optimization/gradient_descent.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is all taken from Joel Grus book Data Science from Scratch!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# todo: move linear_algebra file into Linear Algebra library and import\n",
    "\n",
    "from linear_algebra import Vector, dot "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sum_of_squares(v: Vector) -> float:\n",
    "    \"\"\"Computes the sum of squared elements in v\"\"\"\n",
    "    return dot(v, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from typing import Callable\n",
    "\n",
    "def difference_quotient(f: Callable[[float], float], x: float, h: float) -> float:\n",
    "    \"\"\"Approximation of the derivative\"\"\"\n",
    "    return (f(x + h) - f(x)) / h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def square(x: float) -> float:\n",
    "    return x * x\n",
    "\n",
    "def derivative(x: float) -> float: # derivative of only X ^ 2\n",
    "    return 2 * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20.000999999993496"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "difference_quotient(square, 10, h=.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = range(-10, 11)\n",
    "actuals = [derivative(x) for x in xs]\n",
    "estimates = [difference_quotient(square, x, h=.001) for x in xs]\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "plt.title('Actual Derivatives vs. Estimates')\n",
    "plt.plot(xs, actuals, 'rx', label = 'Actual')\n",
    "plt.plot(xs, estimates, 'b+', label='Estimate')\n",
    "plt.legend(loc=9)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When `f` is function of many variables, it has multiple *partial derivatives*, each indicating how f changes when we make smalle changes in just one of hte input variables\n",
    "\n",
    "We calculate its *i*th partial derivative by treating it as a function of just its *i*th variable, holding the other variables fixed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def partial_difference_quotient(f: Callable[[Vector], float],\n",
    "                                v: Vector,\n",
    "                                i: int,\n",
    "                                h: float) -> float:\n",
    "    \"\"\"Returns the -ith partial diffference quotient of f at v\"\"\"\n",
    "    w = [v_j + (h if j == i else 0) # add h to just the ith element of v\n",
    "         for j, v_j in enumerate(v)]\n",
    "    \n",
    "    return(f(w) - f(v)) / h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def estimate_graient(f: Callable[[Vector], float],\n",
    "                     v: Vector,\n",
    "                     h: float = 0.0001):\n",
    "    return [partial_difference_quotient(f, v, i, h)\n",
    "            for i in range(len(v))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "from linear_algebra import distance, add, scalar_multiply"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`sum_of_squares` (i^2 + .... i_n^2) finding the minimum among all three-dimensional vectors. We know this will be [0,0,0] but assuming we don't know it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 [1.2252058650893745, -8.87457228770524, -3.9374172349775645]\n",
      "1 [1.200701747787587, -8.697080841951134, -3.8586688902780133]\n",
      "2 [1.1766877128318352, -8.52313922511211, -3.781495512472453]\n",
      "3 [1.1531539585751984, -8.35267644060987, -3.705865602223004]\n",
      "4 [1.1300908794036943, -8.185622911797672, -3.631748290178544]\n",
      "5 [1.1074890618156206, -8.02191045356172, -3.559113324374973]\n",
      "6 [1.085339280579308, -7.861472244490486, -3.4879310578874736]\n",
      "7 [1.0636324949677218, -7.704242799600676, -3.418172436729724]\n",
      "8 [1.0423598450683673, -7.550157943608663, -3.3498089879951296]\n",
      "9 [1.021512648167, -7.399154784736489, -3.282812808235227]\n",
      "10 [1.00108239520366, -7.25117168904176, -3.2171565520705223]\n",
      "11 [0.9810607472995868, -7.1061482552609245, -3.1528134210291117]\n",
      "12 [0.961439532353595, -6.964025290155706, -3.0897571526085295]\n",
