From 6c1839ada36e2c63a7e95edc9a8f469b89f30ad2 Mon Sep 17 00:00:00 2001 From: Matt Marcha Date: Thu, 29 Nov 2018 18:28:38 +0100 Subject: [PATCH] Prise de notes cours 1 --- .../Exercice 1-checkpoint.ipynb | 399 ++++++++++++++++++ .../Exercice 2-checkpoint.ipynb | 319 ++++++++++++++ .../Exercices/Exercice 1.ipynb | 366 ++++++++++++++++ .../Exercices/Exercice 2.ipynb | 319 ++++++++++++++ .../Introduction au Machine Learning.md | 148 +++++++ Archis Applications/Ressources/cours_1.pdf | Bin 0 -> 5731766 bytes Archis Applications/Ressources/tp_1.pdf | Bin 0 -> 162936 bytes 7 files changed, 1551 insertions(+) create mode 100644 Archis Applications/Exercices/.ipynb_checkpoints/Exercice 1-checkpoint.ipynb create mode 100644 Archis Applications/Exercices/.ipynb_checkpoints/Exercice 2-checkpoint.ipynb create mode 100644 Archis Applications/Exercices/Exercice 1.ipynb create mode 100644 Archis Applications/Exercices/Exercice 2.ipynb create mode 100644 Archis Applications/Introduction au Machine Learning.md create mode 100644 Archis Applications/Ressources/cours_1.pdf create mode 100644 Archis Applications/Ressources/tp_1.pdf diff --git a/Archis Applications/Exercices/.ipynb_checkpoints/Exercice 1-checkpoint.ipynb b/Archis Applications/Exercices/.ipynb_checkpoints/Exercice 1-checkpoint.ipynb new file mode 100644 index 0000000..b3f5d67 --- /dev/null +++ b/Archis Applications/Exercices/.ipynb_checkpoints/Exercice 1-checkpoint.ipynb @@ -0,0 +1,399 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 114, + "metadata": {}, + "outputs": [], + "source": [ + "# On importe les librairies\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 1 - Créer une fonction J(theta qui implémente J(theta). Quelle est la valeur de de J(4) ?" + ] + }, + { + "cell_type": "code", + "execution_count": 115, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "25" + ] + }, + "execution_count": 115, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def J(theta_i):\n", + " return (theta_i + 1)**2\n", + "\n", + "J(4)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 2" + ] + }, + { + "cell_type": "code", + "execution_count": 116, + "metadata": {}, + "outputs": [], + "source": [ + "# On génère les points dans theta\n", + "theta = np.arange(-10.00, 10.00, 0.01)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 3\n", + "Plot theta vs J(theta). When do we reach the minimum?" + ] + }, + { + "cell_type": "code", + "execution_count": 117, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 117, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Plot de theta\n", + "plt.plot(theta,theta)" + ] + }, + { + "cell_type": "code", + "execution_count": 120, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 120, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# plot de J(theta)\n", + "plt.plot(theta,J(theta))" + ] + }, + { + "cell_type": "code", + "execution_count": 121, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-1.0" + ] + }, + "execution_count": 121, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "J_theta = [J(x) for x in theta]\n", + "# Valeur de theta à laquelle on obtien le plus petit J(theta)\n", + "round(theta[J_theta.index(min(J_theta))], 3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 4\n", + "Create a function dJ_dtheta(x_i) which is computing the gradient\n", + "?" + ] + }, + { + "cell_type": "code", + "execution_count": 122, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 122, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# On calcule la fonction dérivée. D'après J(theta). Merci dcode.fr\n", + "def dJ_theta(x_i):\n", + " return 2 * (x_i + 1)\n", + "# Affichage de la valeur à 0\n", + "dJ_theta(-1)" + ] + }, + { + "cell_type": "code", + "execution_count": 123, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 123, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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f92aRRsCb1toPHB5TpM7weCxvfLuV2e9nYIGZlwzmurFqdibucDTwrbWbgWFOjiFSV23efYCExBS+yy7g9H4deGhSLD3aqdmZuEeXZYoEWEWlhxc+38ITKzbQtFEDHr0ijitGdldbBHGdAl8kgNJ2FjItMZnUHUVcMKQzsyYOpZOanUkdocAXCYCS8kqe+ngjz3+6mbbNI3ju2hFcGBvldlkiP6HAF/HTmq0FTJ2XzKbdB5k8ojv3TBhEm+ZqdiZ1jwJf5AQdLK3g0WWZvPZ1Nl1bN+O1m0ZzZv+ObpclUiMFvsgJ+GzDbqYnpbCz8DDXn9yTO8cPpGUT/TlJ3aY9VOQ4FB4qZ9aSdOatyaF3xxa8e9tYRsW0c7sskVpR4IvU0gepudyzMI2Cg2X89qw+/P5cNTuT+kWBL3IM+cUl3LswjfdTdzGkayteuXEUQ7up2ZnUPwp8kRpYa5m3JocHlqzncHklU8cP4NbTe6vZmdRbCnyRamwvOMRd81P4fOMeRsW0ZfbkOPp0bOl2WSJ+UeCLVOHxWF7/Ops5yzIxwP0Th/CrMT1poGZnEgIU+CI+WfkHSEhMZvXWfZzRvyMPTRpK97ZqdiahQ4EvYa+80sPczzbz5IqNNItoyONXDuPyEd3U7ExCjgJfwlrqjkKmzksmPbeIi2OjmHnpEDpGNnG7LBFHOB74xpjxwJNAQ+BFa+1sp8cUOZaS8kqe/Ggjcz/bTLsWETz/q5GMH9rF7bJEHOX0Rxw2BJ4BzgNygFXGmEXW2nQnxxU5mlXZBUybl8zmPQf5RXx3Zlw0mNbNG7tdlojjnD7CHw1k+T75CmPM28BEQIEvQXegtII5H2Tw+tdb6d62Gf++eQyn9evgdlkiQeN04HcDtlf5OQcY4/CYIj+zMjOfGUkp5BaV8OtTY/jr+QNooWZnEmac3uOru8zB/mQBY6YAUwCio6MdLkfCzb6DZcxanE7SDzvo26kl824/hZE927