main.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Compte Rendu du TP2 de Système de Recommandations\n",
    "\n",
    "### Auteurs : Boubacar TRAORE & Zakarya JARRAYA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 278,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The autoreload extension is already loaded. To reload it, use:\n",
      "  %reload_ext autoreload\n"
     ]
    }
   ],
   "source": [
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 273,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "    <style>\n",
       "    \n",
       "    .rendered_html {\n",
       "         font-size: 22px; \n",
       "         font-family: Garamond;\n",
       "         line-height: 140%;\n",
       "         text-align: justify;\n",
       "         text-justify: inter-word;\n",
       "    }\n",
       "\n",
       "    div.text_cell_render h1 { /* Main titles bigger, centered */\n",
       "        text-align:center;\n",
       "    # }\n",
       "    \n",
       "    </style>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Changement de la police utilisée et de sa taille\n",
    "from IPython.core.display import HTML, display\n",
    "from tp_tools import change_font\n",
    "display(HTML(change_font()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 274,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div id=\"my_id_menu_nb\">run previous cell, wait for 2 seconds</div>\n",
       "<script>\n",
       "function repeat_indent_string(n){\n",
       "    var a = \"\" ;\n",
       "    for ( ; n > 0 ; --n)\n",
       "        a += \"    \";\n",
       "    return a;\n",
       "}\n",
       "// look up into all sections and builds an automated menu //\n",
       "var update_menu_string = function(begin, lfirst, llast, sformat, send, keep_item, begin_format, end_format) {\n",
       "    var anchors = document.getElementsByClassName(\"section\");\n",
       "    if (anchors.length == 0) {\n",
       "        anchors = document.getElementsByClassName(\"text_cell_render rendered_html\");\n",
       "    }\n",
       "    var i,t;\n",
       "    var text_menu = begin;\n",
       "    var text_memo = \"<pre>\\nlength:\" + anchors.length + \"\\n\";\n",
       "    var ind = \"\";\n",
       "    var memo_level = 1;\n",
       "    var href;\n",
       "    var tags = [];\n",
       "    var main_item = 0;\n",
       "    var format_open = 0;\n",
       "    for (i = 0; i <= llast; i++)\n",
       "        tags.push(\"h\" + i);\n",
       "\n",
       "    for (i = 0; i < anchors.length; i++) {\n",
       "        text_memo += \"**\" + anchors[i].id + \"--\\n\";\n",
       "\n",
       "        var child = null;\n",
       "        for(t = 0; t < tags.length; t++) {\n",
       "            var r = anchors[i].getElementsByTagName(tags[t]);\n",
       "            if (r.length > 0) {\n",
       "child = r[0];\n",
       "break;\n",
       "            }\n",
       "        }\n",
       "        if (child == null) {\n",
       "            text_memo += \"null\\n\";\n",
       "            continue;\n",
       "        }\n",
       "        if (anchors[i].hasAttribute(\"id\")) {\n",
       "            // when converted in RST\n",
       "            href = anchors[i].id;\n",
       "            text_memo += \"#1-\" + href;\n",
       "            // passer à child suivant (le chercher)\n",
       "        }\n",
       "        else if (child.hasAttribute(\"id\")) {\n",
       "            // in a notebook\n",
       "            href = child.id;\n",
       "            text_memo += \"#2-\" + href;\n",
       "        }\n",
       "        else {\n",
       "            text_memo += \"#3-\" + \"*\" + \"\\n\";\n",
       "            continue;\n",
       "        }\n",
       "        var title = child.textContent;\n",
       "        var level = parseInt(child.tagName.substring(1,2));\n",
       "\n",
       "        text_memo += \"--\" + level + \"?\" + lfirst + \"--\" + title + \"\\n\";\n",
       "\n",
       "        if ((level < lfirst) || (level > llast)) {\n",
       "            continue ;\n",
       "        }\n",
       "        if (title.endsWith('¶')) {\n",
       "            title = title.substring(0,title.length-1).replace(\"<\", \"&lt;\")\n",
       "         .replace(\">\", \"&gt;\").replace(\"&\", \"&amp;\");\n",
       "        }\n",
       "        if (title.length == 0) {\n",
       "            continue;\n",
       "        }\n",
       "\n",
       "        while (level < memo_level) {\n",
