{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "import random\n",
    "from scipy import stats\n",
    "from scipy.optimize import fmin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gradient Descent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b>Gradient descent</b>, also known as <b>steepest descent</b>, is an optimization algorithm for finding the local minimum of a function. To find a local minimum, the function \"steps\" in the  direction of the negative of the gradient. <b>Gradient ascent</b> is the same as gradient descent, except that it steps in the direction of the positive of the gradient and therefore finds local maximums instead of minimums. The algorithm of gradient descent can be outlined as follows:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp; 1: &nbsp; Choose initial guess $x_0$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    2: &nbsp; <b>for</b> k = 0, 1, 2, ... <b>do</b> <br>\n",
    "&nbsp;&nbsp;&nbsp;    3:   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $s_k$ = -$\\nabla f(x_k)$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    4:   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; choose $\\alpha_k$ to minimize $f(x_k+\\alpha_k s_k)$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    5:   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $x_{k+1} = x_k + \\alpha_k s_k$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    6: &nbsp;  <b>end for</b>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a simple example, let's find a local minimum for the function $f(x) = x^3-2x^2+2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambda x: x**3-2*x**2+2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-1,2.5,1000)\n",
    "plt.plot(x,f(x))\n",
    "plt.xlim([-1,2.5])\n",
    "plt.ylim([0,3])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see from plot above that our local minimum is gonna be near around 1.4 or 1.5 (on the x-axis), but let's pretend that we don't know that, so we set our starting point (arbitrarily, in this case) at $x_0 = 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Local minimum occurs at: 1.3334253508453249\n",
      "Number of steps: 17\n"
     ]
    }
   ],
   "source": [
    "x_old = 0\n",
    "x_new = 2 # The algorithm starts at x=2\n",
    "n_k = 0.1 # step size\n",
    "precision = 0.0001\n",
    "\n",
    "x_list, y_list = [x_new], [f(x_new)]\n",
    "\n",
    "# returns the value of the derivative of our function\n",
    "def f_prime(x):\n",
    "    return 3*x**2-4*x\n",
    " \n",
    "while abs(x_new - x_old) > precision:\n",
    "    x_old = x_new\n",
    "    s_k = -f_prime(x_old)\n",
    "    x_new = x_old + n_k * s_k\n",
    "    x_list.append(x_new)\n",
    "    y_list.append(f(x_new))\n",
    "print(\"Local minimum occurs at:\", x_new)\n",
    "print(\"Number of steps:\", len(x_list))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The figures below show the route that was taken to find the local minimum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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aZQVGOiwh45xzhVm82KY+mDIFnngCbrwRpNjFpiLnyZRzaWDUKOtYWbNm2JE451z8TZ4Mp58Oa9fCyJFw6qnRfX/vPeFcips3D6ZPh9NOCzsS55yLv6FD4ZhjrHb+yy+jn0iBJ1POpbwRI+znmWeGG4dzzsWTKjz4IJx9NrRta/NJHXxwbPblzXzOpbgRI6wgadYs7Eiccy4+srPh8sshKwt69YJXX4XKlWO3P6+Zci6F/fGHTUTntVLOuXSxcKE162VlwQMPwBtvxDaRAq+Zci6lvf++VXV7MuWcSweffmrNeps3W0fzePUV9Zop51LYiBGw777QunXYkTjnXOyoQt++8Le/wR57WP+oeA668WTKuRS1Zg2MG2e1UtGaS8U55xJNdjZcfDFcd52N1Js4sWxLw+yKEpMpEWkkIuNFZJaIzBSR6wrZRkSkr4jMEZFpInJobMJ1zkUkK4tR+13L1q1w5hvdrfNAGvLyy7kUlJUFTZtCuXIs3PsIjmm1ktdfh3vvheHDYbfd4h9SJH2mcoCbVHWKiNQAJovIWFWdlW+bk4Hmwe1w4MXgp3Mu3rKyoE8fhm8cTH2W0GHpe9BnjD3Xq1e4scWfl1/OpZKgfGPjRsbShV6Ls9hMeT64aQKn3t05tLBKTKZUdQmwJLi/TkR+BBoC+QujbsDrqqrANyJSS0QaBK91Jfj9d5g6FWbOtNFXy5fbwrQZGVC9Ouy1FzRqZMPbW7aEihXDjtgltDvvZN3GcoziFC5lAOVQW5DqzjvTLpny8su5FHPnnWzbuIkHuJt7uYeWzGIYZ9Fi6GZ44rfQwirVaD4RaQocAkws8FRDYGG+3xcFj+1QGIlIH6APQOPGjUsXaQpRtREHI0bA6NEwZ87256pUgbp1LWHKzbWp71es2P58xYrQvr2tdN21q01A5v1h3A4WLOADerKJKvTkrR0eT2dlLb+C9/AyzLmwrFzJ8vkb6MVHjOUEejOYF7iKamyEBeH+I4y4A7qIVAeGAder6tpd2Zmq9lfVTFXNrFu37q68RVLbsAGefhoOOACOPRb694f997fHPv0UVq2yCoT582H2bJg712qpNm2Cn3+Gt96Ca6+1be64w2qqDj4YnnwSli4N++hcwth7b96iJw1ZxJF8tf3xNP7nH43yC7wMcy4Uy5fDbbfxZaOeHML3fEZHXuEyBnGxJVIQevkWUTIlIhWwgihLVYcXsslioFG+3/cOHnPYfBfPPAP77GOrVNetC4MHw8qVtgDt9ddDx45Qu3bhr69UyZKuc8+Fxx+3BRuXLIGXXoIaNeCf/7RmwEsvhZ9+iu+xucTz58Gd+C8ncS5vWxMfQNWqtq5CGvLyy7kktXQp3Hwz2qQpTz6aQ6dN/6XyHtX4pvKxXMYA/qqLSoDyLZLRfAIMAH5U1aeK2Gwk0DsYFdMBWOP9DcwXX1gN0g03wEEH2WzUX3wBvXvb339X1a8P//d/9n4//mj98f7zH+tTdfbZVrPl0tCKFbz3SXW2UpFz639mbcBNmlg1aJr1lwIvv5xLSkuWWM1Ds2asenIg3ff4lH/yBN3OzGDy3Nq0ffUaK9cSqXxT1WJvwNGAAtOAqcGtK3AFcEWwjQDPA3OB6UBmSe/brl07TWWbNqlee60qqDZtqjp6dOz3uWyZ6r/+pVqtmmr58qrXXGOPuTRy0016Iv/VZntv1tzcsIPZGTBJSygbonmLVfmlaVCGORd3ixbZP87KlVUzMnTCCQ/q3vW3aIUKqk8/raGXacWVX3Er1AreUrkgmjdPNTPTzu6116quXx/f/S9ZonrFFaoZGaq1aqm++mr4H0IXB4sX67JKe2uG5Ojtt4cdTOHinUzF8pbKZZhzcbVggerVV6tWqqSakaFbL7pU77p6pZYrp9q8ueqkSWEHaIorv3wG9Cj78kvIzLRmthEj4NlnoVq1+MZQvz68+CJMn24d1C+7zDq8//xzfONwcfbAAwzbejrbNINzzw07GOecK8H8+XDllbbm1csvw4UXMn/8PDrNfpX7n9+d3r1hyhRo1y7sQEvmyVQUDRtm6wLVqWOdxM84I9x4DjwQxo+HV16BH36ANm0suVMNNy4XA/PmwSuv8Hrdm2jZ0pJo55xLSL/+CpdfDvvtBwMG2OipOXN494RXaHNaY2bMsD7AAwfaXIvJwJOpKBk82Dp+H3qo1U7tu2/YEZly5axm6scf4fjjbeRg164+lULKufdeZpdrwddL9+Gii3zuMedcApo7F/7+d2jeHF5/3UZRzZ3Lusde5LL7m3DOOVYJMHUqnHde2MGWjidTUfDmm3DJJdClC3zyidVMJZr69WHkSHj+eZgwwUYWfvRR2FG5qJg1C958k9cPfYZy5eCCC8IOyDnn8vnlF7joIlt9eMgQuPpqq03v14/Pf2tEmzZWC3XHHfDZZ9CsWdgBl54nU2U0dKh9Ro49Ft57r2zTHcSaCFx1FUyaBA0awCmnwP3320zrLondfTe5Vavz+qJjOf54W37IOedC9+OPdnV34IHw7rs26/S8efDss2zaoyE33wydOlkLymef2VRRFSqEHfSu8WSqDL780j4nRxxhtT6JnEjl16oVfP21xX733da3a/XqsKNyu2TKFBg2jE+7P8uCRRlcdFHYATnn0t7MmdZO16qVjcS68UbrJ/XUU9CgAd9/bwO1nnjCWvqmToWjjgo76LLxZGoXzZ4N3brZDPbvvx//EXtlVbWq9fN67jlr7jvsMGstcknmX/+C3XdncM757LZb+IMenHNpbNo0OOcc60fy4Ydw663w22+2dEe9euTkwEMPweGH2/Jpo0fbyPNk6WReHE+mdsHatXDaadZsNno07LFH2BHtGhG45hob8bduHRx5pPX5ckniiy/go49Yf8NdDH2/IuecYwtlO+dcXE2dCt2725Dx//7XOj/99hs8/LCtn4ZVVh11FNx5p206YwacfHK4YUeTJ1OlpGqj4+bMsf5S++0XdkRld/TRMHGire938sk2UtUlOFUrlerX5906V7JhA97E55yLr8mTrYnmkENg3DjrN/Lbb/DAA3/VMmzZAvfdZ5vMm2f9z996C3bfPdzQo6182AEkm+ees350jz5qHedSRZMmVtFxzjmWLM6da9+Hcp5uJ6axY63HZr9+vDyoEgcemPx9DpxzSeLbby1DGjUKatWCe++1zuW1au2w2aRJNoXUtGnQsyf07ftXRVXK8X+VpfDtt3DTTZaI33xz2NFEX82a1szdp4/VzvbqZVcVLsHk1Uo1bcrU9n2YONH+Zj63lHMupr7+2povDj/c7j/wgM1ifvfdOyRS2dnWXerww2H5cutXPGRI6iZS4DVTEdu4ES680IadDxqUuv+4KlSAl16ySUdvvdU6CQ4blhodBFPGe+/ZJd/Agbz8WgUqV4bevcMOyjmXsr74wmqixo61iRQfecTm2alRY6dNP/vMJjf/5RerlXriiZ0qrFKS10xF6Pbb7cMxcGDqfzBE4JZb4LXXrEN6ly6wcmXYUTkAtm2Du+6CFi1Yf8YFZGVZ02yq9T9wziWATz+F446DY46xNckef9ymOLj11p0SqRUrbHLzTp2sRWPsWHj11dT/f5nHk6kIjBtnbb3/+Id9rtLFJZdYrdTUqdCxIyxeHHZEjiFDbFjMffcx5N3yrFsHV1wRdlDOuZShav/0OnWCzp1t4s2nnrIk6p//3KmZIjfXLrxbtIA33rA8a8YMuwhPJ55MlWDDhu1LCT3ySNjRxN8ZZ9g8VAsXWgfn2bPDjiiNbd0K99wDbdtCjx689JJN59KhQ9iBOeeSnqpVJ3XsCH/7mw1Zf/ZZG4J3ww2Fzko9Y4blXJdeCi1bwvff2//JZJt3MRo8mSpBXv+6V19NnhnOo+3YY20uqg0bbBqFGTPCjihNDRxoBdsDD/DNt+WYMsVmD07V/nvOuThQtbmhjjwSTjjBpjbo18+GdF97baGT123YALfdZtMdzJpl0+l8+im0bh3/8BOFJ1PFmDXLOs9dfLEl6+msXTv4/HMoX95qfqdODTuiNLNpk3UAPeII6NqVp5+20Zc+t5Rzbpeo2vDtww+3EXq//27Tkc+ZYwsRV65c6EveegsOOMCmB+rdG37+2Vpv0n0anTQ//KKpbh+s8NhjYUeTGA44wK4+qla1vmOTJoUdURp58UXrtPbQQyxYKAwbZtMh+ChL51ypqNpcBZmZtpTH8uXwyivWh+OKK6BSpUJf9v331qR33nk2xcHnn1uNVJ06cY4/QZWYTInIayKyTEQKbdwRkc4iskZEpga3u6MfZvwNGWKJw6OPpvbcGKW133429LVmTWtW//rrsCNKA+vW2cRfXbpA584895w9/I9/hBtWskjXMsy5HeTmwvDh1jZ3xhmwZo31HP/lF5upuWLFQl+2fLl1J2jXzvqi9+8P331nXT7cdpHUTA0CTiphm89VtW1wu6/sYYUrO9umQmjXzjrWuR01bWoJ1Z57WhP7Z5+FHVGKe/ZZK9EefJD16+0iskcPW/7HRWQQaVaGOfeX3FxbtqNtWzjrLJs0cfBg+OknG7JdoUKhL9u61Yqe/fe3nOu666zy6vLLISMjzseQBEpMplT1M2BVHGJJGH37woIF1l8q3duBi9KokdXc7b23Nbf/739hR5Si/vzTPojdukH79gwcaBeUN9wQdmDJIx3LMOfYts06OB10kE1Gt3UrvPmmdQbu3ds6wBZC1eYFbt0arr8e2re35WCefjp95ozaFdFKFY4QkR9E5CMRaVXURiLSR0Qmicik5cuXR2nX0bV8OTz0EJx+unW0dkXbay+YMAH22QdOPRXGjAk7ohT0+OOwdi3cfz85OfDMM9YH/fDDww4s5aRMGebSXE4OZGVZNnTeefbYkCE2DLtXryKTKIBvvrHBVmeeabVPI0faQL8DD4xT7EksGsnUFKCJqrYBngPeK2pDVe2vqpmqmlk3QTsi3XefDft89NGwI0kO9erZtAkHHGCVJ6NHhx1RCvnjD6tn79kTDjqIt9+2mRFuuSXswFJOSpVhLk3l5FjzXcuWcMEF1nz37rswfbqVIcW0zc2eDWefbRdqs2fDyy9bbdRpp/nUK5EqczKlqmtVdX1wfzRQQUSSsn//b7/ZunSXX27JgYtMnTrWzNeqlV3RfPBB2BGliIcfhs2b4d57yc21GtODDrJaUxc9qVSGuTS0dev2KcgvvthmzBw+3Oav6dGj2L4qy5bZVFItW9rkzP/+t82M0KdPsRVYrhBlTqZEpL6I5a4i0j54z6Rcye3BBy15/9e/wo4k+ey+u63j16aN9XEcMSLsiJLcggWW2V9yCTRvznvvWVeHO+7wfnzRlkplmEsjW7bYaJT997eRUrVr25QHU6bYVW0xBcWqVVaW7LMPvPCCvXzOHFtgwadb2TUl5p4iMgToDNQRkUXAPUAFAFV9CegBXCkiOUA20FNVNWYRx8ivv8KgQXDlldCwYdjRJKfatW01gpNOsv6OQ4bYhZHbBfffbz/vugtVm4m/eXOrinelky5lmEsTmzfbaggPP2wXXYcdZjOWd+1aYpvc2rXW7/LJJ23GlXPPtdqoFi3iE3oqKzGZUtXzSni+H9AvahGFJK9W6tZbw44kudWsaR3Ru3a1ZvqsLPvCulKYPdsKy6uvhsaNGT3KJswbONCHJO+KdCnDXIrbtMlmyXzkEVi0yBblfPllOPHEEpOoDRvg+eetL/CqVVZxde+91m3ARYc3GGCdegcNsnZir5Uqu912sxEgRx0F559vCZUrhXvusVmI77iD3Fy4806rju/VK+zAnHNxl51t8/Xsuy9cc41N9Dd2LHz1lTUDFJNIrV9vtVD77msVBYcfbhNuDh/uiVS0eRczrFaqfHlbuNFFR/XqNrLvtNPgwgttoImvIxeBadNsbpjbboN69XjnLfjhB5sepoi59ZxzqWjjRqt5euwxG9nbqZMVBJ07l1gTtXq11UQ9/TSsXGmrVQwbZhe4LjbSPpn6/Xd44w0bwbfXXmFHk1qqVbN1NLt1s37UOTk+o3yJ7rrLqvZuvpmtW20wxMEHb58uxjmX4tavt7U4n3jChtsdd5xdYHXqVOJLV6ywPlHPPWf9o0491Wq2O3SIQ9xpLu2TqbJzN2gAABrhSURBVL59baLYG28MO5LUVLWqTfzWvbst/5STY+s8uUJMnGgn64EHoHZtBrwEc+daQuoj+JxLcevWWXXSk09aVnT88XD33REtgrdokdVCvfSStQqedZaN1jvkkDjE7YA0T6bWrbMPX/fu1qbsYqNKFZsqoUcPW5Q8J8f6VrsC/vUvW1X7uutYt846iB51lHXmd86lqDVrbDTeU09Z7/CTT7Ya6iOOKPGl339vudfbb9syMOefb+vK+ozl8ZfWydSAAfY5/uc/w44k9VWubG32555rfShzcmzhTBcYP94m6nrqKahenYdut24SI0b4DMTOpaTVq61p5Omn7f6pp1oS1b59sS9TtQE+TzwB48ZZ/9R//MPK0yZN4hS720naJlNbt9pn+JhjfJ2zeKlUCd55x/r/XH+9JVQ33RR2VAlA1To27L03XHklc+ZYTtW7t/d1cC7lrFply0Q9+6xdzXfrZklUu3bFviw7G/7zHysbZs2ykeePPWb9fX0B4vClbTI1dKjNd9bPZ5eJq4oVrS9lr15WI5iT43N7MXo0fP21jdypXJkbb7Tz9MgjYQfmnIualSvtCr5vX+tj0r27JVFt2xb7sl9/tf7oAwZYHtamjQ2aOuccKydcYkjbZKpvX5uF/5RTwo4k/VSoYFdYedNR5I1aS0t5E0ntuy9ccgkffWRrGz72GDRoEHZwzrkyW77cOjY9/7zNnnn22VbgFTPRU26uTSXVrx+MGmUDUM44w/qaRjAzggtBWiZTU6bAN9/YEFIfJRWO8uXt6qp8ebs4y8mxuSrTrpAYOvSviaTWb67AlVfa0g7en8y5JLd0qXVseuEFa6Pr2dMunFq1KvIlK1ZYufjCC7ZW3p572kv+7/+sF4BLXGmZTL34og3Z90kkw5WRsX2JlHvvtYTq/vvTKKHKybGhz61aQc+e3HmjNT1//rlX3zuXtJYsgccft6HimzfbELs774QDDih089xcG3syYAC8956tX3zEEVYmnnWW9TV1iS/tkqk//7TlTS64wDvtJYKMDCtEype3mejXrLEaw7RYg+6NN+Dnn2H4cL7+NoPnnrNqfJ+l2LkktHixtc/37299Fy64wCZ72n//QjefP98uJgcOtIuo3Xe3qWMuvdQm6nXJJe2SqcGDrcb1qqvCjsTlKVfO+l7XrGldC5YutTwjpa/INm+2S8/MTLJPPINLM6FRI3joobADc86VysKFtoLwq6/aDNC9e1sSVcjkhdnZNi/va69ZnyiALl0sB+vWzaaQcckprZKp3Fxriz7iiBIHULg4K1fOuhc0aGCj/JYvtyrvmjXDjixGXn3VLk379+fmW4Qff4QxY6BGjbADc85FZP58G3I7YID9fvHFNmNms2Y7bJaTY/NBZWXZvHHr1tmF01132TJbTZvGPXIXA2mVTI0bB7NnW0dnl5huugnq1bNCplMn+OijFBzVtnGjLRnTsSPvbzye55+35YxOOCHswJxzJfr1V6tCHjTIrgIvu8zmd8k3Y6YqfPedJVBvv2217TVr2kC+Xr2sbEuLrgxpJK2SqQEDoHZt69TnEtcFF9iqKmedBUceaQlVEX03k1O/fvDHHyx6/n3+fqlwyCHevOdcwps71zp2vv66dfK84gq45RarZsISqGnTbKWHIUNsNF7Fijaxea9etiyUN+OlrrRJpv7806pYL7/cP9DJ4MQTYcIEmwesQwe7ujvxxLCjioI1a+DRR9l0wul0f6Q9W7ZYwZvS/cOcS2a//GJJVFaWTZJ3zTWWRO21F7m5MPFrGD7cbvPmWWVV587W4te9uw90Shdpk0wNGWJ9fv/+97AjcZHKzIRvv4XTT7eruqeftjWoknrqhKeeQlet4oqKr/Hdd5bgt2gRdlDOuZ38+KMlUXlXO9ddBzffTE6d+nz6KQx/0L6/S5ZYjtWliyVQp59u80O59FLilJUi8pqILBORGUU8LyLSV0TmiMg0ETk0+mGW3Wuv2TT8hxwSdiSuNJo0gS+/hNNOs7Lsiits1HFSWrECnnqKZw4eyOAP9+Cee2xWYxdbqVKGuTiZOdMWEG3VyrKlm25i6bfzGXzwk5x7XX3q1rXEadAg64aQlWUDZkaPtu5Tnkilp0hqpgYB/YDXi3j+ZKB5cDsceDH4mTCmTYPJk21dSZd8qle3KvQ777TBM7Nm2fp+DRuGHVkpPfoob204jRunXUz37jZfp4uLQSR5GebiYNo0GxgydCjbqtZg0oX9GF33IkaPr8akx22T+vWt6e7UU63bQdWq4YbsEkeJyZSqfiYiTYvZpBvwuqoq8I2I1BKRBqq6JEoxltnAgVYNe/75YUfidlW5cvDwwzaZ3eWX29QWWVlJMgIuKwtuvZWxiw+kN6PoeMBSsrLq+VJGcZIKZZiLoqwsuzJbsAAaN4Y+fdDvJvHre1MZV/kUxrWcytg/WrPi9XKUK2d9Nh94wLoatG2b5N0MXMxEo89UQ2Bhvt8XBY/tVBCJSB+gD0Djxo2jsOuSbdkCb75pE6LVqROXXboYOu88a6rt0QNOOsnWC73nngQeZpyVBX368L+NHTiD9ziQH3l//slUHva4DfFxiSChyzAXRcH3kY0b+Z0GjJt/NOPurM84eZr5NIFNUH8lnHSyJU8nnAB77BF20C4ZxLUDuqr2B/oDZGZmajz2OWaMdVW5+OJ47M3FwwEHwMSJtvTK/ffD//5n/ReaNw87skLccQejN3aiO8PZn1/4mBOolb3Mrow9mUo6YZRhroy2bUNnz2HOx/P46rYv+DL7aT6jIz9j863szkqOrTyRW55ownHH2YAQr31ypRWNZGox0Cjf73sHjyWE//zHriySojnIRaxaNUugunSxEX5t2tiKDldfTeI0ny1fzuAFnbmcVziI6XzMCezBKntuwYJQQ3M7SOgyzJXCqlUwfTrZ381g8oR1fPlDdb76vSlf5R7OCloAJ1OLPzmSr7icVziOcbThB8ptAq7KDTt6l8SikUyNBK4RkbewTptrEqWvwfr18P77VitVoULY0bhYuOACOPZY60d17bU2Yd7zz9tAnDBt+/Aj7jh3Lo8xmL/xCUPpQS3WbN/Am4gSScKWYa4IOTk2/9O0aWyeMpPpX61j8szKTF69D5Npx3QuZysVAWheaxmntl7NkZ2zOfLVv3PgH+MoR4FKxcZNCtmJc5ErMZkSkSFAZ6COiCwC7gEqAKjqS8BooCswB9gIXBKrYEvr/fdtYcnzzgs7EhdLDRvCqFE2/cXNN1st1T/+YX2p4j5h3saNLL7qQS4afCz/4xquaD+FvtPPokL22u3bVK1q89e4uEjmMsxh/TSmTYNp01j97S/MnLyJ6fOqMjmnDZNpxwy6/5U41a6SzaEtNnLjUVs48vgKHHGkULfunkAwX8EBl0Cfr21Jpzz+fXTRoKqh3Nq1a6ex1rWraqNGqtu2xXxXLkEsX67ap4+qiOqee6o+84zqxo0x2tmbb6o2aWI7a9JEc++7X99qeIPuzgqtWn6TvvLCFs3N3Xk7ffPNGAWU+IBJGlKZE+1bPMqwlBDp53/LFtXp01WzsnTd9f/SiR2u1ddqXa838oSeyEfakIVqi7bYbfdqm/T4Dmv1tptz9N13VefNU/u+RSse5woorvwSez7+MjMzddKkSTF7/xUrbIHcG2+0vjQuvUyebIsmf/qpfQ5uv91mv69WLUo7yDcqCGAGrbiWvoznODL3X0PWBzXZf/8o7SuFiMhkVc0MO45oiHUZlhIKfE8Aqwl6/HE2NmrBnAmLmDN5DbN/yWX2H7sxR/dhNs35ne2TyFUuv5WWzbJp1aYCrQ+rQuvW0Lq1LYnnHcVdPBVXfqXscjJDh1qzus8tlZ7atbO1/SZMsOa+a6+Fu+6Ciy6yWdQPPLCEN7jqKnj5Zcgt0Ck1IwO2bfvr52QO5WFuZzjdqc2fvFDrDi6f+RDlU/ab5RJe3jxK8+dv/7zmje9ftcr66+U1a+Wfb+nBB3ceYVpwTqbCtilIFV2zlpU/LWfBtYNYuLELC2i8/baxMfOvbsIS9trhZXtWXUfzvbM54cAK7HfoNlodnEHr1tCsWQUyMrzTq0tsKVsz1bEjrFwJM2b41YuzJWleeAHefdeWozn4YJurqmtX62O1Q/KTkbFzEpXPbzThPc7gTS5gMpnUZDVX8zw38hR7yJ/FvjbdpVLNlEim1q8/iT33ZKdb3bpQu7bdatXafqtZM0ZzotWuDatXR759xYrWWpZ/baaqVaF//+3JUr5aJQU2UI3Vleqzuve1LK9/EEsXbGbZkhyWLhWWrqrA0rVVWJpdg2VbarOUPdlElR12WYlNf6VUjVjIfn/vyH4d6tK83W7stx/stlvZT4NzsVRc+ZWSydTvv1un5Pvus9oI5/IsXWrTZQwbBl99Zf9PatSAww+32qr9XrqJBlsXUJ31VGSL/QOhFvPYh59pwdccwXyaAtCOSVzAm1zCQGoSdDBv0gR++y2040t0qZRMNWiQqaedNolly9jhtm5d8a+rUWPH5KpaNctjirtVrmwJf/nyloztcP+c7pTfsBpByaUcuZRjGxmF3t9CRbKpUvQtowbr6jRj9abKrF4rrNHdWE0t1lCTbUU0ZGSQw57lV1Gvylr2rJ5NvdpbqFc3l70bQeMPnqfxmuk0ZgF1WMFf17X+PXFJKO2a+UaMsJ9nnRVuHC7x1KsHN9xgtyVLrE/V55/DN9/YvFXrtj5Z6OuEXJown3ZM5p88wfGMpQW/7LiRjwpKKw0bWkVOQdnZtvDt6tXw55/2s+At7/E1a+z+4sXWrSjvlp1dmgW9h5fpOMqzdXs6tS2bGmtyqFVxI3vpAloyk1qspiZrqMXqv+7XHfAoezavSb0Wtdi9TnnKlcs3Yi6/rOOhz7s+es6lvJRMpoYNs1myW7YMOxKXyBo0gJ497QZWS7WsXD2WU5f1VGcLFanOemqwjkYspDKbd36TvCbBSPuTuJRXpYp9HMo6ldjWrZZU5SVX27ZZP9CcnB3v53Q46q/apwy2BXVQuYXer5A/cQpu5dm2faf5a4yanm79rgpq0gT+3iyyg8j7PpS235VzSSblkqkVK6y24fbbw47EJRsRqMcy6rEsshcU7GPiXBRVqGC3kvsSfVX6N/+rz1S+RKpgjdGDDxY+Eq+0tUq9evl3xKW8RFl4I2ref98qCryJz+2SKlWKfz6v93CTJp5IucQQycy0e+xhNxH77L72GgwcaPfzHiv4ee7Vyx4rbhvnHJCCNVPDhkGzZtC2bdiRuKS0caNdfWdn7/j4lVfacEDnEs2ffxY+mq9aNZveo7jkp6TEyGuVnItISiVTq1fDJ5/Addf5dAiuDPI3aziXDP78M+wInEtrKdXM9+GH1mnTm/icc845Fy8plUwNG2bDldu3DzsS55xzzqWLlEmmNm6EMWPgzDOhXMoclXPOOecSXcqkHf/7n/UZ7tYt7Eicc845l05SJpn68ENbpqFjx7Ajcc4551w6SYlkStWSqRNPtLnonHPOOefiJSWSqe+/t8WNTz017Eicc845l25SIpn64AObV6pr17Ajcc4551y6iSiZEpGTRORnEZkjIrcV8vzFIrJcRKYGt8uiH2rRPvwQOnSAunXjuVfnXDJI9PLLOZf8SpwBXUQygOeB44FFwHciMlJVZxXY9G1VvSYGMRbr999h0iR46KF479k5l+gSvfxyzqWGSGqm2gNzVHWeqm4B3gISZgKCUaPsp/eXcs4VIqHLL+dcaogkmWoILMz3+6LgsYLOEpFpIjJURBpFJboIfPihLWbeunW89uicSyIJXX4551JDtDqgfwA0VdWDgbHA4MI2EpE+IjJJRCYtX768zDvdtMkWNj71VF/Y2Dm3yyIqvyD6ZZhzLjVEkkwtBvJfqe0dPPYXVV2pqpuDX18F2hX2RqraX1UzVTWzbhR6i3/+uS0jc/LJZX4r51xqilr5FWwb1TLMOZcaIkmmvgOai0gzEakI9ARG5t9ARBrk+/V04MfohVi0MWNsks7OneOxN+dcEkrY8ss5lzpKHM2nqjkicg0wBsgAXlPVmSJyHzBJVUcC14rI6UAOsAq4OIYx/+Xjj+Hoo6FatXjszTmXbBK5/HLOpQ5R1VB2nJmZqZMmTdrl1//+OzRsCI8+CrfcEsXAnHMxIyKTVTUz7DiioaxlmHMuuRRXfiXtDOgff2w/Tzwx3Dicc845l96SNpkaMwbq1YODDw47Euecc86ls6RMprZtg7Fj4YQTfEoE55xzzoUrKZOpKVNg5Upv4nPOOedc+JIymcrrL3X88eHG4ZxzzjmXlMnUmDFw6KGw555hR+Kcc865dJd0ydTatfD1197E55xzzrnEkHTJ1OefQ04OdOkSdiTOOeecc0mYTI0fD5UqwRFHhB2Jc84551wSJlPjxsGRR0KVKmFH4pxzzjmXZMnUypUwdSoce2zYkTjnnHPOmaRKpj79FFThuOPCjsQ555xzziRVMjV+PFSrBocdFnYkzjnnnHMmqZKpcePgmGOgYsWwI3HOOeecM0mTTP3xB8ya5U18zjnnnEssSZNMjR9vPz2Zcs4551wiSapkqlYtaNs27Eicc84557ZLmmRq3Djo1AkyMsKOxDnnnHNuu4iSKRE5SUR+FpE5InJbIc9XEpG3g+cnikjTaAa5cCHMnevzSznnSi/s8ss5l/pKTKZEJAN4HjgZaAmcJyItC2x2KfCnqu4HPA08Gs0gP/vMfnbqFM13dc6lukQov5xzqS+Smqn2wBxVnaeqW4C3gG4FtukGDA7uDwX+JiISrSA//xx22w0OOiha7+icSxOhl1/OudQXSTLVEFiY7/dFwWOFbqOqOcAaYI9oBAiWTB11lPeXcs6VWujll3Mu9ZWP585EpA/QJ/h1s4jMiPS1s2ZBjK8V6wArYrqH0ku0mDye4iVaPJB4MbUIO4CyKEsZloQS7bMTbal+fJD6xxjv42tS1BORJFOLgUb5ft87eKywbRaJSHmgJrCy4Bupan+gP4CITFLVzAj2HxeJFg8kXkweT/ESLR5IvJhEZFKcdxm18gsSuwyLNj++5Jfqx5hIxxdJM993QHMRaSYiFYGewMgC24wELgru9wDGqapGL0znnNslXn4552KuxJopVc0RkWuAMUAG8JqqzhSR+4BJqjoSGAC8ISJzgFVYgeWcc6Hy8ss5Fw8R9ZlS1dHA6AKP3Z3v/ibg7FLuu38pt4+1RIsHEi8mj6d4iRYPJF5McY8nRuUXJN65jTY/vuSX6seYMMcnXpvtnHPOObfrkmY5Geecc865RBS3ZEpEzhaRmSKSKyJF9r4vaemHKMazu4iMFZHZwc/aRWy3TUSmBreCHVejEUfCLXURQUwXi8jyfOflshjG8pqILCtqCLqYvkGs00Tk0FjFEmE8nUVkTb5zc3dh20UxnkYiMl5EZgXfr+sK2Sbe5yiSmOJ6nnZVBH/vXsE5nS4iX4lIm3jHWBYlHV++7Q4TkRwR6RGv2KIhkuMLPotTg8/qp/GMLxoi+IzWFJEPROSH4BgviXeMZZGIZVyhVDUuN+BAbI6ZCUBmEdtkAHOBfYCKwA9AyxjF8xhwW3D/NuDRIrZbH8NzUuLxAlcBLwX3ewJvx/jvFElMFwP94vS56QgcCswo4vmuwEeAAB2AiSHH0xn4MB7nJthfA+DQ4H4N4JdC/l7xPkeRxBTX8xTDv/eRQO3g/smxPrfxPr5gmwxgHNbvrEfYMUf571cLmAU0Dn7fM+yYY3CMd+T9fwPqYoMsKoYddymOL+HKuMJucauZUtUfVfXnEjaLZOmHaMm/hMRg4IwY7ac4ibjURTz/BiVS1c+wL39RugGvq/kGqCUiDUKMJ65UdYmqTgnurwN+ZOcZvuN9jiKJKSmU9PdW1a9U9c/g12+weaySRoSf538Aw4BlsY8ouiI4vvOB4aq6INg+FY9RgRrB/43qwbY58YgtGhKxjCtMovWZimTph2ipp6pLgvt/APWK2K6yiEwSkW9EJNoJVyIudRHp3+CsoDp1qIg0KuT5eInnZyZSRwRV6h+JSKt47TRoAj4EmFjgqdDOUTExQUjnKYYuxa6OU4aINATOBF4MO5YY2R+oLSITRGSyiPQOO6AY6Ie1DP0OTAeuU9XccEPaNYlYxuWJ6nIyIvIJUL+Qp+5U1fejua+yxpP/F1VVESlqWGMTVV0sIvsA40RkuqrOjXasSeYDYIiqbhaR/8Nqzo4LOaZEMQX7zKwXka7Ae0DzWO9URKpjtQfXq+raWO8vEiXEFMp5ihURORZLpo4OO5Yoewa4VVVzY1shHpryQDvgb0AV4GsR+UZVfwk3rKg6EZiKldH7AmNF5PNEKScilYhlXH5RTaZUtUsZ3yKSpR+iEo+ILBWRBqq6JKgOLLR6V1UXBz/nicgELCuOVjIV1aUu4hWTqubf/6tY/7OwRPUzU1b5v+SqOlpEXhCROqoas/WjRKQCVshkqerwQjaJ+zkqKaYwzlOsiMjB2Pfg5ALfjVSQCbwVJFJ1gK4ikqOq74UbVtQsAlaq6gZgg4h8BrTB+uWkikuAR9Q6F80RkV+BA4Bvww0rcolYxhWUaM18kSz9EC35l5C4CNip5kxEaotIpeB+HeAorLNitCTiUhclxlSgLfp0rA07LCOB3sFojg7AmnzNt3EnIvXz+rSJSHvsOxazf7DBvgYAP6rqU0VsFtdzFElM8T5PsSIijYHhwIUpVpsBgKo2U9WmqtoU67N5VQolUmDl/tEiUl5EqgKHE255FgsLsJo3RKQeNhBsXqgRlUIilnGFiWrNVHFE5EzgOWw0wSgRmaqqJ4rIXsCrqtpVi1j6IUYhPQK8IyKXAvOBc4I4M4ErVPUyrJ35ZRHJxQr7R1Q1aslUUccrIS51EWFM14rI6VgnxlXY6L6YEJEh2MivOiKyCLgHqBDE+hI2wqgrMAfYiF2FxUwE8fQArhSRHCAb6Bnj5Pco4EJguohMDR67A2icL6a4nqMIY4r3edolEfy978b6ML4Q5IY5miALr0YiguNLaiUdn6r+KCL/BaYBudj/omKniUg0EfwN7wcGich0bLTbrUlWA5yIZdxOfAZ055xzzrkySLRmPuecc865pOLJlHPOOedcGXgy5ZxzzjlXBp5MOeecc86VgSdTzjnnnHNl4MmUc84551wZeDLlnHPOOVcGnkw555xzzpXB/wPm4SQzhX5NcAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 720x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=[10,3])\n",
