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"# Making Complex Decisions\n",
"---\n",
"This Jupyter notebook acts as supporting material for topics covered in **Chapter 17 Making Complex Decisions** of the book* Artificial Intelligence: A Modern Approach*. We make use of the implementations in mdp.py module. This notebook also includes a brief summary of the main topics as a review. Let us import everything from the mdp module to get started."
"cell_type": "code",
"metadata": {},
"from notebook import psource, pseudocode, plot_pomdp_utility"
"## CONTENTS\n",
"\n",
"* Overview\n",
"* MDP\n",
"* Grid MDP\n",
"* Value Iteration\n",
" * Value Iteration Visualization\n",
"* Policy Iteration\n",
"* POMDPs\n",
"* POMDP Value Iteration\n",
" - Value Iteration Visualization"
"metadata": {},
"source": [
"## OVERVIEW\n",
SnShine
a validé
"\n",
"Before we start playing with the actual implementations let us review a couple of things about MDPs.\n",
"\n",
"- A stochastic process has the **Markov property** if the conditional probability distribution of future states of the process (conditional on both past and present states) depends only upon the present state, not on the sequence of events that preceded it.\n",
"\n",
" -- Source: [Wikipedia](https://en.wikipedia.org/wiki/Markov_property)\n",
"\n",
"Often it is possible to model many different phenomena as a Markov process by being flexible with our definition of state.\n",
" \n",
"\n",
"- MDPs help us deal with fully-observable and non-deterministic/stochastic environments. For dealing with partially-observable and stochastic cases we make use of generalization of MDPs named POMDPs (partially observable Markov decision process).\n",
"\n",
"Our overall goal to solve a MDP is to come up with a policy which guides us to select the best action in each state so as to maximize the expected sum of future rewards."
]
},
{
"metadata": {},
"source": [
"## MDP\n",
"\n",
"To begin with let us look at the implementation of MDP class defined in mdp.py The docstring tells us what all is required to define a MDP namely - set of states, actions, initial state, transition model, and a reward function. Each of these are implemented as methods. Do not close the popup so that you can follow along the description of code below."
Aman Deep Singh
a validé
"metadata": {},
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"outputs": [
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"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">class</span> <span class=\"nc\">MDP</span><span class=\"p\">:</span>\n",
"\n",
" <span class=\"sd\">"""A Markov Decision Process, defined by an initial state, transition model,</span>\n",
"<span class=\"sd\"> and reward function. We also keep track of a gamma value, for use by</span>\n",
"<span class=\"sd\"> algorithms. The transition model is represented somewhat differently from</span>\n",
"<span class=\"sd\"> the text. Instead of P(s' | s, a) being a probability number for each</span>\n",
"<span class=\"sd\"> state/state/action triplet, we instead have T(s, a) return a</span>\n",
"<span class=\"sd\"> list of (p, s') pairs. We also keep track of the possible states,</span>\n",
"<span class=\"sd\"> terminal states, and actions for each state. [page 646]"""</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">init</span><span class=\"p\">,</span> <span class=\"n\">actlist</span><span class=\"p\">,</span> <span class=\"n\">terminals</span><span class=\"p\">,</span> <span class=\"n\">transitions</span> <span class=\"o\">=</span> <span class=\"p\">{},</span> <span class=\"n\">reward</span> <span class=\"o\">=</span> <span class=\"bp\">None</span><span class=\"p\">,</span> <span class=\"n\">states</span><span class=\"o\">=</span><span class=\"bp\">None</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"o\">=.</span><span class=\"mi\">9</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"p\">(</span><span class=\"mi\">0</span> <span class=\"o\"><</span> <span class=\"n\">gamma</span> <span class=\"o\"><=</span> <span class=\"mi\">1</span><span class=\"p\">):</span>\n",
" <span class=\"k\">raise</span> <span class=\"ne\">ValueError</span><span class=\"p\">(</span><span class=\"s2\">"An MDP must have 0 < gamma <= 1"</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">if</span> <span class=\"n\">states</span><span class=\"p\">:</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span> <span class=\"o\">=</span> <span class=\"n\">states</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"c1\">## collect states from transitions table</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">get_states_from_transitions</span><span class=\"p\">(</span><span class=\"n\">transitions</span><span class=\"p\">)</span>\n",
" \n",
" \n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">init</span> <span class=\"o\">=</span> <span class=\"n\">init</span>\n",
" \n",
" <span class=\"k\">if</span> <span class=\"nb\">isinstance</span><span class=\"p\">(</span><span class=\"n\">actlist</span><span class=\"p\">,</span> <span class=\"nb\">list</span><span class=\"p\">):</span>\n",
" <span class=\"c1\">## if actlist is a list, all states have the same actions</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">actlist</span> <span class=\"o\">=</span> <span class=\"n\">actlist</span>\n",
