POMDP_practical_solution.ipynb

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      "name": "POMDP_practical_solution.ipynb",
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      "name": "python3",
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  "cells": [
    {
      "cell_type": "markdown",
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        "id": "view-in-github",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/maximilianigl/DVRL/blob/master/POMDP_practical_solution.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "metadata": {
        "id": "WG8IxawUds-v",
        "colab_type": "text"
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      "cell_type": "markdown",
      "source": [
        "# Getting ready #\n",
        "\n",
        "The goal of this exercise is to understand *Value Iteration* in POMDPs.\n",
        "As there's quite a lot going on, we'll slowly walk through every logical step involved, hopefully developing some intuitive understanding of why POMDPs are difficult and how they can be solved.\n",
        "\n",
        "You should already understand the idea of *Value Iteration* in the fully observable case, i.e. in MDPs. Furthermore, we'll need *Bayes Rule* \n",
        "\n",
        "$$ P(A|B) = \\frac{P(B|A) P(A)}{P(B)},$$\n",
        "\n",
        "so make sure you know what's going on there."
      ]
    },
    {
      "metadata": {
        "id": "UJ0AUPl7iSnk",
        "colab_type": "text"
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      "cell_type": "markdown",
      "source": [
        "# Chapter 0: The illustrative example #\n",
        "\n",
        "In this exercise, we will be using the following example POMDP, taken from [here](http://ais.informatik.uni-freiburg.de/teaching/ws11/robotics2/pdfs/rob2-XX-POMDPs.pdf)\n",
        "\n",
        "![Graphical Model of example POMDP](https://github.com/maximilianigl/pomdp-practical/blob/master/images/Graph_model.png?raw=true)\n",
        "\n",
        "So, for rewards $r$, states $x_i$, actions $u_i$ and observations $z_i$ we have for the rewards in state $x_1$: \n",
        "$r(x_1, u_1) = -100$, $r(x_1, u_2) = 100$, $r(x_1, u_3) = -1$. \n",
        "\n",
        "The transition dynamics into state $x_1$ are given by: $p(x_1'|x_1, u_3) = 0.2$ and $p(x_1'|x_2, u_3)= 0.8$, i.e. there's only a chance to switch the state if we pick action $u_3$. Importantly, $u_3$ is the *sensing* action, i.e. we only get observations when we pick $u_3$. \n",
        "\n",
        "The observation probabilities in $x_1$ are given by $p(z_1|x_1) = 0.7$, $p(z_2|x_1) = 0.3$.\n",
        "Lastly, once we pick $u_1$ or $u_2$, we either get a big positive or negative reward *and the episode ends*.\n",
        "\n",
        "Below I already implemented the corresponding functions to represent that POMDP:"
      ]
    },
    {
      "metadata": {
        "id": "oCPz-DZklgxT",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "rewards = [\n",
        "    [-100, 100, -1],\n",
        "    [100,  -50, -1]\n",
        "]\n",
        "def reward(state, action):\n",
        "  \"\"\"\n",
        "  Returns the reward for an `action` in a certain `state`. \n",
        "  \n",
        "  Args: \n",
        "    state (int): The state for which to return the reward\n",
        "    action (int): The action taken in state.\n",
        "\n",
        "  Returns:\n",
        "    int: The reward\n",
        "  \"\"\"\n",
        "  \n",
        "  return rewards[state-1][action-1]\n",
        "\n",
        "transitions = [\n",
        "    [0.2, 0.8],\n",
        "    [0.8, 0.2]\n",
        "]\n",
        "\n",
        "def transition_probs(state, action, new_state):\n",
        "  \"\"\"\n",
        "  Returns the probability to transition to `new_state` when taking action `action` in state `state`.\n",
        "  \n",
        "  Args:\n",
        "    state (int): The current state\n",
        "    action (int): Action taken in `state`\n",
        "    new_state (int): The new state in which to transition to\n",
        "    \n",
        "  Returns:\n",
        "    float: Probability to transition into new_state\n",
        "  \"\"\"\n",
        "  if action != 3:\n",
        "    return state == new_state\n",
        "  else:\n",
        "    return transitions[state-1][new_state-1]\n",
        "  \n",
        "observations = [\n",
        "    [0.7, 0.3],\n",
        "    [0.3, 0.7]\n",
        "]\n",
        "def observation_probs(state, action, obs):\n",
        "  \"\"\"\n",
        "  Returns the probability of observing `obs` in state `state` when taking `action`.\n",
        "  When not taking u_3 as action, set p(z_1|x_i) = 1 for both states, i.e. we don't learn anything from the observation.\n",
        "  \n",
        "  Args:\n",
        "    state (int): The current state\n",
        "    action (int): Action taken in `state`\n",
        "    obs (int): The observation whose probability we evaluate\n",
        "    \n",
        "  Returns:\n",
        "    float: Probability of observation\n",
        "  \"\"\"\n",
        "  if action != 3:\n",
        "    # Set p(z_1|x_i)=1 in both states if we're not using u_3\n",
        "    return obs == 1\n",
        "  else:\n",
        "    return observations[state-1][obs-1]\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "-o4FCA0Lgn-_",
        "colab_type": "text"
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      "cell_type": "markdown",
      "source": [
        "# Chapter 1: Planning in the belief MDP #\n",
        "\n",
        "As you should have learned in the lecture this morning, we can see the POMDP as a *Believe MDP*, e.g. as explained [here](https://en.wikipedia.org/wiki/Partially_observable_Markov_decision_process#Belief_MDP):\n",
        "\n",
        "*A Markovian belief state allows a POMDP to be formulated as a Markov decision process where every belief is a state. The resulting belief MDP will thus be defined on a continuous state space (even if the \"originating\" POMDP has a finite number of states: there are infinite belief states (in $B$) because there are an infinite number of mixtures of the originating states (of $S$), since there are infinite beliefs for any given POMDP.*\n",
        "\n",
        "Before reasoning about how to solve the POMDP by using Value Iteration to find the best action for *every* possible belief state, let's first try to find the best action to take for *one* starting belief state."
