line.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Gradient Descent by hand"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T17:53:20.853837",
     "start_time": "2017-02-15T17:53:20.851136"
    }
   },
   "source": [
    "## Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T19:58:46.808564",
     "start_time": "2017-02-15T19:58:46.486710"
    },
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from __future__ import absolute_import, division, print_function\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt, animation\n",
    "%matplotlib inline\n",
    "\n",
    "# The following is just for the final animation\n",
    "from matplotlib import pyplot as plt, rcParams, animation, rc\n",
    "from ipywidgets import interact, interactive, fixed\n",
    "from ipywidgets.widgets import *\n",
    "rc('animation', html='html5')\n",
    "rcParams['figure.figsize'] = 7, 7\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Definitions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Define Line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T19:59:17.298082",
     "start_time": "2017-02-15T19:59:17.011231"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5fb4ff36d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def line(x, a, b):\n",
    "    return a * x + b\n",
    "\n",
    "x = np.arange(10).astype(float)\n",
    "x += np.random.normal(5)\n",
    "\n",
    "# We will try to guess these values\n",
    "a = 2.5\n",
    "b = 30\n",
    "y = line(x, a, b)\n",
    "\n",
    "# Visualize our new points (on a slightly larger canvas)\n",
    "def plot_points():\n",
    "    plt.plot(x, y, 'ro')\n",
    "    plt.axis((0, x.max() * 1.3, 0, y.max() * 1.3))\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel('y')\n",
    "    plt.title('Line')\n",
    "plot_points()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Define loss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T19:59:18.397906",
     "start_time": "2017-02-15T19:59:18.390635"
    },
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2882.0503293551164"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Use mean squared loss https://en.wikipedia.org/wiki/Mean_squared_error\n",
    "def loss(y, y_pred):\n",
    "    n = y.shape[0]\n",
    "    return np.sum(np.square(y - y_pred)) / n\n",
    "\n",
    "# Test the loss\n",
    "y_pred = np.arange(10) + 100\n",
    "loss(y, y_pred)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T18:16:09.670923",
     "start_time": "2017-02-15T18:16:09.668025"
    }
   },
   "source": [
    "#### Define derivatives\n",
    "The partial derivatives show how to change a and b to make the line closer to the initial y."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T19:59:18.725296",
     "start_time": "2017-02-15T19:59:18.721553"
    },
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Partial derivative of loss wrt a\n",
    "# d[(y - (a * x + b)) ** 2, a] = 2 * x * (a * x + b - y)\n",
    "# dyda = 2 * x * (y_pred - y)\n",
    "# Partial derivative of loss wrt b\n",
    "# d[(y - (a * x + b)) ** 2, a] = 2 * (a * x + b - y)\n",
    "# dydb = 2 * (y_pred - y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Define Gradient Descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T19:59:19.056465",
     "start_time": "2017-02-15T19:59:19.049147"
    },
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "def gradient_descent(a, b, x, y, learning_rate=0.01):\n",
    "    # Prediction of y given current a & b\n",
    "    y_pred = line(x, a, b)\n",
    "    # Compute the partial derivatives\n",
    "    dyda = 2 * x * (y_pred - y)\n",
    "    dydb = 2 * (y_pred - y)\n",
    "    # Adjust the predicted a and b\n",
    "    a = a - dyda.mean() * learning_rate\n",
    "    b = b - dydb.mean() * learning_rate\n",
    "    return a, b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Animate Gradient Descent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T19:59:19.866933",
     "start_time": "2017-02-15T19:59:19.863695"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Initial guess for a & b\n",
    "a_pred = 1\n",
    "b_pred = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2017-02-15T19:59:25.977129",
     "start_time": "2017-02-15T19:59:20.423036"
    },
    "collapsed": false
   },
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x7f5fb2eaa950>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Plot the initial y\n",
    "fig = plt.figure()\n",
    "plot_points()\n",
    "# This is the line we will draw again on each frame\n",
    "draw_line, = plt.plot(x, line(x, a_pred, b_pred))\n",
    "plt.close()\n",
    "\n",
    "# Each fram runs the Gradient Descent for a few step\n",
    "\n",
    "\n",
    "def animate(frame):\n",
    "    steps = 15\n",
    "    global a_pred, b_pred\n",
    "    for i in range(steps):\n",
    "        draw_line.set_ydata(line(x, a_pred, b_pred))\n",
    "        (a_pred, b_pred) = gradient_descent(a_pred, b_pred, x, y)\n",
    "    return draw_line,\n",
    "\n",
    "# Compile the animation\n",
    "anim = animation.FuncAnimation(fig, animate, interval=50)\n",
    "# Display the animation\n",
    "anim"
   ]
  }
 ],
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