       "            text_menu += end_format + \"</ul>\\n\";\n",
       "            format_open -= 1;\n",
       "            memo_level -= 1;\n",
       "        }\n",
       "        if (level == lfirst) {\n",
       "            main_item += 1;\n",
       "        }\n",
       "        if (keep_item != -1 && main_item != keep_item + 1) {\n",
       "            // alert(main_item + \" - \" + level + \" - \" + keep_item);\n",
       "            continue;\n",
       "        }\n",
       "        while (level > memo_level) {\n",
       "            text_menu += \"<ul>\\n\";\n",
       "            memo_level += 1;\n",
       "        }\n",
       "        text_menu += repeat_indent_string(level-2);\n",
       "        text_menu += begin_format + sformat.replace(\"__HREF__\", href).replace(\"__TITLE__\", title);\n",
       "        format_open += 1;\n",
       "    }\n",
       "    while (1 < memo_level) {\n",
       "        text_menu += end_format + \"</ul>\\n\";\n",
       "        memo_level -= 1;\n",
       "        format_open -= 1;\n",
       "    }\n",
       "    text_menu += send;\n",
       "    //text_menu += \"\\n\" + text_memo;\n",
       "\n",
       "    while (format_open > 0) {\n",
       "        text_menu += end_format;\n",
       "        format_open -= 1;\n",
       "    }\n",
       "    return text_menu;\n",
       "};\n",
       "var update_menu = function() {\n",
       "    var sbegin = \"\";\n",
       "    var sformat = '<a href=\"#__HREF__\">__TITLE__</a>';\n",
       "    var send = \"\";\n",
       "    var begin_format = '<li>';\n",
       "    var end_format = '</li>';\n",
       "    var keep_item = -1;\n",
       "    var text_menu = update_menu_string(sbegin, 2, 4, sformat, send, keep_item,\n",
       "       begin_format, end_format);\n",
       "    var menu = document.getElementById(\"my_id_menu_nb\");\n",
       "    menu.innerHTML=text_menu;\n",
       "};\n",
       "window.setTimeout(update_menu,2000);\n",
       "            </script>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 274,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from jyquickhelper import add_notebook_menu\n",
    "add_notebook_menu()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Comparaison des 3 algorithmes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 291,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
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       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Packages\n",
    "from exp import games, cumulative_regret, cumulative_reward, logarithmic_indices\n",
    "from random import shuffle\n",
    "from player import Oracle, EpsilonNGreedy, ThompsonSamplingBernoulli, UCB1, ExploreThenCommit\n",
    "from arm import MyBeta, Bernoulli\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "# Librairies de visualisation des résultats\n",
    "from plotly.offline import download_plotlyjs, init_notebook_mode, plot, iplot\n",
    "import plotly.graph_objs as go\n",
    "from plotly import tools\n",
    "init_notebook_mode(connected=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.1. Création de l'environnement"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous récupérons les informations des 10 bras sur Moodle. Nous fixons le nombre d'essais total de chaque jeu à $T = 10K$, un nombre assez important pour constater l'évolution du regret cumulé lorsque T devient très important (**n_iter dans le code ci-dessous**). Pour l'instant on garde le nombre de jeu à effectuer par chaque algorithme à **n_games = 10** : c'est la courbe moyenne sur l'ensemble des 10 jeux qui sera représenté pour chaque algorithme."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**NB** : Les bras sont tous simulés selon une loi **beta** ($\\beta$) dans cette cette section."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "environment = [MyBeta(mean=0.44), MyBeta(mean=0.52), MyBeta(mean=0.87), MyBeta(mean=0.69), MyBeta(mean=0.32), \n",
    "               MyBeta(mean=0.01), MyBeta(mean=0.24), MyBeta(mean=0.65), MyBeta(mean=0.49), MyBeta(mean=0.68)]\n",
    "\n",
    "# Odonner aléatoirement les bras\n",
    "shuffle(environment)\n",
    "\n",
    "#Paramètres\n",
    "n_iter = 1000\n",
    "n_games = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.2. Implémentation et représentation des algorithmes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Oracle\n",
    "log_oracle = games(Oracle(np.argmax([a.mean() for a in environment])), environment, n_iter, n_games)\n",
