FYS-STK4155/doc/pub/Splines/ipynb/Splines.ipynb

{
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    "<!-- dom:TITLE: Data Analysis and Machine Learning  Lectures: Optimization and  Gradient Methods -->\n",
    "# Data Analysis and Machine Learning  Lectures: Optimization and  Gradient Methods\n",
    "<!-- dom:AUTHOR: Morten Hjorth-Jensen at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University -->\n",
    "<!-- Author: -->  \n",
    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
    "\n",
    "Date: **Sep 27, 2018**\n",
    "\n",
    "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "## Optimization, the central part of any Machine Learning algortithm\n",
    "\n",
    "Almost every problem in machine learning and data science starts with\n",
    "a dataset $X$, a model $g(\\beta)$, which is a function of the\n",
    "parameters $\\beta$ and a cost function $C(X, g(\\beta))$ that allows\n",
    "us to judge how well the model $g(\\beta)$ explains the observations\n",
    "$X$. The model is fit by finding the values of $\\beta$ that minimize\n",
    "the cost function. Ideally we would be able to solve for $\\beta$\n",
    "analytically, however this is not possible in general and we must use\n",
    "some approximative/numerical method to compute the minimum.\n",
    "\n",
    "\n",
    "## Revisiting our Logistic Regression case\n",
    "\n",
    "In our discussion on Logistic Regression we studied the \n",
    "case of\n",
    "two classes, with $y_i$ either\n",
    "$0$ or $1$. Furthermore we assumed also that we have only two\n",
    "parameters $\\beta$ in our fitting, that is we\n",
    "defined probabilities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align*}\n",
    "p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n",
    "p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n",
    "\n",
    "## The equations to solve\n",
    "\n",
    "Our compact equations used a definition of a vector $\\hat{y}$ with $n$\n",
    "elements $y_i$, an $n\\times p$ matrix $\\hat{X}$ which contains the\n",
    "$x_i$ values and a vector $\\hat{p}$ of fitted probabilities\n",
    "$p(y_i\\vert x_i,\\hat{\\beta})$. We rewrote in a more compact form\n",
    "the first derivative of the cost function as"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}} = -\\hat{X}^T\\left(\\hat{y}-\\hat{p}\\right).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we in addition define a diagonal matrix $\\hat{W}$ with elements \n",
    "$p(y_i\\vert x_i,\\hat{\\beta})(1-p(y_i\\vert x_i,\\hat{\\beta})$, we can obtain a compact expression of the second derivative as"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T} = \\hat{X}^T\\hat{W}\\hat{X}.\n",
    "$$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This defines what is called  the Hessian matrix.\n",
    "\n",
    "## Solving using Newton-Raphson's method\n",
    "\n",
    "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. \n",
    "\n",
    "Our iterative scheme is then given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{\\beta}^{\\mathrm{new}} = \\hat{\\beta}^{\\mathrm{old}}-\\left(\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T}\\right)^{-1}_{\\hat{\\beta}^{\\mathrm{old}}}\\times \\left(\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}}\\right)_{\\hat{\\beta}^{\\mathrm{old}}},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or in matrix form as"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{\\beta}^{\\mathrm{new}} = \\hat{\\beta}^{\\mathrm{old}}-\\left(\\hat{X}^T\\hat{W}\\hat{X} \\right)^{-1}\\times \\left(-\\hat{X}^T(\\hat{y}-\\hat{p}) \\right)_{\\hat{\\beta}^{\\mathrm{old}}}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The right-hand side is computed with the old values of $\\beta$. \n",
    "\n",
    "If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. \n",
    "\n",
    "\n",
    "## Brief reminder on Newton-Raphson's method\n",
    "\n",
    "Let us quickly remind ourselves how we derive the above method.\n",
    "\n",
    "Perhaps the most celebrated of all one-dimensional root-finding\n",
    "routines is Newton's method, also called the Newton-Raphson\n",
    "method. This method  requires the evaluation of both the\n",
    "function $f$ and its derivative $f'$ at arbitrary points. \n",
    "If you can only calculate the derivative\n",
    "numerically and/or your function is not of the smooth type, we\n",
    "normally discourage the use of this method.\n",
    "\n",
    "## The equations\n",
    "\n",
    "The Newton-Raphson formula consists geometrically of extending the\n",
    "tangent line at a current point until it crosses zero, then setting\n",
    "the next guess to the abscissa of that zero-crossing.  The mathematics\n",
    "behind this method is rather simple. Employing a Taylor expansion for\n",
    "$x$ sufficiently close to the solution $s$, we have"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- Equation labels as ordinary links -->\n",
    "<div id=\"eq:taylornr\"></div>\n",
    "\n",
    "$$\n",
    "f(s)=0=f(x)+(s-x)f'(x)+\\frac{(s-x)^2}{2}f''(x) +\\dots.\n",
    "    \\label{eq:taylornr} \\tag{1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For small enough values of the function and for well-behaved\n",
    "functions, the terms beyond linear are unimportant, hence we obtain"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f(x)+(s-x)f'(x)\\approx 0,\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "yielding"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "s\\approx x-\\frac{f(x)}{f'(x)}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having in mind an iterative procedure, it is natural to start iterating with"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}.\n",
    "$$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple geometric interpretation\n",
    "\n",
    "The above is Newton-Raphson's method. It has a simple geometric\n",
    "interpretation, namely $x_{n+1}$ is the point where the tangent from\n",
    "$(x_n,f(x_n))$ crosses the $x$-axis.  Close to the solution,\n",
    "Newton-Raphson converges fast to the desired result. However, if we\n",
    "are far from a root, where the higher-order terms in the series are\n",
    "important, the Newton-Raphson formula can give grossly inaccurate\n",
    "results. For instance, the initial guess for the root might be so far\n",
