heavytail.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Cost function for heavy-tailed noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from __future__ import print_function\n",
    "\n",
    "import numpy as np\n",
    "from sklearn.linear_model import Ridge\n",
    "from flexible_linear import FlexibleLinearRegression\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "matplotlib.style.use('ggplot')\n",
    "%matplotlib inline\n",
    "\n",
    "np.random.seed(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generate some data\n",
    "\n",
    "Our \"ground truth\" data is a simple linear function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10f6524a8>]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f5e5470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 500\n",
    "A = 50\n",
    "B = 3\n",
    "x = np.linspace(0, 100, N, dtype=float)\n",
    "X = x.reshape(-1, 1)  # scikit-learn wants 2d arrays\n",
    "y = A + B * x\n",
    "plt.plot(x, y, '-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Let's try adding Gaussian (normal) noise..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10f62c7f0>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hlKYotXBrr8qBacUJq4F2pIIyzBXbY8iqj3dblERPSx63v2UhAClNic07HWmF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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f62c9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_gauss = y + 20*np.random.randn(N)\n",
    "plt.plot(x, y, '-', x, y_gauss, '-')\n",
    "plt.legend(['True', 'Noisy'], loc='upper left')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ... or some Cauchy (heavy-tailed) noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x111faddd8>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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gyt3fd/dm4HFgQZ7n1G/c/QN3fy26vQ/4E1BBqgaro9VWA1flZ4b9J3oXdzmp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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111fb30f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_cauchy = y + 20*np.random.standard_cauchy(N)\n",
    "plt.plot(x, y, '-', x, y_cauchy, '.')\n",
    "plt.legend(['True', 'Noisy'], loc='lower right')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Trying to recover the linear function using Ridge regression\n",
    "\n",
    "### Gaussian noise\n",
    "\n",
    "For Gaussian noise the recovery works fairly well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     True: 50.000 + 3.000 * x\n",
      "Recovered: 51.598 + 2.989 * x\n"
     ]
    },
    {
     "data": {
      "image/png": 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OXi58+K562fc/HoWAiMgRduQU8uGiJHJy9vJg0Sauued2XC2iAz2sWqEQEBHxyztYymdL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\nFXsKmLLSTVTeHl5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CSFygBxBE4qqzcY2dGBYRkfonmI4ERESkjikERERCWFBcRdQY0xt4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\nQEQkhCkERERCmEJARCSE/X9Qq5wird2zHAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111fd35c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clf = Ridge()\n",
    "clf.fit(X, y_gauss)\n",
    "pred = clf.predict(X)\n",
    "plt.plot(x, y, '-', x, pred, '-')\n",
    "plt.legend(['True', 'Recovered'], loc='upper left')\n",
    "print(\"     True: %.3f + %.3f * x\" % (A, B))\n",
    "print(\"Recovered: %.3f + %.3f * x\" % (clf.intercept_, clf.coef_[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cauchy noise\n",
    "\n",
    "Let's try the same with the Cauchy noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     True: 50.000 + 3.000 * x\n",
      "Recovered: -362.909 + 8.056 * x\n"
     ]
    },
    {
     "data": {
      "image/png": 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ZVGgIm0KhswUVrHLzq1ica2dAj478deoAkrpqm0mFlpB819TZggoFhZX1PLe2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\nlFKnpIVCKaXUKf1/dkiBxmIzRAMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b34c2e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clf = Ridge()\n",
    "clf.fit(X, y_cauchy)\n",
    "pred = clf.predict(X)\n",
    "plt.plot(x, y, '-', x, pred, '-')\n",
    "plt.legend(['True', 'Recovered'], loc='upper left')\n",
    "print(\"     True: %.3f + %.3f * x\" % (A, B))\n",
    "print(\"Recovered: %.3f + %.3f * x\" % (clf.intercept_, clf.coef_[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That looks much less impressive. The problem is that the squared-$\\ell^2$-norm minimized by Ridge regression is too sensitive to outliers."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Using a less sensitive cost function\n",
    "\n",
    "Let\n",
    "$$z_i = (X \\cdot W - y)_i$$\n",