    "plt.subplot(1,2,1)\n",
    "plt.scatter(x_list,y_list,c=\"r\")\n",
    "plt.plot(x_list,y_list,c=\"r\")\n",
    "plt.plot(x,f(x), c=\"b\")\n",
    "plt.xlim([-1,2.5])\n",
    "plt.ylim([0,3])\n",
    "plt.title(\"Gradient descent\")\n",
    "plt.subplot(1,2,2)\n",
    "plt.scatter(x_list,y_list,c=\"r\")\n",
    "plt.plot(x_list,y_list,c=\"r\")\n",
    "plt.plot(x,f(x), c=\"b\")\n",
    "plt.xlim([1.2,2.1])\n",
    "plt.ylim([0,3])\n",
    "plt.title(\"Gradient descent (zoomed in)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You'll notice that the step size (also called learning rate) in the implementation above is constant, unlike the algorithm in the pseudocode. Doing this makes it easier to implement the algorithm. However, it also presents some issues: If the step size is too small, then convergence will be very slow, but if we make it too large, then the method may fail to converge at all. \n",
    "\n",
    "A solution to this is to use adaptive step sizes as the algorithm below does (using scipy's fmin function to find optimal step sizes):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Local minimum occurs at  1.3333333284505209\n",
      "Number of steps: 4\n"
     ]
    }
   ],
   "source": [
    "# we setup this function to pass into the fmin algorithm\n",
    "def f2(n,x,s):\n",
    "    x = x + n*s\n",
    "    return f(x)\n",
    "\n",
    "x_old = 0\n",
    "x_new = 2 # The algorithm starts at x=2\n",
    "precision = 0.0001\n",
    "\n",
    "x_list, y_list = [x_new], [f(x_new)]\n",
    "\n",
    "# returns the value of the derivative of our function\n",
    "def f_prime(x):\n",
    "    return 3*x**2-4*x\n",
    "\n",
    "while abs(x_new - x_old) > precision:\n",
    "    x_old = x_new\n",
    "    s_k = -f_prime(x_old)\n",
    "    \n",
    "    # use scipy fmin function to find ideal step size.\n",
    "    n_k = fmin(f2,0.1,(x_old,s_k), full_output = False, disp = False)\n",
    "\n",
    "    x_new = x_old + n_k * s_k\n",
    "    x_list.append(x_new)\n",
    "    y_list.append(f(x_new))\n",
    "    \n",
    "print(\"Local minimum occurs at \", float(x_new))\n",
    "print(\"Number of steps:\", len(x_list))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With adaptive step sizes, the algorithm converges in just 4 iterations rather than 17. Of course, it takes time to compute the appropriate step size at each iteration. Here are some plots of the path taken below. You can see that it converges very quickly to a point near the local minimum, so it's hard to even discern the dots after the first two steps until we zoom in very close in the third frame below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x216 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=[15,3])\n",
    "plt.subplot(1,3,1)\n",
    "plt.scatter(x_list,y_list,c=\"r\")\n",
    "plt.plot(x_list,y_list,c=\"r\")\n",
    "plt.plot(x,f(x), c=\"b\")\n",
    "plt.xlim([-1,2.5])\n",
    "plt.title(\"Gradient descent\")\n",
    "plt.subplot(1,3,2)\n",
    "plt.scatter(x_list,y_list,c=\"r\")\n",
    "plt.plot(x_list,y_list,c=\"r\")\n",
    "plt.plot(x,f(x), c=\"b\")\n",
    "plt.xlim([1.2,2.1])\n",
    "plt.ylim([0,3])\n",
    "plt.title(\"zoomed in\")\n",
    "plt.subplot(1,3,3)\n",
    "plt.scatter(x_list,y_list,c=\"r\")\n",
    "plt.plot(x_list,y_list,c=\"r\")\n",
    "plt.plot(x,f(x), c=\"b\")\n",
    "plt.xlim([1.3333,1.3335])\n",
    "plt.ylim([0,3])\n",
    "plt.title(\"zoomed in more\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another approach to update the step size is choosing a decrease constant $d$ that shrinks the step size over time:\n",
    "$\\eta(t+1) = \\eta(t) / (1+t \\times d)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Local minimum occurs at: 1.3308506740900838\n",
      "Number of steps: 6\n"
     ]
    }
   ],
   "source": [
    "x_old = 0\n",
    "x_new = 2 # The algorithm starts at x=2\n",
    "n_k = 0.17 # step size\n",
    "precision = 0.0001\n",
