6. Modular Network.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"width: 100%; overflow: hidden;\">\n",
    "    <div style=\"width: 150px; float: left;\"> <img src=\"data/D4Sci_logo_ball.png\" alt=\"Data For Science, Inc\" align=\"left\" border=\"0\"> </div>\n",
    "    <div style=\"float: left; margin-left: 10px;\"> <h1>Deep Learning From Scratch</h1>\n",
    "<h1>Modular Network</h1>\n",
    "        <p>Bruno Gonçalves<br/>\n",
    "        <a href=\"http://www.data4sci.com/\">www.data4sci.com</a><br/>\n",
    "            @bgoncalves, @data4sci</p></div>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings('ignore')\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "\n",
    "import watermark\n",
    "\n",
    "%load_ext watermark\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.8.5\n",
      "IPython version      : 7.19.0\n",
      "\n",
      "Compiler    : Clang 10.0.0 \n",
      "OS          : Darwin\n",
      "Release     : 21.2.0\n",
      "Machine     : x86_64\n",
      "Processor   : i386\n",
      "CPU cores   : 16\n",
      "Architecture: 64bit\n",
      "\n",
      "Git hash: 5bcc9b20907e7704347c2e9d705481e2ae74c98c\n",
      "\n",
      "matplotlib: 3.3.2\n",
      "numpy     : 1.19.2\n",
      "watermark : 2.1.0\n",
      "pandas    : 1.1.3\n",
      "seaborn   : 0.11.0\n",
      "json      : 2.0.9\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%watermark -i -n -v -m -g -iv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.style.use('./d4sci.mplstyle')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_train = np.load('input/X_train.npy')\n",
    "X_test = np.load('input/X_test.npy')\n",
    "y_train = np.load('input/y_train.npy')\n",
    "y_test = np.load('input/y_test.npy')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Preprocessing and Normalization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "input_layer_size = X_train.shape[1]\n",
    "\n",
    "X_train /= 255.\n",
    "X_test /= 255."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Initialize weights"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the initializatino function as we'll have to call it more than once"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def init_weights(L_in, L_out, epsilon = 0.12):\n",
    "    return 2*np.random.rand(L_out, L_in+1)*epsilon - epsilon"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the layer sizes we'll be using"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "hidden_layer_size = 50\n",
    "num_labels = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialize the weights. In this case we use a array of weight matrices so that we can easily add/remove layers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "Thetas = []\n",
    "Thetas.append(init_weights(input_layer_size, hidden_layer_size))\n",
    "Thetas.append(init_weights(hidden_layer_size, num_labels))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Utility functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One-hot encoding to define the labels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def one_hot(K, pos):\n",
    "    y0 = np.zeros(K)\n",
    "    y0[pos] = 1\n",
    "\n",
    "    return y0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Activation function base class. Here we must provide an interface to both the activation function and its derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Activation(object):\n",
    "    def f(z):\n",
    "        pass\n",
    "\n",
    "    def df(z):\n",
    "        pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The various activation functions simply extend the base class"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Linear(Activation):\n",
    "    def f(z):\n",
    "        return z\n",
    "\n",
    "    def df(z):\n",
    "        return np.ones(z.shape)\n",
    "\n",
    "class ReLu(Activation):\n",
    "    def f(z):\n",
    "        return np.where(z > 0, z, 0)\n",
    "\n",
    "    def df(z):\n",
    "        return np.where(z > 0, 1, 0)\n",
    "\n",
    "class Sigmoid(Activation):\n",
    "    def f(z):\n",
    "        return 1./(1+np.exp(-z))\n",
    "    \n",
    "    def df(z):\n",
    "        h = Sigmoid.f(z)\n",
