Implementation of the gradient descent method to find the minimum of simple functions of one variable. The training code.

gradient.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Метод градиентного спуска"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# для нахождения производной\n",
    "from scipy.misc import derivative\n",
    "\n",
    "# для работы графиков\n",
    "%matplotlib inline\n",
    "from matplotlib import pylab as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-10.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# определяем функцию\n",
    "def f(x):\n",
    "    return (x-5)**2\n",
    "\n",
    "# проверяем нахождение f'(x)\n",
    "derivative(f, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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uTwVarp4trNCrZz2RMYbnvsrknU2HuW1qItdN0qteXZUWvToj/SO78fp14yg9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\nLXqllHJz/x8s8lab/x0SoQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x8840a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# рисуем график функции\n",
    "x = range(-5,15)\n",
    "y = [f(xn) for xn in x]\n",
    "plt.plot(x, y)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 25,\n",
       " 1.0: 16.0,\n",
       " 1.8: 10.240000000000002,\n",
       " 2.4400000000000004: 6.5535999999999976,\n",
       " 2.9520000000000004: 4.194303999999998,\n",
       " 3.3616000000000001: 2.6843545599999996,\n",
       " 3.6892800000000001: 1.7179869183999996,\n",
       " 3.9514240000000003: 1.0995116277759995,\n",
       " 4.1611392: 0.70368744177663989,\n",
       " 4.3289113600000002: 0.45035996273704931,\n",
       " 4.4631290880000005: 0.28823037615171121,\n",
       " 4.5705032704000006: 0.18446744073709501,\n",
       " 4.6564026163200003: 0.11805916207174093,\n",
       " 4.7251220930560001: 0.07555786372591429,\n",
       " 4.7800976744448: 0.048357032784585148,\n",
       " 4.8240781395558399: 0.030948500982134555,\n",
       " 4.8592625116446717: 0.019807040628566166,\n",
       " 4.8874100093157375: 0.012676506002282305,\n",
       " 4.9099280074525904: 0.0081129638414606121,\n",
       " 4.9279424059620727: 0.0051922968585347406,\n",
       " 4.942353924769658: 0.0033230699894622544,\n",
       " 4.9538831398157264: 0.0021267647932558427,\n",
       " 4.9631065118525814: 0.0013611294676837131,\n",
       " 4.9704852094820655: 0.00087112285931755548,\n",
       " 4.9763881675856521: 0.00055751862996325226,\n",
       " 4.981110534068522: 0.00035681192317646801}"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# подготовка к Град. спуску\n",
    "\n",
    "# начальн. значение\n",
    "xn = 0 \n",
    "yn = f(xn)\n",
    "\n",
    "# заводим словарь, где будем хранить все найденные значения функции\n",
    "Y = {xn: yn}\n",
    "\n",
    "# шаг ГС (произвольный подибраем)\n",
    "step = 0.1\n",
    "\n",
    "# по формуле градиентного спуска получаем все значения x y\n",
    "for _ in range(25):\n",
    "    xn = xn - step*derivative(f, xn)\n",
    "    yn = f(xn)\n",
    "    Y[xn] = yn\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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pVEQ8Mu56RsF2zfYflO9/T8WN//8ab1XVioi/i4ibI6Ku4r/l/4iIvxlzWZWyfX15o1+2\nr5f0KUmVzGDb8oE+jYtR235K0k8lfcj2Gdv3jbumEbhd0j0qemwnytfBcRdVsb2Sjth+RUXH5XBE\nTMU0vimzR9JR2y9LeknSjyLix1WcaMtPWwQA9GbL99ABAL0h0AEgCQIdAJIg0AEgCQIdAJIg0AEg\nCQIdAJIg0AEgif8HcPpxTGxW0xcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9beb828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# наносим найденные точки на график\n",
    "plt.plot(list(Y.keys()), list(Y.values()), 'ro')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x8320b70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# масштабируем\n",
    "plt.plot(list(Y.keys()), list(Y.values()), 'ro')\n",
    "plt.axis([4.95, 5.05, -0.05, 0.05])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "мин. Х = 4.98111053407\n",
      "мин. Y = 0.000356811923176\n"
     ]
    }
   ],
   "source": [
    "# словарь значений, где key = х (просто инвертируем Y)\n",
    "X = {}\n",
    "for i in range(len(Y)):\n",
    "    X[list(Y.values())[i]] = list(Y.keys())[i]\n",
    "\n",
    "# выводим пару искомых минимальных X и Y (близких к минимумам)\n",
    "print ('мин. Х =', min(X.items())[1])\n",
    "print ('мин. Y =', min(X.items())[0])"
   ]
  }
 ],
 "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": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}