08-MarkovDecisionProcesses.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Markov Decision Processes"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "using Interact\n",
      "include(\"gridworld.jl\")\n",
      "include(\"helpers.jl\")\n",
      "g = GridWorld();"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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        "\tthis.id = id;\n",
        "\tthis.elem = elem;\n",
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        "\n",
        "\tInputWidgets.commInitializer(this); // Initialize communication\n",
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        "    Slider.prototype = Widget;\n",
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        "\n",
        "\tInputWidgets.commInitializer(this);\n",
        "    }\n",
        "    Checkbox.prototype = Widget;\n",
        "\n",
        "    var Button = function (typ, id, init) {\n",
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        "\t\t     value:   init.label },\n",
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        "\tthis.getState = function () {\n",
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        "\t}\n",
        "\n",
        "\tthis.id = id;\n",
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        "\tthis.label = init.label;\n",
        "\n",
        "\tInputWidgets.commInitializer(this);\n",
        "    }\n",
        "    Button.prototype = Widget;\n",
        "\n",
        "    var Text = function (typ, id, init) {\n",
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        "\t\t     placeholder: init.label,\n",
        "\t\t     value: init.value },\n",
        "\t    elem = createElem(\"input\", attr),\n",
        "\t    self = this;\n",
        "\tthis.getState = function () {\n",
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        "\t}\n",
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        "\t}\n",
        "\n",
        "\tthis.id = id;\n",
        "\tthis.elem = elem;\n",
        "\tthis.label = init.label;\n",
        "\n",
        "\tInputWidgets.commInitializer(this);\n",
        "    }\n",
        "    Text.prototype = Widget;\n",
        "\n",
        "    var Textarea = function (typ, id, init) {\n",
        "\tvar attr = { placeholder: init.label },\n",
        "\t    elem = createElem(\"textarea\", attr, init.value),\n",
        "\t    self = this;\n",
        "\tthis.getState = function () {\n",
        "\t    return elem.value;\n",
        "\t}\n",
        "\telem.onchange = function () {\n",
        "\t    self.sendUpdate();\n",
        "\t}\n",
        "\n",
        "\tthis.id = id;\n",
        "\tthis.elem = elem;\n",
        "\tthis.label = init.label;\n",
        "\n",
        "\tInputWidgets.commInitializer(this);\n",
        "    }\n",
        "    Textarea.prototype = Widget;\n",
        "\n",
        "    // RadioButtons\n",
        "    // Dropdown\n",
        "    // HTML\n",
        "    // Latex\n",
        "\n",
        "    var InputWidgets = {\n",
        "\tSlider: Slider,\n",
        "\tCheckbox: Checkbox,\n",
        "\tButton: Button,\n",
        "\tText: Text,\n",
        "\tTextarea: Textarea,\n",
        "\tdebug: false,\n",
        "\tlog: function () {\n",
        "\t    if (InputWidgets.debug) {\n",
        "\t\tconsole.log.apply(console, arguments);\n",
        "\t    }\n",
        "\t},\n",
        "\t// a central way to initalize communication\n",
        "\t// for widgets.\n",
        "\tcommInitializer: function (widget) {\n",
        "\t    widget.sendUpdate = function () {};\n",
        "\t}\n",
        "    };\n",
        "\n",
        "    window.InputWidgets = InputWidgets;\n",
        "\n",
        "})(jQuery, undefined);\n",
        "</script>"
       ],
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        "\tWidgets.debug = false; // log messages etc in console.\n",
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        "\t\tcomm.on_msg(function (msg) {\n",
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        "\t\t\t\tevents: IPython.events,\n",
        "\t\t\t\tkeyboard_manager: IPython.keyboard_manager\n",
        "\t\t\t    })); // Hack to work with IPython 2.1.0\n",