    "from the true root as to let the search interval include a local\n",
    "maximum or minimum of the function.  If an iteration places a trial\n",
    "guess near such a local extremum, so that the first derivative nearly\n",
    "vanishes, then Newton-Raphson may fail totally\n",
    "\n",
    "\n",
    "## Extending to more than one variable\n",
    "\n",
    "Newton's method can be generalized to systems of several non-linear equations\n",
    "and variables. Consider the case with two equations"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{array}{cc} f_1(x_1,x_2) &=0\\\\\n",
    "                     f_2(x_1,x_2) &=0,\\end{array}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which we Taylor expand to obtain"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1\n",
    "                     \\partial f_1/\\partial x_1+h_2\n",
    "                     \\partial f_1/\\partial x_2+\\dots\\\\\n",
    "                     0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1\n",
    "                     \\partial f_2/\\partial x_1+h_2\n",
    "                     \\partial f_2/\\partial x_2+\\dots\n",
    "                       \\end{array}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining the Jacobian matrix ${\\bf \\hat{J}}$ we have"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "{\\bf \\hat{J}}=\\left( \\begin{array}{cc}\n",
    "                         \\partial f_1/\\partial x_1  & \\partial f_1/\\partial x_2 \\\\\n",
    "                          \\partial f_2/\\partial x_1     &\\partial f_2/\\partial x_2\n",
    "             \\end{array} \\right),\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "we can rephrase Newton's method as"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\left(\\begin{array}{c} x_1^{n+1} \\\\ x_2^{n+1} \\end{array} \\right)=\n",
    "\\left(\\begin{array}{c} x_1^{n} \\\\ x_2^{n} \\end{array} \\right)+\n",
    "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right),\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where we have defined"
   ]
  },
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   "metadata": {},
   "source": [
    "$$\n",
    "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right)=\n",
    "   -{\\bf \\hat{J}}^{-1}\n",
    "   \\left(\\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\\\ f_2(x_1^{n},x_2^{n}) \\end{array} \\right).\n",
    "$$"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need thus to compute the inverse of the Jacobian matrix and it\n",
    "is to understand that difficulties  may\n",
    "arise in case ${\\bf \\hat{J}}$ is nearly singular.\n",
    "\n",
    "It is rather straightforward to extend the above scheme to systems of\n",
    "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. \n",
    "\n",
    "\n",
    "\n",
    "## Steepest descent\n",
    "\n",
    "The basic idea of gradient descent is\n",
    "that a function $F(\\mathbf{x})$, \n",
    "$\\mathbf{x} \\equiv (x_1,\\cdots,x_n)$, decreases fastest if one goes from $\\bf {x}$ in the\n",
    "direction of the negative gradient $-\\nabla F(\\mathbf{x})$.\n",
    "\n",
    "It can be shown that if"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k),\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with $\\gamma_k > 0$.\n",
    "\n",
    "For $\\gamma_k$ small enough, then $F(\\mathbf{x}_{k+1}) \\leq\n",
    "F(\\mathbf{x}_k)$. This means that for a sufficiently small $\\gamma_k$\n",
    "we are always moving towards smaller function values, i.e a minimum.\n",
    "\n",
    "<!-- !split  -->\n",
    "## More on Steepest descent\n",
    "\n",
    "The previous observation is the basis of the method of steepest\n",
    "descent, which is also referred to as just gradient descent (GD). One\n",
    "starts with an initial guess $\\mathbf{x}_0$ for a minimum of $F$ and\n",
    "computes new approximations according to"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k), \\ \\ k \\geq 0.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The parameter $\\gamma_k$ is often referred to as the step length or\n",
    "the learning rate within the context of Machine Learning.\n",
    "\n",
    "<!-- !split  -->\n",
    "## The ideal\n",
    "\n",
    "Ideally the sequence $\\{\\mathbf{x}_k \\}_{k=0}$ converges to a global\n",
    "minimum of the function $F$. In general we do not know if we are in a\n",
    "global or local minimum. In the special case when $F$ is a convex\n",
    "function, all local minima are also global minima, so in this case\n",
    "gradient descent can converge to the global solution. The advantage of\n",
    "this scheme is that it is conceptually simple and straightforward to\n",
    "implement. However the method in this form has some severe\n",
    "limitations:\n",
    "\n",
    "In machine learing we are often faced with non-convex high dimensional\n",
    "cost functions with many local minima. Since GD is deterministic we\n",
    "will get stuck in a local minimum, if the method converges, unless we\n",
    "have a very good intial guess. This also implies that the scheme is\n",
    "sensitive to the chosen initial condition.\n",
    "\n",
    "Note that the gradient is a function of $\\mathbf{x} =\n",
    "(x_1,\\cdots,x_n)$ which makes it expensive to compute numerically.\n",
    "\n",
    "\n",
    "<!-- !split  -->\n",
    "## The sensitiveness of the gradient descent\n",
    "\n",
    "The gradient descent method \n",
    "is sensitive to the choice of learning rate $\\gamma_k$. This is due\n",
    "to the fact that we are only guaranteed that $F(\\mathbf{x}_{k+1}) \\leq\n",
    "F(\\mathbf{x}_k)$ for sufficiently small $\\gamma_k$. The problem is to\n",
    "determine an optimal learning rate. If the learning rate is chosen too\n",
    "small the method will take a long time to converge and if it is too\n",
    "large we can experience erratic behavior.\n",
    "\n",
    "Many of these shortcomings can be alleviated by introducing\n",
    "randomness. One such method is that of Stochastic Gradient Descent\n",
    "(SGD), see below.\n",
    "\n",
    "\n",
    "<!-- !split  -->\n",
    "## Convex functions\n",
    "\n",
    "Ideally we want our cost/loss function to be convex(concave).\n",
    "\n",
    "First we give the definition of a convex set: A set $C$ in\n",
    "$\\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and\n",
    "all $t \\in (0,1)$ , the point $(1 − t)x + ty$ also belongs to\n",
