project_m2_optim_data_science_2017_ocsvm.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PROJECT: Course Optimization for Data Science\n",
    "## Optimization strategies for anomaly detection\n",
    "## with One class Support Vector Machines (OCSVM)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Author: Alexandre Gramfort\n",
    "\n",
    "If you have questions or if something is not clear in the text below please contact us\n",
    "by email.\n",
    "\n",
    "## Aim:\n",
    "\n",
    "- derive the dual for the one-class SVM model\n",
    "- implement a one-class SVM using a blackbox convex toolbox (cvxopt in Python)\n",
    "- implement your own solvers with: Proximal gradient, Coordinate Descent, Quasi-Newton\n",
    "- Present a clear benchmark of the different strategies on small and medium scale datasets\n",
    "\n",
    "\n",
    "## VERY IMPORTANT\n",
    "\n",
    "This work must be done by pairs of students.\n",
    "Each student must send their work before the 7th of January at 23:59, using the moodle platform.\n",
    "This means that **each student in the pair sends the same file**\n",
    "\n",
    "On the moodle, in the \"Optimization for Data Science\" course, you have a \"devoir\" section called \"Project\".\n",
    "This is where you submit your jupyter notebook file.\n",
    "\n",
    "The name of the file must be constructed as in the next cell\n",
    "\n",
    "### Gentle reminder: no evaluation if you don't respect this EXACTLY\n",
    "\n",
    "#### How to construct the name of your file"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "project_gower_robert_and_gramfort_alexandre.ipynb\n"
     ]
    }
   ],
   "source": [
    "# Change here using YOUR first and last names\n",
    "fn1 = \"robert\"\n",
    "ln1 = \"gower\"\n",
    "fn2 = \"alexandre\"\n",
    "ln2 = \"gramfort\"\n",
    "\n",
    "filename = \"_\".join(map(lambda s: s.strip().lower(), \n",
    "                        [\"project\", ln1, fn1, \"and\", ln2, fn2])) + \".ipynb\"\n",
    "print(filename)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from scipy import linalg\n",
    "import time\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Important:\n",
    "\n",
    "For Part 0 to Part 2 of the project you will need a working install of `cvxopt`.\n",
    "You may struggle a bit to set it up.\n",
    "The simplest way of getting it is by typing \n",
    "\n",
    "`conda install cvxopt`\n",
    "\n",
    "or\n",
    "\n",
    "`pip install cvxopt`\n",
    "\n",
    "if you have `conda` or `pip` installed on your laptop.\n",
    "If you **struggle too much please\n",
    "contact us**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 0: One-class SVM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The original formulation of the One-Class SVM introduced by Schölkopf [1] is given by\n",
    "\n",
    "$$\n",
    "    (\\mathcal{P}): \\left\\{\n",
    "\t\\begin{aligned}\n",
    "\t\\min_{w, \\rho, \\xi} & \\quad \\frac{1}{2}\\Vert w \\Vert^2 - \\rho + \\frac{1}{\\nu n} \\sum_{i=1}^n \\xi_i \\\\\n",
    "\t\\text{s.t.} & \\quad \\langle w, x_i \\rangle \\geq \\rho - \\xi_i \\quad 1 \\leq i \\leq n \\\\\n",
    "\t& \\quad \\xi_i \\geq 0 \\quad 1 \\leq i \\leq n\n",
    "\t\\end{aligned}\n",
    "    \\right.\n",
    "$$\n",
    "\n",
    "where $\\nu \\in (0, 1]$ is a user-specified parameter controlling the proportion of outliers and the proportion of support vectors.\n",
    "\n",
    "Getting rid of the slack variables $\\xi_i, 1 \\leq i \\leq n$ one can also obtain the following equivalent optimization problem\n",
    "\n",
    "$$\n",
    "\t\\min_{w, \\rho} \\frac{1}{2}\\Vert w \\Vert^2 - \\rho + \\frac{1}{\\nu n} \\sum_{i=1}^n \\max(0, \\rho - \\langle w, x_i \\rangle) \\, .\n",
    "$$\n",
    "\n",
    "The idea is that the lower is $\\langle w, x_i \\rangle$ is the more $x_i$ is an anomaly and if $\\langle w, x_i \\rangle \\geq \\rho$ then the model predicts the sample is normal. $\\langle w, x_i \\rangle$ corresponds to the decision function and $\\textrm{sign}(\\langle w, x_i \\rangle - \\rho)$ as the prediction (1 means normal sample).\n",
    "\n",
    "[1] Schölkopf et al. Estimating the support of a high-dimensional distribution. Neural Computation. 2001"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 1:</b>\n",
    "     <ul>\n",
    "      <li>Justify that the problem $(\\mathcal{P})$ is convex.</li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(\\mathcal{P})$ s'écrit de la forme (si $w \\in \\mathbb{R}^p$)\n",
    "$$\n",
    "\\left\\lbrace\n",
    "\\begin{aligned}\n",
    "\\min_{(w,\\rho,\\xi) \\in \\mathbb{R}^p \\times \\mathbb{R} \\times \\mathbb{R}^n} & \\quad f(w, \\rho, \\xi) \\\\\n",
    "\\text{s.t.} & \\quad g(x) \\preceq 0\n",
    "\\end{aligned}\n",
    "\\right.\n",
    "$$\n",
    "avec \n",
    "$$\n",
    "\\begin{aligned}\n",
    "& f : (w,\\rho,\\xi) \\mapsto \\frac{1}{2}\\Vert w \\Vert^2 - \\rho + \\frac{1}{\\nu n} 1_n^\\top \\xi \\\\\n",
    "& g : (w,\\rho,\\xi) \\mapsto \\begin{bmatrix} \n",
    "\\rho - \\xi_1 - \\langle w, x_1 \\rangle \\\\\n",
    "\\vdots \\\\\n",
    "\\rho - \\xi_n - \\langle w, x_n \\rangle \\\\\n",
    "-\\xi_1 \\\\\n",
    "\\vdots \\\\\n",
    "-\\xi_n \n",
    "\\end{bmatrix} \\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "De plus  \n",