    "\n",
    "\n",
    "# Epsilon Greedy\n",
    "c_EG = 1\n",
    "log_EG1    = games(EpsilonNGreedy(nb_arms=len(environment), c=c_EG), environment, n_iter, n_games)\n",
    "\n",
    "## Thompson Sampling\n",
    "log_TS = games(ThompsonSamplingBernoulli(nb_arms=len(environment), prior_s=0.25, prior_f=0.75), environment, n_iter, n_games)\n",
    "\n",
    "#UCB 1\n",
    "c_ucb = 2\n",
    "log_UCB1 = games(UCB1(nb_arms=len(environment), c=c_ucb), environment, n_iter, n_games)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous définissons une fonction globale qui permet de représenter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_cumulative_regret(logs, labels, n_iter=1000):\n",
    "    \"\"\"Représentation du regret cumulé des algorithmes données en input.\n",
    "\n",
    "    Arguments:\n",
    "        logs {list} -- Liste des logs des algorithmes de bandits manchots.\n",
    "        labels {list} -- Liste des labels pour la représentation graphique.\n",
    "\n",
    "    Keyword Arguments:\n",
    "        n_iter {int} -- Nombre d'essais par jeu (default: {1000})\n",
    "        sub_plots {tuple}  -- Nombre de sous-graphes à faire. Nombre de fois à répéter tous les logs (default: {(1,1)})\n",
    "\n",
    "    Returns:\n",
    "        plot -- plotly object for graph\n",
    "    \"\"\"\n",
    "    \n",
    "    inds = logarithmic_indices(n_iter, 100) #Afficher seulement 100 points au toytal\n",
    "\n",
    "    fig = go.Figure() #Figure plotly\n",
    "    for i in range(len(logs)):\n",
    "        plot = go.Scatter(x=inds + 1, y=cumulative_regret(logs[i])[inds], mode = 'lines', name = labels[i])\n",
    "        fig.add_trace(plot)\n",
    "    #Mise en forme    \n",
    "    fig['layout'] = go.Layout(title=\"Evolution du Regret cumulé selon le nombre d'itérations\",\n",
    "                              xaxis=go.layout.XAxis(title=\"Nombre d'essais\"),\n",
    "                              yaxis=go.layout.YAxis(title='Regret Cumulé'))\n",
    "\n",
    "    return iplot(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exécutons la fonction avec les informations requises."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "logs   = [log_oracle, log_EG1, log_UCB1, log_TS]\n",
    "labels = [\"Oracle\", \"EG, c=1\", \"UCB1, c=2\", \"TS\"]\n",
    "plot_cumulative_regret(logs=logs, labels=labels)"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A première vue, l'algorithme de Thompson Sampling semble nettement meilleur que ceux de l'UCB1 (avec c=2) et d'Epsilon-Greedy (avec c=1)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3. Choix de l'hyper-paramètre \"c\" dans Epsilon Greedy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'algorithme choisit de faire de l'exploration souvent selon une probabilité faible fixé généralement à $\\epsilon = 0.05$. Dans notre cas spécifique, nous écrivons $\\epsilon$ comme une fonction décroissante du nombre d'essais afin d'exploiter au mieux le bras ayant la meilleure espérance de gain lorsque le jeu avance : $\\epsilon \\approx \\frac{c}{t}$ où $t \\in [|1,nb\\_trials|]$. Trouver une bonne valeur de $c$ est donc important pour la performance de l'algorithme. C'est un hyper-paramètre dont la valeur optimale dépend de l'environnement dans lequel nous jouns (l'ensemble des bras utilisés). Ici, nous proposons une méthode de sélection de cette valeur."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dans un premier temps, nous n'allons pas représenter l'Oracle et les autres algorithmes (TS et UCB1), nous nous focalisons juste sur l'epsilon-greedy en faisant varier le paramètre $c$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 1.3.1. Choix aléatoire selon une valeur maximale de \"c=50\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\epsilon \\approx \\frac{c}{t}$ signifie que l'algorithme ferra toujours de l'exploration avant que le nombre d'essais $t$ n'atteigne la valeur de $c$ (on suppose, au moment de l'implémentation de l'algorithme, que si $\\epsilon >1$ l'algorithme ne fait qu'explorer, et donc choisit aléatoirement parmi les bras). Etant donné que nous qu'un jeu se fait jusqu'à 1000 essais, une valeur de $c=50$ semble tout fait grand pour explorer. L'algorithme se décrit donc comme suit :\n",
    "* Si $t\\leq50$, on fait de l'exploration (choix aléatoire entre les 10 bras disponibles)\n",
    "* Si $t>50$, on fait de l'exploitation avec une probilité de $p = 1 - \\frac{50}{t}$."