" <span class=\"k\">elif</span> <span class=\"nb\">isinstance</span><span class=\"p\">(</span><span class=\"n\">actlist</span><span class=\"p\">,</span> <span class=\"nb\">dict</span><span class=\"p\">):</span>\n",
" <span class=\"c1\">## if actlist is a dict, different actions for each state</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">actlist</span> <span class=\"o\">=</span> <span class=\"n\">actlist</span>\n",
" \n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">terminals</span> <span class=\"o\">=</span> <span class=\"n\">terminals</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">transitions</span> <span class=\"o\">=</span> <span class=\"n\">transitions</span>\n",
" <span class=\"k\">if</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">transitions</span> <span class=\"o\">==</span> <span class=\"p\">{}:</span>\n",
" <span class=\"k\">print</span><span class=\"p\">(</span><span class=\"s2\">"Warning: Transition table is empty."</span><span class=\"p\">)</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">gamma</span> <span class=\"o\">=</span> <span class=\"n\">gamma</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">reward</span><span class=\"p\">:</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">reward</span> <span class=\"o\">=</span> <span class=\"n\">reward</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">reward</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">s</span> <span class=\"p\">:</span> <span class=\"mi\">0</span> <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span><span class=\"p\">}</span>\n",
" <span class=\"c1\">#self.check_consistency()</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">R</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return a numeric reward for this state."""</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">reward</span><span class=\"p\">[</span><span class=\"n\">state</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">T</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Transition model. From a state and an action, return a list</span>\n",
"<span class=\"sd\"> of (probability, result-state) pairs."""</span>\n",
" <span class=\"k\">if</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">transitions</span> <span class=\"o\">==</span> <span class=\"p\">{}):</span>\n",
" <span class=\"k\">raise</span> <span class=\"ne\">ValueError</span><span class=\"p\">(</span><span class=\"s2\">"Transition model is missing"</span><span class=\"p\">)</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">transitions</span><span class=\"p\">[</span><span class=\"n\">state</span><span class=\"p\">][</span><span class=\"n\">action</span><span class=\"p\">]</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">actions</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Set of actions that can be performed in this state. By default, a</span>\n",
"<span class=\"sd\"> fixed list of actions, except for terminal states. Override this</span>\n",
"<span class=\"sd\"> method if you need to specialize by state."""</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">state</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">terminals</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[</span><span class=\"bp\">None</span><span class=\"p\">]</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">actlist</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">get_states_from_transitions</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">transitions</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">isinstance</span><span class=\"p\">(</span><span class=\"n\">transitions</span><span class=\"p\">,</span> <span class=\"nb\">dict</span><span class=\"p\">):</span>\n",
" <span class=\"n\">s1</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"n\">transitions</span><span class=\"o\">.</span><span class=\"n\">keys</span><span class=\"p\">())</span>\n",
" <span class=\"n\">s2</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">([</span><span class=\"n\">tr</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">]</span> <span class=\"k\">for</span> <span class=\"n\">actions</span> <span class=\"ow\">in</span> <span class=\"n\">transitions</span><span class=\"o\">.</span><span class=\"n\">values</span><span class=\"p\">()</span> \n",
" <span class=\"k\">for</span> <span class=\"n\">effects</span> <span class=\"ow\">in</span> <span class=\"n\">actions</span><span class=\"o\">.</span><span class=\"n\">values</span><span class=\"p\">()</span> <span class=\"k\">for</span> <span class=\"n\">tr</span> <span class=\"ow\">in</span> <span class=\"n\">effects</span><span class=\"p\">])</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">s1</span><span class=\"o\">.</span><span class=\"n\">union</span><span class=\"p\">(</span><span class=\"n\">s2</span><span class=\"p\">)</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">print</span><span class=\"p\">(</span><span class=\"s1\">'Could not retrieve states from transitions'</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">None</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">check_consistency</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">):</span>\n",