    "- $\\forall (w,\\rho,\\xi) \\in \\mathbb{R}^d \\times \\mathbb{R} \\times \\mathbb{R}^n$,\n",
    "$$\n",
    "\\nabla f(w, \\rho, \\xi) = \n",
    "\\begin{bmatrix}\n",
    "w \\\\ -1 \\\\ \\frac{1}{\\nu n} 1_n\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "et \n",
    "$$\n",
    "\\nabla^2 f(w, \\rho, \\xi) = \n",
    "\\begin{bmatrix}\n",
    "I_p & 0_{p, 1} & 0_{p,n} \\\\\n",
    "0_{1,p}  & 0 & 0_{1,n} \\\\\n",
    "0_{n,p} & 0_{n,1} & 0_n \n",
    "\\end{bmatrix}\n",
    "\\succeq 0\n",
    "$$\n",
    "Donc $f$ est convexe\n",
    "- $g = [g_1, \\cdots, g_{2n}]^\\top$ avec $\\forall (w,\\rho,\\xi) \\in \\mathbb{R}^d \\times \\mathbb{R} \\times \\mathbb{R}^n$,\n",
    "\\begin{eqnarray*}\n",
    "\\forall 1 \\leq i \\leq n, & \\quad g_i : (w,\\rho, \\xi) &\\mapsto \\rho - \\xi_i - \\langle w, x_i \\rangle & \\text{ est linéaire donc convexe} \\\\\n",
    "\\forall n+1 \\leq i \\leq 2n, & \\quad g_i : (w,\\rho, \\xi) &\\mapsto - \\xi_i & \\text{ est linéaire donc convexe}\n",
    "\\end{eqnarray*}\n",
    "\n",
    "Donc $(\\mathcal{P})$ est un problème d'optimisation convexe."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 2:</b>\n",
    "     <ul>\n",
    "      <li>\n",
    "      By looking at the saddle points of the Lagrangian $\\mathcal{L}(w, \\rho, \\mu)$ of $\\mathcal{P}$, $\\mu \\in \\mathbb{R}_+^n$, show that the dual problem $(\\mathcal{D})$ can be written as:\n",
    "$$\n",
    "(\\mathcal{D}): \n",
    " \\left\\{\n",
    " \\begin{array}{lll}\n",
    " \\min_{\\mu} &\\frac{1}{2}\\mu^{\\top}GG^{t}\\mu\n",
    " \\\\\n",
    " \\mathrm{s.t.}& \\mu^{\\top}1_n = 1\n",
    " \\\\\n",
    " \\mathrm{and}& 0 \\leq \\mu \\leq \\frac{1}{\\nu n}\n",
    " \\end{array}\n",
    " \\right .\n",
    "$$\n",
    "\n",
    "with\n",
    "\n",
    "$$\n",
    " G = \\begin{bmatrix}x_{1}^{\\top} \\\\ \\vdots \\\\ x_{n}^{\\top}\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "and $1_n = (1, \\dots, 1) \\in \\mathbb{R}^n$.\n",
    "    </li>\n",
    "    </ul>\n",
    "</div>\n",
    "\n",
    "We will **assume here qualification of the contraints** and that **strong duality holds**.\n",
    "\n",
    "**Remark:** The problem $(\\mathcal{D})$ is a *quadratic program* (QP) for which their exist off-the-shelf techniques. See below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Lagrangien \n",
    "$\\forall (w, \\rho, \\xi, \\mu, \\phi) \\in \\mathbb{R}^p \\times \\mathbb{R} \\times \\mathbb{R}^n \\times \\mathbb{R}_+^n \\times \\mathbb{R}_+^n$,\n",
    "\\begin{eqnarray*}\n",
    "\\mathcal{L}(w, \\rho, \\xi, \\phi, \\mu) \n",
    "&=&  \\frac{1}{2}\\Vert w \\Vert^2 - \\rho + \\frac{1}{\\nu n} \\sum_{i=1}^n \\xi_i + \\sum_{i=1}^{n} \\mu_i (\\rho - \\xi_i - \\langle w, x_i \\rangle) - \\sum_{i=1}^n \\phi_i \\xi_i \\\\\n",
    "&=& \\frac{1}{2}\\Vert w \\Vert^2 + \\left (\\sum_{i=1}^n \\mu_i - 1 \\right) \\rho + \\sum_{i=1}^n \\left( \\frac{1}{\\nu n} - \\mu_i - \\phi_i \\right) \\xi_i  - \\left\\langle w, \\sum_{i=1}^{n} \\mu_i x_i \\right\\rangle \\\\\n",
    "&=& \\frac{1}{2}\\Vert w \\Vert^2 + \\left (\\mu^\\top 1_n  - 1 \\right) \\rho + \\left\\langle \\frac{1}{\\nu n} 1_n - \\mu - \\phi , \\xi \\right\\rangle  - \\left\\langle w, \\sum_{i=1}^{n} \\mu_i x_i \\right\\rangle\n",
    "\\end{eqnarray*}\n",
    "\n",
    "#### Fonction duale\n",
    "La fonction duale associée à $(\\mathcal{P})$ est $\\delta : (\\phi, \\mu) \\in \\mathbb{R}_+^n \\times \\mathbb{R}_+^n \\mapsto \\inf_{w, \\rho, \\xi} \\mathcal{L}(w, \\rho, \\xi, \\phi, \\mu)$. Or\n",
    "$$\n",
    "\\nabla_{(w,\\rho,\\xi)} \\mathcal{L}(w,\\rho,\\xi,\\mu,\\phi) =\n",
    "\\begin{bmatrix}\n",
    "w - \\sum_{i=1}^n \\mu_i x_i \\\\ \\mu^\\top 1_n - 1 \\\\ \\frac{1}{\\nu n} 1_n - \\mu - \\phi \n",
    "\\end{bmatrix} \n",
    "$$\n",
    "et \n",
    "$$\n",
    "\\nabla_{(w,\\rho,\\xi)}^2 \\mathcal{L}(w,\\rho,\\xi,\\mu,\\phi) =\n",
    "\\begin{bmatrix}\n",
    "I_p & 0_{p, 1} & 0_{p,n} \\\\\n",
    "0_{1,p}  & 0 & 0_{1,n} \\\\\n",
    "0_{n,p} & 0_{n,1} & 0_n \n",
    "\\end{bmatrix}\n",
    "\\succeq 0\n",
    "$$\n",
    "Donc pour tout $(\\mu, \\phi) \\in \\mathbb{R}_+^n \\times \\mathbb{R}_+^n, (w,\\rho,\\xi) \\mapsto \\mathcal{L}(w,\\rho,\\xi,\\mu,\\phi)$ est convexe donc $\\delta(\\mu, \\phi) = \\mathcal{L}(w^*,\\rho^*,\\xi^*,\\mu,\\phi)$ avec \n",
    "$$\n",
    "\\nabla_{(w,\\rho,\\xi)} \\mathcal{L}(w^*,\\rho^*,\\xi^*,\\mu,\\phi) = 0 \n",
    "\\iff\n",
    "\\left\\lbrace \n",
    "\\begin{array}{l}\n",
    "w = \\sum_{i=1}^n \\mu_i x_i \\\\\n",
    "\\mu^\\top 1_n = 1 \\\\\n",
    "\\phi = \\frac{1}{\\nu n} 1_n - \\mu\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "On remarque notamment que pour que le lagrangien ait un minimum, il faut que $\\phi = \\frac{1}{\\nu n} 1_n - \\mu$ et $\\mu^\\top 1_n = 1$, dans le cas contraire $\\delta(\\phi, \\mu) = -\\infty$ car le lagrangien n'est pas minoré. Donc finalement $\\forall (\\mu, \\phi) \\in \\mathbb{R}_+^n \\times \\mathbb{R}_+^n$,\n",
    "\n",
    "$$\n",
    "\\delta(\\phi, \\mu) = \n",
    "\\left\\lbrace \n",
    "\\begin{array}{ll}\n",
    "- \\frac{1}{2} \\Vert \\sum_{i=1}^n \\mu_i x_i \\Vert^2 \n",
    "& \\text{si }  \\phi = \\frac{1}{\\nu n} 1_n - \\mu \\text{ et } \\mu^\\top 1_n = 1 \\\\\n",
    "- \\infty  & \\text{sinon}\n",
    "\\end{array}\n",
    "\\right.\n",
    "=\n",
    "\\left\\lbrace \n",
    "\\begin{array}{ll}\n",
    "- \\frac{1}{2} \\mu^\\top G G^\\top \\mu\n",