    "t, d = 0, 1\n",
    "\n",
    "x_list, y_list = [x_new], [f(x_new)]\n",
    "\n",
    "# returns the value of the derivative of our function\n",
    "def f_prime(x):\n",
    "    return 3*x**2-4*x\n",
    " \n",
    "while abs(x_new - x_old) > precision:\n",
    "    x_old = x_new\n",
    "    s_k = -f_prime(x_old)\n",
    "    x_new = x_old + n_k * s_k\n",
    "    x_list.append(x_new)\n",
    "    y_list.append(f(x_new))\n",
    "    n_k = n_k / (1 + t * d)\n",
    "    t += 1\n",
    "\n",
    "print(\"Local minimum occurs at:\", x_new)\n",
    "print(\"Number of steps:\", len(x_list))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now consider an example which is a little bit more complicated. Consider a simple linear regression where we want to see how the temperature affects the noises made by crickets. We have a data set of cricket chirp rates at various temperatures. First we'll load that data set in and plot it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Load the dataset\n",
    "data = np.loadtxt('SGD_data.txt', delimiter=',')\n",
    " \n",
    "#Plot the data\n",
    "plt.scatter(data[:, 0], data[:, 1], marker='o', c='b')\n",
    "plt.title('cricket chirps vs temperature')\n",
    "plt.xlabel('chirps/sec for striped ground crickets')\n",
    "plt.ylabel('temperature in degrees Fahrenheit')\n",
    "plt.xlim([13,21])\n",
    "plt.ylim([65,95])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our goal is to find the equation of the straight line $h_\\theta(x) = \\theta_0 + \\theta_1 x$ that best fits our data points. The function that we are trying to minimize in this case is:\n",
    "\n",
    "$J(\\theta_0,\\theta_1) = {1 \\over 2m} \\sum\\limits_{i=1}^m (h_\\theta(x_i)-y_i)^2$\n",
    "\n",
    "In this case, our gradient will be defined in two dimensions:\n",
    "\n",
    "$\\frac{\\partial}{\\partial \\theta_0} J(\\theta_0,\\theta_1) = \\frac{1}{m}  \\sum\\limits_{i=1}^m (h_\\theta(x_i)-y_i)$\n",
    "\n",
    "$\\frac{\\partial}{\\partial \\theta_1} J(\\theta_0,\\theta_1) = \\frac{1}{m}  \\sum\\limits_{i=1}^m ((h_\\theta(x_i)-y_i) \\cdot x_i)$\n",
    "\n",
    "Below, we set up our function for h, J and the gradient:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = lambda theta_0,theta_1,x: theta_0 + theta_1*x\n",
    "\n",
    "def J(x,y,m,theta_0,theta_1):\n",
    "    returnValue = 0\n",
    "    for i in range(m):\n",
    "        returnValue += (h(theta_0,theta_1,x[i])-y[i])**2\n",
    "    returnValue = returnValue/(2*m)\n",
    "    return returnValue\n",
    "\n",
    "def grad_J(x,y,m,theta_0,theta_1):\n",
    "    returnValue = np.array([0.,0.])\n",
    "    for i in range(m):\n",
    "        returnValue[0] += (h(theta_0,theta_1,x[i])-y[i])\n",
    "        returnValue[1] += (h(theta_0,theta_1,x[i])-y[i])*x[i]\n",
    "    returnValue = returnValue/(m)\n",
    "    return returnValue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we'll load our data into the x and y variables;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = data[:, 0]\n",
    "y = data[:, 1]\n",
    "m = len(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we run our gradient descent algorithm (without adaptive step sizes in this example):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Local minimum occurs where:\n",
      "theta_0 = 25.128552558595363\n",
      "theta_1 = 3.297264756251897\n",
      "This took 565859 steps to converge\n"
     ]
    }
   ],
   "source": [
    "theta_old = np.array([0.,0.])\n",
    "theta_new = np.array([1.,1.]) # The algorithm starts at [1,1]\n",
    "n_k = 0.001 # step size\n",
    "precision = 0.001\n",
    "num_steps = 0\n",
    "s_k = float(\"inf\")\n",
    "\n",
    "while np.linalg.norm(s_k) > precision:\n",
    "    num_steps += 1\n",
    "    theta_old = theta_new\n",
    "    s_k = -grad_J(x,y,m,theta_old[0],theta_old[1])\n",
    "    theta_new = theta_old + n_k * s_k\n",
    "\n",
    "print(\"Local minimum occurs where:\")\n",
    "print(\"theta_0 =\", theta_new[0])\n",
    "print(\"theta_1 =\", theta_new[1])\n",
    "print(\"This took\",num_steps,\"steps to converge\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For comparison, let's get the actual values for $\\theta_0$ and $\\theta_1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Actual values for theta are:\n",
      "theta_0 = 25.232304983426026\n",
      "theta_1 = 3.2910945679475647\n"
     ]
    }
   ],
   "source": [
    "actualvalues = sp.stats.linregress(x,y)\n",