        "\t\t\t    var toinsert = IPython.OutputArea.append_map[type].apply(\n",
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        "\n",
        "\t    // coordingate with Comm and redraw Signals\n",
        "\t    // XXX: Test using Reactive here to improve performance\n",
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        "\t\t\ttoinsert.bind(\"destroyed\", function() {\n",
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        "\t\t    // `this` is a widget here.\n",
        "\t\t    // TODO: I have a feeling there's some\n",
        "\t\t    //       IPython bookkeeping to be done here.\n",
        "\t\t    // Widgets.log(\"State changed\", this, this.getState());\n",
        "\t\t    comm.send({value: this.getState()});\n",
        "\t\t}\n",
        "\t    };\n",
        "\t}\n",
        "\n",
        "\ttry {\n",
        "\t    // try to initialize right away. otherwise, wait on the status_started event.\n",
        "\t    initComm(undefined, IPython.notebook);\n",
        "\t} catch (e) {\n",
        "\t    $([IPython.events]).on('status_started.Kernel', initComm);\n",
        "\t}\n",
        "    });\n",
        "})(IPython, jQuery, _, MathJax, InputWidgets);\n",
        "</script>"
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the grid world problem we have a $10 \\times 10$ grid. Each cell in the grid represents a state in an MDP. The available actions are up, down, left, and right. The effects of these actions are stochastic. We move one step in the specified direction with probability 0.7, and we move one step in one of the three other directions, each with probability 0.1. If we bump against the outer border of the grid, we do not move at all.\n",
      "\n",
      "We receive a cost of 1 for bumping against the outer border of the grid. There are four cells in which we receive rewards upon entering:\n",
      "* $(8,9)$ has a reward of $+10$\n",
      "* $(3,8)$ has a reward of $+3$\n",
      "* $(5,4)$ has a reward of $-5$\n",
      "* $(8,4)$ has a reward of $-10$\n",
      "\n",
      "The coordinates are specified using the matrix convention in which the first coordinate is the row starting from the top and the second coordinate is the column starting from the left. The cells with rewards of $+10$ and $+3$ are absorbing states where no additional reward is ever received from that point onward."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Compute the maximum immediate reward for each state and plot. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@manipulate for a in actions(g)\n",
      "    policy = s -> a\n",
      "    plot(g, s -> reward(g, s, a), policy)\n",
      "end"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Policy Evaluation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$U^\\pi_k(s) = R(s, \\pi(s)) + \\gamma \\sum_{s'} T(s' \\mid s, \\pi(s))U^\\pi_{k+1}(s')$"
     ]
    },
    {
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     "input": [
      "function iterativePolicyEvaluation(mdp::MDP, policy::Function, numIterations::Integer)\n",
      "    (S, A, T, R, discount) = locals(mdp)\n",
      "    U = @array (s0 in S) 0.\n",
      "    for t = 1:numIterations\n",
      "        U = @array (s0 in S) begin\n",
      "            R(s0, policy(s0)) + discount*@sum (s1 in S) (T(s0, policy(s0), s1)*U[s1])\n",
      "            end\n",
      "    end\n",
      "    U\n",
      "end;"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
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    {
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     "collapsed": false,
     "input": [
      "@manipulate for numIterations = 0:10, a = actions(g), discount = 0.1:0.1:0.9\n",
      "    policy = s -> a\n",
      "    g.discount = discount\n",
      "    U = iterativePolicyEvaluation(g, policy, numIterations)\n",
      "    plot(g, U, policy)\n",
      "end"
     ],
     "language": "python",
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      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Value Iteration"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$U^*_k(s) = \\max_a\\left[R(s, a) + \\gamma \\sum_{s'} T(s' \\mid s, a)U^*_{k+1}(s')\\right]$  \n",