    "& \\text{si }  \\phi = \\frac{1}{\\nu n} 1_n - \\mu \\text{ et } \\mu^\\top 1_n = 1 \\\\\n",
    "- \\infty  & \\text{sinon}\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "car $\\Vert \\sum_{i=1}^n \\mu_i x_i \\Vert^2 = \\sum_{i=1}^n \\sum_{j=1}^n \\mu_i \\mu_j x_i^\\top x_j = \\mu^\\top G G^\\top \\mu$\n",
    "\n",
    "#### Problème dual\n",
    "Le problème dual est donc \n",
    "\\begin{eqnarray*}\n",
    "(\\mathcal{D}) : \\max_{\\phi,\\mu \\in \\mathbb{R}_+^n} \\delta(\\phi,\\mu)\n",
    "& \\iff & \n",
    " \\left\\{\n",
    " \\begin{array}{lll}\n",
    " \\min_{\\mu \\in \\mathbb{R}_+^n} &\\frac{1}{2}\\mu^{\\top}GG^\\top \\mu\n",
    " \\\\\n",
    " \\mathrm{s.t.} & \\mu^{\\top}1_n = 1\n",
    " \\\\\n",
    " &\\frac{1}{\\nu n} 1_n - \\mu \\in \\mathbb{R}_+^n\n",
    " \\end{array}\n",
    " \\right. \\\\\n",
    " & \\iff & \n",
    " \\left\\{\n",
    " \\begin{array}{lll}\n",
    " \\min_{\\mu \\in \\mathbb{R}^n} &\\frac{1}{2}\\mu^{\\top}GG^\\top \\mu\n",
    " \\\\\n",
    " \\mathrm{s.t.} & \\mu^{\\top}1_n = 1\n",
    " \\\\\n",
    " & 0 \\leq \\mu \\leq \\frac{1}{\\nu n}\n",
    " \\end{array}\n",
    " \\right.\n",
    "\\end{eqnarray*}\n",
    "\n",
    "#### Points selles du lagrangien\n",
    "Les points selles du lagrangien $(w,\\rho,\\xi,\\phi,\\mu)$ sont tels que $(w,\\rho,\\xi)$ est solution de $(\\mathcal{P})$ et $(\\phi,\\mu)$ est solution de $(\\mathcal{D})$. Ils vérifient les conditions KKT : \n",
    "-  contraintes primales : $\\forall 1 \\leq i \\leq n$,\n",
    "\\begin{align*}\n",
    "\\rho - \\xi_i - \\langle w, x_i \\rangle & \\leq 0 \\\\\n",
    "\\xi_i & \\geq 0\n",
    "\\end{align*}\n",
    "- complémentarité : $\\forall 1 \\leq i \\leq n$,\n",
    "\\begin{align*}\n",
    "\\mu_i(\\rho - \\xi_i - \\langle w, x_i \\rangle) & = 0 \\\\\n",
    "\\phi_i \\xi_i & = 0\n",
    "\\end{align*}\n",
    "- stationnarité : \n",
    "\\begin{align*}\n",
    "& w = \\sum_{i=1}^n \\mu_i x_i \\\\\n",
    "& \\mu^\\top 1_n = 1 \\\\\n",
    "& \\phi = \\frac{1}{\\nu n} 1_n - \\mu\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 3:</b>\n",
    "     <ul>\n",
    "     <li>Justify that given the estimated $\\mu$, the prediction function for a new sample $x$ (-1 if anomaly else 1) is given by:\n",
    "        $$\n",
    "        y = \\mathrm{sign}(\\sum_{i=1}^{n} \\mu_i x_i^\\top x - \\rho) \\enspace .\n",
    "        $$\n",
    "        The vector $w$ is therefore equal to $\\sum_{i=1}^{n} \\mu_i x_i$\n",
    "    </li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si $\\mu$ est solution du dual alors $(w,\\rho, \\xi, \\mu)$ est un point selle du lagrangian et donc notamment $w = w^*, \\rho = \\rho^*$ et $\\xi = \\xi^*$. Alors la fonction de prédiction définit $y = {\\rm sign}(\\langle w^*, x \\rangle - \\rho^*) = \\mathrm{sign}(\\sum_{i=1}^{n} \\mu_i x_i^\\top x - \\rho)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 1: Implementation of solver using cvxopt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's make a toy dataset in dimension 2. When using a linear kernel we need all samples to have positive values and the outliers to be close to the origin."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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4cGkzYTtZxCYCa1n6Ncb7WSAdxOK31SMnJozQHTmi9X33pfsgZbIuL4etdil5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\nzAkhpAAo5oQQUgD/Hw/SQ1x6HihPAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f66b6f79f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def make_dataset(n_samples, contamination=0.05, random_state=42):\n",
    "    rng = np.random.RandomState(random_state)\n",
    "    X_inliers = rng.uniform(low=2, high=5, size=(int(n_samples * (1 - contamination)), 2))\n",
    "    X_outliers = rng.uniform(low=0, high=1, size=(int(n_samples * contamination), 2))\n",
    "    X = np.concatenate((X_inliers, X_outliers), axis=0)\n",
    "    rng.shuffle(X)\n",
    "    return X\n",
    "\n",
    "X = make_dataset(500)\n",
    "plt.scatter(X[:, 0], X[:, 1], color='b');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The following function qp allows to solve a quadratic problem of the form:\n",
    "\n",
    "$$\n",
    "\\left\\{\n",
    "  \\begin{array}{cll}\n",
    "  &\\min_{(x)} \\frac{1}{2}x^{\\top} P x + q^\\top x\n",
    "  \\\\\n",
    "   & \\textrm{s.t.}\\; A x = b, 0 \\leq x \\leq C.\n",
    "  \\end{array}\n",
    "  \\right.\n",
    "$$\n",
    "\n",
    "See documentation: http://cvxopt.org/userguide/coneprog.html#quadratic-programming"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import cvxopt\n",
    "\n",
    "\n",
    "def qp(P, q, A, b, C, verbose=True):\n",
    "    # Gram matrix\n",
    "    n = P.shape[0]\n",
    "    P = cvxopt.matrix(P)\n",
    "    q = cvxopt.matrix(q)\n",
    "    A = cvxopt.matrix(A)\n",
    "    b = cvxopt.matrix(b)\n",
    "    G = cvxopt.matrix(np.concatenate([np.diag(np.ones(n) * -1),\n",
    "                                     np.diag(np.ones(n))], axis=0))\n",
    "    h = cvxopt.matrix(np.concatenate([np.zeros(n), C * np.ones(n)]))\n",
    "\n",
    "    # Solve QP problem\n",
    "    cvxopt.solvers.options['show_progress'] = verbose\n",
    "    solution = cvxopt.solvers.qp(P, q, G, h, A, b, solver='mosec')\n",
    " \n",