    "        return h*(1-h)\n",
    "\n",
    "class TanH(Activation):\n",
    "    def f(z):\n",
    "        return np.tanh(z)\n",
    "\n",
    "    def df(z):\n",
    "        return 1-np.power(np.tanh(z), 2.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Forward Propagation and Prediction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The forward and predict functions are also generalized"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def forward(Theta, X, active):\n",
    "    N = X.shape[0]\n",
    "\n",
    "    # Add the bias column\n",
    "    X_ = np.concatenate((np.ones((N, 1)), X), 1)\n",
    "\n",
    "    # Multiply by the weights\n",
    "    z = np.dot(X_, Theta.T)\n",
    "\n",
    "    # Apply the activation function\n",
    "    a = active.f(z)\n",
    "\n",
    "    return a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The predict function now takes the entire model as input and it must loop over the various layers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def predict(model, X):\n",
    "    h = X.copy()\n",
    "\n",
    "    for i in range(0, len(model), 2):\n",
    "        theta = model[i]\n",
    "        activation = model[i+1]\n",
    "\n",
    "        h = forward(theta, h, activation)\n",
    "\n",
    "    return np.argmax(h, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The accuracy function is just the same as before"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def accuracy(y_, y):\n",
    "    return np.mean((y_ == y.flatten()))*100."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Back propagation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def backprop(model, X, y):\n",
    "    M = X.shape[0]\n",
    "\n",
    "    Thetas=[0]\n",
    "    Thetas.extend(model[0::2])\n",
    "    activations = [0]\n",
    "    activations.extend(model[1::2])\n",
    "\n",
    "    layers = len(Thetas)\n",
    "\n",
    "    K = Thetas[-1].shape[0]\n",
    "    J = 0\n",
    "\n",
    "    Deltas = [0]\n",
    "\n",
    "    for i in range(1, layers):\n",
    "        Deltas.append(np.zeros(Thetas[i].shape))\n",
    "\n",
    "    deltas = [0]*(layers+1)\n",
    "\n",
    "    for i in range(M):\n",
    "        As = [0]\n",
    "        Zs = [0, 0]\n",
    "        Hs = [0, X[i]]\n",
    "\n",
    "        # Forward propagation, saving intermediate results\n",
    "        As.append(np.concatenate(([1], Hs[1])))  # Input layer\n",
    "\n",
    "        for l in range(2, layers+1):\n",
    "            Zs.append(np.dot(Thetas[l-1], As[l-1]))\n",
    "            Hs.append(activations[l-1].f(Zs[l]))\n",
    "            As.append(np.concatenate(([1], Hs[l])))\n",
    "\n",
    "        y0 = one_hot(K, y[i])\n",
    "\n",
    "        # Cross entropy\n",
    "        J -= np.dot(y0.T, np.log(Hs[-1]))+np.dot((1-y0).T, np.log(1-Hs[-1]))\n",
    "\n",
    "        deltas[layers] = Hs[layers]-y0\n",
    "\n",
    "        # Calculate the weight deltas\n",
    "        for l in range(layers-1, 1, -1):\n",
    "            deltas[l] = np.dot(Thetas[l].T, deltas[l+1])[1:]*activations[l].df(Zs[l])\n",
    "\n",
    "        Deltas[2] += np.outer(deltas[3], As[2])\n",
    "        Deltas[1] += np.outer(deltas[2], As[1])\n",
    "\n",
    "    J /= M\n",
    "\n",
    "    grads = []\n",
    "\n",
    "    grads.append(Deltas[1]/M)\n",
    "    grads.append(Deltas[2]/M)\n",
    "\n",
    "    return [J, grads]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Definition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = []\n",
    "\n",
    "# Layer 1\n",
    "model.append(Thetas[0])\n",
    "model.append(Sigmoid)\n",
    "\n",
    "# Layer 2\n",
    "model.append(Thetas[1])\n",
    "model.append(Sigmoid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Training procedure\n",
    "The same basic idea as before"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 3.1320942216417107 3.1234621323173486 45.68 41.699999999999996\n",
      "20 2.942178872062528 2.9416078268740358 58.74 54.300000000000004\n",
      "30 2.6498403186823842 2.6648267397672445 64.36 60.4\n",
      "40 2.3150585457000514 2.353600561363548 69.19999999999999 65.3\n",
