Backpropagation Tutorial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A Small Explanation of Backpropagation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Initialization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from IPython.display import Image"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Want first to train one layer network which tries to predict  \n",
    "output \n",
    "$$\n",
    " y = f(\\vec x) = f(x_1, x_2, x_3)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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efGDTfkkv4/ffp8m9bxyX8/TyTVC3Cngm5LZLHo9/M+M1nVfA/brA9XUWOM+b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\n5uPk3yeN04/3sLmNVZtXENe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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 2,
     "metadata": {
      "image/png": {
       "width": 600
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image( filename = \"one_layer_net.png\", width = 600 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# input \n",
    "X = np.array([ [0,0,2],\n",
    "              [0,1,1],\n",
    "              [1,0,1],\n",
    "              [1,1,1] ])\n",
    "\n",
    "y = np.array([[0,0,1,1]]).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the rows of the matrix $X$ are 4 \"datapotins\":\n",
    "$$\n",
    "    X = \\begin{pmatrix}\n",
    "    x^{(1)}_1 \\; \\; x^{(1)}_2 \\; \\; x^{(1)}_3 \\\\\n",
    "    x^{(2)}_1 \\; \\; x^{(2)}_2 \\; \\; x^{(2)}_3 \\\\\n",
    "    x^{(3)}_1 \\; \\; x^{(3)}_2 \\; \\; x^{(3)}_3 \\\\\n",
    "    x^{(4)}_1 \\; \\; x^{(4)}_2 \\; \\; x^{(4)}_3 \\\\\n",
    "    \\end{pmatrix}\n",
    "      =\\begin{pmatrix} \\vec x^{(1)} \\\\ \\vec x^{(2)} \\\\ \\vec x^{(3)} \\\\ \\vec x^{(4)} \\\\ \\end{pmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X = \n",
      "[[0 0 2]\n",
      " [0 1 1]\n",
      " [1 0 1]\n",
      " [1 1 1]]\n",
      "\n",
      "\n",
      "y = \n",
      "[[0]\n",
      " [0]\n",
      " [1]\n",
      " [1]]\n"
     ]
    }
   ],
   "source": [
    "print 'X = \\n', X\n",
    "print '\\n'\n",
    "print 'y = \\n', y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pay attention: obviously $f(\\vec{x})$ *might be* linear function of inputs, obviously with the first weight 1 and two others 0:\n",
    "$$\n",
    "    f(\\vec{x}) = 1 \\cdot x_1 + 0 \\cdot x_2 + 0 \\cdot x_3\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We utilize sigmoid activation function:\n",
    "$$\n",
    "    h(a) = \\frac{1}{1 + e^{-a}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# sigmoid function\n",
    "def nonlin( x, deriv = False ):\n",
    "    if( deriv == True ):\n",
    "        return x * ( 1 - x )\n",
    "    return 1/( 1 + np.exp( -x ) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# seed random numbers to make calculation deterministic\n",
    "np.random.seed( 1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One Layer Back-Propagation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# initialize weights uniformly randomly with mean 0\n",
    "syn0 = 2*np.random.random( (3,1) ) - 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the weights are a vector: \n",
    "$$\n",
    "\\vec{w} = \\begin{pmatrix} w_{11}\\\\ w_{12} \\\\ w_{13} \\end{pmatrix}  \\qquad \\text{notation $w_{ji} = z_i \\to a_j$}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.16595599]\n",
      " [ 0.44064899]\n",
      " [-0.99977125]]\n"
     ]
    }
   ],
   "source": [
    "# weights (syn stands for 'synaps' here)\n",
    "print syn0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Prerequisites/Refresher\n",
    "\n",
    "We assume quadratc loss: $E = \\frac{1}{2} \\Bigl(h(a_1) - y\\Bigr)^2$, this means that the initial **error** looks like\n",
    "$$\n",
    "    \\delta_1 = \\frac{\\partial E}{\\partial a_1} = h'(a_1) \\cdot \\Bigl(h(a_1) - y\\Bigr)\n",
    "$$\n",
    "\n",
    "The respecive gradient component\n",
    "$$\n",
    "    \\nabla_{ji} E \\equiv \\frac{\\partial E}{\\partial w_{ji}} = \\delta_j \\cdot z_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Now Run Back-Propagation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Back-propagation\n",
    "for iter in xrange( 10000 ):\n",
    "\n",
    "    # forward propagation of the result for all four \n",