    "    return np.ravel(solution['x'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definition of the linear kernel:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def kernel(X1, X2):\n",
    "    n1 = X1.shape[0]\n",
    "    n2 = X2.shape[0]\n",
    "    K = np.empty((n1, n2))\n",
    "    for i in range(n1):\n",
    "        for j in range(n2):\n",
    "            K[i, j] = np.dot(X1[i], X2[j])\n",
    "    return K"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Remark:** You can use `sklearn.metrics.pairwise_distances` to avoid the for loops and make it faster."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 4:</b>\n",
    "     <ul>\n",
    "      <li>Modify the following cell to solve the One-Class SVM dual problem.</li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of support vectors: 50\n"
     ]
    }
   ],
   "source": [
    "# TODO\n",
    "def ocsvm_solver(K, nu=0.1):\n",
    "    n = len(K)\n",
    "    P = K\n",
    "    q = np.zeros((n, 1))\n",
    "    A = np.ones((1, n))\n",
    "    b = np.array([1.])\n",
    "    C = 1. / (nu * n)\n",
    "    mu = qp(P, q, A, b, C, verbose=False)\n",
    "    idx_support = np.where(np.abs(mu) > 1e-5)[0]\n",
    "    mu_support = mu[idx_support]\n",
    "    return mu_support, idx_support\n",
    "\n",
    "\n",
    "K = kernel(X, X)\n",
    "\n",
    "# Uncomment the following lines when your ocsvm_solver is completed:\n",
    "mu_support, idx_support = ocsvm_solver(K)\n",
    "print(\"Number of support vectors: %s\" % idx_support.size)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 5:</b>\n",
    "     <ul>\n",
    "      <li>Compute w using mu.</li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En théorie $w = \\sum_{i=1}^n \\mu_i x_i$ mais cette somme n'est effectuée que sur les vecteurs de support i.e. les vecteurs tels que $\\mu_i > 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.37631398  1.37252753]\n"
     ]
    }
   ],
   "source": [
    "# TODO\n",
    "w = np.dot(X[idx_support, :].T, mu_support)\n",
    "print(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 6:</b>\n",
    "     <ul>\n",
    "      <li>Using complementary slackness, explain how to obtain $\\rho$ from $\\mu$.</li>\n",
    "    </ul>\n",
    "</div>\n",
    "\n",
    "**HINT:** Use the fact that for all support vectors for which $0 < \\mu_i < \\frac{1}{\\nu n}$ one has $w^{t}x_{i} = \\rho$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Les conditions KKT donnent\n",
    "- Si $\\rho - \\xi_i - \\langle w, x_i \\rangle < 0$ alors on a forcément $\\mu_i = 0$ \n",
    "- Si $0 < \\mu_i < \\frac{1}{\\nu n}$ alors $\\phi_i > 0$ et donc $\\xi_i = 0$ et $\\rho = \\langle w, x_i \\rangle $\n",
    "- Si $\\xi_i > 0$ alors on a forcément $\\mu_i = \\frac{1}{\\nu n}$\n",
    "\n",
    "Les vecteurs de support correspondent aux deux derniers points où $\\mu_i > 0$. On peut alors calculer $\\rho$ par $\\rho = \\langle w, x_j \\rangle$ où $j$ est tel que $\\mu_j = \\min \\{ \\mu_i > 0 \\}$. Remarquons cependant qu'il se peut que $\\forall i, \\mu_i \\approx \\frac{1}{\\nu n}$ et alors en théorie il est alors impossible de calculer $\\rho$ car on n'est jamais dans le deuxième cas ci-dessus. Le calcul de $\\rho$ proposé donnera tout de même une valeur mais celle-ci peut être fausse.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# TODO\n",
    "\n",
    "\n",
    "def compute_rho(K, mu_support, idx_support):\n",
    "    # TODO\n",
    "    K_support = K[idx_support][:, idx_support]\n",
    "    idx = np.argmin(mu_support)\n",
    "    rho = np.mean(np.dot(mu_support, K_support[:, idx]))\n",
    "    return rho\n",
    "\n",
    "\n",
    "rho = compute_rho(K, mu_support, idx_support)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 7:</b>\n",
    "     <ul>\n",
    "      <li>Verify that you find the same values as what scikit-learn gives. You should have the same support vectors and:\n",
    "\n",
    "\n",
    "`rho = -ocsvm.intercept_[0] / (ocsvm.nu * len(K))`\n",
    "\n",
    "and `mu_support = ocsvm.dual_coef_.ravel() / (ocsvm.nu * len(K))`</li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  4  23  66  72  90  92 109 113]\n",
      "[  4  23  66  72  90  92 109 113]\n",
      "2.16400352023\n",
      "2.16400355501\n",
      "[ 0.13422819  0.13422819  0.06040268  0.13422819  0.13422819  0.13422819\n",
      "  0.13422819  0.13422819]\n",
      "[ 0.13422819  0.13422819  0.06040268  0.13422819  0.13422819  0.13422819\n",
      "  0.13422819  0.13422819]\n"
     ]
    }
   ],
   "source": [
    "from sklearn.utils.testing import assert_array_almost_equal\n",
    "from sklearn.utils.testing import assert_almost_equal\n",
    "from sklearn.svm import OneClassSVM\n",
    "\n",
    "X = make_dataset(150)\n",
    "\n",
    "ocsvm = OneClassSVM(kernel='linear', nu=0.05)\n",
    "ocsvm.fit(X)\n",
    "\n",
    "K = kernel(X, X)\n",
    "mu_support, idx_support = ocsvm_solver(K, nu=ocsvm.nu)\n",
    "rho = compute_rho(K, mu_support, idx_support)\n",
    "\n",
    "# Check indices of support vectors\n",
    "print(ocsvm.support_)\n",
    "print(idx_support)\n",
    "\n",
    "# Check rho\n",
    "print(rho)\n",
    "print(-ocsvm.intercept_[0] / (ocsvm.nu * len(K)))\n",
    "\n",
    "# Check dual variables\n",
    "print(mu_support)\n",
    "print(ocsvm.dual_coef_.ravel() / (ocsvm.nu * len(K)))\n",
    "\n",
    "# Run formal tests:\n",
    "assert_array_almost_equal(ocsvm.support_, idx_support)\n",