      "13 [0.9422107417065231, -6.824744784352592, -3.027962009556359]\n",
      "14 [0.9233665268723926, -6.68824988866554, -2.967402769365232]\n",
      "15 [0.9048991963349448, -6.55448489089223, -2.9080547139779274]\n",
      "16 [0.8868012124082458, -6.423395193074385, -2.849893619698369]\n",
      "17 [0.8690651881600809, -6.294927289212898, -2.7928957473044016]\n",
      "18 [0.8516838843968794, -6.1690287434286395, -2.7370378323583138]\n",
      "19 [0.8346502067089417, -6.045648168560067, -2.6822970757111473]\n",
      "20 [0.8179572025747629, -5.924735205188866, -2.6286511341969243]\n",
      "21 [0.8015980585232676, -5.806240501085089, -2.576078111512986]\n",
      "22 [0.7855660973528023, -5.690115691063387, -2.524556549282726]\n",
      "23 [0.7698547754057462, -5.57631337724212, -2.4740654182970716]\n",
      "24 [0.7544576798976312, -5.464787109697277, -2.4245841099311303]\n",
      "25 [0.7393685262996786, -5.3554913675033315, -2.3760924277325075]\n",
      "26 [0.7245811557736851, -5.2483815401532645, -2.3285705791778573]\n",
      "27 [0.7100895326582114, -5.143413909350199, -2.2819991675943]\n",
      "28 [0.6958877420050472, -5.040545631163195, -2.236359184242414]\n",
      "29 [0.6819699871649463, -4.939734718539931, -2.191632000557566]\n",
      "30 [0.6683305874216473, -4.840940024169132, -2.1477993605464145]\n",
      "31 [0.6549639756732144, -4.74412122368575, -2.1048433733354863]\n",
      "32 [0.6418646961597502, -4.649238799212035, -2.0627465058687764]\n",
      "33 [0.6290274022365552, -4.556254023227794, -2.021491575751401]\n",
      "34 [0.6164468541918241, -4.465128942763238, -1.981061744236373]\n",
      "35 [0.6041179171079876, -4.375826363907973, -1.9414405093516456]\n",
      "36 [0.5920355587658278, -4.288309836629813, -1.9026116991646127]\n",
      "37 [0.5801948475905112, -4.202543639897217, -1.8645594651813204]\n",
      "38 [0.568590950638701, -4.118492767099273, -1.827268275877694]\n",
      "39 [0.557219131625927, -4.036122911757287, -1.79072291036014]\n",
      "40 [0.5460747489934085, -3.9554004535221416, -1.7549084521529372]\n",
      "41 [0.5351532540135403, -3.876292444451699, -1.7198102831098785]\n",
      "42 [0.5244501889332694, -3.798766595562665, -1.685414077447681]\n",
      "43 [0.5139611851546041, -3.7227912636514118, -1.6517057958987273]\n",
      "44 [0.503681961451512, -3.6483354383783837, -1.6186716799807528]\n",
      "45 [0.49360832222248174, -3.575368729610816, -1.5862982463811377]\n",
      "46 [0.4837361557780321, -3.5038613550185995, -1.554572281453515]\n",
      "47 [0.4740614326624714, -3.4337841279182277, -1.5234808358244447]\n",
      "48 [0.464580204009222, -3.365108445359863, -1.4930112191079559]\n",
      "49 [0.4552885999290376, -3.2978062764526657, -1.4631509947257968]\n",
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      "51 [0.43725917137184767, -3.16721314790514, -1.4052102153346553]\n",
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      "993 [2.375211185303238e-09, -1.7204442096759027e-08, -7.633164127109588e-09]\n",
      "994 [2.3277069615971734e-09, -1.6860353254823847e-08, -7.480500844567397e-09]\n",
      "995 [2.2811528223652298e-09, -1.652314618972737e-08, -7.330890827676049e-09]\n",
      "996 [2.2355297659179253e-09, -1.6192683265932823e-08, -7.184273011122528e-09]\n",
      "997 [2.190819170599567e-09, -1.5868829600614165e-08, -7.040587550900078e-09]\n",
      "998 [2.1470027871875755e-09, -1.555145300860188e-08, -6.899775799882076e-09]\n",
      "999 [2.104062731443824e-09, -1.5240423948429844e-08, -6.7617802838844346e-09]\n"
     ]
    }
   ],
   "source": [
    "def gradient_step(v: Vector, gradient: Vector, step_size: float) -> Vector:\n",
    "    \"\"\"Moves `step_size` in the `gradient` direction from `v`\"\"\"\n",
    "    assert len(v) == len(gradient)\n",
    "    step = scalar_multiply(step_size, gradient)\n",
    "    return add(v, step)\n",
    "\n",
    "def sum_of_squares_gradient(v: Vector) -> Vector:\n",
    "    return [2 * v_i for v_i in v] # explicitly define gradient\n",
    "\n",
    "#pick a random starting point\n",
    "v = [random.uniform(-10, 10) for i in range(3)]\n",
    "\n",