pdlogrnA78HKBHlZ+7AzurLmCtnQvMBYiPj//Jk4HIibLWsjRlF/cuSmX/oXJ+f05f7jinL00aqdmZhC+nA38V0M8Y0wvYAVwFXOPwmBLm8otKuHtBKsvT84jt1prXbxrD4K6t3C5LxHWOBr61tsIY8ztgGd7LMl+21qY5OaaEL2st/1mdw6wl6ZRVeJh+4UBuPq0XjdTsTAQIwnX41tqlwFKnx5Hwtm2vt9nZF1l7GN2rHbMvj6W3mp2J/IQuU5B6rdJjefWrbB5blknDBoYHLhvKNaOj1exMpBoKfKm3NuYVMzUxmR+27efsAR15cFIsXds0c7sskTpLgS/1TlmFh+c/3cTTH2fRoklD/vHL4Uwc3lXNzkSOQYEv9Upyzn6mzksmY1cxlwzryr2XDKZDSzU7E6kNBb7UCyXllTzx4QZe+HwzHSOb8ML18Zw3uLPbZYnUKwp8qfO+2byXhMRksvce4urRPUi4cBCtm6nZmcjxUuBLnVVcUs7s9zN449ttRLdrzpu3jOGUvmp2JnKiFPhSJ32ckceM+ankFZVwy2m9+PP5/Wkeod1VxB/6C5I6peBgGfe/l8aCtTvp37klz157CidFq9mZSCAo8KVOsNbyXnIuMxelUVxSzh/O7ccdZ/clopHaIogEigJfXLer0NvsbMX6PIZ1b80jV4xhYBc1OxMJNAW+uMZay9urtvPQkvWUezzMuGgQN53Wi4ZqiyDiCAW+uGLr3oMkJKbw9ea9nNy7HbMvjyOmQwu3yxIJaQp8CapKj+WVL7fw2PJMGjdowMOXx/LL+B5qdiYSBAp8CZrMXd5mZ+u272fcoE48cFksXVo3dbsskbChwBfHlVV4ePaTLJ5ZmUVk08b879UncUlclJqdiQSZY4FvjJkJ3Ars9k26y/dhKBJG1m7fz7R5yWTmFTNxeFfuvWQI7VpEuF2WSFhy+gj/CWvtYw6PIXXQ4bJKHl+eyctfbqFTZFNeuiGecwep2ZmIm3RKRwLuq017SEhMYVvBIa4ZE03ChQNp1VTNzkTc5nTg/84Ycz2wGviLtXafw+OJi4pKynl46Xre+m47Me2b89atJzO2T3u3yxIRH78C3xizAuhSzawZwHPALMD6vj8O3FTNOqYAUwCio6P9KUdctCI9jxkLUthdXMptZ/Tmj+P60yyiodtliUgVxlrr/CDGxACLrbVDj7ZcfHy8Xb16teP1SODsPVDKzPfSeW/dTgZ2iWTOFXHEdW/jdlkiYcUYs8ZaG3+s5Zy8SifKWpvr+3ESkOrUWBJ81loWrt3Jfe+lcaC0gj+f15/bz+yjZmcidZiT5/DnGGOG4z2lkw3c5uBYEkQ79x/m7gWpfJyRz/AebZhzRRz9O0e6XZaIHINjgW+tvc6pdYs7PB7Lm99tY/b7GVR6LPdMGMyNp8So2ZlIPaHLMqVWtuw5SEJiMt9uKeDUvu15eFIc0e2bu12WiBwHBb4cVUWlh5e+2MLfP9xARKMGzJkcx5Xx3dUWQaQeUuBLjdJ3FjEtMZmUHYWcN7gzD1w2lM6t1OxMpL5S4MvPlFZU8vTHWTz3ySbaNG/MM9eM4KLYLjqqF6nnFPjyE2u27mNaYjJZ+Qe4/KRu3DNhMG3V7EwkJCjwBYBDZRU8uiyTV7/KJqpVU1759SjOHtDJ7bJEJIAU+MIXG/eQkJRMzr7DXD+2J1PHD6RlE+0aIqFGf9VhrPBwOQ8uSefd1Tn06tCCd28by+he7dwuS0QcosAPU8vSdnHPglT2HizjN2f14Q/n9qNpYzU7EwllCvwws7u4lJmL0liSksugqFa8dMMoYru3drssEQkCBX6YsNaS9P0O7l+czuGySu68YABTzuhN44ZqdiYSLhT4YWDH/sPclZTCpxt2M7JnWx6ZHEvfTmp2JhJuFPghzOOx/PvbrTzyfgYWmHnJYK4fG0MDNTsTCUsK/BC1afcBEhKTWZW9j9P7deChSbH0aKdmZyLhTIEfYsorPbzw+Wb+sWIjTRs14NEr4rhipJqdiYgCP6Sk7ihkWmIyaTuLGD+kC/dfNoROkWp2JiJefl2iYYy50hiTZozxGGPij5g33RiTZYzJNMZc4F+ZcjQl5ZU8uiyDic98SV5RKc9dO4LnrxupsBeRn/D3CD8VuBz4Z9WJxpjBwFXAEKArsMIY099aW+nneHKE1dkFTE1MZvPug1wxsjt3XzyINs3V7ExEfs6vwLfWrgeqOz88EXjbWlsKbDHGZAGjga/9GU/+38FSb7Oz177OpmvrZrx+02jO6N/R7bJEpA5z6hx+N+CbKj/n+Kb9jDFmCjAFIDo62qFyQsunG3ZzV1IKOwsPc8PYGO68YAAt1OxMRI7hmClhjFkBdKlm1gxr7cKa7lbNNFvdgtbaucBcgPj4+GqXEa/9h8qYtXg9id/n0LtjC/5z21jiY9TsTERq55iBb60ddwLrzQF6VPm5O7DzBNYjPu+n5HLPwjT2HSrjjrP78D/nqNmZiBwfp84DLALeNMb8He+Ltv2A7xwaK6TlF5Xwt4VpfJC2iyFdW/HaTaMY0lXNzkTk+PkV+MaYScBTQEdgiTFmrbX2AmttmjHmXSAdqADu0BU6x8day7w1OcxanE5JhYdp4wdy6+m9aKRmZyJygvy9Smc+ML+GeQ8CD/qz/nC1veAQd81P4fONexgV05bZk+Po07Gl22WJSD2nSzvqkEqP5fWvs3l0WSYGmDVxCNeO6almZyISEAr8OiIrv5hpiSms2bqPM/t35MFJQ+neVs3ORCRwFPguK6/08M9PN/G/H2XRvElD/v6LYUw6qZuanYlIwCnwXZS6o5A75yWzPreIi+OimHnJEDpGNnG7LBEJUQp8F5SUV/KPFRt54fPNtGsRwT+vG8kFQ6p7b5uISOAo8IPsuy0FJCQms3nPQX4Z34O7LhpE6+aN3S5LRMKAAj9IikvKmfNBJv/6Zivd2zbj3zeP4bR+HdwuS0TCiAI/CFZm5jMjKYXcohJuOrUXf72gP80jtOlFJLiUOg7ad7CMWYvTSfphB307tWTe7acwsmdbt8sSkTClwHeAtZYlKbncuzCNwsPl/P6cvtxxTl+aNFKzMxFxjwI/wPKKSrhnQSrL0/OI7daaf98yhkFRrdwuS0REgR8o1lreXb2dB5asp6zCw/QLB3LzaWp2JiJ1hwI/ALbtPURCUjJfbdrL6F7teGRyHL06tHC7LBGRn1Dg+6HSY3n1q2weW5ZJwwaGBy4byjWjo9XsTETqJAX+CdqQV8zUecms3b6fswd05MFJsXRt08ztskREauTvB6BcCcwEBgGjrbWrfdNjgPVApm/Rb6y1t/szVl1RVuHh+U838dTHG2nZpBFPXjWcS4d1VbMzEanz/D3CTwUuB/5ZzbxN1trhfq6/Tlm3fT/TEpPJ2FXMJcO6MvOSwbRvqWZnIlI/+PuJV+uBkD+6PVxWyRMrNvDi55vpGNmEF66P57zBnd0uS0TkuDh5Dr+XMeYHoAi421r7uYNjOebrTXuZnpRM9t5DXD26B9MvGkSrpmp2JiL1zzED3xizAqiud+8Ma+3CGu6WC0Rba/caY0YCC4wxQ6y1RdWsfwowBSA6Orr2lTusqKSc2e9n8Oa324hu15w3bxnDKX3V7ExE6q9jBr61dtzxrtRaWwqU+m6vMcZsAvoDq6tZdi4wFyA+Pt4e71hO+Dgjj7uSUskvLuHW03vx5/MG0CxCbRFEpH5z5JSOMaYjUGCtrTTG9Ab6AZudGCuQ9h4o5f7F6Sxcu5MBnSN5/rqRDO/Rxu2yREQCwt/LMicBTwEdgSXGmLXW2guAM4D7jTEVQCVwu7W2wO9qHWKtZdG6ndz3XjrFJeX8cVw/fntWXyIaqS2CiIQOf6/SmQ/Mr2Z6IpDoz7qDJbfwMHfPT+WjjHyG9WjDnMlxDOgS6XZZIiIBF7bvtPV4LG+v2s7DS9dT7vFw98WD+PWpvWiotggiEqLCMvCz9xwkISmZbzYXMLZ3e2ZPjqVnezU7E5HQFlaBX+mxvPzFFh7/MJPGDRrw8OWxXDWqR8i/cUxEBMIo8DN2FTFtXjLrcgoZN6gTD1wWS5fWTd0uS0QkaEI+8EsrKnlm5SaeXZlF62aNeerqk5gQF6WjehEJOyEd+D9s28e0xGQ25B3gsuFd+dslQ2jXIsLtskREXBGSgX+orILHl2/g5S+30KVVU16+MZ5zBqrZmYiEt5AL/K+y9pCQlMK2gkNcOyaahAsHEqlmZyIioRP4hYfLeXjpet5etZ2Y9s15e8rJnNy7vdtliYjUGSER+Mk5+7n19dXsLi7ltjN786dx/WnaWM3ORESqConAj27XnP6dI3nh+njiuqvZmYhIdUIi8Ns0j+BfN49xuwwRkTpN7SBFRMKEAl9EJEwo8EVEwoQCX0QkTPgV+MaYR40xGcaYZGPMfGNMmyrzphtjsowxmcaYC/wvVURE/OHvEf6HwFBrbRywAZgOYIwZDFwFDAHGA88aY3RhvIiIi/wKfGvtcmtthe/Hb4DuvtsTgbettaXW2i1AFjDan7FERMQ/gTyHfxPwvu92N2B7lXk5vmkiIuKSY77xyhizAuhSzawZ1tqFvmVmABXAGz/erZrlbQ3rnwJM8f14wBiTeayajqIDsMeP+ztFdR0f1XV8VNfxCcW6etZmoWMGvrV23NHmG2NuACYA51prfwz1HKBHlcW6AztrWP9cYG5tij0WY8xqa218INYVSKrr+Kiu46O6jk841+XvVTrjgWnApdbaQ1VmLQKuMsY0Mcb0AvoB3/kzloiI+MffXjpPA02AD30fGfiNtfZ2a22aMeZdIB3vqZ47rLWVfo4lIiJ+8CvwrbV9jzLvQeBBf9Z/AgJyasgBquv4qK7jo7qOT9jWZf7/tLuIiIQytVYQEQkT9SrwjTFXGmPSjDEeY0z8EfOO2crBGNPLGPOtMWajMeYdY0yEQ3W+Y4xZ6/vKNsasrWG5bGNMim+51U7UcsR4M40xO6rUdlENy433bccsY0xCEOqqsUXHEcsFZXsd6/f3XYzwjm/+t8aYGKdqqTJmD2PMSmPMet/fwB+qWeYsY0xhlcf3b07X5Rv3qI+L8fpf3/ZKNsaMCEJNA6psh7XGmCJjzB+PWCYo28sY87INDdJdAAAEs0lEQVQxJt8Yk1plWjtjzIe+LPrQGNO2hvve4Ftmo++KSP9Ya+vNFzAIGAB8AsRXmT4YWIf3BeRewCagYTX3fxe4ynf7eeA3Qaj5ceBvNczLBjoEcfvNBP56jGUa+rZfbyDCt10HO1zX+UAj3+1HgEfc2l61+f2B3wLP+25fBbwThMcuChjhux2Jt5XJkXWdBSwO1v5U28cFuAjvmzINcDLwbZDrawjsAnq6sb2AM4ARQGqVaXOABN/thOr2eaAdsNn3va3vdlt/aqlXR/jW2vXW2uremHXMVg7GexnROcA836TXgMucrNc35i+At5wcJ8BGA1nW2s3W2jLgbbzb1zG25hYdbqjN7z8R7/4D3v3pXN9j7Rhrba619nvf7WJgPfXn3esTgdet1zdAG2NMVBDHPxfYZK3dGsQx/8ta+xlQcMTkqvtQTVl0AfChtbbAWrsPb++y8f7UUq8C/yhq08qhPbC/SrAEo93D6UCetXZjDfMtsNwYs8b3juNg+J3v3+qXa/g30u22GFVbdBwpGNurNr//f5fx7U+FePevoPCdQjoJ+Laa2WONMeuMMe8bY4YEqaRjPS5u71NXUfNBlxvbC6CztTYXvE/mQKdqlgn4dqtzn2lratHKobq7VTPtyMuPat3uoTZqWefVHP3o/lRr7U5jTCe872XI8B0NnLCj1QU8B8zC+3vPwnu66aYjV1HNff2+lKs228v8vEXHkQK+vaortZppju5Lx8MY0xJIBP5orS06Yvb3eE9bHPC9PrMA75senXasx8XN7RUBXIqvk+8R3NpetRXw7VbnAt8eo5VDDWrTymEP3n8lG/mOymps91Abx6rTGNMIuBwYeZR17PR9zzfGzMd7OsGvAKvt9jPGvAAsrmZWrdtiBLIuU32LjiPXEfDtVY3a/P4/LpPje5xb8/N/2QPOGNMYb9i/Ya1NOnJ+1ScAa+1SY8yzxpgO1lpH+8bU4nFxZJ+qpQuB7621eUfOcGt7+eQZY6Kstbm+01v51SyTg/d1hh91x/v65QkLlVM6x2zl4AuRlcAVvkk3ADX9xxAI44AMa21OdTONMS2MMZE/3sb7wmVqdcsGyhHnTSfVMN4qoJ/xXtEUgfff4UUO11VTi46qywRre9Xm91+Ed/8B7/70cU1PUoHie43gJWC9tfbvNSzT5cfXEowxo/H+fe91uK7aPC6LgOt9V+ucDBT+eDojCGr8L9uN7VVF1X2opixaBpxvjGnrO/16vm/aiXP6FepAfuENqRygFMgDllWZNwPv1RWZwIVVpi8Fuvpu98b7RJAF/Ado4mCtrwK3HzGtK7C0Si3rfF9peE9tOL39/gWkAMm+HS7qyLp8P1+E9yqQTUGqKwvvucq1vq/nj6wrmNurut8fuB/vExJAU9/+k+Xbn3oHYRudhvff+eQq2+ki4PYf9zPgd75tsw7vi9+nBKGuah+XI+oywDO+7ZlClSvsHK6tOd4Ab11lWtC3F94nnFyg3JdfN+N9zecjYKPvezvfsvHAi1Xue5NvP8sCfu1vLXqnrYhImAiVUzoiInIMCnwRkTChwBcRCRMKfBGRMKHAFxEJEwp8EZEwocAXEQkTCnwRkTDxfxoy281xBJeTAAAAAElFTkSuQmCC\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.plot(theta, dJ_theta(theta))\n", + "# dJ_t = [dJ_theta(x) for x in theta]\n", + "# plt.plot(theta, dJ_t)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 5\n", + "Create a function gradient_descent(theta_0, lr, nb_iters, df_dx, f) which returns the solution\n", + "of argmin f (θ). \n", + "x_0, lr, nb_iters, df_dx, f correspond respectively to the initial value of θ, the\n", + "θ\n", + "learning rate, the number of iterations allowed for solving the problem, the gradient of θ and the\n", + "function J(.).\n", + "\n", + "What is the solution θ̂ found by gradient descent to our problem? Note: assume that\n", + "x_0, lr, nb_iters = -7, 0.1, 100 while debugging" + ] + }, + { + "cell_type": "code", + "execution_count": 147, + "metadata": {}, + "outputs": [], + "source": [ + "def gradient_descent(theta_0, lr, nb_iters, dJ_t, J_theta):\n", + " # puis on reprend les étapes vues en cours\n", + " theta = theta_0\n", + " \n", + " for t in range(nb_iters): \n", + " # calcul de theta\n", + " theta = theta - lr * dJ_t(theta)\n", + "\n", + " return theta\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 148, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-1.0000000100978041" + ] + }, + "execution_count": 148, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# On appelle la fonction \n", + "gradient_descent(-7, 0.01, 1000, dJ_theta, J)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 6\n", + "Update your function gradient_descent for printing the optimization path (i.e. print the line between\n", + "θ t and θ t+1 and saving the figure at the end of the optimization process)." + ] + }, + { + "cell_type": "code", + "execution_count": 192, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "-1.0000000100978041" + ] + }, + "execution_count": 192, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "def gradient_descent(theta_0, lr, nb_iters, dJ_t, J_theta):\n", + " # puis on reprend les étapes vues en cours\n", + " theta = theta_0\n", + " \n", + " # on initialise les séries pour le plot\n", + " x = [theta]\n", + " y = [J_theta(theta)]\n", + " \n", + " for t in range(nb_iters): \n", + " # calcul de theta\n", + " theta = theta - lr * dJ_t(theta)\n", + " # ajout de la valeur au plot\n", + " x.append(theta)\n", + " y.append(J_theta(theta))\n", + " \n", + " # affichage du plot\n", + " plt.plot(x,y)\n", + " \n", + " return theta\n", + "\n", + "# On appelle la fonction \n", + "gradient_descent(-7, 0.01, 1000, dJ_theta, J)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 7\n", + "Assuming that you are setting nb_iters equals to 100 what is the solution θ̂ when varying nb_iters in\n", + "[-8, -1, 7] and lr in [0.1, 0.01, 0.001, -0.01, 0.8, 1.01]. Does the estimated solution always\n", + "the same? What are pros and cons about these hyperparameters and GD in general?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 8\n", + "Now assume the function J(θ) = sin(2θ + 1), what is the solution θ̂ given by GD?