    "assert_almost_equal(rho, -ocsvm.intercept_[0] / (ocsvm.nu * len(K)))\n",
    "assert_array_almost_equal(mu_support, ocsvm.dual_coef_.ravel() / (ocsvm.nu * len(K)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En changeant X = make_dataset(150) par X = make_dataset(100), on observe que toutes les valeurs de $\\mu_i$ sont les même (elles sont égales à $\\frac{1}{\\nu n}$ du moins pour la précision de l'ordinateur) ainsi on ne peut pas calculer $\\rho$ par la formule donnée. Dans ce cas, le résultat donné par ocsvm n'est pas le même que celui donné par notre calcul. ocsvm a donc sûrement un méthode pour gérer le cas où tous les $\\mu_i$ atteignent la borne maximale.  \n",
    "\n",
    "En changeant X = make_dataset(150) par X = make_dataset(300), on observe un autre problème : un vecteur de support est gardé par ocsvm mais pas par notre algorithme ce qui donne ici aussi des résultats différents. ocsvm semble donc mieux optimisé pour résoudre le problème.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check your code by running the following cell:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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lZXz32GPk5+V59DdIcac/cWOXsgFs/XzAvjbBk7nLcqeNARIjgOTxAJx7D0nn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\nBEopleL+H3oTc4XLXFwZAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f66a9e81ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = make_dataset(300)\n",
    "\n",
    "K = kernel(X, X)\n",
    "mu_support, idx_support = ocsvm_solver(K, nu=0.05)\n",
    "rho = compute_rho(K, mu_support, idx_support)\n",
    "\n",
    "def plot_ocsvm(X, mu_support, idx_support, rho, kernel=kernel):\n",
    "    # Compute decision function on a grid\n",
    "    X1, X2 = np.mgrid[-1:5.1:0.2, -1:5.1:0.2]\n",
    "    na, nb = X1.shape\n",
    "    X_test = np.c_[np.reshape(X1, (na * nb, 1)),\n",
    "                   np.reshape(X2, (na * nb, 1))]\n",
    "\n",
    "    # Compute dot products\n",
    "    X_support = X[idx_support]\n",
    "    G = kernel(X_test, X_support)\n",
    "    # Compute decision function\n",
    "    decision = G.dot(mu_support) - rho\n",
    "\n",
    "    # Compute predict label\n",
    "    y_pred = np.sign(decision)\n",
    "\n",
    "    # Plot decision boundary\n",
    "    plt.plot(X[:,0], X[:, 1], 'or', linewidth=2)\n",
    "    Z = np.reshape(decision, (na, nb))\n",
    "    plt.contourf(X1, X2, Z, 20, cmap=plt.cm.gray)\n",
    "    cs = plt.contour(X1, X2, Z, [0], colors='y', linewidths=2, zorder=10)\n",
    "    plt.xlabel('x1')\n",
    "    plt.ylabel('x2')\n",
    "    plt.xlim([-1, 5])\n",
    "    plt.ylim([-1, 5])\n",
    "\n",
    "plot_ocsvm(X, mu_support, idx_support, rho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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o9cfo0eNbfH0rASgvH0xBwZ00N7vPmN3VGQhw3+drhr35KzEY+PuCBQwuKcGb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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f66a9d56828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Test avec ocsvm\n",
    "ocsvm = OneClassSVM(kernel='linear', nu=0.05)\n",
    "ocsvm.fit(X)\n",
    "K = kernel(X, X)\n",
    "\n",
    "plot_ocsvm(X, ocsvm.dual_coef_.ravel() / (ocsvm.nu * len(K)), \n",
    "           ocsvm.support_,\n",
    "           -ocsvm.intercept_[0] / (ocsvm.nu * len(K)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "support ocsvm : [  2  27  46 109 124 132 141 154 160 164 165 218 222 234 254 264]\n",
      "support ocsvm_solver : [  2  27  46 109 124 132 141 154 160 164 165 218 234 254 264]\n",
      "rho ocsvm : 0.806819272281\n",
      "rho ocsvm_solver : 1.86717663606\n",
      "mu_support ocsvm : [ 0.06666667  0.06666667  0.06666667  0.06666667  0.06666667  0.06666667\n",
      "  0.06666667  0.06666667  0.06666667  0.06666667  0.06666667  0.06666667\n",
      "  0.06666667  0.06666667  0.06666667]\n",
      "mu_support ocsvm : [  6.66666667e-02   6.66666667e-02   6.66666667e-02   6.66666667e-02\n",
      "   6.66666667e-02   6.66666667e-02   6.66666667e-02   6.66666667e-02\n",
      "   6.66666667e-02   6.66666667e-02   6.66666667e-02   6.66666667e-02\n",
      "   5.21064673e-15   6.66666667e-02   6.66666667e-02   6.66666667e-02]\n"
     ]
    }
   ],
   "source": [
    "# Check indices of support vectors\n",
    "print(\"support ocsvm : %s\"%ocsvm.support_)\n",
    "print(\"support ocsvm_solver : %s\"%idx_support)\n",
    "\n",
    "# Check rho\n",
    "print(\"rho ocsvm : %s\"%rho)\n",
    "print(\"rho ocsvm_solver : %s\"%(-ocsvm.intercept_[0] / (ocsvm.nu * len(K))))\n",
    "\n",
    "# Check dual variables\n",
    "print(\"mu_support ocsvm : %s\"%mu_support)\n",
    "print(\"mu_support ocsvm : %s\"%(ocsvm.dual_coef_.ravel() / (ocsvm.nu * len(K))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comme souligné précédemment, dans le cas du dataset 300, ocsvm et ocsvm_solver ne donnent pas les même vecteurs de support. On observe que cela donne deux résultats différents. Cependant aucun des deux résultats ne semble clairement meilleur que l'autre, on aimerait avoir une droite au milieu des deux clusters. Cela peut être dû au fait que les vecteurs de support sont soit tels que $\\mu_i \\approx \\frac{1}{\\nu n}$ soit $\\mu_i \\approx 10^{-15}$ (très faible). Il se peut alors qu'on n'ait pas réellement $\\xi_i \\approx 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 8:</b>\n",