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   "source": [
    "Nous allons choisir aléatoirement 10 valeurs de $c$ selon une loi uniforme [0,50] et représenter l'évolution du regret cumulé pour chacun d'entre eux."
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    "**Remarque**: Nous allons beaucoup augmenter le nombre de jeu pour chaque valeur de $c$ afin que les courbes soient les plus stables que possible (loi des grands nombres). On prendra **nb_games = 100**. Les courbes tracées pour chaque valeur de $c$ ne seront autre que les moyennes empiriques des 100 courbes de jeu pour chacun d'entre eux. Augmenter le nombre de jeux assure une certaine fiabilité des résultats mais demande aussi un temps de calcul non négligeable (nous avons voulu en prendre plus mais ça prenait beaucoup de temps, vaudrait donc mieux se limiter à 100)."
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "n_games = 100 #Augmentation du nombre de jeux\n",
    "np.random.seed(26) #Fixer la graine générateur pour avoir un résultat réplicable\n",
    "c_range = sorted(np.random.randint(low=1, high=50, size=10))\n",
    "logs_EG = []\n",
    "labels  = []\n",
    "for c in c_range:\n",
    "    logs_EG.append(games(EpsilonNGreedy(nb_arms=len(environment), c=c), environment, n_iter, n_games))\n",
    "    labels.append('EG, c = %d'%c)\n",
    "plot_cumulative_regret(logs=logs_EG, labels=labels)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous remarquons que pour $c=2$, il n'y a vraiment pas assez d'exploration, ceci est interprété par une courbe presque linéaire. Cette valeur de $c$ est donc très faible. D'un autre côté, pour les valeurs de $c\\ge27$, il y a trop d'exploration et les courbes de regret cumulées décollent très vite et on du mal à se stabiliser lorsque le nombre d'essais grandit. Avec un nombre d'essais $T = 1000$, ce sont les valeurs $c=5$ et $c=7$ qui enregistrent les meilleures performances pour cet algorithme (même si $c=5$ gagne légèrement)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous allons maintenant appliquer deux grandes opérations :\n",
    "* Zoomer dans la zone des entiers compris entre 3 et 25 (valeurs assez raisonables d'après le graphique précédent) comme valeurs possibles de $c$\n",
    "* Doubler le nombre d'essais à $T=2000$ pour mieux distinguer la performance à long terme de l'algorirhme pour ces différentes valeurs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 1.3.2. Choix avec une recherche quasi-exhaustive des valeurs entières entre [3,25]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dans ce problème, nous nous limitons aux nombres entiers comme valeurs possibles de $c$ afin de ne pas complexifier le problème, sinon le raisonnement peut bien s'étendre aux nombres réels. Nous allons donc parcourir toutes les valeurs de $c$ comprises entre 3 et 25 avec un pas de 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "n_games = 100 #Augmentation du nombre de jeux\n",
    "n_iter  = 2000 # Doubler le nombre d'itérations\n",
    "c_range = range(3,26,2)\n",
    "logs_EG = []\n",
    "labels  = []\n",
    "for c in c_range:\n",
    "    logs_EG.append(games(EpsilonNGreedy(nb_arms=len(environment), c=c), environment, n_iter, n_games))\n",
    "    labels.append('EG, c = %d'%c)\n",
    "plot_cumulative_regret(logs=logs_EG, labels=labels, n_iter=n_iter)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Visiblement, les valeurs de $c\\leq5$ ne sont pas bonnes (vu leurs pentes). $c=11$ semble se distinguer des autres et possède la meilleure performance (nous augmeterons le nombre de jeu. Les autres valeurs ne semblent pas mal non plus. Pour finaliser cette recherche, nous allons encore raffiner l'intervalle en nous foclisant uniquement sur les valeurs de $c$ qui ont cumulé moins de regret et dont les pentes sont quasi-nulles au bout de 2000 essais."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 1.3.3. Dernier raffinage sur [10,20]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Une fois de plus nous allons zommer sur [10,20] et augmenter le nombre d'itérations. Pour encore être plus précis, nous fixons le nombre de jeux 200 (2 fois plus que la dernière fois)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