      "$\\pi^*(s) = \\arg\\max_a\\left[R(s, a) + \\gamma \\sum_{s'} T(s' \\mid s, a)U^*(s')\\right]$  "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "function valueIteration(mdp::MDP, numIterations::Integer; discount = 0.9)\n",
      "    (S, A, T, R) = locals(mdp)\n",
      "    U = @array (s0 in S) 0.\n",
      "    for t = 1:numIterations\n",
      "        Q(s0, a) = R(s0, a) + discount*@sum (s1 in S) T(s0, a, s1)*U[s1]\n",
      "        U = @array s0 in S begin\n",
      "                @max (a in A) Q(s0,a)\n",
      "            end\n",
      "    end\n",
      "    U\n",
      "end\n",
      "function extractPolicy(mdp::MDP, U::Vector)\n",
      "    (S, A, T, R, discount) = locals(mdp)    \n",
      "    Q(s0, a) = R(s0, a) + discount*@sum (s1 in S) T(s0, a, s1)*U[s1]\n",
      "    [@argmax (a in A) Q(s0,a) for s0 in S]\n",
      "end;"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@manipulate for numIterations = 0:10, discount = 0.1:0.1:0.9\n",
      "    U = valueIteration(g, numIterations, discount = discount)\n",
      "    policy = extractPolicy(g, U)\n",
      "    plot(g, U, policy)\n",
      "end"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [],
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     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Gauss-Seidel Value Iteration"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$U^*(s) \\gets \\max_a\\left[R(s, a) + \\gamma \\sum_{s'} T(s' \\mid s, a)U^*(s')\\right]$  \n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "function valueIterationGaussSeidel(mdp::MDP, numIterations::Integer)\n",
      "    (S, A, T, R, discount) = locals(mdp)\n",
      "    U = @array (s in S) 0.\n",
      "    for t = 1:numIterations\n",
      "        for s0 in S\n",
      "            Q(s0, a) = R(s0, a) + discount*@sum (s1 in S) T(s0, a, s1)*U[s1]\n",
      "            U[s0] = @max (a in A) Q(s0,a)\n",
      "        end\n",
      "    end\n",
      "    U\n",
      "end;"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "@manipulate for numIterations = 0:10, discount = 0.1:0.1:0.9\n",
      "    U = valueIterationGaussSeidel(g, numIterations)\n",
      "    policy = extractPolicy(g, U)\n",
      "    plot(g, U, policy)\n",
      "end"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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{\\$0.47\\$};\\n\\\\fill (8.55, 5.5) -- (8.45, 5.5) -- (8.5, 5.0) -- cycle;\\n\\\\node[above right] at (8.0, 5.0) {\\$0.53\\$};\\n\\\\fill (8.55, 4.5) -- (8.45, 4.5) -- (8.5, 4.0) -- cycle;\\n\\\\node[above right] at (8.0, 4.0) {\\$1.39\\$};\\n\\\\fill (8.55, 3.5) -- (8.45, 3.5) -- (8.5, 3.0) -- cycle;\\n\\\\node[above right] at (8.0, 3.0) {\\$3.72\\$};\\n\\\\fill (8.5, 2.45) -- (8.5, 2.55) -- (8.0, 2.5) -- cycle;\\n\\\\node[above right] at (8.0, 2.0) {\\$10.00\\$};\\n\\\\fill (8.45, 1.5) -- (8.55, 1.5) -- (8.5, 2.0) -- cycle;\\n\\\\node[above right] at (8.0, 1.0) {\\$3.72\\$};\\n\\\\fill (8.45, 0.5) -- (8.55, 0.5) -- (8.5, 1.0) -- cycle;\\n\\\\node[above right] at (8.0, 0.0) {\\$1.32\\$};\\n\\\\fill (9.55, 9.5) -- (9.45, 9.5) -- (9.5, 9.0) -- cycle;\\n\\\\node[above right] at (9.0, 9.0) {\\$-0.19\\$};\\n\\\\fill (9.5, 8.45) -- (9.5, 8.55) -- (9.0, 8.5) -- cycle;\\n\\\\node[above right] at (9.0, 8.0) {\\$0.07\\$};\\n\\\\fill (9.5, 7.45) -- (9.5, 7.55) -- (9.0, 7.5) -- cycle;\\n\\\\node[above right] at (9.0, 7.0) {\\$0.31\\$};\\n\\\\fill (9.5, 6.45) -- (9.5, 6.55) -- (9.0, 6.5) -- cycle;\\n\\\\node[above right] at (9.0, 6.0) {\\$0.09\\$};\\n\\\\fill (9.5, 5.45) -- (9.5, 5.55) -- (9.0, 5.5) -- cycle;\\n\\\\node[above right] at (9.0, 5.0) {\\$0.12\\$};\\n\\\\fill (9.55, 4.5) -- (9.45, 4.5) -- (9.5, 4.0) -- cycle;\\n\\\\node[above right] at (9.0, 4.0) {\\$0.52\\$};\\n\\\\fill (9.55, 3.5) -- (9.45, 3.5) -- (9.5, 3.0) -- cycle;\\n\\\\node[above right] at (9.0, 3.0) {\\$1.49\\$};\\n\\\\fill (9.5, 2.45) -- (9.5, 2.55) -- (9.0, 2.5) -- cycle;\\n\\\\node[above right] at (9.0, 2.0) {\\$3.74\\$};\\n\\\\fill (9.45, 1.5) -- (9.55, 1.5) -- (9.5, 2.0) -- cycle;\\n\\\\node[above right] at (9.0, 1.0) {\\$1.49\\$};\\n\\\\fill (9.45, 0.5) -- (9.55, 0.5) -- (9.5, 1.0) -- cycle;\\n\\\\node[above right] at (9.0, 0.0) {\\$0.43\\$};\\n\\\\end{scope}\\n\\\\draw[black] grid(10,10);\\n\",\"scale=1.25\",\"\",true,true)"
       ]
      }
     ],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}