    "C. Geometrically this means that every point on the line segment\n",
    "connecting $x$ and $y$ is in $C$ as discussed below.\n",
    "\n",
    "The convex subsets of $\\mathbb{R}$ are the intervals of\n",
    "$\\mathbb{R}$. Examples of convex sets of $\\mathbb{R}^2$ are the\n",
    "regular polygons (triangles, rectangles, pentagons, etc...).\n",
    "\n",
    "## Convex function\n",
    "\n",
    "**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.\n",
    "\n",
    "## Conditions on convex functions\n",
    "\n",
    "In the following we state first and second-order conditions which\n",
    "ensures convexity of a function $f$. We write $D_f$ to denote the\n",
    "domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more\n",
    "details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004).\n",
    "\n",
    "**First order condition.**\n",
    "\n",
    "Suppose $f$ is differentiable (i.e $\\nabla f(x)$ is well defined for\n",
    "all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$\n",
    "is a convex set and $$f(y) \\geq f(x) + \\nabla f(x)^T (y-x) $$ holds\n",
    "for all $x,y \\in D_f$. This condition means that for a convex function\n",
    "the first order Taylor expansion (right hand side above) at any point\n",
    "a global under estimator of the function. To convince yourself you can\n",
    "make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and\n",
    "note that it is always below the graph.\n",
    "\n",
    "\n",
    "\n",
    "**Second order condition.**\n",
    "\n",
    "Assume that $f$ is twice\n",
    "differentiable, i.e the Hessian matrix exists at each point in\n",
    "$D_f$. Then $f$ is convex if and only if $D_f$ is a convex set and its\n",
    "Hessian is positive semi-definite for all $x\\in D_f$. For a\n",
    "single-variable function this reduces to $f''(x) \\geq 0$. Geometrically this means that $f$ has nonnegative curvature\n",
    "everywhere.\n",
    "\n",
    "\n",
    "\n",
    "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.\n",
    "\n",
    "## More on convex functions\n",
    "\n",
    "The next result is of great importance to us and the reason why we are\n",
    "going on about convex functions. In machine learning we frequently\n",
    "have to minimize a loss/cost function in order to find the best\n",
    "parameters for the model we are considering. \n",
    "\n",
    "Ideally we want the\n",
    "global minimum (for high-dimensional models it is hard to know\n",
    "if we have local or global minimum). However, if the cost/loss function\n",
    "is convex the following result provides invaluable information:\n",
    "\n",
    "**Any minimum is global for convex functions.**\n",
    "\n",
    "Consider the problem of finding $x \\in \\mathbb{R}^n$ such that $f(x)$\n",
    "is minimal, where $f$ is convex and differentiable. Then, any point\n",
    "$x^*$ that satisfies $\\nabla f(x^*) = 0$ is a global minimum.\n",
    "\n",
    "\n",
    "\n",
    "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.\n",
    "\n",
    "## Some simple problems\n",
    "\n",
    "1. Show that $f(x)=x^2$ is convex for $x \\in \\mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \\in D_f$ and any $\\lambda \\in [0,1]$ $\\lambda f(x)+(1-\\lambda)f(y)-f(\\lambda x + (1-\\lambda) y ) \\geq 0$.\n",
    "\n",
    "2. Using the second order condition show that the following functions are convex on the specified domain.\n",
    "\n",
    " * $f(x) = e^x$ is convex for $x \\in \\mathbb{R}$.\n",
    "\n",
    " * $g(x) = -\\ln(x)$ is convex for $x \\in (0,\\infty)$.\n",
    "\n",
    "\n",
    "3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \\in \\mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.\n",
    "\n",
    "4. A norm is any function that satisfy the following properties\n",
    "\n",
    " * $f(\\alpha x) = |\\alpha| f(x)$ for all $\\alpha \\in \\mathbb{R}$.\n",
    "\n",
    " * $f(x+y) \\leq f(x) + f(y)$\n",
    "\n",
    " * $f(x) \\leq 0$ for all $x \\in \\mathbb{R}^n$ with equality if and only if $x = 0$\n",
    "\n",
    "\n",
    "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).\n",
    "\n",
    "\n",
    "## Standard steepest descent\n",
    "\n",
    "\n",
    "Before we proceed, we would like to discuss the approach called the\n",
    "**standard Steepest descent**, which again leads to us having to be able\n",
    "to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).\n",
    "\n",
    "[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)\n",
    "for finding solutions of non-linear problems is based on the theory\n",
    "of conjugate gradients for linear systems of equations. It belongs to\n",
    "the class of iterative methods for solving problems from linear\n",
    "algebra of the type"
   ]
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   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{A}\\hat{x} = \\hat{b}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the iterative process we end up with a problem like"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{r}= \\hat{b}-\\hat{A}\\hat{x},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\hat{r}$ is the so-called residual or error in the iterative process.\n",
    "\n",
    "When we have found the exact solution, $\\hat{r}=0$.\n",
    "\n",
    "## Gradient method\n",
    "\n",
    "The residual is zero when we reach the minimum of the quadratic equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "P(\\hat{x})=\\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T\\hat{b},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with the constraint that the matrix $\\hat{A}$ is positive definite and\n",
    "symmetric.  This defines also the Hessian and we want it to be  positive definite.  \n",
    "\n",
    "\n",
    "## Steepest descent  method\n",
    "\n",
    "We denote the initial guess for $\\hat{x}$ as $\\hat{x}_0$. \n",
    "We can assume without loss of generality that"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{x}_0=0,\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or consider the system"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{A}\\hat{z} = \\hat{b}-\\hat{A}\\hat{x}_0,\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "instead.\n",
    "\n",
    "\n",
    "## Steepest descent  method\n",