" <span class=\"c1\"># check that all states in transitions are valid</span>\n",
" <span class=\"k\">assert</span> <span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">get_states_from_transitions</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">transitions</span><span class=\"p\">)</span>\n",
" <span class=\"c1\"># check that init is a valid state</span>\n",
" <span class=\"k\">assert</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">init</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span>\n",
" <span class=\"c1\"># check reward for each state</span>\n",
" <span class=\"c1\">#assert set(self.reward.keys()) == set(self.states)</span>\n",
" <span class=\"k\">assert</span> <span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">reward</span><span class=\"o\">.</span><span class=\"n\">keys</span><span class=\"p\">())</span> <span class=\"o\">==</span> <span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span><span class=\"p\">)</span>\n",
" <span class=\"c1\"># check that all terminals are valid states</span>\n",
" <span class=\"k\">assert</span> <span class=\"nb\">all</span><span class=\"p\">([</span><span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span> <span class=\"k\">for</span> <span class=\"n\">t</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">terminals</span><span class=\"p\">])</span>\n",
" <span class=\"c1\"># check that probability distributions for all actions sum to 1</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">s1</span><span class=\"p\">,</span> <span class=\"n\">actions</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">transitions</span><span class=\"o\">.</span><span class=\"n\">items</span><span class=\"p\">():</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">a</span> <span class=\"ow\">in</span> <span class=\"n\">actions</span><span class=\"o\">.</span><span class=\"n\">keys</span><span class=\"p\">():</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">o</span> <span class=\"ow\">in</span> <span class=\"n\">actions</span><span class=\"p\">[</span><span class=\"n\">a</span><span class=\"p\">]:</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">+=</span> <span class=\"n\">o</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]</span>\n",
" <span class=\"k\">assert</span> <span class=\"nb\">abs</span><span class=\"p\">(</span><span class=\"n\">s</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span> <span class=\"o\"><</span> <span class=\"mf\">0.001</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
Aman Deep Singh
a validé
"psource(MDP)"
"metadata": {},
"source": [
"The **_ _init_ _** method takes in the following parameters:\n",
"\n",
"- init: the initial state.\n",
"- actlist: List of actions possible in each state.\n",
"- terminals: List of terminal states where only possible action is exit\n",
"- gamma: Discounting factor. This makes sure that delayed rewards have less value compared to immediate ones.\n",
"\n",
"**R** method returns the reward for each state by using the self.reward dict.\n",
"\n",
"**T** method is not implemented and is somewhat different from the text. Here we return (probability, s') pairs where s' belongs to list of possible state by taking action a in state s.\n",
"\n",
"**actions** method returns list of actions possible in each state. By default it returns all actions for states other than terminal states.\n"
]
},
{
"metadata": {},
"source": [
"Now let us implement the simple MDP in the image below. States A, B have actions X, Y available in them. Their probabilities are shown just above the arrows. We start with using MDP as base class for our CustomMDP. Obviously we need to make a few changes to suit our case. We make use of a transition matrix as our transitions are not very simple.\n",
"<img src=\"files/images/mdp-a.png\">"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true
},
"# Transition Matrix as nested dict. State -> Actions in state -> List of (Probability, State) tuples\n",
" \"X\": [(0.3, \"A\"), (0.7, \"B\")],\n",
" \"Y\": [(1.0, \"A\")]\n",
" \"X\": {(0.8, \"End\"), (0.2, \"B\")},\n",
" \"Y\": {(1.0, \"A\")}\n",
" },\n",
" \"End\": {}\n",
"}\n",
"\n",
"init = \"A\"\n",
"\n",
"terminals = [\"End\"]\n",
"\n",
"rewards = {\n",
" \"A\": 5,\n",
" \"B\": -10,\n",
" \"End\": 100\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 4,
"source": [
"class CustomMDP(MDP):\n",
" def __init__(self, init, terminals, transition_matrix, reward = None, gamma=.9):\n",
" # All possible actions.\n",
" actlist = []\n",
" for state in transition_matrix.keys():\n",
" actlist.extend(transition_matrix[state])\n",
" MDP.__init__(self, init, actlist, terminals, transition_matrix, reward, gamma=gamma)\n",
"\n",
" def T(self, state, action):\n",
" if action is None:\n",
" return [(0.0, state)]\n",
" else: \n",
" return self.t[state][action]"
"metadata": {},
"source": [
"Finally we instantize the class with the parameters for our MDP in the picture."
]
},
{
"cell_type": "code",
"execution_count": 5,
"our_mdp = CustomMDP(init, terminals, t, rewards, gamma=.9)"
"With this we have successfully represented our MDP. Later we will look at ways to solve this MDP."