    "     <ul>\n",
    "      <li>Without any implementation, can you justify if you can apply any of the algorithm, among proximal gradient, coordinate descent, stochastic gradient descent and L-BFGS to the dual problem?\n",
    "      </li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ANSWER HERE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 1: non-linear case with kernels"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another scenario is when the boundary between normal samples and outliers is not linear.\n",
    "\n",
    "To cope with this the idea is to use kernels.\n",
    "\n",
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 9:</b>\n",
    "     <ul>\n",
    "      <li>Denoting by $K(x_i, x_j)$ the dot product between samples show that dual problem and the decision function f(x) can be reformulated just using calls to $K$\n",
    "      </li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Si $K(x_i,x_j)$ désigne le produit scalaire entre $x_i$ et $x_j$, \n",
    "Comme $$\n",
    " G = \\begin{bmatrix}x_{1}^{\\top} \\\\ \\vdots \\\\ x_{n}^{\\top}\\end{bmatrix}\n",
    "$$\n",
    "on peut remplacer les produits scalaires dans le problème dual par :\n",
    "$$\n",
    "(\\mathcal{D}): \n",
    " \\left\\{\n",
    " \\begin{array}{lll}\n",
    " \\min_{\\mu} &\\frac{1}{2}\\mu^{\\top}K\\mu\n",
    " \\\\\n",
    " \\mathrm{s.t.}& \\mu^{\\top}1_n = 1\n",
    " \\\\\n",
    " \\mathrm{and}& 0 \\leq \\mu \\leq \\frac{1}{\\nu n}\n",
    " \\end{array}\n",
    " \\right .\n",
    "$$\n",
    "avec $K=[K(x_i,x_j)]_{i,j}$\n",
    "\n",
    "Ainsi que dans la fonction de prédiction :\n",
    "$$\n",
    "        y = \\mathrm{sign}(\\sum_{i=1}^{n} \\mu_i K(x_i,x) - \\rho) \\enspace .\n",
    "        $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will use another dataset to evaluate this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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rV/ZfH7t3S5ddJh04kD0MMguLeGwPpGVQqXbA7ncseltWj3XSTU9Ll14q/exnmx9r7z9/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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f0ae9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def make_blob_dataset(n_samples, contamination=0.05, random_state=42):\n",
    "    rng = np.random.RandomState(random_state)\n",
    "    X_inliers = 0.3 * rng.normal(size=(int(n_samples * (1. - contamination)), 2)) + 2\n",
    "    X_outliers = rng.uniform(low=-1, high=5, size=(int(n_samples * contamination), 2))\n",
    "    X = np.concatenate((X_inliers, X_outliers), axis=0)\n",
    "    rng.shuffle(X)\n",
    "    return X\n",
    "\n",
    "X = make_blob_dataset(300)\n",
    "plt.scatter(X[:, 0], X[:, 1], color='b');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 10:</b>\n",
    "     <ul>\n",
    "      <li>Update your kernel function so it computes the Gaussian kernel:\n",
    "      \n",
    "        $$\n",
    "            K(x_i, x_j) = \\exp(-\\gamma \\| x_i - x_j \\|^2)\n",
    "        $$\n",
    "\n",
    "        where $\\gamma > 0$ is the kernel bandwidth.\n",
    "      </li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-16-a0a536046bb8>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m     14\u001b[0m \u001b[0mX1\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     15\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 16\u001b[0;31m \u001b[0mrbf_kernel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mX1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mX1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m<ipython-input-16-a0a536046bb8>\u001b[0m in \u001b[0;36mrbf_kernel\u001b[0;34m(X1, X2)\u001b[0m\n\u001b[1;32m     10\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m             \u001b[0mK\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mgamma\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlinalg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnorm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mX1\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mX2\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m^\u001b[0m\u001b[0;36m2\u001b[0m  \u001b[0;31m# CHANGE THIS\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     13\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     14\u001b[0m \u001b[0mX1\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/staermanguillaume/anaconda2/lib/python2.7/site-packages/numpy/linalg/linalg.pyc\u001b[0m in \u001b[0;36mnorm\u001b[0;34m(x, ord, axis, keepdims)\u001b[0m\n\u001b[1;32m   2159\u001b[0m         if ((ord is None) or\n\u001b[1;32m   2160\u001b[0m             \u001b[0;34m(\u001b[0m\u001b[0mord\u001b[0m \u001b[0;32min\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m'f'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'fro'\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mndim\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mor\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2161\u001b[0;31m             (ord == 2 and ndim == 1)):\n\u001b[0m\u001b[1;32m   2162\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2163\u001b[0m             \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mravel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0morder\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'K'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mValueError\u001b[0m: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()"