    "One can show that the solution $\\hat{x}$ is also the unique minimizer of the quadratic form"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f(\\hat{x}) = \\frac{1}{2}\\hat{x}^T\\hat{A}\\hat{x} - \\hat{x}^T \\hat{x} , \\quad \\hat{x}\\in\\mathbf{R}^n.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This suggests taking the first basis vector $\\hat{r}_1$ (see below for definition) \n",
    "to be the gradient of $f$ at $\\hat{x}=\\hat{x}_0$, \n",
    "which equals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{A}\\hat{x}_0-\\hat{b},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and \n",
    "$\\hat{x}_0=0$ it is equal $-\\hat{b}$.\n",
    "\n",
    "\n",
    "\n",
    "## Final expressions\n",
    "We can compute the residual iteratively as"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{r}_{k+1}=\\hat{b}-\\hat{A}\\hat{x}_{k+1},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which equals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{b}-\\hat{A}(\\hat{x}_k+\\alpha_k\\hat{r}_k),\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "(\\hat{b}-\\hat{A}\\hat{x}_k)-\\alpha_k\\hat{A}\\hat{r}_k,\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which gives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\alpha_k = \\frac{\\hat{r}_k^T\\hat{r}_k}{\\hat{r}_k^T\\hat{A}\\hat{r}_k}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "leading to the iterative scheme"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{x}_{k+1}=\\hat{x}_k-\\alpha_k\\hat{r}_{k},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple codes for  steepest descent and conjugate gradient using a $2\\times 2$ matrix, in c++, Python code to come"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "        #include <cmath>\n",
    "        #include <iostream>\n",
    "        #include <fstream>\n",
    "        #include <iomanip>\n",
    "        #include \"vectormatrixclass.h\"\n",
    "        using namespace  std;\n",
    "        //   Main function begins here\n",
    "        int main(int  argc, char * argv[]){\n",
    "          int dim = 2;\n",
    "          Vector x(dim),xsd(dim), b(dim),x0(dim);\n",
    "          Matrix A(dim,dim);\n",
    "        \n",
    "          // Set our initial guess\n",
    "          x0(0) = x0(1) = 0;\n",
    "          // Set the matrix\n",
    "          A(0,0) =  3;    A(1,0) =  2;   A(0,1) =  2;   A(1,1) =  6;\n",
    "          b(0) = 2; b(1) = -8;\n",
    "          cout << \"The Matrix A that we are using: \" << endl;\n",
    "          A.Print();\n",
    "          cout << endl;\n",
    "          xsd = SteepestDescent(A,b,x0);\n",
    "          cout << \"The approximate solution using Steepest Descent is: \" << endl;\n",
    "          xsd.Print();\n",
    "          cout << endl;\n",
    "        }\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The routine for the steepest descent method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "        Vector SteepestDescent(Matrix A, Vector b, Vector x0){\n",
    "          int IterMax, i;\n",
    "          int dim = x0.Dimension();\n",
    "          const double tolerance = 1.0e-14;\n",
    "          Vector x(dim),f(dim),z(dim);\n",
    "          double c,alpha,d;\n",
    "          IterMax = 30;\n",
    "          x = x0;\n",
    "          r = A*x-b;\n",
    "          i = 0;\n",
    "          while (i <= IterMax){\n",
    "            z = A*r;\n",
    "            c = dot(r,r);\n",
    "            alpha = c/dot(r,z);\n",
    "            x = x - alpha*r;\n",
    "            r =  A*x-b;\n",
    "            if(sqrt(dot(r,r)) < tolerance) break;\n",
    "            i++;\n",
    "          }\n",
    "          return x;\n",
    "        }\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Steepest descent example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x109127ac8>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "\n",
    "import scipy.optimize as sopt\n",
    "\n",
    "import matplotlib.pyplot as pt\n",
    "from mpl_toolkits.mplot3d import axes3d\n",
    "\n",
    "def f(x):\n",
    "    return 0.5*x[0]**2 + 2.5*x[1]**2\n",
    "\n",
    "def df(x):\n",
    "    return np.array([x[0], 5*x[1]])\n",
    "\n",
    "fig = pt.figure()\n",
    "ax = fig.gca(projection=\"3d\")\n",
    "\n",
    "xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]\n",
    "fmesh = f(np.array([xmesh, ymesh]))\n",
    "ax.plot_surface(xmesh, ymesh, fmesh)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And then as countor plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pt.axis(\"equal\")\n",
    "pt.contour(xmesh, ymesh, fmesh)\n",
    "guesses = [np.array([2, 2./5])]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find guesses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = guesses[-1]\n",
    "s = -df(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run it!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.33333333 -0.26666667]\n"
     ]
    }
   ],
   "source": [
    "def f1d(alpha):\n",
    "    return f(x + alpha*s)\n",
    "\n",
    "alpha_opt = sopt.golden(f1d)\n",
    "next_guess = x + alpha_opt * s\n",
    "guesses.append(next_guess)\n",
    "print(next_guess)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happened?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x109c1e438>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pt.axis(\"equal\")\n",
    "pt.contour(xmesh, ymesh, fmesh, 50)\n",
    "it_array = np.array(guesses)\n",
    "pt.plot(it_array.T[0], it_array.T[1], \"x-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- !split  -->\n",
    "## Revisiting our first homework\n",
    "\n",
    "We will use linear regression as a case study for the gradient descent\n",
    "methods. Linear regression is a great test case for the gradient\n",
    "descent methods discussed in the lectures since it has several\n",
    "desirable properties such as:\n",
    "\n",
    "1. An analytical solution (recall homework set 1).\n",
    "\n",
    "2. The gradient can be computed analytically.\n",
    "\n",
    "3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates\n",
    "\n",
    "We revisit the example from homework set 1 where we had"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "y_i = 5x_i^2 + 0.1\\xi_i, \\ i=1,\\cdots,100\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with $x_i \\in [0,1] $ chosen randomly with a uniform distribution. Additionally $\\xi_i$ represents stochastic noise chosen according to a normal distribution $\\cal {N}(0,1)$. \n",