      ]
    },
    {
      "metadata": {
        "id": "eFj_N7cHk-M3",
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      "cell_type": "markdown",
      "source": [
        "## 1.1 The 1-step horizon ##\n",
        "\n",
        "First, let's assume we can only take *one* action and then the episode ends. \n",
        "\n",
        "### The expected reward ###\n",
        "\n",
        "Given a belief $b=[p_1, 1-p_1]$ of being in state 1 or 2 respectively, what is the expected reward for taking action $u_1$ or $u_2$?\n",
        "\n",
        "**Question:**  Fill out the function below.\n",
        "\n",
        "**Note:** For the implementation, because we only have two states, we can represent the belief by $p_1$, i.e. the probability to be in state 1, with the remaining probability to be in state 2 being of course $1-p_1$."
      ]
    },
    {
      "metadata": {
        "id": "5A6BPqrudmAd",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def reward_b(p1, action):\n",
        "  \"\"\"\n",
        "  Compute the expected reward for taking a certain `action`.\n",
        "  \n",
        "  Args:\n",
        "    p1 (float): Belief to be in state 1\n",
        "    action (int): Action taken\n",
        "    \n",
        "  Returns:\n",
        "    float: Expected reward.\n",
        "  \n",
        "  \"\"\"\n",
        "  \n",
        "  \n",
        "  return p1 * reward(state=1, action=action) + (1-p1) * reward(state=2, action=action)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "IRW_-aajB44U",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Let's visualize this for the different actions (just run the code below). \n",
        "\n"
      ]
    },
    {
      "metadata": {
        "id": "VdUi4KxzB4oV",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 347
        },
        "outputId": "136399f8-7424-473f-96d4-98ed5ad41c7d"
      },
      "cell_type": "code",
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "\n",
        "# With numpy, operations like +/-/*// are performed element-wise, \n",
        "# so unless you did something weird above, we can evaluate the values of all \n",
        "# beliefes in parallel.\n",
        "p1s = np.linspace(0,1)\n",
        "\n",
        "plt.plot(p1s, reward_b(p1s, action=1), color='red')\n",
        "plt.plot(p1s, reward_b(p1s, action=2), color='blue')\n",
        "plt.plot(p1s, reward_b(p1s, action=3), color='green')\n",
        "\n",
        "# We'll need this one later\n",
        "# plt.plot(p1s, one_step_value(p1s), color='black', linestyle='--')\n",
        "plt.show()\n",
        "\n"
      ],
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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3duTFi7V06RLLxYs2HDo0RgpaiD+QohapmqLR4NmtJ8Enz/GqXAVaXLtCmVpV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\nIhb2v/m4CVhMAnIgAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 576x396 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "hvuAPzZlLN5J",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "**Question:** Why does this make sense? What's the best action to take in each belief state (assuming we can only take this one action and the episode ends afterwards)? Is the resulting function of 'best possible expected reward we can achieve in each belief state' convex ;)? If yes - why does this make intuitive sense? Is it piecewise linear? "
      ]
    },
    {
      "metadata": {
        "id": "mt7tXJHOLkNd",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Because we'll need this in the 2-step case below, let's write a function which returns us the best achievable 1-step return for each possible belief\n",
        "\n",
        "**Question:**  Fill out the function below.\n",
        "\n",
        "(**Hint:** The numpy function you're looking for is `np.maximum` and not `np.max`.)"
