EoS.dropna() +Energies = EoS['Energy'] +Density = EoS['Density'] +# The design matrix now as function of various polytrops + +Maxpolydegree = 30 +X = np.zeros((len(Density),Maxpolydegree)) +X[:,0] = 1.0 +estimated_mse_sklearn = np.zeros(Maxpolydegree) +polynomial = np.zeros(Maxpolydegree) +k =5 +kfold = KFold(n_splits = k) + +for polydegree in range(1, Maxpolydegree): + polynomial[polydegree] = polydegree + for degree in range(polydegree): + X[:,degree] = Density**(degree/3.0) + OLS = LinearRegression(fit_intercept=False) +# loop over trials in order to estimate the expectation value of the MSE + estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold) +#[:, np.newaxis] + estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds) + +plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error') +plt.xlabel('Polynomial degree') +plt.ylabel('log10[MSE]') +plt.legend() plt.show() !ec -!split -===== Same code but now with momentum gradient descent ===== -!bc pycod -# Using Autograd to calculate gradients for OLS -from random import random, seed -import numpy as np -import autograd.numpy as np -import matplotlib.pyplot as plt -from autograd import grad -def CostOLS(beta): - return (1.0/n)*np.sum((y-X @ beta)**2) - -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x#+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X -theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) -# Hessian matrix -H = (2.0/n)* XT_X -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") - -theta = np.random.randn(2,1) -eta = 1.0/np.max(EigValues) -Niterations = 30 - -# define the gradient -training_gradient = grad(CostOLS) - -for iter in range(Niterations): - gradients = training_gradient(theta) - theta -= eta*gradients - print(iter,gradients[0],gradients[1]) -print("theta from own gd") -print(theta) - -# Now improve with momentum gradient descent -change = 0.0 -delta_momentum = 0.3 -for iter in range(Niterations): - # calculate gradient - gradients = training_gradient(theta) - # calculate update - new_change = eta*gradients+delta_momentum*change - # take a step - theta -= new_change - # save the change - change = new_change - print(iter,gradients[0],gradients[1]) -print("theta from own gd wth momentum") -print(theta) - -!ec - -!split -===== But none of these can compete with Newton's method ===== - -!bc pycod -# Using Newton's method -from random import random, seed -import numpy as np -import autograd.numpy as np -import matplotlib.pyplot as plt -from autograd import grad - -def CostOLS(beta): - return (1.0/n)*np.sum((y-X @ beta)**2) - -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X -beta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(beta_linreg) -# Hessian matrix -H = (2.0/n)* XT_X -# Note that here the Hessian does not depend on the parameters beta -invH = np.linalg.pinv(H) -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") - -beta = np.random.randn(2,1) -Niterations = 5 - -# define the gradient -training_gradient = grad(CostOLS) - -for iter in range(Niterations): - gradients = training_gradient(beta) - beta -= invH @ gradients - print(iter,gradients[0],gradients[1]) -print("beta from own Newton code") -print(beta) -!ec !split -===== Including Stochastic Gradient Descent with Autograd ===== -In this code we include the stochastic gradient descent approach discussed above. Note here that we specify which argument we are taking the derivative with respect to when using _autograd_. - -!bc pycod -# Using Autograd to calculate gradients using SGD -# OLS example -from random import random, seed -import numpy as np -import autograd.numpy as np -import matplotlib.pyplot as plt -from autograd import grad - -# Note change from previous example -def CostOLS(y,X,theta): - return np.sum((y-X @ theta)**2) - -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X -theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) -# Hessian matrix -H = (2.0/n)* XT_X -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") - -theta = np.random.randn(2,1) -eta = 1.0/np.max(EigValues) -Niterations = 1000 - -# Note that we request the derivative wrt third argument (theta, 2 here) -training_gradient = grad(CostOLS,2) - -for iter in range(Niterations): - gradients = (1.0/n)*training_gradient(y, X, theta) - theta -= eta*gradients -print("theta from own gd") -print(theta) - -xnew = np.array([[0],[2]]) -Xnew = np.c_[np.ones((2,1)), xnew] -ypredict = Xnew.dot(theta) -ypredict2 = Xnew.dot(theta_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'Random numbers ') -plt.show() - -n_epochs = 50 -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -t0, t1 = 5, 50 -def learning_schedule(t): - return t0/(t+t1) - -theta = np.random.randn(2,1) - -for epoch in range(n_epochs): -# Can you figure out a better way of setting