    "The linear regression model is given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "h_\\beta(x) = \\hat{y} = \\beta_0 + \\beta_1 x,\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "such that"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{y}_i = \\beta_0 + \\beta_1 x_i.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- !split  -->\n",
    "## Gradient descent example\n",
    "\n",
    "Let $\\mathbf{y} = (y_1,\\cdots,y_n)^T$, $\\mathbf{\\hat{y}} = (\\hat{y}_1,\\cdots,\\hat{y}_n)^T$ and $\\beta = (\\beta_0, \\beta_1)^T$\n",
    "\n",
    "It is convenient to write $\\mathbf{\\hat{y}} = X\\beta$ where $X \\in \\mathbb{R}^{100 \\times 2} $ is the design matrix given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "X \\equiv \\begin{bmatrix}\n",
    "1 &amp; x_1  \\\\\n",
    "\\vdots &amp; \\vdots  \\\\\n",
    "1 &amp; x_{100} &amp;  \\\\\n",
    "\\end{bmatrix}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The loss function is given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "C(\\beta) = ||X\\beta-\\mathbf{y}||^2 = ||X\\beta||^2 - 2 \\mathbf{y}^T X\\beta + ||\\mathbf{y}||^2 = \\sum_{i=1}^{100} (\\beta_0 + \\beta_1 x_i)^2 - 2 y_i (\\beta_0 + \\beta_1 x_i) + y_i^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and we want to find $\\beta$ such that $C(\\beta)$ is minimized.\n",
    "\n",
    "## The derivative of the cost/loss function\n",
    "\n",
    "Computing $\\partial C(\\beta) / \\partial \\beta_0$ and $\\partial C(\\beta) / \\partial \\beta_1$ we can show  that the gradient can be written as"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\nabla_{\\beta} C(\\beta) = (\\partial C(\\beta) / \\partial \\beta_0, \\partial C(\\beta) / \\partial \\beta_1)^T = 2\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n",
    "\\sum_{i=1}^{100}\\left( x_i (\\beta_0+\\beta_1x_i)-y_ix_i\\right) \\\\\n",
    "\\end{bmatrix} = 2X^T(X\\beta - \\mathbf{y}),\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $X$ is the design matrix defined above.\n",
    "\n",
    "## The Hessian matrix\n",
    "The Hessian matrix of $C(\\beta)$ is given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hat{H} \\equiv \\begin{bmatrix}\n",
    "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0^2} &amp; \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1}  \\\\\n",
    "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} &amp; \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_1^2} &amp;  \\\\\n",
    "\\end{bmatrix} = 2X^T X.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This result implies that $C(\\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.\n",
    "\n",
    "## Simple program\n",
    "\n",
    "We can now write a program that minimizes $C(\\beta)$ using the gradient descent method with a constant learning rate $\\gamma$ according to"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\beta_{k+1} = \\beta_k - \\gamma \\nabla_\\beta C(\\beta_k), \\ k=0,1,\\cdots\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the expression we computed for the gradient and let use a\n",
    "$\\beta_0$ be chosen randomly and let $\\gamma = 0.001$. Stop iterating\n",
    "when $||\\nabla_\\beta C(\\beta_k) || \\leq \\epsilon = 10^{-8}$. \n",
    "\n",
    "And finally we can compare our solution for $\\beta$ with the analytic result given by \n",
    "$\\beta= (X^TX)^{-1} X^T \\mathbf{y}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "\"\"\"\n",
    "The following setup is just a suggestion, feel free to write it the way you like.\n",
    "\"\"\"\n",
    "\n",
    "#Setup problem described in the exercise\n",
    "N  = 100 #Nr of datapoints\n",
    "M  = 2 #Nr of features\n",
    "x  = np.random.rand(N) #Uniformly generated x-values in [0,1]\n",
    "y  = 5*x**2 + 0.1*np.random.randn(N)\n",
    "X  = np.c_[np.ones(N),x] #Construct design matrix\n",
    "\n",
    "#Compute beta according to normal equations to compare with GD solution\n",
    "Xt_X_inv = np.linalg.inv(np.dot(X.T,X))\n",
    "Xt_y     = np.dot(X.transpose(),y)\n",
    "beta_NE = np.dot(Xt_X_inv,Xt_y)\n",
    "print(beta_NE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gradient Descent Example\n",
    "\n",
    "Another simple example is here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[3.96812677]\n",
      " [2.93472871]]\n",
      "[[3.96812677]\n",
      " [2.93472871]]\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "# Importing various packages\n",
    "from random import random, seed\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib import cm\n",
    "from matplotlib.ticker import LinearLocator, FormatStrFormatter\n",
    "import sys\n",
    "\n",
    "x = 2*np.random.rand(100,1)\n",
    "y = 4+3*x+np.random.randn(100,1)\n",
    "\n",
    "xb = np.c_[np.ones((100,1)), x]\n",
    "beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)\n",
    "print(beta_linreg)\n",
    "beta = np.random.randn(2,1)\n",
    "\n",
    "eta = 0.1\n",
    "Niterations = 1000\n",
    "m = 100\n",
    "\n",
    "for iter in range(Niterations):\n",
    "    gradients = 2.0/m*xb.T.dot(xb.dot(beta)-y)\n",
    "    beta -= eta*gradients\n",
    "\n",
    "print(beta)\n",
    "xnew = np.array([[0],[2]])\n",
    "xbnew = np.c_[np.ones((2,1)), xnew]\n",
    "ypredict = xbnew.dot(beta)\n",
    "ypredict2 = xbnew.dot(beta_linreg)\n",
    "plt.plot(xnew, ypredict, \"r-\")\n",
    "plt.plot(xnew, ypredict2, \"b-\")\n",
    "plt.plot(x, y ,'ro')\n",
    "plt.axis([0,2.0,0, 15.0])\n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel(r'$y$')\n",
    "plt.title(r'Gradient descent example')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## And a corresponding example using **scikit-learn**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Importing various packages\n",
    "from random import random, seed\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.linear_model import SGDRegressor\n",
    "\n",
    "x = 2*np.random.rand(100,1)\n",
    "y = 4+3*x+np.random.randn(100,1)\n",
    "\n",
    "xb = np.c_[np.ones((100,1)), x]\n",
    "beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(y)\n",
    "print(beta_linreg)\n",
    "sgdreg = SGDRegressor(n_iter = 50, penalty=None, eta0=0.1)\n",
    "sgdreg.fit(x,y.ravel())\n",
    "print(sgdreg.intercept_, sgdreg.coef_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- !split  -->\n",
    "## Gradient descent and Ridge\n",
    "\n",
