Jupyter notebook on adverserial learning on logistic regression

AdverserialLearningLogisticRegression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import tensorflow as tf\n",
    "import numpy as np\n",
    "import time\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from sklearn import linear_model \n",
    "from scipy.stats import logistic\n",
    "\n",
    "def sigmoid(a):\n",
    "    return (1./(1.+np.exp(-1*a)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1046a6f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N=100\n",
    "d1mu = 5.0; d1sig = 0.5\n",
    "d2mu = 4.0; d2sig = 0.3\n",
    "\n",
    "# X = [bias, x]\n",
    "def sample_data():\n",
    "    X = np.concatenate( [np.stack([np.ones(N),np.random.normal(d1mu, d1sig, N)]).T,\n",
    "                         np.stack([np.ones(N),np.random.normal(d2mu, d2sig, N)]).T ])\n",
    "    y = np.concatenate([np.ones(N), np.zeros(N)])\n",
    "    return X,y\n",
    "\n",
    "X,y = sample_data()\n",
    "Xt,yt = sample_data()\n",
    "\n",
    "plt.hist([X[0:N,1],X[N:-1,1]])\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Logistic regression: $\\sigma(\\beta_0 + \\beta_1 x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Train set error: 0.895\n",
      "Test set error:  0.87\n"
     ]
    }
   ],
   "source": [
    "lg = linear_model.LogisticRegression(fit_intercept=False)\n",
    "lg.fit(X,y)\n",
    "print('Train set error: ' + str(np.sum(lg.predict(X)==yt)/len(y)))\n",
    "print('Test set error:  ' + str(np.sum(lg.predict(Xt)==yt)/len(yt)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LiIkxJjrbtQv69TO7V8KbSA1fCC9w4AA89JBRs4+Pr72FwoV3kBq+ED4gPx/e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\nkvCFEMJPSMIXQgg/IQlfCCH8xP8HJ5AalZqWWKYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1146af048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.linspace(2.5,6.5)\n",
    "classes = lg.predict(X).astype(np.bool)\n",
    "plt.plot(t,sigmoid(w.dot(np.vstack([np.ones(len(t)),t]))), label='logistic fn')\n",
    "plt.scatter(X[classes,1], y[classes], color=['darkblue'], label='classified 1')\n",
    "plt.scatter(X[~classes,1], y[~classes], color=['darkred'], label='classified 0')\n",
    "plt.legend(loc=2)\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Extract coefficients and manually classify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\n",
       "       1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\n",
       "       0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1,\n",
       "       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\n",
       "       1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0,\n",
       "       0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n",
       "       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n",
       "       0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n",
       "       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = np.array(*lg.coef_)\n",
    "c = (sigmoid(w.dot(X.T)) > 0.5).astype(int)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-5.56724366,  1.27707096])"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Add perturbations to the input\n",
    "\n",
    "$\\tilde{x} = x + \\eta$\n",
    "\n",
    "The linear part of the logistic regression formula is characterized by weight vector $w$ with $\\beta_0$ being the intercept.\n",
    "\n",
    "$w = (\\beta_0, \\beta_1, \\ldots)$\n",
    "\n",
    "Then the perturbed output of the linear regression function adds perturbation term $\\eta$. It adds up to increase the activation and thus trick the activation function $\\sigma$ of the logistic regression into yielding the wrong output.\n",
    "\n",
    "$w^T\\tilde{x} = w^Tx + w^T\\eta$\n",
    "\n",
    "A choice to maximize the activation with respect to the max norm is to use the sign function. \n",
    "\n",
    "Note: Activation will grow by $\\epsilon \\, m \\, n$  ($m \\ldots$ average magnitude, $n \\ldots$ dimension of input vector $x$)\n",
    "\n",
    "$\\eta = sign(w)$\n",
    "\n",
    "A constraint on $\\eta$ is to be lower than the precision of the sensor. This is important that the argument that the perturbed input $\\tilde{x}$ is _essentially_ the same as the original input $x$ holds. A bitwise comparison of the original $x$ and the perturbed input $\\tilde{x}$ would, of course, fail. Restricting the perturbation to be smaller then the maximum sensor error $\\epsilon$ makes it plausible that both inputs $x,\\tilde{x}$ could have been measured by same sensor.\n",
    "\n",
    "$\\Vert \\eta \\Vert_\\infty < \\epsilon$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.1,  0.1])"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon = 0.1\n",
    "eta = np.sign(w) * epsilon\n",
    "eta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.1450863489592109"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.68443146232189256"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w.dot(eta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def check_eta(eta, epsilon):\n",
    "    return np.linalg.norm(eta, np.inf) <= epsilon\n",
    "\n",
    "check_eta(eta, epsilon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\n",
       "       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\n",
       "       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\n",
       "       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\n",
       "       1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0,\n",
       "       0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0,\n",
       "       1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1,\n",
       "       1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1,\n",
       "       1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1])"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ct = (sigmoid(w.dot(Xt.T)) > 0.5).astype(int)\n",
    "cta = (sigmoid(w.dot(Xt.T) + w.dot(eta)) > 0.5).astype(int)\n",
    "ca = (sigmoid(w.dot(X.T) + w.dot(eta)) > 0.5).astype(int)\n",
    "ca"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Accuracy of the classifier on (i) train data set and (ii) test data set\n",
    "\n",
    "$accuracy = \\frac{correctly\\ classified\\ examples}{all\\ examples}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Train set original:  0.94\n",
      "Train set perturbed: 0.66\n"
     ]
    }
   ],
   "source": [
    "print('Train set original:  %.2f\\nTrain set perturbed: %.2f' % (np.sum(c==y)/len(y),np.sum(ca==y)/len(y)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Test set original:  0.91\n",
      "Test set perturbed: 0.64\n"
     ]
    }
   ],
   "source": [
    "print('Test set original:  %.2f\\nTest set perturbed: %.2f' % (np.sum(ct==y)/len(y), np.sum(cta==y)/len(y)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Count the occurences of examples that were misclassified because of the perturbation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([63,  7])"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.bincount(y[ca!=c].astype(np.int64))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([61,  7])"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.bincount(y[cta!=ct].astype(np.int64))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Random noise doesn't seem to have a similar impact."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Worst accuracy for randomly perturbed input: 0.82 (Test set original: 0.91)\n"
     ]
    }
   ],
   "source": [
    "res = []\n",
    "for _ in range(100000):\n",
    "    etar = np.random.rand(2,)*epsilon\n",
    "    ct = (sigmoid(w.dot(Xt.T) + w.dot(etar)) > 0.5).astype(int)\n",
    "    res.append(np.sum(ct==y)/len(y))\n",
    "print('Worst accuracy for randomly perturbed input: %.2f (Test set original: %.2f)' % (np.min(res),(np.sum(ct==y)/len(y))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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