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        "title": "Evolution du Regret cumulé selon le nombre d'itérations",
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        "yaxis": {
         "title": "Regret Cumulé"
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   "source": [
    "n_games = 200 #Doubler du nombre de jeux par rapport à la dernière fois\n",
    "n_iter  = 3000 # Augmenter le nombre d'itérations de 1000 encore.\n",
    "c_range = range(10,21)\n",
    "logs_EG = []\n",
    "labels  = []\n",
    "for c in c_range:\n",
    "    logs_EG.append(games(EpsilonNGreedy(nb_arms=len(environment), c=c), environment, n_iter, n_games))\n",
    "    labels.append('EG, c = %d'%c)\n",
    "plot_cumulative_regret(logs=logs_EG, labels=labels, n_iter=n_iter)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comme le résultat varie souvent lorsque nous réexécutons la même cellule, nous avons stockés les 5 meilleurs valeurs de $c$ pour 5 exécutions différentes. Les valeurs sont recenssés ci-dessous:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<style type=\"text/css\">\n",
    ".tg  {border-collapse:collapse;border-spacing:0;}\n",
    ".tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;}\n",
    ".tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;}\n",
    ".tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top}\n",
    ".tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top}\n",
    ".tg .tg-dvpl{border-color:inherit;text-align:right;vertical-align:top}\n",
    "</style>\n",
    "<table class=\"tg\">\n",
    "  <tr>\n",
    "    <th class=\"tg-0pky\"></th>\n",
    "    <th class=\"tg-c3ow\" colspan=\"5\">Rang</th>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td class=\"tg-dvpl\">i-ème exécution</td>\n",
    "    <td class=\"tg-0pky\">1er</td>\n",
    "    <td class=\"tg-0pky\">2e</td>\n",
    "    <td class=\"tg-0pky\">3e</td>\n",
    "    <td class=\"tg-0pky\">4e</td>\n",
    "    <td class=\"tg-0pky\">5e</td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td class=\"tg-dvpl\">1</td>\n",
    "    <td class=\"tg-0pky\">16</td>\n",
    "    <td class=\"tg-0pky\">14</td>\n",
    "    <td class=\"tg-0pky\">13</td>\n",
    "    <td class=\"tg-0pky\">12</td>\n",
    "    <td class=\"tg-0pky\">17</td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td class=\"tg-dvpl\">2</td>\n",
    "    <td class=\"tg-0pky\">16</td>\n",
    "    <td class=\"tg-0pky\">18</td>\n",
    "    <td class=\"tg-0pky\">17</td>\n",
    "    <td class=\"tg-0pky\">14</td>\n",
    "    <td class=\"tg-0pky\">13</td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td class=\"tg-dvpl\">3</td>\n",
    "    <td class=\"tg-0pky\">13</td>\n",
    "    <td class=\"tg-0pky\">17</td>\n",
    "    <td class=\"tg-0pky\">18</td>\n",
    "    <td class=\"tg-0pky\">14</td>\n",
    "    <td class=\"tg-0pky\">15</td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td class=\"tg-dvpl\">4</td>\n",
    "    <td class=\"tg-0pky\">13</td>\n",
    "    <td class=\"tg-0pky\">14</td>\n",
    "    <td class=\"tg-0pky\">10</td>\n",
    "    <td class=\"tg-0pky\">15</td>\n",
    "    <td class=\"tg-0pky\">17</td>\n",
    "  </tr>\n",
    "  <tr>\n",
    "    <td class=\"tg-dvpl\">5</td>\n",
    "    <td class=\"tg-0pky\">12</td>\n",
    "    <td class=\"tg-0pky\">14</td>\n",
    "    <td class=\"tg-0pky\">15</td>\n",
    "    <td class=\"tg-0pky\">8</td>\n",
    "    <td class=\"tg-0pky\">13</td>\n",
    "  </tr>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous voyons bien que ce sont les chiffres 13 et 14 qui se répètent beaucoup. Nous pouvons choisir 13 comme valeur à retenir. On retient qu'à long terme le 13 est plus performant que le 11 retenu dans la section précédente."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.4. Nouvelle comparaison des algorithmes avec la valeur optimale de c trouvée précédemment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