    "\n",
    "denote the residuals. Instead of the usual least-squares approach of minimizing\n",
    "\n",
    "$$\\min_W ||z||_{\\ell^2}^2 = \\frac12 \\sum_i |z_i|^2 \\,, $$\n",
    "\n",
    "let's try to minimize the $\\ell^1$ norm\n",
    "\n",
    "$$\\min_W ||z||_{\\ell^1} = \\sum_i |z_i|$$\n",
    "\n",
    "instead. Alternatively, since that is not smooth, we could try to minimize an objective like\n",
    "\n",
    "$$ \\min_W \\eta^2 \\sum_i \\left( \\sqrt{ 1 + \\left( \\frac{|z_i|}{\\eta} \\right)^2 } - 1 \\right) \\,, $$\n",
    "\n",
    "where $\\eta$ is a positive scale parameter.\n",
    "This is a *smooth* convex cost function based component-wise on the _Japanese bracket_:\n",
    "\n",
    "$$ \\langle z \\rangle := \\sqrt{1 + |z|^2} \\,. $$\n",
    "\n",
    "It behaves like the least-squares cost function for $|z| \\ll \\eta$,\n",
    "but grows like $O(|z|)$ for $|z| \\gg \\eta$.\n",
    "So it gives us the best of $\\ell^1$ and squared-$\\ell^2$:\n",
    "The smoothness of squared-$\\ell^2$ for small values\n",
    "and the insensitivity to outliers of $\\ell^1$ for large values.\n",
    "\n",
    "We use a custom `FlexibleLinearRegression` scikit-learn estimator that allows us to try these different cost functions.\n",
    "\n",
    "**Note:** For simplicity, we omit regularization in the above (and below by setting `C=0.0`). It really does not matter for this toy problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### squared $\\ell^2$ cost\n",
    "\n",
    "This is pretty much the same as `Ridge` above (minus the regularization). It performs horribly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     True: 50.000 + 3.000 * x\n",
      "Recovered: -362.910 + 8.056 * x\n"
     ]
    },
    {
     "data": {
      "image/png": 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ZVGgIm0KhswUVrHLzq1ica2dAj478deoAkrpqm0mFlpB819TZggoFhZX1PLe2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\nlFKnpIVCKaXUKf1/dkiBxmIzRAMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111fb3908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clf = FlexibleLinearRegression(cost_func='l2', C=0.0)\n",
    "clf.fit(X, y_cauchy)\n",
    "pred = clf.predict(X)\n",
    "plt.plot(x, y, '-', x, pred, '-')\n",
    "plt.legend(['True', 'Recovered'], loc='upper left')\n",
    "print(\"     True: %.3f + %.3f * x\" % (A, B))\n",
    "print(\"Recovered: %.3f + %.3f * x\" % (clf.coef_[0], clf.coef_[1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\ell^1$ cost\n",
    "\n",
    "If we switch to the $\\ell^1$ cost, we start to get very good results!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     True: 50.000 + 3.000 * x\n",
      "Recovered: 47.083 + 3.007 * x\n"
     ]
    },
    {
     "data": {
      "image/png": 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bCul2HB4dnELrGM33F/EXNQGpVc7cf9Mob+7v+cnDvRkJpLWO9vfwRIJehU3A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\nogAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b9956a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clf = FlexibleLinearRegression(cost_func='l1', C=0.0)\n",
    "clf.fit(X, y_cauchy)\n",
    "pred = clf.predict(X)\n",
    "plt.plot(x, y, '-', x, pred, '-')\n",
    "plt.legend(['True', 'Recovered'], loc='upper left')\n",
    "print(\"     True: %.3f + %.3f * x\" % (A, B))\n",
    "print(\"Recovered: %.3f + %.3f * x\" % (clf.coef_[0], clf.coef_[1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Japanese cost\n",
    "\n",
    "If we switch to the *Japanese bracket* cost, we get results just as good. For this simple example, the $\\ell^1$ cost already gave us good answers, but for more complex problems the smoothness of the *Japanese bracket* may make it a numerically safer choice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     True: 50.000 + 3.000 * x\n",
      "Recovered: 47.135 + 2.998 * x\n"
     ]
    },
    {
     "data": {
      "image/png": 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3F4lqCV5vxLxHe/Pz88M9hogQSYe6wfP932zVi9fW7eFXmc0xPdrQLCm08/0j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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1044e6a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clf = FlexibleLinearRegression(cost_func='japanese', C=0.0, cost_opts={'eta': 10.0})\n",
    "clf.fit(X, y_cauchy)\n",
    "pred = clf.predict(X)\n",
    "plt.plot(x, y, '-', x, pred, '-')\n",
    "plt.legend(['True', 'Recovered'], loc='upper left')\n",
    "print(\"     True: %.3f + %.3f * x\" % (A, B))\n",
    "print(\"Recovered: %.3f + %.3f * x\" % (clf.coef_[0], clf.coef_[1]))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}