      ]
    },
    {
      "metadata": {
        "id": "r5nb_gaDECRd",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def one_step_value(p1):\n",
        "  \"\"\"\n",
        "  Compute the 1-step value for a given belief.\n",
        "  \n",
        "  Args:\n",
        "    p1 (float): Belief to be in state 1\n",
        "    \n",
        "  Returns:\n",
        "    float: Expected reward.\n",
        "  \n",
        "  \"\"\"\n",
        "  \n",
        "  return np.maximum(reward_b(p1, action=1), reward_b(p1, action=2), reward_b(p1, action=3))\n",
        "  "
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "jPpAjPOZHYoX",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Go back up to the plot, uncomment the remaining line and plot this function as well, just to check it does what it's supposed to do."
      ]
    },
    {
      "metadata": {
        "id": "i1VIwY1pEC6E",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## 1.2 Planning for with a 2 step horizon ## \n",
        "\n",
        "If we take $u_1$ or $u_2$, the episode ends immediately. However, when we take $u_3$ we actually get an observation (i.e. more information) and can take another action. So the value in the above plot for taking $u_3$ is not true if we look at an horizon of length $>1$. So let's look at next closest thing and fix the horizon length to 2. \n"
      ]
    },
    {
      "metadata": {
        "id": "quq4ufjQrq1-",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "### Updating the belief ###\n",
        "\n",
        "The first thing we need to do is figure out how to update our belief when we take an action and get an observation back from the environment.\n",
        "\n",
        "**Question:** Please fill out the function below to compute the updated belief given the past belief, the action taken and the observation recieved from the environment. \n",
        "\n",
        "**Note:** We need to get the posterior $p(s'|a,o,b) = \\sum_s p(s'|a,o,s) b(s)$, i.e. not only do we need to use Bayes rule to go from our model $p(o|s)$ to the posterior, but we also need to marginalise over our previous belief distribution. Try figuring out the formula by yourself (or in groups) and check with for example slide 16 [here](https://www.cs.cmu.edu/~ggordon/780-fall07/lectures/POMDP_lecture.pdf). \n",
        "\n"
      ]
    },
    {
      "metadata": {
        "id": "sEeuff0BsASV",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Helper function\n",
        "def prior_next_state(p1, action, next_state):\n",
        "  \"\"\"\n",
        "  Compute the probability p(next_state|action, p1), i.e. the prior probability\n",
        "  (before taking any observation into account)\n",
        "  to be in state `next_state` when taking `action`, given our current belief \n",
        "  as represented by p1\n",
        "  \n",
        "  Args:\n",
        "    p1 (float): Belief to be in state 1\n",
        "    action (int): Action taken\n",
        "    next_staet (int): Next state\n",
        "    \n",
        "  Returns:\n",
        "    float: Prior probability to be in state `next_state`\n",
        "  \n",
        "  \"\"\"\n",
        "  return (p1 * transition_probs(state=1, action=action, new_state=next_state) +\n",
        "          (1-p1) * transition_probs(state=2, action=action, new_state=next_state))\n",
        "\n",
        "\n",
        "\n",
        "def belief_update(p1, action, obs):\n",
        "  \"\"\"\n",
        "  Compute the new probability `p1` to be in state 1.\n",
        "  \n",
        "  Args:\n",
        "    p1 (float): Belief to be in state 1\n",
        "    action (int): Action taken\n",
        "    obs (int): Observation recieved from the environment\n",
        "    \n",
        "  Returns:\n",
        "    float: Updated belief to be in state 1\n",
        "  \n",
        "  \"\"\"\n",
        "  \n",
        "  p_o_marginal = (prior_next_state(p1, action, next_state=1) * observation_probs(state=1, action=action, obs=obs) +\n",
        "                  prior_next_state(p1, action, next_state=2) * observation_probs(state=2, action=action, obs=obs))\n",
        "  \n",
        "  \n",
        "  p1_new = observation_probs(state=1, action=action, obs=obs) * prior_next_state(p1, action, next_state=1) / p_o_marginal\n",
        "  return p1_new\n",
        "  "
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "LpRw3k81PHBF",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Ok, let's just visualize that again. Just execute the field below and think about whether the result makes sense."