    "    #  data rows simultaneously\n",
    "    l0 = X\n",
    "    l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "\n",
    "    # how much did we miss?\n",
    "    l1_error = l1 - y\n",
    "\n",
    "    # multiply how much we missed by the \n",
    "    # slope of the sigmoid at the values in l1\n",
    "    l1_delta = l1_error * nonlin( l1, True )\n",
    "\n",
    "    # update weights in the direction of gradient\n",
    "    syn0 += - np.dot( l0.T, l1_delta )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The output "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 9.67299303]\n",
      " [-0.2078435 ]\n",
      " [-4.62963669]]\n"
     ]
    }
   ],
   "source": [
    "# weights\n",
    "print syn0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.009664  ]\n",
      " [ 0.00786466]\n",
      " [ 0.99358931]\n",
      " [ 0.99211997]]\n"
     ]
    }
   ],
   "source": [
    "# output\n",
    "l0 = X\n",
    "l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "print l1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Works! Additional things to see:**\n",
    "- Change the speed of convergence?\n",
    "\n",
    "**Question**\n",
    "Remember that the guessed that the weights are $(1,0,0)$, why have we obtained smth else?\n",
    "\n",
    "**Answer**\n",
    "Sigmoid function as activation function. Could we use linear (i.e. identity) $h(a) = a$?\n",
    "\n",
    "Let's try!\n",
    "\n",
    "### Back-Prop with identity activation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# identity function\n",
    "def ident_activation( x, deriv = False ):\n",
    "    if( deriv == True ):\n",
    "        return 1\n",
    "    return x\n",
    "\n",
    "# seed random numbers to make calculation\n",
    "# deterministic (just a good practice)\n",
    "np.random.seed(1)\n",
    "\n",
    "# initialize weights randomly with mean 0\n",
    "syn0 = 2*np.random.random((3,1)) - 1\n",
    "\n",
    "for iter in xrange(10000):\n",
    "\n",
    "    # forward propagation\n",
    "    l0 = X\n",
    "    l1 = ident_activation( np.dot( l0, syn0 ) )\n",
    "    \n",
    "    # how much did we miss?\n",
    "    l1_error = l1 - y\n",
    "        \n",
    "    # multiply how much we missed by the \n",
    "    # slope of the activation at the values in l1\n",
    "    l1_delta = l1_error * ident_activation( l1, True )\n",
    "        \n",
    "    # update weights \n",
    "    syn0 += - 0.1 * np.dot( l0.T, l1_delta ) # don't ask why we need 0.1! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  1.00000000e+00]\n",
      " [  5.12930596e-31]\n",
      " [ -4.44089210e-16]]\n"
     ]
    }
   ],
   "source": [
    "# weights\n",
    "print syn0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ -4.44089210e-16]\n",
      " [ -4.44089210e-16]\n",
      " [  1.00000000e+00]\n",
      " [  1.00000000e+00]]\n"
     ]
    }
   ],
   "source": [
    "# output\n",
    "l0 = X\n",
    "l1 = ident_activation( np.dot( l0, syn0 ) )\n",
    "print l1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Works!**\n",
    "\n",
    "So replacing the sigmoid function by identity worked **in this case** (JUST FOR DEMONSTRATION). Obviously this type of output is too unflexible!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question** \n",
    "Which type of Gradient Descent was used here?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer**\n",
    "\n",
    "See how the gradients are computed and weights are updated:\n",
    "\n",
    "    l0 = X  # -----> here we use ALL data rows\n",
    "    l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "\n",
    "    syn0 += - np.dot( l0.T, l1_delta )  # -----> we actually summed up all the rows!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This product of matrix $X$ and vector $\\vec{\\delta}$ means\n",
    "\n",
    "$$\n",
    "    \\vec{w}_{k+1} = \\vec{w}_k - \\Biggl[ \n",
    "        \\nabla E \\bigl(\\vec{x}^{(1)} \\bigr) \n",
    "        + \\nabla E\\bigl(\\vec{x}^{(2)} \\bigr) \n",
    "        + \\nabla E\\bigl(\\vec{x}^{(3)} \\bigr) \n",
    "        + \\nabla E\\bigl(\\vec{x}^{(4)} \\bigr) \\Biggr]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Now what if we do SGD (randomly choose online Grad Descent)?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for iter in xrange( 50000 ):\n",