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    "n_games = 200\n",
    "n_iter  = 3000\n",
    "\n",
    "#Oracle\n",
    "log_oracle = games(Oracle(np.argmax([a.mean() for a in environment])), environment, n_iter, n_games)\n",
    "\n",
    "# Epsilon Greedy\n",
    "c_EG = 13\n",
    "log_EG = games(EpsilonNGreedy(nb_arms=len(environment), c=c_EG), environment, n_iter, n_games)\n",
    "\n",
    "## Thompson Sampling\n",
    "log_TS = games(ThompsonSamplingBernoulli(nb_arms=len(environment), prior_s=0.25, prior_f=0.75), environment, n_iter, n_games)\n",
    "\n",
    "#UCB1\n",
    "c_ucb = 1\n",
    "log_UCB1 = games(UCB1(nb_arms=len(environment), c=c_ucb), environment, n_iter, n_games)\n",
    "\n",
    "logs   = [log_oracle, log_EG, log_UCB1, log_TS]\n",
    "labels = [\"Oracle\", \"EG, c=11\", \"UCB1, c=1\", \"TS\"]\n",
    "plot_cumulative_regret(logs=logs, labels=labels, n_iter=n_iter)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous pouvons remarquer que l'algorithme d'Epsilon Greedy a drastiquement diminué sa différence de regret cumulé face à l'algorithme de Thompson Sampling qui reste toujours meilleur. De la même manière, il possible de bien tuner l'hyper-paramètre $c$ de l'algorithme UCB1 pour atteindre de meilleurs performances."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Comparaisons générales**\n",
    "\n",
    "* Le Thompson Sampling reste globalement le meilleur algorithme. Elle fait assez d'exploration au départ et sa phase d'exploitation est excellente.\n",
    "* L'UCB1 est aussi réputé pour avoir de bons résultats malgré qu'il n'a quasiment aucum paramètre à tuner (on peut généralement fixer la constante d'UCB1 à $c\\_ucb=2$, quoique dans notre cas elle ne donne pas d'assez bons résultats).\n",
    "* Epsilon Greedy, avec de bons paramètres, a quasiment la même performance que Thompson Sampling. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Bonus : Bernouilli arms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'implémentation est effectuée dans la classe correspondante. Nous redéfinissons npotre environnement avec toujours 10 bras, mais cette fois-ci les résultats des bras sont tirés selon une loi bernoulli."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 289,
   "metadata": {},
   "outputs": [],
   "source": [
    "environment = [Bernoulli(p=0.2), Bernoulli(p=0.75), Bernoulli(p=0.9), Bernoulli(p=0.3), Bernoulli(p=0.7), \n",
    "               Bernoulli(p=0.1), Bernoulli(p=0.8), Bernoulli(p=0.4), Bernoulli(p=0.6), Bernoulli(p=0.5)]"
   ]
  },
  {
   "cell_type": "code",
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_games = 50\n",
    "n_iter  = 1000\n",
    "\n",
    "#Oracle\n",
    "log_oracle = games(Oracle(np.argmax([a.mean() for a in environment])), environment, n_iter, n_games)\n",
    "\n",
    "# Epsilon Greedy\n",
    "c_EG = 13\n",
    "log_EG = games(EpsilonNGreedy(nb_arms=len(environment), c=c_EG), environment, n_iter, n_games)\n",
    "\n",
    "## Thompson Sampling\n",
    "log_TS = games(ThompsonSamplingBernoulli(nb_arms=len(environment), prior_s=0.25, prior_f=0.75), environment, n_iter, n_games)\n",
    "\n",
    "##ETC\n",
    "log_ETC = games(ExploreThenCommit(nb_arms=len(environment), n0=300), environment, n_iter, n_games)\n",
    "\n",
    "#UCB1\n",
    "c_ucb = 1\n",
    "log_UCB1 = games(UCB1(nb_arms=len(environment), c=c_ucb), environment, n_iter, n_games)\n",
    "\n",
    "logs   = [log_oracle, log_EG, log_UCB1, log_TS, log_ETC]\n",
    "labels = [\"Oracle\", \"EG, c=11\", \"UCB1, c=1\", \"TS\", \"ETC, n0 = 300\"]\n",
    "plot_cumulative_regret(logs=logs, labels=labels, n_iter=n_iter)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Avec cette configuration, nous remarquons que tous les algorithmes sont plus performants que l'ETC. Ceci est dû au fait qu'on lui donne trop de temps d'exploration, il passe donc $\\frac{9}{10}$ de son temps avant $n_0$ à chosir un mauvais bras."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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