      ]
    },
    {
      "metadata": {
        "id": "O2FTMb3AOMt7",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 364
        },
        "outputId": "49a59050-142e-4059-c09f-b4e5133d02e0"
      },
      "cell_type": "code",
      "source": [
        "p1s = np.linspace(0,1)\n",
        "\n",
        "# Let's plot the belief to be in state 1 for both possible observations that we\n",
        "# Could get when taking action 3\n",
        "\n",
        "plt.plot(p1s, belief_update(p1s, action=3, obs=1), color='red')\n",
        "plt.plot(p1s, belief_update(p1s, action=3, obs=2), color='blue')\n",
        "plt.plot()"
      ],
      "execution_count": 33,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f3d80857898>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 33
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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jwGxWu7pzgrqHFURQTUdXNVdf7WPJEjupqUXY7QrDhoXy4IOy7aUQVV5oKPaH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\nRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 576x396 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "KDmVbswiIE1c",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "### 2 step planning for actions $u_3$###\n",
        "\n",
        "**Question:** Ok, now, let's write a function that computes the *expected* two step return for taking action $u_3$. To help, I've provided the `simulate` function, that returns the probability for observation 1 given a certain belief $p_1$."
      ]
    },
    {
      "metadata": {
        "id": "SuNu_DskGcAc",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def simulate(p1):\n",
        "  \"\"\"\n",
        "  Returns the probability for observation 1 when our current belief is given\n",
        "  by p1 and we exectue the sensing action u3\n",
        "  \n",
        "  Args:\n",
        "    p1 (float): Belief to be in state 1\n",
        "    \n",
        "  Returns:\n",
        "    float: probability to observe o1 when taking action 3\n",
        "  \"\"\"\n",
        "  p1_prime = (p1 * transition_probs(state=1, action=3, new_state=1) +\n",
        "             (1-p1) * transition_probs(state=2, action=3, new_state=1))\n",
        "  \n",
        "  p_o1 = (p1_prime * observation_probs(state=1, action=3, obs=1) + \n",
        "          (1-p1_prime) * observation_probs(state=2, action=3, obs=1))\n",
        "          \n",
        "  return p_o1\n",
        "  \n",
        "  \n",
        "\n",
        "def value_taking_u3_first(p1):\n",
        "  \"\"\"\n",
        "  Compute the 2-step value for a given belief when we take action u_3\n",
        "  \n",
        "  Args:\n",
        "    p1 (float): Belief to be in state 1\n",
        "    \n",
        "  Returns:\n",
        "    float: Expected reward.\n",
        "  \"\"\"\n",
        "  p_o1 = simulate(p1)\n",
        "          \n",
        "  new_belief_o1 = belief_update(p1, action=3, obs=1)\n",
        "  new_belief_o2 = belief_update(p1, action=3, obs=2)\n",
        "          \n",
        "  return (p_o1 * one_step_value(new_belief_o1) + \n",
        "          (1-p_o1) * one_step_value(new_belief_o2))\n",
        "\n",
        "\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "bcbLWM5pWLWf",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 347
        },
        "outputId": "d37597a3-0d2c-40d8-f617-dc39ecc28981"
      },
      "cell_type": "code",
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "\n",
        "# With numpy, operations like +/-/*// are performed element-wise, \n",
        "# so unless you did something weird above, we can evaluate the values of all \n",
        "# beliefes in parallel.\n",
        "p1s = np.linspace(0,1)\n",
        "\n",
        "plt.plot(p1s, reward_b(p1s, action=1), color='red')\n",
        "plt.plot(p1s, reward_b(p1s, action=2), color='blue')\n",
        "plt.plot(p1s, reward_b(p1s, action=3), color='green')\n",
        "\n",
        "# We'll need this one later\n",
        "plt.plot(p1s, one_step_value(p1s), color='black', linestyle='--')\n",
        "\n",
        "plt.plot(p1s, value_taking_u3_first(p1s), color='black')\n",
        "plt.show()"
      ],
      "execution_count": 39,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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2ZSK9SxQi4s5tpUsTQiQTR4/+SalS/jRoUISbN5ej1eamc+f+HDigo25dA2rJZ5uQZB/P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\nQmCC\n",
            "text/plain": [
              "<Figure size 576x396 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "gALU0G_JXtbi",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "**Question:** Again, does that make intuitive sense? Why? And is the resulting value function of 'best choices' piecewise linear and convex :)? "
      ]
    },
    {
      "metadata": {
        "id": "HYVQrRh-Y0GO",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "# Chapter 2: From planning to Value Iteration #\n",
        "\n",
        "Now, those nice plots we have made so far are what we would like to get out of Value Iteration: Telling us exactly what the expected value for each action is: Each line corresponds to one action we can take now. This allows us to, for each possible belief, pick the action with the highest expected return. \n",
        "\n",
        "So far, we've created those plots by just picking a bunch of points (50 to be exact) and evaluate the value for all possible actions at all of those points. Of course, that scales terribly. For one, planning more and more into the future becomes costly. But more importantly, our simple POMDP has only two states. If it had more, the resulting belief distribution would become higher dimensional, requiring more and more points to cover it with sufficient accuracy. So what can we do?\n",
        "\n",
        "Well, we can realise that the resulting value function for any horizon length will always be convex (why?) and consist of a *finite* number of linear (why linear?) segments, in other words it is *piecewise linear and convex* (PWLC)."