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "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": "3.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Archis Applications/Exercices/.ipynb_checkpoints/Exercice 2-checkpoint.ipynb b/Archis Applications/Exercices/.ipynb_checkpoints/Exercice 2-checkpoint.ipynb new file mode 100644 index 0000000..86b82e3 --- /dev/null +++ b/Archis Applications/Exercices/.ipynb_checkpoints/Exercice 2-checkpoint.ipynb @@ -0,0 +1,319 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 56, + "metadata": {}, + "outputs": [], + "source": [ + "# On importe les librairies\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We want to estimate parameters of a simple linear regression (y i = wx i + b) by closed-form and GD. \n", + "\n", + "The loss function is defined on a data point by l(ŷ i , y i ) = (ŷ i − y i )² where the prediction is given by ŷ i = ŵx + b̂. The\n", + "P N\n", + "optimization problem we want to solve is min i=1 l(ŷ i , y i ). \n", + "\n", + "We assume the following data generation process:\n", + "w,b\n", + "Y ∼ wX + b + \\eta where X ∼ U[20; 40] and \\eta ∼ N (0, 1).\n", + "\n", + "# 1\n", + "\n", + "Generate N data points (x i , y i ) using the data generation process given above and store them into x vs\n", + "N\n", + "y. Note: N = 100 and start by first generating {x i } N\n", + "1 and then {y i } 1 . Do not forget to add noise!\n", + "\n", + "D'abord on va générer les x" + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "metadata": {}, + "outputs": [], + "source": [ + "# On génère les x\n", + "list_x = np.random.uniform(20, 40, 100)\n", + "\n", + "# Pis les y\n", + "\n", + "def y(x):\n", + " w = 1.5\n", + " b = 5\n", + " return w * x + b + np.random.normal(0,1)\n", + "\n", + "list_y = [y(x) for x in list_x ]\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 2\n", + "Plot the data points x vs y." + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 58, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.scatter(list_x, list_y, s=6)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 3\n", + "Estimate parameters of the simple linear regression by closed-form. Note: store these values into w_cf\n", + "and b_cf." + ] + }, + { + "cell_type": "code", + "execution_count": 100, + "metadata": {}, + "outputs": [], + "source": [ + "# D'abord, on doit transposer nos donner pour pouvoir les utiliser dans le calcul matriciel\n", + "# Donc on génère une liste de 1 qu'on va nsuit coller à notre liste de x\n", + "\n", + "list_1 = np.ones(len(list_x))\n", + "\n", + "\n", + "# Pis on colle les deux\n", + "\n", + "X = np.stack((list_1, list_x), axis=1)\n", + "\n", + "# Enfin, on fait notre calcul de ouf\n", + "\n", + "def B(y): \n", + " X_T = np.transpose(X)\n", + " prod = np.dot(X_T, X)\n", + " inv = np.linalg.inv(prod)\n", + " return np.dot(inv, np.dot(X_T,y))\n", + "\n", + "Y = np.asarray(list_y)\n", + "beta = B(Y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 8\n", + "Create a function loss(x_i, y_i, w, b) which is computing the regression loss on the full dataset.\n", + "What is the value of loss([1], [3], 1, 2) ?" + ] + }, + { + "cell_type": "code", + "execution_count": 101, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 101, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# On détermine la fonction de coût\n", + "\n", + "def loss(x_i, y_i, w, b):\n", + " return ((w * x_i) + b - y_i)**2\n", + "\n", + "loss(1, 3, 1, 2)" + ] + }, + { + "cell_type": "code", + "execution_count": 102, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "8" + ] + }, + "execution_count": 102, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def dl_dw(xi, yi, w, b):\n", + " return -2 * xi * ( yi - (w * xi + b))\n", + "\n", + "dl_dw(4, -1, 0, 0)" + ] + }, + { + "cell_type": "code", + "execution_count": 103, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 103, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def dl_db(xi, yi, w, b):\n", + " return -2 * ( yi - (w * xi + b))\n", + "\n", + "dl_db(4, -1, 0, 0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 11\n", + "Implement a function update_w_and_b(x_i, y_i, w, b, lr) which is updating w and b according to the gradient compute on the full data points. What is the output of the following command of update_w_and_b([0], [3], 5, 3, 0.1) and why?" + ] + }, + { + "cell_type": "code", + "execution_count": 109, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(5.0, 3.0)" + ] + }, + "execution_count": 109, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def update_w_and_b(x, y, w, b, lr):\n", + " grad_w = 0\n", + " grad_b = 0\n", + " \n", + " N = len(y)\n", + " \n", + " for t in range(N):\n", + " grad_w += dl_dw(x[t], y[t], w, b)\n", + " grad_b += dl_db(x[t], y[t], w, b)\n", + " \n", + " # update\n", + " w -= (1 / float(N)) * grad_w * lr\n", + " b -= (1 / float(N)) * grad_b * lr\n", + " \n", + " return w, b\n", + " \n", + "update_w_and_b([0], [3], 5, 3, 0.1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 12 \n", + "Estimate parameters of the simple linear regression by gradient descent from a random initialization of\n", + "w and b and with a leanring rate equals to 0.001. Note: store these values into w_gd and b_gd." + ] + }, + { + "cell_type": "code", + "execution_count": 106, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[array([ 8.46732643, -634.60911809]), array([ 8.46732643, -20.74489117])]" + ] + }, + "execution_count": 106, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Maintenant on voit comment régler ça en 3 lignes\n", + "\n", + "from sklearn.linear_model import LinearRegression\n", + "...\n" + ] + } + ], + "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": "3.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Archis Applications/Exercices/Exercice 1.ipynb b/Archis Applications/Exercices/Exercice 1.ipynb new file mode 100644 index 0000000..9d0a4e4 --- /dev/null +++ b/Archis Applications/Exercices/Exercice 1.ipynb @@ -0,0 +1,366 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 114, + "metadata": {}, + "outputs": [], + "source": [ + "# On importe les librairies\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 1 - Créer une fonction J(theta qui implémente J(theta). Quelle est la valeur de de J(4) ?" + ] + }, + { + "cell_type": "code", + "execution_count": 115, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "25" + ] + }, + "execution_count": 115, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def J(theta_i):\n", + " return (theta_i + 1)**2\n", + "\n", + "J(4)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 2" + ] + }, + { + "cell_type": "code", + "execution_count": 116, + "metadata": {}, + "outputs": [], + "source": [ + "# On génère les points dans theta\n", + "theta = np.arange(-10.00, 10.00, 0.01)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 3\n", + "Plot theta vs J(theta). When do we reach the minimum?" + ] + }, + { + "cell_type": "code", + "execution_count": 120, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 120, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# plot de J(theta)\n", + "plt.plot(theta,J(theta))" + ] + }, + { + "cell_type": "code", + "execution_count": 121, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-1.0" + ] + }, + "execution_count": 121, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "J_theta = [J(x) for x in theta]\n", + "# Valeur de theta à laquelle on obtien le plus petit J(theta)\n", + "round(theta[J_theta.index(min(J_theta))], 3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 4\n", + "Create a function dJ_dtheta(x_i) which is computing the gradient\n", + "?" + ] + }, + { + "cell_type": "code", + "execution_count": 122, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 122, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# On calcule la fonction dérivée. D'après J(theta). Merci dcode.fr\n", + "def dJ_theta(x_i):\n", + " return 2 * (x_i + 1)\n", + "# Affichage de la valeur à 0\n", + "dJ_theta(-1)" + ] + }, + { + "cell_type": "code", + "execution_count": 123, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 123, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.plot(theta, dJ_theta(theta))\n", + "# dJ_t = [dJ_theta(x) for x in theta]\n", + "# plt.plot(theta, dJ_t)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 5\n", + "Create a function gradient_descent(theta_0, lr, nb_iters, df_dx, f) which returns the solution\n", + "of argmin f (θ). \n", + "x_0, lr, nb_iters, df_dx, f correspond respectively to the initial value of θ, the\n", + "θ\n", + "learning rate, the number of iterations allowed for solving the problem, the gradient of θ and the\n", + "function J(.).\n", + "\n", + "What is the solution θ̂ found by gradient descent to our problem? Note: assume that\n", + "x_0, lr, nb_iters = -7, 0.1, 100 while debugging" + ] + }, + { + "cell_type": "code", + "execution_count": 147, + "metadata": {}, + "outputs": [], + "source": [ + "def gradient_descent(theta_0, lr, nb_iters, dJ_t, J_theta):\n", + " # puis on reprend les étapes vues en cours\n", + " theta = theta_0\n", + " \n", + " for t in range(nb_iters): \n", + " # calcul de theta\n", + " theta = theta - lr * dJ_t(theta)\n", + "\n", + " return theta\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 148, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-1.0000000100978041" + ] + }, + "execution_count": 148, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# On appelle la fonction \n", + "gradient_descent(-7, 0.01, 1000, dJ_theta, J)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 6\n", + "Update your function gradient_descent for printing the optimization path (i.e. print the line between\n", + "θ t and θ t+1 and saving the figure at the end of the