    "for epoch in range(1000):\n",
    "    grad = sum_of_squares_gradient(v) # compute gradient at v\n",
    "    v = gradient_step(v, grad, -0.01) # take a negative gradient step\n",
    "    print(epoch, v)\n",
    "    \n",
    "assert distance(v, [0, 0, 0]) < 0.001"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Using Gradient Descent to Fit Models**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# x range from -50 to 49, y is always 20 * x + 5\n",
    "inputs = [(x, 20 * x + 5) for x in range(-50,50)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case we *Know* the parameters of the linear relationship between x and y. but imagine we'd like to learn them from the data. We'll use gradient descent to find the slope adn intercept that minimize the average squared error\n",
    "\n",
    "We'll start off with a function that determins the gradient based on the errror from a single data point:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linear_gradient(x: float, y: float, theta: Vector) -> Vector:\n",
    "    slope, intercept = theta\n",
    "    predicted = slope * x + intercept\n",
    "    error = (predicted - y)\n",
    "    squared_error = error ** 2        # we'll minimize squared error\n",
    "    grad = [2 * error * x, 2 * error] # using its gradient\n",
    "    return grad\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "1936 [19.999937514988407, 4.895962418174019]\n",
      "1937 [19.99993763992091, 4.896170430852659]\n",
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      "1939 [19.999937889037025, 4.896585209340216]\n",
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      "1941 [19.999938137157972, 4.896998330870174]\n",
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      "1943 [19.999938384287727, 4.897409802061747]\n",
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      "1945 [19.999938630430247, 4.89781962950771]\n",
      "1946 [19.999938753132533, 4.898023928879125]\n",
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      "1950 [19.999939241493248, 4.898837049763285]\n",
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      "1958 [19.999940206563682, 4.900443892619722]\n",
      "1959 [19.999940326114643, 4.900642945041047]\n",
      "1960 [19.999940445426574, 4.900841599477079]\n",
      "1961 [19.99994056449995, 4.901039856723552]\n",
      "1962 [19.999940683335257, 4.901237717574604]\n",
      "1963 [19.999940801932958, 4.90143518282279]\n",
      "1964 [19.99994092029354, 4.901632253259077]\n",
      "1965 [19.999941038417468, 4.9018289296728526]\n",
      "1966 [19.999941156305223, 4.902025212851925]\n",
      "1967 [19.99994127395727, 4.902221103582526]\n",
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      "1969 [19.999941508556137, 4.9026117108353935]\n",
      "1970 [19.99994162550389, 4.902806428922279]\n",
      "1971 [19.99994174221783, 4.903000757689938]\n",
      "1972 [19.9999418586984, 4.903194697916776]\n",
      "1973 [19.999941974946083, 4.903388250379641]\n",
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      "1975 [19.999942206744638, 4.9037741951130815]\n",
      "1976 [19.99994232229644, 4.9039665889296]\n",
      "1977 [19.999942437617204, 4.904158598074037]\n",
      "1978 [19.9999425527074, 4.9043502233155065]\n",
      "1979 [19.99994266756748, 4.904541465421583]\n",
      "1980 [19.999942782197913, 4.9047323251583075]\n",
      "1981 [19.99994289659915, 4.904922803290189]\n",
      "1982 [19.999943010771656, 4.905112900580208]\n",
      "1983 [19.999943124715887, 4.905302617789819]\n",
      "1984 [19.999943238432294, 4.905491955678955]\n",
      "1985 [19.99994335192134, 4.905680915006029]\n",
      "1986 [19.999943465183474, 4.905869496527939]\n",
      "1987 [19.999943578219153, 4.9060577010000666]\n",
      "1988 [19.999943691028825, 4.906245529176285]\n",
      "1989 [19.99994380361295, 4.906432981808962]\n",
      "1990 [19.999943915971972, 4.906620059648957]\n",
      "1991 [19.999944028106345, 4.906806763445631]\n",
      "1992 [19.999944140016513, 4.906993093946846]\n",
      "1993 [19.99994425170293, 4.907179051898969]\n",