    "We have also discussed Ridge regression where the loss function contains a regularized given by the $L_2$ norm of $\\beta$,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "C_{\\text{ridge}}(\\beta) = ||X\\beta -\\mathbf{y}||^2 + \\lambda ||\\beta||^2, \\ \\lambda \\geq 0.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to minimize $C_{\\text{ridge}}(\\beta)$ using GD we only have adjust the gradient as follows"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\nabla_\\beta C_{\\text{ridge}}(\\beta)  = 2\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n",
    "\\sum_{i=1}^{100}\\left( x_i (\\beta_0+\\beta_1x_i)-y_ix_i\\right) \\\\\n",
    "\\end{bmatrix} + 2\\lambda\\begin{bmatrix} \\beta_0 \\\\ \\beta_1\\end{bmatrix} = 2 (X^T(X\\beta - \\mathbf{y})+\\lambda \\beta).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now extend our program to minimize $C_{\\text{ridge}}(\\beta)$ using gradient descent and compare with the analytical solution given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\beta_{\\text{ridge}} = \\left(X^T X + \\lambda I_{2 \\times 2} \\right)^{-1} X^T \\mathbf{y},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "for $\\lambda = {0,1,10,50,100}$ ($\\lambda = 0$ corresponds to ordinary least squares). \n",
    "We can then compute $||\\beta_{\\text{ridge}}||$ for each $\\lambda$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "\"\"\"\n",
    "The following setup is just a suggestion, feel free to write it the way you like.\n",
    "\"\"\"\n",
    "\n",
    "#Setup problem described in the exercise\n",
    "N  = 100 #Nr of datapoints\n",
    "M  = 2   #Nr of features\n",
    "x  = np.random.rand(N)\n",
    "y  = 5*x**2 + 0.1*np.random.randn(N)\n",
    "\n",
    "\n",
    "#Compute analytic beta for Ridge regression \n",
    "X    = np.c_[np.ones(N),x]\n",
    "XT_X = np.dot(X.T,X)\n",
    "\n",
    "l  = 0.1 #Ridge parameter lambda\n",
    "Id = np.eye(XT_X.shape[0])\n",
    "\n",
    "Z = np.linalg.inv(XT_X+l*Id)\n",
    "beta_ridge = np.dot(Z,np.dot(X.T,y))\n",
    "\n",
    "print(beta_ridge)\n",
    "print(np.linalg.norm(beta_ridge)) #||beta||"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Automatic differentiation\n",
    "Python has tools for so-called **automatic differentiation**.\n",
    "Consider the following example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f(x) = \\sin\\left(2\\pi x + x^2\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which has the following derivative"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f'(x) = \\cos\\left(2\\pi x + x^2\\right)\\left(2\\pi + 2x\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using **autograd** we have"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The max absolute difference is: 1.77636e-15\n"
     ]
    }
   ],
   "source": [
    "import autograd.numpy as np\n",
    "\n",
    "# To do elementwise differentiation:\n",
    "from autograd import elementwise_grad as egrad \n",
    "\n",
    "# To plot:\n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "\n",
    "def f(x):\n",
    "    return np.sin(2*np.pi*x + x**2)\n",
    "\n",
    "def f_grad_analytic(x):\n",
    "    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)\n",
    "\n",
    "# Do the comparison:\n",
    "x = np.linspace(0,1,1000)\n",
    "\n",
    "f_grad = egrad(f)\n",
    "\n",
    "computed = f_grad(x)\n",
    "analytic = f_grad_analytic(x)\n",
    "\n",
    "plt.title('Derivative computed from Autograd compared with the analytical derivative')\n",
    "plt.plot(x,computed,label='autograd')\n",
    "plt.plot(x,analytic,label='analytic')\n",
    "\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.legend()\n",
    "\n",
    "plt.show()\n",
    "\n",
    "print(\"The max absolute difference is: %g\"%(np.max(np.abs(computed - analytic))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- !split  -->\n",
    "## Using autograd\n",
    "\n",
    "Here we\n",
    "experiment with what kind of functions Autograd is capable\n",
    "of finding the gradient of. The following Python functions are just\n",
    "meant to illustrate what Autograd can do, but please feel free to\n",
    "experiment with other, possibly more complicated, functions as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The gradient of f1 evaluated at a = 1 using autograd is: 3\n",
      "The gradient of f1 evaluated at a = 1 by finding the analytic expression is: 3\n"
     ]
    }
   ],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "\n",
    "def f1(x):\n",
    "    return x**3 + 1\n",
    "\n",
    "f1_grad = grad(f1)\n",
    "\n",
    "# Remember to send in float as argument to the computed gradient from Autograd!\n",
    "a = 1.0\n",
    "\n",
    "# See the evaluated gradient at a using autograd:\n",
    "print(\"The gradient of f1 evaluated at a = %g using autograd is: %g\"%(a,f1_grad(a)))\n",
    "\n",
    "# Compare with the analytical derivative, that is f1'(x) = 3*x**2 \n",
    "grad_analytical = 3*a**2\n",
    "print(\"The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g\"%(a,grad_analytical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Autograd with more complicated functions\n",
    "\n",
    "To differentiate with respect to two (or more) arguments of a Python\n",
    "function, Autograd need to know at which variable the function if\n",
    "being differentiated with respect to."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "def f2(x1,x2):\n",
    "    return 3*x1**3 + x2*(x1 - 5) + 1\n",
    "\n",
    "# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1\n",
    "f2_grad_x1 = grad(f2,0)\n",
    "\n",
    "# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad\n",
    "f2_grad_x2 = grad(f2,1)\n",
    "\n",
    "x1 = 1.0\n",
    "x2 = 3.0 \n",
    "\n",
    "print(\"Evaluating at x1 = %g, x2 = %g\"%(x1,x2))\n",
    "print(\"-\"*30)\n",
    "\n",
    "# Compare with the analytical derivatives:\n",
    "\n",
    "# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:\n",
    "f2_grad_x1_analytical = 9*x1**2 + x2\n",
    "\n",
    "# Derivative of f2 w.r.t x2 is: x1 - 5:\n",
    "f2_grad_x2_analytical = x1 - 5\n",
    "\n",
    "# See the evaluated derivations:\n",
    "print(\"The derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n",
    "print(\"The analytical derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n",
    "\n",
    "print()\n",
    "\n",