    "\n",
    "    rand_index = np.random.randint( 4 )\n",
    "    \n",
    "    # forward propagation with ONLY ONE datapoint X\n",
    "    l0 = X[rand_index].reshape(1,3)\n",
    "    l1 = nonlin( np.dot( l0, syn0 ) )  # now one-dim\n",
    "    \n",
    "    # how much did we miss?\n",
    "    l1_error = l1 - y[rand_index]  # now one-dim\n",
    "    \n",
    "    # multiply how much we missed by the \n",
    "    # slope of the sigmoid at the values in l1\n",
    "    l1_delta = l1_error * nonlin( l1, True )\n",
    "\n",
    "    # update weights\n",
    "    syn0 += -np.dot( l0.T, l1_delta )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 9.90263399]\n",
      " [-0.20864662]\n",
      " [-4.74026187]]\n"
     ]
    }
   ],
   "source": [
    "# weights\n",
    "print syn0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.0086607 ]\n",
      " [ 0.00704121]\n",
      " [ 0.99430453]\n",
      " [ 0.99299238]]\n"
     ]
    }
   ],
   "source": [
    "# output\n",
    "l0 = X\n",
    "l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "print l1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Works!**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Harder Problem — No Easy Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# input \n",
    "X = np.array([ [0,0,1],\n",
    "              [0,1,1],\n",
    "              [1,0,1],\n",
    "              [1,1,1] ])\n",
    "\n",
    "y = np.array([[0,1,1,0]]).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X = \n",
      "[[0 0 1]\n",
      " [0 1 1]\n",
      " [1 0 1]\n",
      " [1 1 1]]\n",
      "\n",
      "\n",
      "y = \n",
      "[[0]\n",
      " [1]\n",
      " [1]\n",
      " [0]]\n"
     ]
    }
   ],
   "source": [
    "print 'X = \\n', X\n",
    "print '\\n'\n",
    "print 'y = \\n', y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Try to run the **one-layer** code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# seed random numbers to make calculation\n",
    "# deterministic (just a good practice)\n",
    "np.random.seed(1)\n",
    "\n",
    "# initialize weights randomly with mean 0\n",
    "syn0 = 2 * np.random.random((3,1)) - 1\n",
    "\n",
    "# Back-propagation\n",
    "for iter in xrange( 10000 ):\n",
    "\n",
    "    # forward propagation of the result for all four \n",
    "    #  data rows simultaneously\n",
    "    l0 = X\n",
    "    l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "\n",
    "    # how much did we miss?\n",
    "    l1_error = l1 - y\n",
    "\n",
    "    # multiply how much we missed by the \n",
    "    # slope of the sigmoid at the values in l1\n",
    "    l1_delta = l1_error * nonlin( l1, True )\n",
    "\n",
    "    # update weights in the direction of gradient\n",
    "    syn0 += - np.dot( l0.T, l1_delta )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.5]\n",
      " [ 0.5]\n",
      " [ 0.5]\n",
      " [ 0.5]]\n"
     ]
    }
   ],
   "source": [
    "# output\n",
    "l0 = X\n",
    "l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "print l1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks like he puts all weights to 0?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  2.08166817e-16]\n",
      " [  2.22044605e-16]\n",
      " [ -3.05311332e-16]]\n"
     ]
    }
   ],
   "source": [
    "# weights\n",
    "print syn0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yes, all weights 0: we can't learn such function!\n",
    "Need something more complicated: two layers?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Two Layers for Harder Problem\n",
    "\n",
    "We hope that two layers can learn the necessary function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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COmrYPq4Qd+sKL4wgO8htTXSs6mEKlKM5R33VDLbbT2luCBB3gmpL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\nAAAA6NEiv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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 23,
     "metadata": {
      "image/png": {
       "width": 600
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image( filename = \"two_layer_net.png\", width = 600 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X = \n",
      "[[0 0 1]\n",
      " [0 1 1]\n",
      " [1 0 1]\n",
      " [1 1 1]]\n",
      "\n",