      "1994 [19.999944363166044, 4.907364638046873]\n",
      "1995 [19.999944474406295, 4.907549853133946]\n",
      "1996 [19.999944585424135, 4.907734697902084]\n",
      "1997 [19.999944696220005, 4.907919173091704]\n",
      "1998 [19.99994480679435, 4.9081032794417405]\n",
      "1999 [19.99994491714761, 4.908287017689651]\n",
      "2000 [19.999945027280233, 4.90847038857142]\n",
      "2001 [19.999945137192658, 4.908653392821557]\n",
      "2002 [19.999945246885318, 4.908836031173107]\n",
      "2003 [19.999945356358666, 4.909018304357645]\n",
      "2004 [19.999945465613127, 4.909200213105288]\n",
      "2005 [19.99994557464915, 4.909381758144691]\n",
      "2006 [19.99994568346716, 4.909562940203051]\n",
      "2007 [19.999945792067606, 4.909743760006112]\n",
      "2008 [19.999945900450914, 4.909924218278167]\n",
      "2009 [19.999946008617517, 4.910104315742061]\n",
      "2010 [19.99994611656786, 4.910284053119195]\n",
      "2011 [19.999946224302356, 4.910463431129525]\n",
      "2012 [19.99994633182146, 4.910642450491569]\n",
      "2013 [19.99994643912558, 4.910821111922407]\n",
      "2014 [19.99994654621516, 4.910999416137687]\n",
      "2015 [19.999946653090625, 4.911177363851627]\n",
      "2016 [19.999946759752405, 4.911354955777015]\n",
      "2017 [19.999946866200922, 4.911532192625214]\n",
      "2018 [19.99994697243661, 4.911709075106164]\n",
      "2019 [19.999947078459886, 4.911885603928388]\n",
      "2020 [19.999947184271186, 4.912061779798991]\n",
      "2021 [19.999947289870917, 4.912237603423664]\n",
      "2022 [19.99994739525952, 4.912413075506688]\n",
      "2023 [19.999947500437408, 4.912588196750934]\n",
      "2024 [19.999947605405, 4.91276296785787]\n",
      "2025 [19.999947710162722, 4.91293738952756]\n",
      "2026 [19.999947814710993, 4.913111462458668]\n",
      "2027 [19.999947919050225, 4.913285187348461]\n",
      "2028 [19.99994802318085, 4.913458564892815]\n",
      "2029 [19.999948127103266, 4.91363159578621]\n",
      "2030 [19.999948230817907, 4.913804280721741]\n",
      "2031 [19.99994833432518, 4.913976620391115]\n",
      "2032 [19.9999484376255, 4.914148615484658]\n",
      "2033 [19.999948540719277, 4.914320266691314]\n",
      "2034 [19.999948643606935, 4.914491574698651]\n",
      "2035 [19.999948746288872, 4.914662540192861]\n",
      "2036 [19.999948848765516, 4.914833163858765]\n",
      "2037 [19.999948951037258, 4.915003446379813]\n",
      "2038 [19.99994905310453, 4.91517338843809]\n",
      "2039 [19.999949154967716, 4.9153429907143185]\n",
      "2040 [19.999949256627247, 4.915512253887858]\n",
      "2041 [19.999949358083516, 4.915681178636709]\n",
      "2042 [19.999949459336932, 4.915849765637519]\n",
      "2043 [19.999949560387904, 4.916018015565581]\n",
      "2044 [19.999949661236833, 4.916185929094838]\n",
      "2045 [19.999949761884128, 4.916353506897885]\n",
      "2046 [19.999949862330183, 4.916520749645974]\n",
      "2047 [19.999949962575414, 4.916687658009012]\n",
      "2048 [19.999950062620208, 4.916854232655569]\n",
      "2049 [19.999950162464977, 4.917020474252879]\n",
      "2050 [19.999950262110115, 4.917186383466838]\n",
      "2051 [19.99995036155602, 4.917351960962014]\n",
      "2052 [19.999950460803095, 4.917517207401646]\n",
      "2053 [19.999950559851737, 4.917682123447646]\n",
      "2054 [19.99995065870234, 4.917846709760602]\n",
      "2055 [19.9999507573553, 4.918010966999783]\n",
      "2056 [19.999950855811015, 4.918174895823139]\n",
      "2057 [19.999950954069877, 4.918338496887303]\n",
      "2058 [19.99995105213228, 4.918501770847598]\n",
      "2059 [19.99995114999862, 4.918664718358035]\n",
      "2060 [19.99995124766928, 4.918827340071317]\n",
      "2061 [19.999951345144662, 4.918989636638844]\n",
      "2062 [19.99995144242515, 4.9191516087107106]\n",
      "2063 [19.999951539511134, 4.919313256935714]\n",
      "2064 [19.99995163640301, 4.919474581961354]\n",
      "2065 [19.999951733101152, 4.919635584433834]\n",
      "2066 [19.999951829605966, 4.919796264998068]\n",
      "2067 [19.99995192591782, 4.9199566242976775]\n",
      "2068 [19.99995202203711, 4.920116662975]\n",