    "print(\"The derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))\n",
    "print(\"The analytical derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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.\n",
    "\n",
    "\n",
    "## More complicated functions using the elements of their arguments directly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "def f3(x): # Assumes x is an array of length 5 or higher\n",
    "    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2\n",
    "\n",
    "f3_grad = grad(f3)\n",
    "\n",
    "x = np.linspace(0,4,5)\n",
    "\n",
    "# Print the computed gradient:\n",
    "print(\"The computed gradient of f3 is: \", f3_grad(x))\n",
    "\n",
    "# The analytical gradient is: (2, 3, 5, 7, 22*x[4])\n",
    "f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])\n",
    "\n",
    "# Print the analytical gradient:\n",
    "print(\"The analytical gradient of f3 is: \", f3_grad_analytical)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that in this case, when sending an array as input argument, the\n",
    "output from Autograd is another array. This is the true gradient of\n",
    "the function, as opposed to the function in the previous example. By\n",
    "using arrays to represent the variables, the output from Autograd\n",
    "might be easier to work with, as the output is closer to what one\n",
    "could expect form a gradient-evaluting function.\n",
    "\n",
    "<!-- !split  -->\n",
    "## Functions using mathematical functions from Numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "def f4(x):\n",
    "    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)\n",
    "\n",
    "f4_grad = grad(f4)\n",
    "\n",
    "x = 2.7\n",
    "\n",
    "# Print the computed derivative:\n",
    "print(\"The computed derivative of f4 at x = %g is: %g\"%(x,f4_grad(x)))\n",
    "\n",
    "# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi\n",
    "f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi\n",
    "\n",
    "# Print the analytical gradient:\n",
    "print(\"The analytical gradient of f4 at x = %g is: %g\"%(x,f4_grad_analytical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More autograd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "def f5(x):\n",
    "    if x >= 0:\n",
    "        return x**2\n",
    "    else:\n",
    "        return -3*x + 1\n",
    "\n",
    "f5_grad = grad(f5)\n",
    "\n",
    "x = 2.7\n",
    "\n",
    "# Print the computed derivative:\n",
    "print(\"The computed derivative of f5 at x = %g is: %g\"%(x,f5_grad(x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## And  with loops"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1\n",
    "7\n",
    " \n",
    "<\n",
    "<\n",
    "<\n",
    "!\n",
    "!\n",
    "C\n",
    "O\n",
    "D\n",
    "E\n",
    "_\n",
    "B\n",
    "L\n",
    "O\n",
    "C\n",
    "K\n",
    " \n",
    " \n",
    "p\n",
    "y\n",
    "c\n",
    "o\n",
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9\n",
    "# The analytical derivative is: sum(i*x**(i-1)) \n",
    "f6_grad_analytical = 0\n",
    "for i in range(10):\n",
    "    f6_grad_analytical += i*x**(i-1)\n",
    "\n",
    "print(\"The analytical derivative of f6 at x = %g is: %g\"%(x,f6_grad_analytical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using recursion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "\n",
    "def f7(n): # Assume that n is an integer\n",
    "    if n == 1 or n == 0:\n",
    "        return 1\n",
    "    else:\n",
    "        return n*f7(n-1)\n",
    "\n",
    "f7_grad = grad(f7)\n",
    "\n",
    "n = 2.0\n",
    "\n",
    "print(\"The computed derivative of f7 at n = %d is: %g\"%(n,f7_grad(n)))\n",
    "\n",
    "# The function f7 is an implementation of the factorial of n.\n",
    "# By using the product rule, one can find that the derivative is:\n",
    "\n",
    "f7_grad_analytical = 0\n",
    "for i in range(int(n)-1):\n",
    "    tmp = 1\n",
    "    for k in range(int(n)-1):\n",
    "        if k != i:\n",
    "            tmp *= (n - k)\n",
    "    f7_grad_analytical += tmp\n",
    "\n",
    "print(\"The analytical derivative of f7 at n = %d is: %g\"%(n,f7_grad_analytical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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.\n",
    "\n",
    "## Unsupported functions\n",
    "Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.\n",
    "\n",
    "Assigning a value to the variable being differentiated with respect to"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "def f8(x): # Assume x is an array\n",
    "    x[2] = 3\n",
    "    return x*2\n",
    "\n",
    "f8_grad = grad(f8)\n",
    "\n",
    "x = 8.4\n",
    "\n",
    "print(\"The derivative of f8 is:\",f8_grad(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.\n",
    "\n",
    "## The syntax a.dot(b) when finding the dot product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "def f9(a): # Assume a is an array with 2 elements\n",
    "    b = np.array([1.0,2.0])\n",
    "    return a.dot(b)\n",
    "\n",
    "f9_grad = grad(f9)\n",
    "\n",
    "x = np.array([1.0,0.0])\n",
    "\n",
    "print(\"The derivative of f9 is:\",f9_grad(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we are told that the 'dot' function does not belong to Autograd's\n",
    "version of a Numpy array.  To overcome this, an alternative syntax\n",
    "which also computed the dot product can be used:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "import autograd.numpy as np\n",
    "from autograd import grad\n",
    "def f9_alternative(x): # Assume a is an array with 2 elements\n",
    "    b = np.array([1.0,2.0])\n",
    "    return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2\n",
    "\n",
    "f9_alternative_grad = grad(f9_alternative)\n",
    "\n",
    "x = np.array([3.0,0.0])\n",
    "\n",
    "print(\"The gradient of f9 is:\",f9_alternative_grad(x))\n",
    "\n",
    "# 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\n",
    "# w.r.t x is (b_1, b_2)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Recommended to avoid\n",
    "The documentation recommends to avoid inplace operations such as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "a += b\n",
    "a -= b\n",
    "a*= b\n",
    "a /=b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stochastic Gradient Descent\n",
    "\n",
    "Stochastic gradient descent (SGD) and variants thereof address some of\n",