      ]
    },
    {
      "metadata": {
        "id": "9fMDNpnG76bB",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## 2.1 From Point to Lines ##\n",
        "\n",
        "Let's recap what we did above: How did we move from the 1-step case to 2 steps?\n",
        "\n",
        "1. For each belief $b$, we computed the new belief $b'$ *for each possible observation $z_i$ that we could receive* \n",
        "2. We evaluated the new belief $b'$ using our old `one_step_value` function\n",
        "2. We summed all those values up, weighted by how likely each observation was (as given by the `simulate` function). \n",
        "\n",
        "Now, we know that the value function is represented by a finite number of *lines*. Can we apply the same 3 steps to lines instead of points? Why? \n",
        "\n",
        "This is shown wonderfully in the [POMDP tutorial](http://pomdp.org/tutorial/pomdp-vi-example.html):\n",
        "\n",
        "We start with the Horizon 1 value function:\n",
        "\n",
        "![Horizon 1 value function](https://github.com/maximilianigl/pomdp-practical/blob/master/images/Tutorial_1.png?raw=true)\n",
        "\n",
        "Note that here the colors on the bottom represent which action is optimal.\n",
        "\n",
        "Next, for each action and for each possible observation, we transform this one step value function based on how likely this observation is and how this would affect our belief state. In this example, there are 3 possible observations, so we get three transformed value functions (for action $a_1$):\n",
        "\n",
        "![Transformed horizon 1  value function](https://github.com/maximilianigl/pomdp-practical/blob/master/images/Tutorial_2.png?raw=true)\n",
        "\n",
        "Now we need to sum those up. But how? Add all blue lines together? That would mean always taking $a_1$, independenlty of the observation. So instead, we can come up with different strategies like $(z_1:a_2, z_2:a_1, z_3:a_1)$, i.e. take $a_2$ when we see $z_1$, take $a_1$ for $z_2$ and so on. Here that would mean add the left green line, the middle blue line and the right blue line. But which strategy is best? Above, when we did it for a fixed initial belief, we just took the `max` operation given the new belief. However, now we need to think about *all* possible initial believes, so the optimal strategy depends on our *original* belief $b$ which is still shown on the $x$-axis. We can easily read off the best strategies like so: \n",
        "\n",
        "![Best response strategy](https://github.com/maximilianigl/pomdp-practical/blob/master/images/Tutorial_3.png?raw=true)\n",
        "\n",
        "where we see that there are 4 best strategies reacting to observations, depending on what is our starting belief. \n",
        "Each of those strategies tells us which lines to add up, given the resulting value function for taking action $a_1$:\n",
        "\n",
        "![Value function for a1](https://github.com/maximilianigl/pomdp-practical/blob/master/images/Tutorial_4.png?raw=true).\n",
        "\n",
        "Now we can do the same thing again with $a_2$ and take the max between the two and voila, we have the horizon 2 value function.  \n",
        "\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "\n"
      ]
    },
    {
      "metadata": {
        "id": "FTM47EHyDt_a",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## 2.2 Playing around with Lines ##\n",
        "\n",
        "But grey is all theory, so let's just play around with Value Iteration. Because we're lazy and don't want to implement it from scratch, let's use [this](https://github.com/pemami4911/POMDPy) repo with 98 stars that implements Value Iteration for the Tiger problem, which is very similar to our setting: A tiger is behind one of two doors and you can either open one of the doors, or listen for a *roar*, giving you a noisy observation."
      ]
    },
    {
      "metadata": {
        "id": "vM0vShxjD4ga",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 156
        },
        "outputId": "976570d5-a673-4cf7-9b48-ce3acad29f35"
      },
      "cell_type": "code",
      "source": [
        "# Install a POMDP Value Iteration solver because \n",
        "# implementing that in half an afternoon is a bit ambitious\n",
        "\n",
        "!pip install git+https://github.com/pemami4911/POMDPy.git"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Collecting git+https://github.com/pemami4911/POMDPy.git\n",
            "  Cloning https://github.com/pemami4911/POMDPy.git to /tmp/pip-req-build-z725dlef\n",
            "Requirement already satisfied (use --upgrade to upgrade): POMDPy==2.0.0 from git+https://github.com/pemami4911/POMDPy.git in /usr/local/lib/python3.6/dist-packages\n",
            "Building wheels for collected packages: POMDPy\n",
            "  Building wheel for POMDPy (setup.py) ... \u001b[?25ldone\n",
            "\u001b[?25h  Stored in directory: /tmp/pip-ephem-wheel-cache-0add2j7f/wheels/b0/95/04/cadab923d0dcd09126efc00b0a9d8cfc7cabe27aa22fef4340\n",
            "Successfully built POMDPy\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {
        "id": "PbeN9H27gmlK",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "### Running Value Iteration ###\n",
        "\n",
        "In the field below, set the planning horizon and run it. I'd recommend starting with one and working your way up ;). The next field allows you to plot the resulting value function."
