learning-week-04.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Learning from Data Week 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Questions provided here:\n",
    "https://work.caltech.edu/homework/hw4.pdf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy.optimize import fsolve\n",
    "from __future__ import division"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 2\n",
    "There are a number of bounds on the generalization error ε, all holding with probability at least 1−δ. Fix dvc = 50 and δ = 0.05 and plot these bounds as a function of N. Which bound is the smallest for very large N, say N = 10,000? Note that [c] and [d] are implicit bounds in ε."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Investigate different bounds on generalization error\n",
    "dvc = 50\n",
    "delta = 0.05\n",
    "\n",
    "# Original VC bound\n",
    "Original_VC = lambda N: np.sqrt(8/N*(np.log(4) + dvc*np.log(2*N) - np.log(delta)))\n",
    "\n",
    "# Rademacher Penalty bound\n",
    "Rademacher = lambda N: np.sqrt(2/N*(np.log(2*N)+dvc*np.log(N))) + np.sqrt(2/N*np.log(1/delta)) + 1/N\n",
    "\n",
    "# Parrondo and Van den Broek\n",
    "def PVdB(N):\n",
    "    a = 1\n",
    "    b = -2/N\n",
    "    c = -1/N * (np.log(6/delta) + dvc*np.log(2*N))\n",
    "    return (-b + np.sqrt(np.square(b)-4*a*c))/(2*a)\n",
    "\n",
    "# Devroye\n",
    "def Devroye(N):\n",
    "    a = 1-2/N\n",
    "    b = -2/N\n",
    "    c = -1/(2*N) * (np.log(4/delta) + 2*dvc*np.log(N))\n",
    "    return (-b + np.sqrt(np.square(b)-4*a*c))/(2*a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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IixbVoCfbAAAAADWHZ2PvxIEnauc3O3VS0kn6+sfKby0waZK9TW3BgjAOF63q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\nEWut0rH0XYdici0pl9+1bSwWR4dbbR1tZcu9eDxGqWwuq0g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      "text/plain": [
       "<matplotlib.figure.Figure at 0x107766450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plots\n",
    "N = np.arange(1,10001)\n",
    "plt.figure(figsize=(15,10))\n",
    "plt.plot(N, Original_VC(N), 'r-')\n",
    "plt.plot(N, Rademacher(N), 'b-')\n",
    "plt.plot(N, PVdB(N), 'g-')\n",
    "plt.plot(N, Devroye(N), 'purple')\n",
    "plt.yscale('log')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Devroye provides the smallest bound on large N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3\n",
    "For the same values of dvc and δ of Problem 2, but for small N, say N = 5,\n",
    "which bound is the smallest?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10a0a0850>]"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Hfr90/HhouN26JXk8drA99JD9XFzM/jYAmGNEGgAAsLW0hO9vS0oKnW2rrJRycpweKQDE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\nhPw/wksrebETx+YAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ab2c650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plots\n",
    "N = np.arange(1,6)\n",
    "plt.figure(figsize=(15,10))\n",
    "plt.plot(N, Original_VC(N), 'r-')\n",
    "plt.plot(N, Rademacher(N), 'b-')\n",
    "plt.plot(N, PVdB(N), 'g-')\n",
    "plt.plot(N, Devroye(N), 'purple')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Parrondo and Van den Broek provides the smallest bound for small N."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the average hypothesis?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The coefficient from the average hypothesis is 1.40456931112\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "# Define target function\n",
    "target_func = lambda x: np.sin(np.pi * x)\n",
    "\n",
    "# Run experiment\n",
    "def bias_variance_exp():    \n",
    "    # Initialize array of coefficients\n",
    "    coefs = []\n",
    "    # Iterate 10,000 times\n",
    "    for i in range(10000):\n",
    "        # Pick 2 random points from uniform distribution from [-1,1]\n",
    "        x_points = np.random.uniform(-1,1,2)\n",
    "        y_points = target_func(x_points)\n",
    "        # Define beta estimate with 2 input points, given in form of [x1, x2], [y1, y2]\n",
    "        # Model of the form y = ax\n",
    "        beta = np.sum(x_points * y_points) / np.sum(x_points**2)\n",
    "        coefs.append(beta)\n",
    "    return np.mean(coefs)\n",
    "print (\"The coefficient from the average hypothesis is {}\".format(bias_variance_exp()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Questions 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the bias?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The bias is 0.270251936638\n"
     ]
    }
   ],
   "source": [
    "avg_coef = bias_variance_exp()\n",
    "\n",
    "def bias():\n",
    "    # Create 10,000 points\n",
    "    x = np.arange(-1, 1, 2/10000)\n",
    "    # Calculate target values and average hypothesis values\n",
    "    y = target_func(x)\n",
    "    g_avg = avg_coef * x\n",
    "    return np.mean((g_avg - y)**2)\n",
    "\n",
    "print (\"The bias is {}\".format(bias()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the variance?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The variance is 0.236485750147\n"
     ]
    }
   ],
   "source": [
    "avg_coef = bias_variance_exp()\n",
    "\n",
    "def variance():\n",
    "    # Create 10,000 points\n",
    "    x = np.arange(-1, 1, 2/10000)\n",
    "    g_avg = avg_coef * x\n",
    "    # Initialize array to hold variance of each dataset\n",
    "    var_data = []\n",
    "    # Iterate over 10,000 datasets\n",
    "    for i in range(10000):\n",
    "        # Pick 2 random points from uniform distribution from [-1,1] estimate beta\n",
    "        x_points = np.random.uniform(-1,1,2)\n",
    "        y_points = target_func(x_points)\n",
    "        beta = np.sum(x_points * y_points) / np.sum(x_points**2)\n",
    "        # Calculate g of x\n",
    "        g = beta * x\n",
    "        # Take variance of g and add to var_data\n",
    "        var_data.append(np.mean((g - g_avg)**2))\n",
    "    # Take expected value over all datasets\n",
    "    return np.mean(var_data)\n",
    "\n",
    "print (\"The variance is {}\".format(variance()))"
   ]
  }
 ],
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