    "the shortcomings of the Gradient descent method discussed above.\n",
    "\n",
    "The underlying idea of SGD comes from the observation that the cost\n",
    "function, which we want to minimize, can almost always be written as a\n",
    "sum over $n$ data points $\\{\\mathbf{x}_i\\}_{i=1}^n$,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "C(\\mathbf{\\beta}) = \\sum_{i=1}^n c_i(\\mathbf{x}_i,\n",
    "\\mathbf{\\beta}).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computation of gradients\n",
    "\n",
    "This in turn means that the gradient can be\n",
    "computed as a sum over $i$-gradients"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\nabla_\\beta C(\\mathbf{\\beta}) = \\sum_i^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n",
    "\\mathbf{\\beta}).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stochasticity/randomness is introduced by only taking the\n",
    "gradient on a subset of the data called minibatches.  If there are $n$\n",
    "data points and the size of each minibatch is $M$, there will be $n/M$\n",
    "minibatches. We denote these minibatches by $B_k$ where\n",
    "$k=1,\\cdots,n/M$.\n",
    "\n",
    "## SGD example\n",
    "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n",
    "and we choose to have $M=5$ minibathces,\n",
    "then each minibatch contains two data points. In particular we have\n",
    "$B_1 = (\\mathbf{x}_1,\\mathbf{x}_2), \\cdots, B_5 =\n",
    "(\\mathbf{x}_9,\\mathbf{x}_{10})$. Note that if you choose $M=1$ you\n",
    "have only a single batch with all data points and on the other extreme,\n",
    "you may choose $M=n$ resulting in a minibatch for each datapoint, i.e\n",
    "$B_k = \\mathbf{x}_k$.\n",
    "\n",
    "The idea is now to approximate the gradient by replacing the sum over\n",
    "all data points with a sum over the data points in one the minibatches\n",
    "picked at random in each gradient descent step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\nabla_{\\beta}\n",
    "C(\\mathbf{\\beta}) = \\sum_{i=1}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n",
    "\\mathbf{\\beta}) \\rightarrow \\sum_{i \\in B_k}^n \\nabla_\\beta\n",
    "c_i(\\mathbf{x}_i, \\mathbf{\\beta}).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The gradient step\n",
    "\n",
    "Thus a gradient descent step now looks like"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\beta_{j+1} = \\beta_j - \\gamma_j \\sum_{i \\in B_k}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n",
    "\\mathbf{\\beta})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $k$ is picked at random with equal\n",
    "probability from $[1,n/M]$. An iteration over the number of\n",
    "minibathces (n/M) is commonly referred to as an epoch. Thus it is\n",
    "typical to choose a number of epochs and for each epoch iterate over\n",
    "the number of minibatches, as exemplified in the code below.\n",
    "\n",
    "## Simple example code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np \n",
    "\n",
    "n = 100 #100 datapoints \n",
    "M = 5   #size of each minibatch\n",
    "m = int(n/M) #number of minibatches\n",
    "n_epochs = 10 #number of epochs\n",
    "\n",
    "j = 0\n",
    "for epoch in range(1,n_epochs+1):\n",
    "    for i in range(m):\n",
    "        k = np.random.randint(m) #Pick the k-th minibatch at random\n",
    "        #Compute the gradient using the data in minibatch Bk\n",
    "        #Compute new suggestion for \n",
    "        j += 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Taking the gradient only on a subset of the data has two important\n",
    "benefits. First, it introduces randomness which decreases the chance\n",
    "that our opmization scheme gets stuck in a local minima. Second, if\n",
    "the size of the minibatches are small relative to the number of\n",
    "datapoints ($M <  n$), the computation of the gradient is much\n",
    "cheaper since we sum over the datapoints in the $k-th$ minibatch and not\n",
    "all $n$ datapoints.\n",
    "\n",
    "## When do we stop?\n",
    "\n",
    "A natural question is when do we stop the search for a new minimum?\n",
    "One possibility is to compute the full gradient after a given number\n",
    "of epochs and check if the norm of the gradient is smaller than some\n",
    "threshold and stop if true. However, the condition that the gradient\n",
    "is zero is valid also for local minima, so this would only tell us\n",
    "that we are close to a local/global minimum. However, we could also\n",
    "evaluate the cost function at this point, store the result and\n",
    "continue the search. If the test kicks in at a later stage we can\n",
    "compare the values of the cost function and keep the $\\beta$ that\n",
    "gave the lowest value.\n",
    "\n",
    "## Slightly different approach\n",
    "\n",
    "Another approach is to let the step length $\\gamma_j$ depend on the\n",
    "number of epochs in such a way that it becomes very small after a\n",
    "reasonable time such that we do not move at all.\n",
    "\n",
    "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* $t$.\n",
    "\n",
    "In this way we can fix the number of epochs, compute $\\beta$ and\n",
    "evaluate the cost function at the end. Repeating the computation will\n",
    "give a different result since the scheme is random by design. Then we\n",
    "pick the final $\\beta$ that gives the lowest value of the cost\n",
    "function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np \n",
    "\n",
    "def step_length(t,t0,t1):\n",
    "    return t0/(t+t1)\n",
    "\n",
    "n = 100 #100 datapoints \n",
    "M = 5   #size of each minibatch\n",
    "m = int(n/M) #number of minibatches\n",
    "n_epochs = 500 #number of epochs\n",
    "t0 = 1.0\n",
    "t1 = 10\n",
    "\n",
    "gamma_j = t0/t1\n",
    "j = 0\n",
    "for epoch in range(1,n_epochs+1):\n",
    "    for i in range(m):\n",
    "        k = np.random.randint(m) #Pick the k-th minibatch at random\n",
    "        #Compute the gradient using the data in minibatch Bk\n",
    "        #Compute new suggestion for beta\n",
    "        t = epoch*m+i\n",
    "        gamma_j = step_length(t,t0,t1)\n",
    "        j += 1\n",
    "\n",
    "print(\"gamma_j after %d epochs: %g\" % (n_epochs,gamma_j))"
   ]
  }
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