      "50 2.0256068929597157 2.0858112394703796 73.61999999999999 68.7\n",
      "60 1.8040358894698654 1.8790895464102773 77.24 71.8\n",
      "70 1.634514466332663 1.7194182050879576 79.86 75.0\n",
      "80 1.500379839109838 1.5925277313527253 81.78 76.9\n",
      "90 1.3910946383657832 1.4892481107899804 83.34 79.10000000000001\n",
      "100 1.3001646136165266 1.4037441697416067 84.66 81.10000000000001\n",
      "110 1.223245919869348 1.3319585807531926 85.66 82.6\n",
      "120 1.1572302049126235 1.270878245662349 86.53999999999999 83.39999999999999\n",
      "130 1.0998339235853507 1.218224398599894 87.1 84.2\n",
      "140 1.0493730862286945 1.1722861763185946 87.78 84.39999999999999\n",
      "150 1.0046031777494882 1.1317920041437493 88.26 84.8\n",
      "160 0.9645942104547188 1.0957976816498334 88.68 85.39999999999999\n",
      "170 0.928635705290855 1.0635933528315655 89.08 85.9\n",
      "180 0.8961685486721342 1.0346320605024562 89.46 86.4\n",
      "190 0.8667388258543945 1.0084788208324593 89.66 86.6\n",
      "200 0.8399679030591409 0.984776553151143 89.8 87.0\n",
      "210 0.8155335309537786 0.9632243772640603 90.14 86.9\n",
      "220 0.7931579000089969 0.943564169153674 90.3 87.2\n",
      "230 0.772599822409826 0.9255721895796613 90.4 87.1\n",
      "240 0.7536492522947034 0.909053604165577 90.72 87.1\n",
      "250 0.736123102729512 0.8938385535509157 90.86 87.2\n",
      "260 0.7198618006235481 0.8797790310014575 91.0 87.1\n",
      "270 0.7047263049613789 0.8667462012025926 91.16 87.2\n",
      "280 0.690595465414547 0.8546280039391856 91.34 87.3\n",
      "290 0.6773636701584357 0.8433269886567286 91.5 87.4\n",
      "300 0.6649387599294202 0.8327583667088921 91.60000000000001 87.4\n",
      "310 0.653240192649887 0.82284827828515 91.84 87.6\n",
      "320 0.6421974420179059 0.8135322686582488 91.92 87.6\n",
      "330 0.6317486106063448 0.8047539627781447 92.08 87.7\n",
      "334 0.6277229503365784 0.8047539627781447 92.08 87.7\n",
      "DONE\n"
     ]
    }
   ],
   "source": [
    "step = 0\n",
    "tol = 1e-3\n",
    "J_old = 1/tol\n",
    "diff = 1\n",
    "\n",
    "acc_train = []\n",
    "acc_test = []\n",
    "J_val = []\n",
    "steps = []\n",
    "\n",
    "alpha = 0.5\n",
    "\n",
    "while diff > tol:\n",
    "    J_train, grads = backprop(model, X_train, y_train)\n",
    "\n",
    "    diff = abs(J_old-J_train)\n",
    "    J_old = J_train\n",
    "    J_val.append(J_train)\n",
    "\n",
    "    step += 1\n",
    "\n",
    "    for i in range(len(Thetas)):\n",
    "        Thetas[i] -= alpha*grads[i]\n",
    "    \n",
    "    if step % 10 == 0:\n",
    "        pred_train = predict(model, X_train)\n",
    "        pred_test = predict(model, X_test)\n",
    "\n",
    "        J_test, grads = backprop(model, X_test, y_test)\n",
    "\n",
    "        acc_train.append(accuracy(pred_train, y_train))\n",
    "        acc_test.append(accuracy(pred_test, y_test))\n",
    "        steps.append(step)\n",
    "        \n",
    "        print(step, J_train, J_test, acc_train[-1], acc_test[-1])\n",
    "        \n",
    "print(step, J_train, J_test, acc_train[-1], acc_test[-1])\n",
    "print(\"DONE\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Accuracy during training"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Cost function')"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      "text/plain": [
       "<Figure size 3840x2640 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.arange(1, len(J_val)+1), J_val)\n",
    "plt.xlabel(\"Iterations\")\n",
    "plt.ylabel(\"Cost function\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fbf707e73a0>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 3840x2640 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(steps, acc_train, label='Training dataset')\n",
    "plt.plot(steps, acc_test, label='Testing dataset')\n",
    "plt.xlabel(\"iterations\")\n",
    "plt.ylabel(\"Accuracy (%)\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"width: 100%; overflow: hidden;\">\n",
    "     <img src=\"data/D4Sci_logo_full.png\" alt=\"Data For Science, Inc\" align=\"center\" border=\"0\" width=300px> \n",
    "</div>"
   ]
  }
 ],
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