optimization process)." + ] + }, + { + "cell_type": "code", + "execution_count": 192, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "-1.0000000100978041" + ] + }, + "execution_count": 192, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "def gradient_descent(theta_0, lr, nb_iters, dJ_t, J_theta):\n", + " # puis on reprend les étapes vues en cours\n", + " theta = theta_0\n", + " \n", + " # on initialise les séries pour le plot\n", + " x = [theta]\n", + " y = [J_theta(theta)]\n", + " \n", + " for t in range(nb_iters): \n", + " # calcul de theta\n", + " theta = theta - lr * dJ_t(theta)\n", + " # ajout de la valeur au plot\n", + " x.append(theta)\n", + " y.append(J_theta(theta))\n", + " \n", + " # affichage du plot\n", + " plt.plot(x,y)\n", + " \n", + " return theta\n", + "\n", + "# On appelle la fonction \n", + "gradient_descent(-7, 0.01, 1000, dJ_theta, J)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 7\n", + "Assuming that you are setting nb_iters equals to 100 what is the solution θ̂ when varying nb_iters in\n", + "[-8, -1, 7] and lr in [0.1, 0.01, 0.001, -0.01, 0.8, 1.01]. Does the estimated solution always\n", + "the same? What are pros and cons about these hyperparameters and GD in general?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 8\n", + "Now assume the function J(θ) = sin(2θ + 1), what is the solution θ̂ given by GD?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "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": "3.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Archis Applications/Exercices/Exercice 2.ipynb b/Archis Applications/Exercices/Exercice 2.ipynb new file mode 100644 index 0000000..86b82e3 --- /dev/null +++ b/Archis Applications/Exercices/Exercice 2.ipynb @@ -0,0 +1,319 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 56, + "metadata": {}, + "outputs": [], + "source": [ + "# On importe les librairies\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We want to estimate parameters of a simple linear regression (y i = wx i + b) by closed-form and GD. \n", + "\n", + "The loss function is defined on a data point by l(ŷ i , y i ) = (ŷ i − y i )² where the prediction is given by ŷ i = ŵx + b̂. The\n", + "P N\n", + "optimization problem we want to solve is min i=1 l(ŷ i , y i ). \n", + "\n", + "We assume the following data generation process:\n", + "w,b\n", + "Y ∼ wX + b + \\eta where X ∼ U[20; 40] and \\eta ∼ N (0, 1).\n", + "\n", + "# 1\n", + "\n", + "Generate N data points (x i , y i ) using the data generation process given above and store them into x vs\n", + "N\n", + "y. Note: N = 100 and start by first generating {x i } N\n", + "1 and then {y i } 1 . Do not forget to add noise!\n", + "\n", + "D'abord on va générer les x" + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "metadata": {}, + "outputs": [], + "source": [ + "# On génère les x\n", + "list_x = np.random.uniform(20, 40, 100)\n", + "\n", + "# Pis les y\n", + "\n", + "def y(x):\n", + " w = 1.5\n", + " b = 5\n", + " return w * x + b + np.random.normal(0,1)\n", + "\n", + "list_y = [y(x) for x in list_x ]\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 2\n", + "Plot the data points x vs y." + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 58, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.scatter(list_x, list_y, s=6)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 3\n", + "Estimate parameters of the simple linear regression by closed-form. Note: store these values into w_cf\n", + "and b_cf." + ] + }, + { + "cell_type": "code", + "execution_count": 100, + "metadata": {}, + "outputs": [], + "source": [ + "# D'abord, on doit transposer nos donner pour pouvoir les utiliser dans le calcul matriciel\n", + "# Donc on génère une liste de 1 qu'on va nsuit coller à notre liste de x\n", + "\n", + "list_1 = np.ones(len(list_x))\n", + "\n", + "\n", + "# Pis on colle les deux\n", + "\n", + "X = np.stack((list_1, list_x), axis=1)\n", + "\n", + "# Enfin, on fait notre calcul de ouf\n", + "\n", + "def B(y): \n", + " X_T = np.transpose(X)\n", + " prod = np.dot(X_T, X)\n", + " inv = np.linalg.inv(prod)\n", + " return np.dot(inv, np.dot(X_T,y))\n", + "\n", + "Y = np.asarray(list_y)\n", + "beta = B(Y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 8\n", + "Create a function loss(x_i, y_i, w, b) which is computing the regression loss on the full dataset.