      "2069 [19.999952117964224, 4.920276381671087]\n",
      "2070 [19.999952213699533, 4.920435781025709]\n",
      "2071 [19.999952309243437, 4.920594861677357]\n",
      "2072 [19.999952404596304, 4.920753624263246]\n",
      "2073 [19.99995249975853, 4.920912069419316]\n",
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      "2080 [19.999953160587843, 4.922012350629502]\n",
      "2081 [19.999953254238537, 4.92216827908883]\n",
      "2082 [19.999953347701986, 4.922323895784891]\n",
      "2083 [19.999953440978558, 4.922479201341023]\n",
      "2084 [19.99995353406864, 4.922634196379319]\n",
      "2085 [19.999953626972594, 4.9227888815206295]\n",
      "2086 [19.9999537196908, 4.922943257384561]\n",
      "2087 [19.99995381222362, 4.923097324589483]\n",
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      "2090 [19.999954088713505, 4.923557680415349]\n",
      "2091 [19.999954180508507, 4.923710519143231]\n",
      "2092 [19.999954272119968, 4.923863052285453]\n",
      "2093 [19.999954363548266, 4.924015280453002]\n",
      "2094 [19.99995445479376, 4.924167204255644]\n",
      "2095 [19.99995454585682, 4.924318824301927]\n",
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      "2097 [19.999954727437085, 4.9246211555535195]\n",
      "2098 [19.999954817955018, 4.92477186796985]\n",
      "2099 [19.999954908291972, 4.924922279051865]\n",
      "2100 [19.999954998448306, 4.925072389402053]\n",
      "2101 [19.999955088424382, 4.925222199621697]\n",
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      "2125 [19.99995719469902, 4.928729148159773]\n",
      "2126 [19.999957280283912, 4.928871647058153]\n",
      "2127 [19.99995736569769, 4.929013861044321]\n",
      "2128 [19.999957450940684, 4.92915579068793]\n",
      "2129 [19.99995753601325, 4.929297436557495]\n",
      "2130 [19.99995762091572, 4.929438799220393]\n",
      "2131 [19.999957705648434, 4.929579879242868]\n",
      "2132 [19.999957790211738, 4.929720677190031]\n",
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      "4983 [19.999999859611503, 4.999766253069806]\n",
      "4984 [19.9999998598922, 4.9997667204232785]\n",
      "4985 [19.999999860172327, 4.999767186842324]\n",
      "4986 [19.9999998604519, 4.999767652328812]\n",
      "4987 [19.99999986073091, 4.9997681168846055]\n",
      "4988 [19.999999861009368, 4.999768580511567]\n",
      "4989 [19.999999861287264, 4.999769043211553]\n",
      "4990 [19.999999861564607, 4.999769504986418]\n",
      "4991 [19.999999861841392, 4.999769965838009]\n",
      "4992 [19.99999986211763, 4.999770425768174]\n",
      "4993 [19.99999986239331, 4.999770884778756]\n",
      "4994 [19.999999862668442, 4.9997713428715915]\n",
      "4995 [19.99999986294302, 4.999771800048517]\n",
      "4996 [19.999999863217052, 4.999772256311363]\n",
      "4997 [19.999999863490537, 4.999772711661958]\n",
      "4998 [19.999999863763474, 4.999773166102124]\n",
      "4999 [19.999999864035864, 4.999773619633683]\n",
      "slope =  19.999999864035864 intercept =  4.999773619633683\n"
     ]
    }
   ],
   "source": [
    "from linear_algebra import vector_mean\n",
    "\n",
    "# random start\n",
    "theta = [random.uniform(-1,1),random.uniform(-1,1)]\n",
    "\n",
    "learning_rate = 0.001\n",
    "\n",
    "for epoch in range(5000):\n",
    "    # compute mean of the gradients (on the full dataset)\n",
    "    grad = vector_mean([linear_gradient(x, y, theta) for x, y in inputs])\n",
    "    # take a step in that direction\n",
    "    theta = gradient_step(theta, grad, -learning_rate)\n",
    "    print(epoch, theta)\n",
    "    \n",
    "slope, intercept = theta\n",
    "print('slope = ', slope, 'intercept = ', intercept)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mini - Batch -> Computes the gradient on a smaller batch rather than full dataset\n",
    "Stochastic Gradient Descent -> Computes on each datapoint\n",
    "\n",
    "Tradeoffs for both\n",
    "\n",
    "Further:\n",
    "* [Sebastian Ruder - Gradient Descent Variations](http://ruder.io/optimizing-gradient-descent/index.html)"
   ]
  }
 ],
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