      "\n",
      "y = \n",
      "[[0]\n",
      " [1]\n",
      " [1]\n",
      " [0]]\n",
      "Error STEP 0:  0.496410031903\n",
      "Error STEP 10000:  0.00858452565325\n",
      "Error STEP 20000:  0.00578945986251\n",
      "Error STEP 30000:  0.00462917677677\n",
      "Error STEP 40000:  0.00395876528027\n",
      "Error STEP 50000:  0.00351012256786\n",
      "Error STEP 60000:  0.00318350238587\n",
      "Error STEP 70000:  0.00293230634228\n",
      "Error STEP 80000:  0.00273150641821\n",
      "Error STEP 90000:  0.00256631724004\n"
     ]
    }
   ],
   "source": [
    "# input \n",
    "X = np.array([ [0,0,1],\n",
    "              [0,1,1],\n",
    "              [1,0,1],\n",
    "              [1,1,1] ])\n",
    "\n",
    "y = np.array([[0,1,1,0]]).T\n",
    "\n",
    "print 'X = \\n', X\n",
    "print '\\n'\n",
    "print 'y = \\n', y\n",
    "\n",
    "np.random.seed(1)\n",
    "\n",
    "number_hidden_neurons = 4\n",
    "\n",
    "# randomly initialize our weights with mean 0\n",
    "syn0 = 2*np.random.random( (3, number_hidden_neurons) ) - 1\n",
    "syn1 = 2*np.random.random( ( number_hidden_neurons,1 ) ) - 1\n",
    "\n",
    "for j in xrange( 100000 ):\n",
    "\n",
    "\t# Feed forward through layers 0, 1, and 2\n",
    "    l0 = X\n",
    "    l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "    l2 = nonlin( np.dot( l1, syn1 ) )\n",
    "\n",
    "    # how much did we miss the target value?\n",
    "    l2_error = l2 - y\n",
    "    \n",
    "    if ( j % 10000 ) == 0:\n",
    "        print \"Error STEP \" + str( j ) + \":  \" +  str( np.mean( np.abs(l2_error) ) )\n",
    "        \n",
    "    # in what direction is the target value?\n",
    "    # were we really sure? if so, don't change too much.\n",
    "    l2_delta = l2_error * nonlin( l2, deriv = True )\n",
    "\n",
    "    # how much did each l1 value contribute to the l2 error (according to the weights)?\n",
    "    l1_error = l2_delta.dot( syn1.T )\n",
    "    \n",
    "    # in what direction is the target l1?\n",
    "    # were we really sure? if so, don't change too much.\n",
    "    l1_delta = l1_error * nonlin( l1, deriv = True )\n",
    "\n",
    "    syn1 += - l1.T.dot( l2_delta )\n",
    "    syn0 += - l0.T.dot( l1_delta )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.00199094]\n",
      " [ 0.99751458]\n",
      " [ 0.99771098]\n",
      " [ 0.00294418]]\n"
     ]
    }
   ],
   "source": [
    "# output\n",
    "print l2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Works!**\n",
    "\n",
    "<img src=\"files/success_baby.jpg\" width = \"400\">\n",
    "\n",
    "### Questions\n",
    "\n",
    "- Try to change the number of hidden neurons to 10? to 2? What happens?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What about Drop-Out?\n",
    "\n",
    "It switches off certain (random on each step) hidden neurons so that they don't propagate computation **forward**. This hopefully reduces some of the computation associated with them (for large nets). **During backward step** we keep them!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X = \n",
      "[[0 0 1]\n",
      " [0 1 1]\n",
      " [1 0 1]\n",
      " [1 1 1]]\n",
      "\n",
      "\n",
      "y = \n",
      "[[0]\n",
      " [1]\n",
      " [1]\n",
      " [0]]\n"
     ]
    }
   ],
   "source": [
    "print 'X = \\n', X\n",
    "print '\\n'\n",
    "print 'y = \\n', y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error STEP 0:  0.49619315376\n",
      "Error STEP 10000:  0.131834498105\n",
      "Error STEP 20000:  0.00660730539334\n",
      "Error STEP 30000:  0.00526422501944\n",
      "Error STEP 40000:  0.00449259810269\n",
      "Error STEP 50000:  0.127981329241\n",
      "Error STEP 60000:  0.375855658433\n",
      "Error STEP 70000:  0.251508012642\n",
      "Error STEP 80000:  0.251371984684\n",
      "Error STEP 90000:  0.12700962344\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(1)\n",
    "\n",
    "number_hidden_neurons = 4\n",
    "\n",
    "# randomly initialize our weights with mean 0\n",
    "syn0 = 2*np.random.random( ( 3, number_hidden_neurons ) ) - 1\n",
    "syn1 = 2*np.random.random( ( number_hidden_neurons, 1 ) ) - 1\n",
    "\n",
    "for j in xrange( 100000 ):\n",
    "\n",
    "    # Feed forward through layers 0, 1, and 2 \n",