      ]
    },
    {
      "metadata": {
        "id": "io9VUfBREVLA",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 850
        },
        "outputId": "b176e8c8-f04f-456d-839d-b267bc1e3271"
      },
      "cell_type": "code",
      "source": [
        "from pomdpy import Agent\n",
        "from pomdpy.solvers import ValueIteration\n",
        "from pomdpy.log import init_logger\n",
        "from examples.tiger import TigerModel\n",
        "\n",
        "planning_horizon = 1\n",
        "\n",
        "args = {\n",
        "    'planning_horizon': planning_horizon,\n",
        "    'n_epochs': 1,\n",
        "    'max_steps': 1,\n",
        "    'env': 'Tiger',\n",
        "    'solver': 'ValueIteration',\n",
        "    'discount': 1,\n",
        "    'save': True,\n",
        "}\n",
        "\n",
        "env = TigerModel(args)\n",
        "env.weight_dir = ('./')\n",
        "agent = Agent(env, ValueIteration)\n",
        "test = agent.discounted_return()"
      ],
      "execution_count": 67,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "{'discount': 1,\n",
            " 'env': 'Tiger',\n",
            " 'max_steps': 1,\n",
            " 'n_epochs': 1,\n",
            " 'planning_horizon': 2,\n",
            " 'save': True,\n",
            " 'solver': 'ValueIteration'}\n",
            "======================================================================\n",
            "\tEpoch #1 RESULTS\n",
            "======================================================================\n",
            "agent - discounted return statistics\n",
            "==========================================\n",
            "Name =  discounted return\n",
            "Running Total =  -1.0\n",
            "Mean =  -1.0\n",
            "Count =  1.0\n",
            "Variance =  0.0\n",
            "Max =  -1.0\n",
            "Min =  -1.0\n",
            "==========================================\n",
            "agent - undiscounted return statistics\n",
            "==========================================\n",
            "Name =  undiscounted return\n",
            "Running Total =  -1.0\n",
            "Mean =  -1.0\n",
            "Count =  1.0\n",
            "Variance =  0.0\n",
            "Max =  -1.0\n",
            "Min =  -1.0\n",
            "==========================================\n",
            "agent - Time\n",
            "==========================================\n",
            "Name =  Time\n",
            "Running Total =  0.19347071647644043\n",
            "Mean =  0.19347071647644043\n",
            "Count =  1.0\n",
            "Variance =  0.0\n",
            "Max =  0.19347071647644043\n",
            "Min =  0.19347071647644043\n",
            "==========================================\n",
            "==========================================\n",
            "  [*] save ./VI_planning_horizon_2.pkl\n",
            "   [-] save_pkl : 0.00057 sec\n",
            "\n",
            "\n",
            "agent - epochs: 1\n",
            "agent - ave undiscounted return/step: -1.0 +- 0.0\n",
            "agent - ave discounted return/step: -1.0 +- 0.0\n",
            "agent - ave time/epoch: 0.19347071647644043\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {
        "id": "rO9rKQY1ctUi",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 466
        },
        "outputId": "06f32e71-4c6c-4aea-d1e9-90901cef8f7b"
      },
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "ps = np.linspace(0,1)\n",
        "\n",
        "line = lambda p, v0, v1: p * v0 + (1-p) * v1\n",
        "\n",
        "colors = ['red', 'blue', 'green']\n",
        "\n",
        "import pickle\n",
        "alpha_vectors = pickle.load(open('VI_planning_horizon_{}.pkl'.format(planning_horizon), 'rb'))\n",
        "\n",
        "\n",
        "print(\"There are {} alpha-vectors\".format(len(alpha_vectors)))\n",
        "for av in alpha_vectors:\n",
        "  print(vars(av))\n",
        "  plt.plot(ps, line(ps, av.v[0], av.v[1]), color = colors[av.action])\n",
        "\n",
        "  \n"
      ],
      "execution_count": 68,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "There are 6 alpha-vectors\n",
            "{'action': 0, 'v': array([-1., -1.])}\n",
            "{'action': 0, 'v': array([ 7.5, -4. ])}\n",
            "{'action': 0, 'v': array([-4. ,  7.5])}\n",
            "{'action': 2, 'v': array([ 10., -20.])}\n",
            "{'action': 2, 'v': array([  9.5, -20.5])}\n",
            "{'action': 1, 'v': array([-20.,  10.])}\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
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yTF3CkBVQFeKpjuZU7MJLNaI+dtTV5mjQAGtkDJbIaKzRsdg7duJ6097jxyXGjTPwxx8a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\n/wAzhSWC9qahXAAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 576x396 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "zWVYW2VbhU4D",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "**Question:** Have you tried the above for several different `planning_horizon`s? Does the result look reasonable? Why not?\n",
        "\n",
        "Looks like there's a bug....\n",
        "\n",
        "**Bonus question:** Fix the bug. Below I copy-pasted his code for his value iteration solver for your convenience. Because the solver is passed into his `Agent` as argument, we can just replace it with our version. Tipp: Look for line 56, marked by `# TODO: Fix BUG`, that's where the magic happens."