     ]
    }
   ],
   "source": [
    "# TODO\n",
    "\n",
    "from scipy import linalg\n",
    "\n",
    "def rbf_kernel(X1, X2):\n",
    "    n1 = X1.shape[0]\n",
    "    n2 = X2.shape[0]\n",
    "    K = np.empty((n1, n2))\n",
    "    gamma = .3\n",
    "    for i in range(n1):\n",
    "        for j in range(n2):\n",
    "            K[i, j] = # CHANGE THIS\n",
    "    return K\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the change above the follwing code should allow you to nicely separate the dense region from the outliers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
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qQYjdu8lffPGsuQp24wl1CvurNZme3mubxhAO9Ui6ONy+jcf2e3ynQeUrgDdK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\nYDAYDIARBIPBYDDU+P+Viqb/k8nfWAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1112279b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = make_blob_dataset(150)\n",
    "\n",
    "K = rbf_kernel(X, X)\n",
    "mu_support, idx_support = ocsvm_solver(K, nu=0.1)\n",
    "rho = compute_rho(K, mu_support, idx_support)\n",
    "\n",
    "plot_ocsvm(X, mu_support, idx_support, rho, kernel=rbf_kernel)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2: One-class SVM without $\\rho$\n",
    "\n",
    "Formulation for example used in:\n",
    "\n",
    "```\n",
    "Consistency of one-class SVM and related algorithms\n",
    "Régis Vert and Jean-philippe Vert\n",
    "NIPS 2006\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The problem of the formuation of SVMs with the variable $\\rho$ is that\n",
    "it leads to an annoying constraint in the dual, namely the $\\mu^{\\top}1_n = 1$.\n",
    "\n",
    "We will now see what we can do about it.\n",
    "\n",
    "Let's consider the problem\n",
    "\n",
    "$$\n",
    "\t\\begin{aligned}\n",
    "\t\\min_{w, \\xi} & \\quad \\frac{1}{2}\\Vert w \\Vert^2 + \\frac{C}{n} \\sum_{i=1}^n \\xi_i \\\\\n",
    "\t\\text{s.t.} & \\quad \\langle w, x_i \\rangle \\geq 1 - \\xi_i \\quad 1 \\leq i \\leq n \\\\\n",
    "\t& \\quad \\xi_i \\geq 0 \\quad 1 \\leq i \\leq n\n",
    "\t\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 10:</b>\n",
    "     <ul>\n",
    "      <li>Show that a dual problem of this problem is given by:\n",
    "      \n",
    "        $$\n",
    "        (\\mathcal{D}):\n",
    "            \\left\\{\n",
    "            \\begin{array}{lll}\n",
    "            \\min_{\\mu} &\\frac{1}{2}\\mu^{\\top}GG^{\\top}\\mu-\\mu^{\\top} 1_n\n",
    "            \\\\\n",
    "            \\mathrm{s.t.}& 0 \\leq \\mu \\leq \\frac{C}{n}\n",
    "            \\end{array}\n",
    "            \\right .\n",
    "        $$\n",
    "      </li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Lagrangien : \n",
    "$\\forall (w, \\xi, \\mu, \\phi) \\in \\mathbb{R}^p \\times \\mathbb{R}^n \\times \\mathbb{R}_+^n \\times \\mathbb{R}_+^n$,\n",
    "\\begin{eqnarray*}\n",
    "\\mathcal{L}(w, \\xi, \\phi, \\mu) \n",
    "&=&  \\frac{1}{2}\\Vert w \\Vert^2 + \\frac{C}{n} \\sum_{i=1}^n \\xi_i + \\sum_{i=1}^{n} \\mu_i (1 - \\xi_i - \\langle w, x_i \\rangle) - \\sum_{i=1}^n \\phi_i \\xi_i \\\\\n",
    "&=& \\frac{1}{2}\\Vert w \\Vert^2  + \\sum_{i=1}^n \\left( \\frac{C}{n} - \\mu_i - \\phi_i \\right) \\xi_i  - \\left\\langle w, \\sum_{i=1}^{n} \\mu_i x_i \\right\\rangle + \\sum_{i=1}^n \\mu_i \\\\\n",
    "&=& \\frac{1}{2}\\Vert w \\Vert^2 + \\left\\langle \\frac{C}{n} 1_n - \\mu - \\phi , \\xi \\right\\rangle  - \\left\\langle w, \\sum_{i=1}^{n} \\mu_i x_i \\right\\rangle + \\mu^\\top 1_n\n",
    "\\end{eqnarray*}\n",
    "\n",
    "#### Fonction duale :\n",
    "$\\delta : (\\phi, \\mu) \\in \\mathbb{R}_+^n \\times \\mathbb{R}_+^n \\mapsto \\inf_{w, \\xi} \\mathcal{L}(w, \\xi, \\phi, \\mu)$. Or\n",
    "$$\n",
    "\\nabla_{(w,\\xi)} \\mathcal{L}(w,\\xi,\\mu,\\phi) =\n",
    "\\begin{bmatrix}\n",
    "w - \\sum_{i=1}^n \\mu_i x_i \\\\ \\frac{C}{n} 1_n - \\mu - \\phi \n",
    "\\end{bmatrix} \n",
    "$$\n",
    "et \n",
    "$$\n",
    "\\nabla_{(w,\\xi)}^2 \\mathcal{L}(w,\\xi,\\mu,\\phi) =\n",
    "\\begin{bmatrix}\n",
    "I_p  & 0_{p,n} \\\\\n",
    "0_{n,p} & 0_n \n",
    "\\end{bmatrix}\n",
    "\\succeq 0\n",
    "$$\n",
    "Donc pour tout $(\\mu, \\phi) \\in \\mathbb{R}_+^n \\times \\mathbb{R}_+^n, (w,\\xi) \\mapsto \\mathcal{L}(w,\\xi,\\mu,\\phi)$ est convexe donc $\\delta(\\mu, \\phi) = \\mathcal{L}(w^*,\\xi^*,\\mu,\\phi)$ avec \n",
    "$$\n",
    "\\nabla_{(w,\\xi)} \\mathcal{L}(w^*,\\xi^*,\\mu,\\phi) = 0 \n",
    "\\iff\n",
    "\\left\\lbrace \n",
    "\\begin{array}{l}\n",
    "w = \\sum_{i=1}^n \\mu_i x_i \\\\\n",
    "\\phi = \\frac{C}{n} 1_n - \\mu\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "Donc finalement $\\forall (\\mu, \\phi) \\in \\mathbb{R}_+^n \\times \\mathbb{R}_+^n$,\n",
    "$$\n",
    "\\delta(\\phi, \\mu) = \n",
    "\\left\\lbrace \n",
    "\\begin{array}{ll}\n",
    "- \\frac{1}{2} \\Vert \\sum_{i=1}^n \\mu_i x_i \\Vert^2  + \\mu^\\top 1_n\n",
    "& \\text{si }  \\phi = \\frac{C}{n} 1_n - \\mu\\\\\n",
    "- \\infty  & \\text{sinon}\n",
    "\\end{array}\n",
    "\\right.\n",
    "=\n",
    "\\left\\lbrace \n",
    "\\begin{array}{ll}\n",
    "- \\frac{1}{2} \\mu^\\top G G^\\top \\mu  + \\mu^\\top 1_n\n",
