FPA/FixedPointTheory.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Make sure we can plot things\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Make sure we load the root containing the algos\n",
    "import os\n",
    "import sys\n",
    "sys.path.insert(0, os.path.abspath('../'))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fixed point iterations\n",
    "=================\n",
    "\n",
    "This notebook is about Banach's fixed point iterations. \n",
    "\n",
    "In a nutshell: \n",
    "* Solves equations of the form $f(x) = x$. \n",
    "* Does not require the evaluation of a gradient\n",
    "* Converges to the unique local minimum (under some conditions)\n",
    "\n",
    "The gist of the algorithms is to reduce the search space little by little, by iterated applications of a constraction. \n",
    "Here, $f: X \\to X$ is a contraction from the non-empty complete metrix space $X$ to itself. \n",
    "A contraction means that the distance between two points in $X$ will always shrink when passing through the mapping $f$. More formally, $f$ is a contraction if there exists a constant $L \\in (0,1)$ such that \n",
    "$$\n",
    "    d(f(x),f(y)) \\leq L\\dot d(x,y)\n",
    "$$\n",
    "for all $x,y \\in X$. \n",
    "\n",
    "Informally, assuming we have picked a point $x^{(0)} \\in X$ (remember that $x$ may be a vector, not necessarily a simple scalar), that $x_*$ is a fixed point (i.e. such that $f(x_*) = x_*$, and that $f$ is a contraction with constant $0 < L < 1$, it follows that \n",
    "$$\n",
    "d(f(x^{(0)}), f(x_*)) \\leq L d(x^{(0)},x_*). \n",
    "$$\n",
    "Similarly, letting $x^{(1)} := f(x^{(0)})$, we have \n",
    "$$\n",
    "d(f^2(x^{(0)}), x_*) = d(f^2(x^{(0)}), f^2(x_*)) = d(f(x^{(1)}), f(x_*))  \\leq L d(x^{(1)},x_*) \\leq L^2 d(x^{(0)},x_*). \n",
    "$$\n",
    "It is easy to verify that $d(x^{(n)}, x_*) \\leq L^n d(x^{(0)},x_*)$ in which we have defined $x^{(n)}$ as the result of the iterated application of the mapping $f$, i.e. $x^{(n)} = f \\circ f \\circ \\cdots \\circ f(x^{(0)})$.\n",
    "\n",
    "What this tells us, is that every time we apply $f$ to the previous results, we get closer to the fixed point. How much closer? Well, $L$ times closer in the sense that \n",
    "$$\n",
    "\\frac{d(x^{(n+1)}, x_*)}{d(x^{(n)}, x_*)} = \\frac{d(f(x^{(n)}), x_*)}{d(x^{(n)}, x_*)} \\leq L\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Some import used for the algos and examples \n",
    "import numpy as np\n",
    "\n",
    "# Display the iterated points getting closer to the fixed point"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As one can see from the graph above, we are indeed getting closer and closer to the solution as we iteratively apply the function (in this case, we have taken $f: $ XXX and $x^{(0)} = YYY$).\n",
    "\n",
    "## Theoretical results: Banach Fixed Point theorem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0000000000000002\n"
     ]
    }
   ],
   "source": [
    "# And keep just the one algo we'll be checking in this notebook\n",
    "from Algos.FixedPointAlgos.BanachFixedPoint import FPAlgo as FPAlgo\n",
    "\n",
    "functionToSolve = lambda x: np.sqrt(x)\n",
    "x0 = 2\n",
    "nbIter = 50\n",
    "\n",
    "myFPIterations = FPAlgo(functionToSolve, x0, nbIter)\n",
    "\n",
    "print(myFPIterations.solve())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
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     "execution_count": 18,
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   "source": [
    "plt.plot(range(0,nbIter+1), myFPIterations.estimates)"
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   "metadata": {},
   "source": [
    "It is usually better to display in log-log plots to understand convergences of algorithms: (conclude on what this tells us on the speed of the convergence?)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fecac161c88>]"
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     "output_type": "execute_result"
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   "source": [
    "# We know what to expect here! \n",
    "xstar = 1\n",
    "\n",
    "plt.plot(range(0,nbIter+1), np.log(np.abs([xstar - xn for xn in myFPIterations.estimates])))"
   ]
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   "execution_count": 20,
   "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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    "sinFunction = lambda x: np.sin(x)\n",
    "x0 = np.pi/2\n",
    "nbIter_sin = 50000\n",
    "\n",
    "sinesFPA = FPAlgo(sinFunction, x0, nbIter_sin)\n",
    "\n",
    "print(sinesFPA.solve())\n",
    "\n",
    "cosFunction = lambda x: np.cos(x)\n",
    "x0 = 0.0\n",
    "nbIter = 50\n",
    "\n",
    "cosFPA = FPAlgo(cosFunction, x0, nbIter)\n",
    "\n",
    "print(cosFPA.solve())\n",
    "\n",
    "x0 = np.pi/2\n",
    "cosFPA = FPAlgo(cosFunction, x0, 10*nbIter)\n",
    "\n",
    "print(cosFPA.solve())\n",
    "\n",
    "x0 = 0.5\n",
    "cosFPA = FPAlgo(cosFunction, x0, nbIter)\n",
    "\n",
    "print(cosFPA.solve())"
   ]
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    "xstar = 0.0\n",
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    "plt.plot(range(0,nbIter_sin+1), np.log(np.abs([xstar - xn for xn in sinesFPA.estimates])))"
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