      ]
    },
    {
      "metadata": {
        "id": "p2XXf5DqSd-T",
        "colab_type": "code",
        "colab": {},
        "cellView": "both"
      },
      "cell_type": "code",
      "source": [
        "from pomdpy.solvers.solver import Solver\n",
        "from pomdpy.solvers.alpha_vector import AlphaVector\n",
        "from scipy.optimize import linprog\n",
        "import numpy as np\n",
        "from itertools import product\n",
        "\n",
        "\n",
        "class ValueIteration(Solver):\n",
        "    def __init__(self, agent):\n",
        "        \"\"\"\n",
        "        Initialize the POMDP exact value iteration solver\n",
        "        :param agent:\n",
        "        :return:\n",
        "        \"\"\"\n",
        "        super(ValueIteration, self).__init__(agent)\n",
        "        self.gamma = set()\n",
        "        self.history = agent.histories.create_sequence()\n",
        "\n",
        "    @staticmethod\n",
        "    def reset(agent):\n",
        "        return ValueIteration(agent)\n",
        "\n",
        "    def value_iteration(self, t, o, r, horizon):\n",
        "        \"\"\"\n",
        "        Solve the POMDP by computing all alpha vectors\n",
        "        :param t: transition probability matrix\n",
        "        :param o: observation probability matrix\n",
        "        :param r: immediate rewards matrix\n",
        "        :param horizon: integer valued scalar represented the number of planning steps\n",
        "        :return:\n",
        "        \"\"\"\n",
        "        discount = self.model.discount\n",
        "        actions = len(self.model.get_all_actions())  # |A| actions\n",
        "        states = self.model.num_states  # |S| states\n",
        "        observations = len(self.model.get_all_observations())  # |Z| observations\n",
        "        first = True\n",
        "\n",
        "        # initialize gamma with a 0 alpha-vector\n",
        "        dummy = AlphaVector(a=-1, v=np.zeros(states))\n",
        "        self.gamma.add(dummy)\n",
        "\n",
        "        # start with 1 step planning horizon, up to horizon-length planning horizon\n",
        "        for k in range(horizon):\n",
        "            # new set of alpha vectors to add to set gamma\n",
        "            gamma_k = set()\n",
        "            # Compute the new coefficients for the new alpha-vectors\n",
        "            v_new = np.zeros(shape=(len(self.gamma), actions, observations, states))\n",
        "            idx = 0\n",
        "            for v in self.gamma:\n",
        "                for u in range(actions):\n",
        "                    for z in range(observations):\n",
        "                        for j in range(states):\n",
        "                            for i in range(states):\n",
        "                                \n",
        "                                # TODO: Fix BUG\n",
        "                                # v_i_k * p(z | x_i, u) * p(x_i | u, x_j)\n",
        "#                                 v_new[idx][u][z][i] += v.v[i] * o[u][i][z] * t[u][j][i]\n",
        "                                v_new[idx][u][z][j] += v.v[i] * o[u][i][z] * t[u][i][j]\n",
        "\n",
        "                idx += 1\n",
        "            # add (|A| * |V|^|Z|) alpha-vectors to gamma, |V| is |gamma_k|\n",
        "            for u in range(actions):\n",
        "                c = self.compute_indices(idx, observations)\n",
        "                for indices in c:  # n elements in c is |V|^|Z|\n",
        "                    for z in range(observations):\n",
        "                        temp = np.zeros(states)\n",
        "                        for i in range(states):\n",
        "                            temp[i] = discount * (r[u][i] + v_new[indices[z]][u][z][i])\n",
        "                        gamma_k.add(AlphaVector(a=u, v=temp))\n",
        "            self.gamma.update(gamma_k)\n",
        "            if first:\n",
        "                # remove the dummy alpha vector\n",
        "                self.gamma.remove(dummy)\n",
        "                first = False\n",
        "            self.prune(states)\n",
        "            #  plot_gamma(title='V(b) for horizon T = ' + str(k + 1), self.gamma)\n",
        "\n",
        "    @staticmethod\n",
        "    def compute_indices(k, m):\n",
        "        \"\"\"\n",
        "        Compute all orderings of m elements with values between [0, k-1]\n",
        "        :param k: Number of alpha-vectors\n",
        "        :param m: Number of observations\n",
        "        :return: list of lists, where each list contains m elements, and each element is in [0, k-1].\n",
        "        Total should be k^m elements\n",