    "print(\"Actual values for theta are:\")\n",
    "print(\"theta_0 =\", actualvalues.intercept)\n",
    "print(\"theta_1 =\", actualvalues.slope)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we see that our values are relatively close to the actual values (even though our method was pretty slow). If you look at the source code of [linregress](https://github.com/scipy/scipy/blob/master/scipy/stats/_stats_mstats_common.py), it uses the convariance matrix of x and y to compute fastly. Below, you can see a plot of the line drawn with our theta values against the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = np.linspace(0,21,1000)\n",
    "plt.scatter(data[:, 0], data[:, 1], marker='o', c='b')\n",
    "plt.plot(xx,h(theta_new[0],theta_new[1],xx))\n",
    "plt.xlim([13,21])\n",
    "plt.ylim([65,95])\n",
    "plt.title('cricket chirps vs temperature')\n",
    "plt.xlabel('chirps/sec for striped ground crickets')\n",
    "plt.ylabel('temperature in degrees Fahrenheit')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that in the method above we need to calculate the gradient in every step of our algorithm. In the example with the crickets, this is not a big deal since there are only 15 data points. But imagine that we had 10 million data points. If this were the case, it would certainly make the method above far less efficient.\n",
    "\n",
    "In machine learning, the algorithm above is often called <b>batch gradient descent</b> to contrast it with <b>mini-batch gradient descent</b> (which we will not go into here) and <b>stochastic gradient descent</b>."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Stochastic gradient descent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we said above, in batch gradient descent, we must look at every example in the entire training set on every step (in cases where a training set is used for gradient descent). This can be quite slow if the training set is sufficiently large. In <b>stochastic gradient descent</b>, we update our values after looking at <i>each</i> item in the training set, so that we can start making progress right away. Recall the linear regression example above. In that example, we calculated the gradient for each of the two theta values as follows:\n",
    "\n",
    "$\\frac{\\partial}{\\partial \\theta_0} J(\\theta_0,\\theta_1) = \\frac{1}{m}  \\sum\\limits_{i=1}^m (h_\\theta(x_i)-y_i)$\n",
    "\n",
    "$\\frac{\\partial}{\\partial \\theta_1} J(\\theta_0,\\theta_1) = \\frac{1}{m}  \\sum\\limits_{i=1}^m ((h_\\theta(x_i)-y_i) \\cdot x_i)$\n",
    "\n",
    "Where $h_\\theta(x) = \\theta_0 + \\theta_1 x$\n",
    "\n",
    "Then we followed this algorithm (where $\\alpha$ was a non-adapting stepsize):\n",
    "\n",
    "&nbsp;&nbsp;&nbsp; 1: &nbsp; Choose initial guess $x_0$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    2: &nbsp; <b>for</b> k = 0, 1, 2, ... <b>do</b> <br>\n",
    "&nbsp;&nbsp;&nbsp;    3:   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $s_k$ = -$\\nabla f(x_k)$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    4:   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $x_{k+1} = x_k + \\alpha s_k$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    5: &nbsp;  <b>end for</b>\n",
    "\n",
    "When the sample data had 15 data points as in the example above, calculating the gradient was not very costly. But for very large data sets, this would not be the case. So instead, we consider a stochastic gradient descent algorithm for simple linear regression such as the following, where m is the size of the data set:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp; 1: &nbsp; Randomly shuffle the data set <br>\n",
    "&nbsp;&nbsp;&nbsp;    2: &nbsp; <b>for</b> k = 0, 1, 2, ... <b>do</b> <br>\n",
    "&nbsp;&nbsp;&nbsp;    3: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <b>for</b> i = 1 to m <b>do</b> <br>\n",
    "&nbsp;&nbsp;&nbsp;    4:   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\\begin{bmatrix}\n",
    " \\theta_{1} \\\\ \n",
    " \\theta_2 \\\\ \n",
    " \\end{bmatrix}=\\begin{bmatrix}\n",
    " \\theta_1 \\\\ \n",
    " \\theta_2 \\\\ \n",
    " \\end{bmatrix}-\\alpha\\begin{bmatrix}\n",
    " 2(h_\\theta(x_i)-y_i) \\\\ \n",
    " 2x_i(h_\\theta(x_i)-y_i) \\\\ \n",
    " \\end{bmatrix}$ <br>\n",
    "&nbsp;&nbsp;&nbsp;    5: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <b>end for</b> <br> \n",
    "&nbsp;&nbsp;&nbsp;    6: &nbsp;  <b>end for</b>\n",
    "\n",