up the contributions to each batch? - for i in range(m): - random_index = M*np.random.randint(m) - xi = X[random_index:random_index+M] - yi = y[random_index:random_index+M] - gradients = (1.0/M)*training_gradient(yi, xi, theta) - eta = learning_schedule(epoch*m+i) - theta = theta - eta*gradients -print("theta from own sdg") -print(theta) - - -!ec - - -!split -===== Same code but now with momentum gradient descent ===== -!bc pycod -# Using Autograd to calculate gradients using SGD -# OLS example -from random import random, seed -import numpy as np -import autograd.numpy as np -import matplotlib.pyplot as plt -from autograd import grad - -# Note change from previous example -def CostOLS(y,X,theta): - return np.sum((y-X @ theta)**2) - -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X -theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) -# Hessian matrix -H = (2.0/n)* XT_X -EigValues, EigVectors = np.linalg.eig(H) -print(f"Eigenvalues of Hessian Matrix:{EigValues}") - -theta = np.random.randn(2,1) -eta = 1.0/np.max(EigValues) -Niterations = 100 - -# Note that we request the derivative wrt third argument (theta, 2 here) -training_gradient = grad(CostOLS,2) - -for iter in range(Niterations): - gradients = (1.0/n)*training_gradient(y, X, theta) - theta -= eta*gradients -print("theta from own gd") -print(theta) - - -n_epochs = 50 -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -t0, t1 = 5, 50 -def learning_schedule(t): - return t0/(t+t1) - -theta = np.random.randn(2,1) - -change = 0.0 -delta_momentum = 0.3 - -for epoch in range(n_epochs): - for i in range(m): - random_index = M*np.random.randint(m) - xi = X[random_index:random_index+M] - yi = y[random_index:random_index+M] - gradients = (1.0/M)*training_gradient(yi, xi, theta) - eta = learning_schedule(epoch*m+i) - # calculate update - new_change = eta*gradients+delta_momentum*change - # take a step - theta -= new_change - # save the change - change = new_change -print("theta from own sdg with momentum") -print(theta) -!ec - - -!split -===== Similar (second order function now) problem but now with AdaGrad ===== -!bc pycod -# Using Autograd to calculate gradients using AdaGrad and Stochastic Gradient descent -# OLS example -from random import random, seed -import numpy as np -import autograd.numpy as np -import matplotlib.pyplot as plt -from autograd import grad - -# Note change from previous example -def CostOLS(y,X,theta): - return np.sum((y-X @ theta)**2) - -n = 1000 -x = np.random.rand(n,1) -y = 2.0+3*x +4*x*x - -X = np.c_[np.ones((n,1)), x, x*x] -XT_X = X.T @ X -theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) - - -# Note that we request the derivative wrt third argument (theta, 2 here) -training_gradient = grad(CostOLS,2) -# Define parameters for Stochastic Gradient Descent -n_epochs = 50 -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -# Guess for unknown parameters theta -theta = np.random.randn(3,1) - -# Value for learning rate -eta = 0.01 -# Including AdaGrad parameter to avoid possible division by zero -delta = 1e-8 -for epoch in range(n_epochs): - Giter = 0.0 - for i in range(m): - random_index = M*np.random.randint(m) - xi = X[random_index:random_index+M] - yi = y[random_index:random_index+M] - gradients = (1.0/M)*training_gradient(yi, xi, theta) - Giter += gradients*gradients - update = gradients*eta/(delta+np.sqrt(Giter)) - theta -= update -print("theta from own AdaGrad") -print(theta) - - -!ec - -Running this code we note an almost perfect agreement with the results from matrix inversion. - -!split -===== RMSprop for adaptive learning rate with Stochastic Gradient Descent ===== -!bc pycod -# Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent -# OLS example -from random import random, seed -import numpy as np -import autograd.numpy as np -import matplotlib.pyplot as plt -from autograd import grad - -# Note change from previous example -def CostOLS(y,X,theta): - return np.sum((y-X @ theta)**2) - -n = 1000 -x = np.random.rand(n,1) -y = 2.0+3*x +4*x*x# +np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x, x*x] -XT_X = X.T @ X -theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) - - -# Note that we request the derivative wrt third argument (theta, 2 here) -training_gradient = grad(CostOLS,2) -# Define parameters for Stochastic Gradient Descent -n_epochs = 50 -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -# Guess for unknown parameters theta -theta = np.random.randn(3,1) - -# Value for learning rate -eta = 0.01 -# Value for parameter rho -rho = 0.99 -# Including AdaGrad parameter to avoid possible division by zero -delta = 1e-8 -for epoch in range(n_epochs): - Giter = 0.0 - for i in range(m): - random_index = M*np.random.randint(m) - xi = X[random_index:random_index+M] - yi = y[random_index:random_index+M] - gradients = (1.0/M)*training_gradient(yi, xi, theta) - # Accumulated gradient - # Scaling with rho the new and the previous results - Giter = (rho*Giter+(1-rho)*gradients*gradients) - # Taking the diagonal only and inverting - update = gradients*eta/(delta+np.sqrt(Giter)) - # Hadamard product - theta -= update -print("theta from own RMSprop") -print(theta) -!ec - -!split -===== And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf" ===== - -!bc pycod -# Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent -# OLS example -from random import random, seed -import numpy as np -import autograd.numpy as np -import matplotlib.pyplot as plt -from autograd import grad - -# Note change from previous example -def CostOLS(y,X,theta): - return np.sum((y-X @ theta)**2) - -n = 1000 -x = np.random.rand(n,1) -y = 2.0+3*x +4*x*x# +np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x, x*x] -XT_X = X.T @ X -theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) - - -# Note that we request the derivative wrt third argument (theta, 2 here) -training_gradient = grad(CostOLS,2) -# Define parameters for Stochastic Gradient Descent -n_epochs = 50 -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -# Guess for unknown parameters theta -theta = np.random.randn(3,1) - -# Value for learning rate -eta = 0.01 -# Value for parameters beta1 and beta2, see https://arxiv.org/abs/1412.6980 -beta1 = 0.9 -beta2 = 0.999 -# Including AdaGrad parameter to avoid possible division by zero -delta = 1e-7 -iter = 0 -for epoch in range(n_epochs): - first_moment = 0.0 - second_moment = 0.0 - iter += 1 - for i in range(m): - random_index = M*np.random.randint(m) - xi = X[random_index:random_index+M] - yi = y[random_index:random_index+M] - gradients = (1.0/M)*training_gradient(yi, xi, theta) - # Computing moments first - first_moment = beta1*first_moment + (1-beta1)*gradients - second_moment = beta2*second_moment+(1-beta2)*gradients*gradients - first_term = first_moment/(1.0-beta1**iter) - second_term = second_moment/(1.0-beta2**iter) - # Scaling with rho the new and the previous results - update = eta*first_term/(np.sqrt(second_term)+delta) - theta -= update -print("theta from own ADAM") -print(theta) -!ec - -!split -===== And Logistic Regression ===== - -!bc pycod -import autograd.numpy as np -from autograd import grad - -def sigmoid(x): - return 0.5 * (np.tanh(x / 2.) + 1) - -def logistic_predictions(weights, inputs): - # Outputs probability of a label being true according to logistic model. - return sigmoid(np.dot(inputs, weights)) - -def training_loss(weights): - # Training loss is the negative log-likelihood of the training labels. - preds = logistic_predictions(weights, inputs) - label_probabilities = preds * targets + (1 - preds) * (1 - targets) - return -np.sum(np.log(label_probabilities)) - -# Build a toy dataset. -inputs = np.array([[0.52, 1.12, 0.77], - [0.88, -1.08, 0.15], - [0.52, 0.06, -1.30], - [0.74, -2.49, 1.39]]) -targets = np.array([True, True, False, True]) - -# Define a function that returns gradients of training loss using Autograd. -training_gradient_fun = grad(training_loss) - -# Optimize weights using gradient descent. -weights = np.array([0.0, 0.0, 0.0]) -print("Initial loss:", training_loss(weights)) -for i in range(100): - weights -= training_gradient_fun(weights) * 0.01 - -print("Trained loss:", training_loss(weights)) -!ec - - -!split -===== Introducing "JAX":"https://jax.readthedocs.io/en/latest/" ===== - -Presently, instead of using _autograd_, we recommend using "JAX":"https://jax.readthedocs.io/en/latest/" - -_JAX_ is Autograd and "XLA (Accelerated Linear Algebra))":"https://www.tensorflow.org/xla", -brought together for high-performance numerical computing and machine learning research. -It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more. - -Here's a simple example on how you can use _JAX_ to compute the derivate of the logistic function. - -!bc pycod -import jax.numpy as jnp -from jax import grad, jit, vmap - -def sum_logistic(x): - return jnp.sum(1.0 / (1.0 + jnp.exp(-x))) - -x_small = jnp.arange(3.) -derivative_fn = grad(sum_logistic) -print(derivative_fn(x_small)) - -!ec - - +===== Material for the lab sessions ===== +This week we will discuss during the first hour of each lab session +some technicalities related to the project and methods for updating +the learning like ADAgrad, RMSprop and ADAM. As teaching material, see +the jupyter-notebook from week 37 (September 12-16). +For the lab session, the following video on cross validation (from 2024), could be helpful, see URL:"https://www.youtube.com/watch?v=T9jjWsmsd1o" +See also video on ADAgrad, RMSprop and ADAM (material from last week not covered during lecture) at URL:"https://youtu.be/J_41Hld6tTU"