        "        \"\"\"\n",
        "        x = list(range(k))\n",
        "        return [p for p in product(x, repeat=m)]\n",
        "\n",
        "    def prune(self, n_states):\n",
        "        \"\"\"\n",
        "        Remove dominated alpha-vectors using Lark's filtering algorithm\n",
        "        :param n_states\n",
        "        :return:\n",
        "        \"\"\"\n",
        "        # parameters for linear program\n",
        "        delta = 0.0000000001\n",
        "        # equality constraints on the belief states\n",
        "        A_eq = np.array([np.append(np.ones(n_states), [0.])])\n",
        "        b_eq = np.array([1.])\n",
        "\n",
        "        # dirty set\n",
        "        F = self.gamma.copy()\n",
        "        # clean set\n",
        "        Q = set()\n",
        "\n",
        "        for i in range(n_states):\n",
        "            max_i = -np.inf\n",
        "            best = None\n",
        "            for av in F:\n",
        "                if av.v[i] > max_i:\n",
        "                    max_i = av.v[i]\n",
        "                    best = av\n",
        "            Q.update({best})\n",
        "            F.remove(best)\n",
        "        while F:\n",
        "            av_i = F.pop()  # get a reference to av_i\n",
        "            F.add(av_i)  # don't want to remove it yet from F\n",
        "            dominated = False\n",
        "            for av_j in Q:\n",
        "                c = np.append(np.zeros(n_states), [1.])\n",
        "                A_ub = np.array([np.append(-(av_i.v - av_j.v), [-1.])])\n",
        "                b_ub = np.array([-delta])\n",
        "\n",
        "                res = linprog(c, A_eq=A_eq, b_eq=b_eq, A_ub=A_ub, b_ub=b_ub, bounds=(0, None))\n",
        "                if res.x[n_states] > 0.0:\n",
        "                    # this one is dominated\n",
        "                    dominated = True\n",
        "                    F.remove(av_i)\n",
        "                    break\n",
        "\n",
        "            if not dominated:\n",
        "                max_k = -np.inf\n",
        "                best = None\n",
        "                for av_k in F:\n",
        "                    b = res.x[0:2]\n",
        "                    v = np.dot(av_k.v, b)\n",
        "                    if v > max_k:\n",
        "                        max_k = v\n",
        "                        best = av_k\n",
        "                F.remove(best)\n",
        "                if not self.check_duplicate(Q, best):\n",
        "                    Q.update({best})\n",
        "        self.gamma = Q\n",
        "\n",
        "    @staticmethod\n",
        "    def check_duplicate(a, av):\n",
        "        \"\"\"\n",
        "        Check whether alpha vector av is already in set a\n",
        "\n",
        "        :param a:\n",
        "        :param av:\n",
        "        :return:\n",
        "        \"\"\"\n",
        "        for av_i in a:\n",
        "            if np.allclose(av_i.v, av.v):\n",
        "                return True\n",
        "            if av_i.v[0] == av.v[0] and av_i.v[1] > av.v[1]:\n",
        "                return True\n",
        "            if av_i.v[1] == av.v[1] and av_i.v[0] > av.v[0]:\n",
        "                return True\n",
        "\n",
        "    @staticmethod\n",
        "    def select_action(belief, vector_set):\n",
        "        \"\"\"\n",
        "        Compute optimal action given a belief distribution\n",
        "        :param belief: dim(belief) == dim(AlphaVector)\n",
        "        :param vector_set\n",
        "        :return:\n",
        "        \"\"\"\n",
        "        max_v = -np.inf\n",
        "        best = None\n",
        "        for av in vector_set:\n",
        "            v = np.dot(av.v, belief)\n",
        "\n",
        "            if v > max_v:\n",
        "                max_v = v\n",
        "                best = av\n",
        "\n",
        "        if best is None:\n",
        "            raise ValueError('Vector set should not be empty')\n",
        "\n",
        "        return best.action, best\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "5iSlsJZzfHKv",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "# Epilogue\n",
        "\n",
        "This is the first version of this tutorial. I assume that whoever will hold the POMDP tutorial next year is likely to re-use the material here. So both for my own curiosity as well as the benefit of all future CDT generations, it would be great if you could provide some feedback in this *very short* [feedback form](https://docs.google.com/forms/d/e/1FAIpQLSdOFYkBDFt2biLdZj6fyvnrjUDhRzDjSD6CYg9CE7A1Ohnq_A/viewform?usp=sf_link):)"
      ]
    }
  ]
}