\n", + "What is the value of loss([1], [3], 1, 2) ?" + ] + }, + { + "cell_type": "code", + "execution_count": 101, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 101, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# On détermine la fonction de coût\n", + "\n", + "def loss(x_i, y_i, w, b):\n", + " return ((w * x_i) + b - y_i)**2\n", + "\n", + "loss(1, 3, 1, 2)" + ] + }, + { + "cell_type": "code", + "execution_count": 102, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "8" + ] + }, + "execution_count": 102, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def dl_dw(xi, yi, w, b):\n", + " return -2 * xi * ( yi - (w * xi + b))\n", + "\n", + "dl_dw(4, -1, 0, 0)" + ] + }, + { + "cell_type": "code", + "execution_count": 103, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 103, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def dl_db(xi, yi, w, b):\n", + " return -2 * ( yi - (w * xi + b))\n", + "\n", + "dl_db(4, -1, 0, 0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 11\n", + "Implement a function update_w_and_b(x_i, y_i, w, b, lr) which is updating w and b according to the gradient compute on the full data points. What is the output of the following command of update_w_and_b([0], [3], 5, 3, 0.1) and why?" + ] + }, + { + "cell_type": "code", + "execution_count": 109, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(5.0, 3.0)" + ] + }, + "execution_count": 109, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def update_w_and_b(x, y, w, b, lr):\n", + " grad_w = 0\n", + " grad_b = 0\n", + " \n", + " N = len(y)\n", + " \n", + " for t in range(N):\n", + " grad_w += dl_dw(x[t], y[t], w, b)\n", + " grad_b += dl_db(x[t], y[t], w, b)\n", + " \n", + " # update\n", + " w -= (1 / float(N)) * grad_w * lr\n", + " b -= (1 / float(N)) * grad_b * lr\n", + " \n", + " return w, b\n", + " \n", + "update_w_and_b([0], [3], 5, 3, 0.1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# 12 \n", + "Estimate parameters of the simple linear regression by gradient descent from a random initialization of\n", + "w and b and with a leanring rate equals to 0.001. Note: store these values into w_gd and b_gd." + ] + }, + { + "cell_type": "code", + "execution_count": 106, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[array([ 8.46732643, -634.60911809]), array([ 8.46732643, -20.74489117])]" + ] + }, + "execution_count": 106, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# Maintenant on voit comment régler ça en 3 lignes\n", + "\n", + "from sklearn.linear_model import LinearRegression\n", + "...\n" + ] + } + ], + "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": "3.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Archis Applications/Introduction au Machine Learning.md b/Archis Applications/Introduction au Machine Learning.md new file mode 100644 index 0000000..3e529ca --- /dev/null +++ b/Archis Applications/Introduction au Machine Learning.md @@ -0,0 +1,148 @@ +# Intro + +## Définition du machine learning + +Principalement à titre prédictif aujourd'hui. On utilise des données existantes pour prédire quelles seront les prochaines. + +Touche à + +- Statistiques +- Maths +- Computer science +- Big data +- Intelligence artificielle + +Aujourd'hui, le ML c'est de l'informatique avec des outils venus des maths. + +## Historique + +### Machine learning + +Alan Turing : début de l'informatique, enigma et surtout test de Turing (qui vise à évaluer une intelligence artificielle en déterminant si l'interlocuteur est une machine ou non). + +Dans les années 50, plusieurs découvertes plutôt cool t basiqeu : Percetron. + +Ensuite, le machine learning a changé : concept de rule-based approach : au leu de faire apprendre à une machine, on fait une succession de "if then". + +Années 2000 on voit l'arrivée du deep learning avec les résaux de neurones. + +Réseau de neurones : fonctions très complexes qui s'appuient sur grosses bdd et qui vont donner des résultats pus précis que les humains + +### Intelligence articifielle + +D'abord approche connexionniste puis symbolique, et là on revient sur la première. + +## Catégories + +3 parties : + +- Apprentissage supervisé (SL) : + On a des images, on connaît leur label (genre ce qu'il y a dans img). On fait une fonction qui détermine ce qui est dans l'image et ressort un résultat. C'est assez coûteux et chronophage, genre il faut mettre un label sur les millions d'images. + Pour des labels d'images ça va mais si on doit estimer la posture d'un humain, ça devient encore plus galère à déterminer sur des millions d'images. +- Apprentissage non supervisé (UL) : + Apprendre à générer des images, juste en regardant des images. On ne dit pas "c'est un chien", il se débrouille à comprendre seul ds concepts en analysant des images. + La prédiction du futur est égalelement bien à la mode. Savoir à partir d'une vidéo quelles seront les 5 secondes suivantes implique une réflexion et connexion entre objets et environnement +- Apprentissage par renforcement (RL) : + on a systématiquement un agent, qui doit comprendre seul les concepts du jeu. au début il ne sait rien à part les actions qu'il peut faire et quand il a perdu. Exemple d'alphago : un jeu de go qui joue contre lui-même pour devenir de plus en plus fort. + +## Lui + +Il travaille sur compréhension des vidéos, et interactions entre les objets dans cette vidéo. Notamment ce qu'ils se passe entre deux images (exemple des bébés, on se fabrique une histoire où les bébés changent d'attitude en fonction de qui a la télécommande) + +## Strong AI + +L'IA d'aujourd'hui est très spécifique : elle peut pas passer du Go au Ping-Pong. Un humain saura s'adapter à un nouveau jeu. La strong IA vise un champ plus large justement + +## Applications dans le monde réel + +On a par exemple les Tesla : marche très bien, notamment sur autoroute. Y a toujours des accidents, mais moins. + +Google translate : y a 20 an des linguistes codaient les enchaînements de termes, ce qui correspondait. Puis Google a appris à des machines à traduire des chaînes en utilisant les données qu'il possédait. + +Système de recommandations de Netflix + +Criteo : un peu bâtard, récupère les cookies et fait un profil utilisateur à filer au site qu'on visite pour savoir quoi nous vendre. + +### Dans l'industrie + +Chez GAFA & co, beaucoup d'investissements et de centres de recherche qui s'ouvrent. + +## Le cours + +On va utiliser Anaconda et notamment Miniconda car plus léger. l + +Librairies à la mode en data Science + +- SciPy +- NumPy +- IPython +- pandas +- matplotlib +- scikit learn + +On aura pas de gros GPU, c'est un peu le nerf de la guerre : elle peuvent faire des calculs très rapides comparé aux CPU. + +# Apprentissage supervisé + +## Régression linéaire + +### Régression linéaire simple + +À partir de données fournies (x1, y1), on va les voir comme des caractéristiques t des targets : on veut prédire y en fonction de x. Exemple de la slide : on veut prédire une droite. + +Là c'est un simple produit scalaire. + +À patir de cet exemple simple qui vient des maths, on part ne informatique. En IT, l'objectif d'optimisation sera tjrs présent. On a un b et un w détemriné à partir de la droite. + +$Droite = wx +b$ + +On détermine une fonction de coût : écart de la valeur prédite par rapport à la vraie valeur. L but st de matcher le plus possible les valeurs réelles. +Coût pour un point : $l(\hat y_1, y_1)$ + +Coût total : $L(_{w, b, x_i, y_i}) = (w * x_i+b) - y_i)$ + +En gros on attribue des valeurs w et b complètement aléatoires et on voit la valeur du coût. Ensuit la machine tâche de minimiser au max ce coût avec des valeurs différentes de w et b. + +#### Solution closed-form + +On fait des fucking matrices pour chaque point de 1 à n + +Avec ça, on peut réécrire notre fonction de coût en quelque chose de matriciel. (calcul dans optimization). Et on trouve les meilleurs cas grâce à ce calcul. Mais ça ne marche que dans ce cas précis, car le calcul de la matrice inverse est parfois trop chiadé pour être réalisable. + +Des fois du coup on cherche des solutions qui vont être "à peu près similaire" à ce résultat pour quand on a des matrice énorme. + +#### Régression linéaire avec descente de gradient + +La descente de gradient donne une courbe montrant la valeur du coût en fonction de la valeur du $\theta$ . La solution optimale est au creux de la courbe, quand on a une valeur minimale et qu'on peut pas aller plus bas. + +On va voir ça aujourd'hui et prendre l’algorithme exact. + +Voici les étapes : + +1. on sait pas où est le minimum. on choisit un $\theta_0$ aléatoire +2. On détermine un "learning rate", on reviendra dessus +3. On choisit ensuite le nombre d'itérations (genre 1000), et pour chaque on met à jour les paramètres en cherchant à aller dans le sens de la pente. + +C'est là qu'intervient la dérivée, qui est le gradient, qui indique l'inverse de la pente. On va donc prendre le sens contraire : on soustraire la dérivée à la valeur actuelle de $\theta$ + +Il y a la problématique de l'initialisation, si on tombe direct sur le le mini ou bien sur les courbe problématiques qui ont des parties horizontales. Une solution est de prendre plusieurs points d'initialisation. + +Comment trouver w et b à partir de cet algo d'optimisation ? Et bien on dit que $J(\theta) = J(w,b)$. On initialise donc w et b et on calcule les gradients pour w et pour b. + +en gros, c'est le calcul de la dérivée qui permet de savoir le sens de la pente. REVOIR LES DÉRIVÉES. La dérivée est une limite de tangente quand un point de la fonction tend vers 0. + + + +Pour reprendre exemple classe, on se pose la question : quand température est x, combien je vais vendre de glacE. On veut rsourdre le problèm en paratn fu principe qu'on va vendre + +wx + b glace où x est la température. Si x = 0 on va vendre b glace. b est appelé l'intercepte. +$$ +\hat o = argmin\ J(\theta) +$$ +Se comprend comme "le parmaètre $\theta$ qui minimise $J(\theta)$" + +le chapeau indique la prédiction. Pour trouver le coût, donc l'écart entre prédiction et réalité, moyen (de tous les points), on va chercher à déterminer $\hat w$ et $\hat b$ + +On mate un graph avec trois méthodes : closed-form, gradient descend et stochastic gradient descent. 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