    "Typically, with stochastic gradient descent, you will run through the entire data set 1 to 10 times (see value for k in line 2 of the pseudocode above), depending on how fast the data is converging and how large the data set is.\n",
    "\n",
    "With batch gradient descent, we must go through the entire data set before we make any progress. With this algorithm though, we can make progress right away and continue to make progress as we go through the data set. Therefore, stochastic gradient descent is often preferred when dealing with large data sets.\n",
    "\n",
    "Unlike gradient descent, stochastic gradient descent will tend to oscillate <i>near</i> a minimum value rather than continuously getting closer. It may never actually converge to the minimum though. One way around this is to slowly decrease the step size $\\alpha$ as the algorithm runs. However, this is less common than using a fixed $\\alpha$.\n",
    "\n",
    "Let's look at another example where we illustrate the use of stochastic gradient descent for linear regression. In the example below, we'll create a set of 500,000 points around the line $y = 2x+17+\\epsilon$, for values of x between 0 and 100:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambda x: x*2+17+np.random.randn(len(x))*10\n",
    "\n",
    "x = np.random.random(500000)*100\n",
    "y = f(x) \n",
    "m = len(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's randomly shuffle around our dataset. Note that in this example, this step isn't strictly necessary since the data is already in a random order. However, that obviously may not always be the case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "from random import shuffle\n",
    "\n",
    "x_shuf = []\n",
    "y_shuf = []\n",
    "index_shuf = list(range(len(x)))\n",
    "shuffle(index_shuf)\n",
    "for i in index_shuf:\n",
    "    x_shuf.append(x[i])\n",
    "    y_shuf.append(y[i])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we'll setup our h function and our cost function, which we will use to check how the value is improving."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = lambda theta_0,theta_1,x: theta_0 + theta_1*x\n",
    "cost = lambda theta_0,theta_1, x_i, y_i: 0.5*(h(theta_0,theta_1,x_i)-y_i)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we'll run our stochastic gradient descent algorithm. To see it's progress, we'll take a cost measurement at every step. Every 10,000 steps, we'll get an average cost from the last 10,000 steps and then append that to our cost_list variable. We will run through the entire list 10 times here:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Local minimum occurs where:\n",
      "theta_0 = 16.955415496837382\n",
      "theta_1 = 1.999169728242732\n"
     ]
    }
   ],
   "source": [
    "theta_old = np.array([0.,0.])\n",
    "theta_new = np.array([1.,1.]) # The algorithm starts at [1,1]\n",
    "n_k = 0.000005 # step size\n",
    "\n",
    "iter_num = 0\n",
    "s_k = np.array([float(\"inf\"),float(\"inf\")])\n",
    "sum_cost = 0\n",
    "cost_list = []\n",
    "\n",
    "for j in range(10):\n",
    "    for i in range(m):\n",
    "        iter_num += 1\n",
    "        theta_old = theta_new\n",
    "        s_k[0] = (h(theta_old[0],theta_old[1],x[i])-y[i])\n",
    "        s_k[1] = (h(theta_old[0],theta_old[1],x[i])-y[i])*x[i]\n",
    "        s_k = (-1)*s_k\n",
    "        theta_new = theta_old + n_k * s_k\n",
    "        sum_cost += cost(theta_old[0],theta_old[1],x[i],y[i])\n",
    "        if (i+1) % 10000 == 0:\n",
    "            cost_list.append(sum_cost/10000.0)\n",
    "            sum_cost = 0   \n",
    "            \n",
    "print(\"Local minimum occurs where:\")\n",
    "print(\"theta_0 =\", theta_new[0])\n",
    "print(\"theta_1 =\", theta_new[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, our values for $\\theta_0$ and $\\theta_1$ are close to their true values of 17 and 2.\n",
    "\n",
    "Now, we plot our cost versus the number of iterations. As you can see, the cost goes down quickly at first, but starts to level off as we go through more iterations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "iterations = np.arange(len(cost_list))*10000\n",
    "plt.plot(iterations,cost_list)\n",
    "plt.xlabel(\"iterations\")\n",
    "plt.ylabel(\"avg cost\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " "
   ]
  }
 ],
 "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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}