    "    # WITH Drop-Out \n",
    "    \n",
    "    l0 = X\n",
    "    l1 = nonlin( np.dot( l0, syn0 ) )\n",
    "    \n",
    "    # Magic\n",
    "    l1_drop = l1\n",
    "    for k in range( len( l1 ) ):\n",
    "        if np.random.random() < 0.2:\n",
    "            l1_drop[k] = 0.0\n",
    "    \n",
    "    l2 = nonlin( np.dot( l1_drop, syn1 ) )\n",
    "\n",
    "    # how much did we miss the target value?\n",
    "    l2_error = l2 - y\n",
    "    \n",
    "    if ( j % 10000 ) == 0:\n",
    "        print \"Error STEP \" + str( j ) + \":  \" +  str( np.mean( np.abs(l2_error) ) )\n",
    "        \n",
    "    # in what direction is the target value?\n",
    "    # were we really sure? if so, don't change too much.\n",
    "    l2_delta = l2_error * nonlin( l2, deriv = True )\n",
    "\n",
    "    # how much did each l1 value contribute to the l2 error (according to the weights)?\n",
    "    l1_error = l2_delta.dot( syn1.T )\n",
    "    \n",
    "    # in what direction is the target l1?\n",
    "    # were we really sure? if so, don't change too much.\n",
    "    l1_delta = l1_error * nonlin( l1, deriv = True )\n",
    "\n",
    "    syn1 += - l1.T.dot( l2_delta )\n",
    "    syn0 += - l0.T.dot( l1_delta )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Notice** Obviously tries to regularize (notice error goes back to some level)! However solves correctly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.0022532 ]\n",
      " [ 0.99727598]\n",
      " [ 0.99737811]\n",
      " [ 0.00336391]]\n"
     ]
    }
   ],
   "source": [
    "# output\n",
    "print l2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*A. Gronskiy, Based on http://iamtrask.github.io/2015/07/12/basic-python-network/, revised, refactored*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Appendix: Additional Stuff \n",
    "\n",
    "## Tanh Activation? \n",
    "Back to one-layer, simple problem, just to demonstrate tanh (shifted and scaled to imitate 0-1 step) activation:\n",
    "\n",
    "$$\n",
    "    h(a) = \\frac{1}{2}\\Bigl(\\tanh(a) + 1\\Bigr)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X = \n",
      "[[0 0 1]\n",
      " [0 1 1]\n",
      " [1 0 1]\n",
      " [1 1 1]]\n",
      "\n",
      "\n",
      "y = \n",
      "[[0]\n",
      " [0]\n",
      " [1]\n",
      " [1]]\n"
     ]
    }
   ],
   "source": [
    "# input \n",
    "X = np.array([ [0,0,1],\n",
    "              [0,1,1],\n",
    "              [1,0,1],\n",
    "              [1,1,1] ])\n",
    "\n",
    "y = np.array([[0,0,1,1]]).T\n",
    "\n",
    "print 'X = \\n', X\n",
    "print '\\n'\n",
    "print 'y = \\n', y\n",
    "\n",
    "# tanh function\n",
    "def tanh_activation( x, deriv = False ):\n",
    "    if( deriv == True ):\n",
    "        return 0.5 * ( 1 - x * x )\n",
    "    return 0.5 * (np.exp(x) - np.exp(-x))/( np.exp(x) + np.exp( -x ) ) + 0.5\n",
    "\n",
    "\n",
    "# seed random numbers to make calculation\n",
    "# deterministic (just a good practice)\n",
    "np.random.seed(1)\n",
    "\n",
    "# initialize weights randomly with mean 0\n",
    "syn0 = 2*np.random.random((3,1)) - 1\n",
    "\n",
    "# Back-propagation\n",
    "for iter in xrange( 30000 ):\n",
    "\n",
    "    # forward propagation of the result for all four \n",
    "    #  data rows simultaneously\n",
    "    l0 = X\n",
    "    l1 = tanh_activation( np.dot( l0, syn0 ) )\n",
    "\n",
    "    # how much did we miss?\n",
    "    l1_error = l1 - y\n",
    "\n",
    "    # multiply how much we missed by the \n",
    "    # slope of the sigmoid at the values in l1\n",
    "    l1_delta = l1_error * tanh_activation( l1, True )\n",
    "\n",
    "    # update weights in the direction of gradient\n",
    "    syn0 += - np.dot( l0.T, l1_delta )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 7.88724422]\n",
      " [-0.1875984 ]\n",
      " [-4.95679531]]\n"
     ]
    }
   ],
   "source": [
    "# Weights\n",
    "print syn0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  4.94965918e-05]\n",
      " [  3.40123090e-05]\n",
      " [  9.97159360e-01]\n",
      " [  9.95871418e-01]]\n"
     ]
    }
   ],
   "source": [
    "# Output\n",
    "print l1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
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