    "& \\text{si }  \\phi = \\frac{C}{n} 1_n - \\mu \\\\\n",
    "- \\infty  & \\text{sinon}\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "#### Problème dual\n",
    "Le problème dual est donc \n",
    "\\begin{eqnarray*}\n",
    "(\\mathcal{D}) : \\max_{\\phi,\\mu \\in \\mathbb{R}_+^n} \\delta(\\phi,\\mu)\n",
    "& \\iff & \n",
    " \\left\\{\n",
    " \\begin{array}{lll}\n",
    " \\min_{\\mu \\in \\mathbb{R}_+^n} &\\frac{1}{2}\\mu^{\\top}GG^\\top \\mu  - \\mu^\\top 1_n\n",
    " \\\\\n",
    " \\mathrm{s.t.} & \\frac{C}{n} 1_n - \\mu \\in \\mathbb{R}_+^n\n",
    " \\end{array}\n",
    " \\right. \\\\\n",
    " & \\iff & \n",
    " \\left\\{\n",
    " \\begin{array}{lll}\n",
    " \\min_{\\mu \\in \\mathbb{R}^n} &\\frac{1}{2}\\mu^{\\top}GG^\\top \\mu  - \\mu^\\top 1_n\n",
    " \\\\\n",
    " \\mathrm{s.t.} & 0 \\leq \\mu \\leq \\frac{C}{n}\n",
    " \\end{array}\n",
    " \\right.\n",
    "\\end{eqnarray*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 11:</b>\n",
    "     <ul>\n",
    "      <li> Rewrite the dual in the form:\n",
    "      \n",
    "            $$\n",
    "            (\\mathcal{D}): \\min_{\\mu} f(\\mu) + g(\\mu) .\n",
    "            $$\n",
    "\n",
    "            where $f$ is here a smooth function of $\\mu$ with L-Liptschitz gradient and $g$ is a non-smooth function that is separable, namely:\n",
    "\n",
    "            $$\n",
    "            g(\\mu) = \\sum_{i=1}^n g_i(\\mu_i)\n",
    "            $$\n",
    "      </li>\n",
    "    </ul>\n",
    "</div>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notons $\\mathbb{I}_{\\mathcal{A}} : x \\mapsto \\left\\lbrace \\begin{array}{ll} 0 & \\text{si } x \\in A \\\\ +\\infty & \\text{sinon} \\end{array} \\right.$ l'indicatrice convexe de l'ensemble $\\mathcal{A}$ alors \n",
    "$$\n",
    "(\\mathcal{D}) \\iff \\min_{\\mu \\in \\mathbb{R}^n} \\frac{1}{2} \\mu^\\top G G^\\top \\mu - \\mu^\\top 1_n - \\mathbb{I}_{[0, C/n]^n}(\\mu) \\iff \\min_{\\mu \\in \\mathbb{R}^n} f(\\mu) + g(\\mu)\n",
    "$$\n",
    "avec \n",
    "- $f : \\mu \\mapsto \\mu^\\top G G^\\top \\mu - 1_n^\\top \\mu$ est une fonction quadratique donc convexe avec un gradient lipschitzien\n",
    "- $g : \\mu \\mapsto \\mathbb{I}_{[0, C/n]^n}(\\mu) = \\sum_{i=1}^n \\mathbb{I}_{[0, C/n]}(\\mu_i)$ est convexe et séparable"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dual in the later form can be readily optimized using the methods that you have been studying in this class:\n",
    "\n",
    "- Proximal gradient method with and without acceleration\n",
    "- L-BFGS-B\n",
    "- Coordinate descent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "    <b>QUESTION 12:</b>\n",
    "    Implement:\n",
    "     <ul>\n",
    "      <li> your own solver based on the QP function in cvxopt (reuse `qp` function above).</li>\n",
    "      <li> your own solver of proximal gradient with and without acceleration</li>\n",
    "      <li> your own version of coordinate descent</li>\n",
    "      <li> an L-BFGS-B solver using `scipy.optimize.fmin_l_bfgs_b`</li>\n",
    "    </ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You will test your implementations on the Gaussian blob corrupted with some uniform noise obtained with `make_blob_dataset` function.\n",
    "\n",
    "You will make sure that all solvers converge towards the same minimum !\n",
    "\n",
    "You will evaluate the performance of your implementations in terms of computation time on problems where the matrix $G G^\\top$ can fit in memory on your machine.\n",
    "\n",
    "You should reuse as much as possible the convergence evaluation code that you used during the labs.\n",
    "\n",
    "**Remark:** For the coordinate descent method there exist some smart updates provided you use a linear kernel.\n",
    "\n",
    "**BONUS (Hard):** Explain without implementing it how such smart updates could work for the coordinate descent solver.\n",
    "\n",
    "**IMPORTANT : This question 12 is the most important and will constitute half of the final grade on the project !**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'make_blob_dataset' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-1-4c293397df37>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# TODO\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mmake_blob_dataset\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      3\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mscipy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0moptimize\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfmin_l_bfgs_b\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'make_blob_dataset' is not defined"
     ]
    }
   ],
   "source": [
    "# TODO\n",
    "\n",
    "from scipy.optimize import fmin_l_bfgs_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x1a1bd01290>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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E+RyA1sUiqn77yHITad11mNqx0eqQrdaE8R7YwiVV1Lh4uXvzec58dpe6sbft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\nm4PqonaEKyWQIQmssfchcTS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BnrDz527C0dMXYke/qKYhOfcvff2kd9SKCVsSmY/N2ZTw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\nBgt2hmGYPoMFO8MwTJ/Bgp1hGKbP+P8RSslVdVp/uAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115677b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}