"Now we look at a concrete implementation that makes use of the MDP as base class. The GridMDP class in the mdp module is used to represent a grid world MDP like the one shown in in **Fig 17.1** of the AIMA Book. We assume for now that the environment is _fully observable_, so that the agent always knows where it is. The code should be easy to understand if you have gone through the CustomMDP example."
Aman Deep Singh
a validé
"metadata": {},
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"outputs": [
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">class</span> <span class=\"nc\">GridMDP</span><span class=\"p\">(</span><span class=\"n\">MDP</span><span class=\"p\">):</span>\n",
"\n",
" <span class=\"sd\">"""A two-dimensional grid MDP, as in [Figure 17.1]. All you have to do is</span>\n",
"<span class=\"sd\"> specify the grid as a list of lists of rewards; use None for an obstacle</span>\n",
"<span class=\"sd\"> (unreachable state). Also, you should specify the terminal states.</span>\n",
"<span class=\"sd\"> An action is an (x, y) unit vector; e.g. (1, 0) means move east."""</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">grid</span><span class=\"p\">,</span> <span class=\"n\">terminals</span><span class=\"p\">,</span> <span class=\"n\">init</span><span class=\"o\">=</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">),</span> <span class=\"n\">gamma</span><span class=\"o\">=.</span><span class=\"mi\">9</span><span class=\"p\">):</span>\n",
" <span class=\"n\">grid</span><span class=\"o\">.</span><span class=\"n\">reverse</span><span class=\"p\">()</span> <span class=\"c1\"># because we want row 0 on bottom, not on top</span>\n",
" <span class=\"n\">reward</span> <span class=\"o\">=</span> <span class=\"p\">{}</span>\n",
" <span class=\"n\">states</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">()</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">rows</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">grid</span><span class=\"p\">)</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">cols</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">grid</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">grid</span> <span class=\"o\">=</span> <span class=\"n\">grid</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">cols</span><span class=\"p\">):</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">y</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">rows</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">grid</span><span class=\"p\">[</span><span class=\"n\">y</span><span class=\"p\">][</span><span class=\"n\">x</span><span class=\"p\">]</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"bp\">None</span><span class=\"p\">:</span>\n",
" <span class=\"n\">states</span><span class=\"o\">.</span><span class=\"n\">add</span><span class=\"p\">((</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">))</span>\n",
" <span class=\"n\">reward</span><span class=\"p\">[(</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">)]</span> <span class=\"o\">=</span> <span class=\"n\">grid</span><span class=\"p\">[</span><span class=\"n\">y</span><span class=\"p\">][</span><span class=\"n\">x</span><span class=\"p\">]</span>\n",
" <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span> <span class=\"o\">=</span> <span class=\"n\">states</span>\n",
" <span class=\"n\">actlist</span> <span class=\"o\">=</span> <span class=\"n\">orientations</span>\n",
" <span class=\"n\">transitions</span> <span class=\"o\">=</span> <span class=\"p\">{}</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">states</span><span class=\"p\">:</span>\n",
" <span class=\"n\">transitions</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"p\">{}</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">a</span> <span class=\"ow\">in</span> <span class=\"n\">actlist</span><span class=\"p\">:</span>\n",
" <span class=\"n\">transitions</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">][</span><span class=\"n\">a</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">calculate_T</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span>\n",
" <span class=\"n\">MDP</span><span class=\"o\">.</span><span class=\"fm\">__init__</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">init</span><span class=\"p\">,</span> <span class=\"n\">actlist</span><span class=\"o\">=</span><span class=\"n\">actlist</span><span class=\"p\">,</span>\n",
" <span class=\"n\">terminals</span><span class=\"o\">=</span><span class=\"n\">terminals</span><span class=\"p\">,</span> <span class=\"n\">transitions</span> <span class=\"o\">=</span> <span class=\"n\">transitions</span><span class=\"p\">,</span> \n",
" <span class=\"n\">reward</span> <span class=\"o\">=</span> <span class=\"n\">reward</span><span class=\"p\">,</span> <span class=\"n\">states</span> <span class=\"o\">=</span> <span class=\"n\">states</span><span class=\"p\">,</span> <span class=\"n\">gamma</span><span class=\"o\">=</span><span class=\"n\">gamma</span><span class=\"p\">)</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">calculate_T</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">action</span> <span class=\"ow\">is</span> <span class=\"bp\">None</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[(</span><span class=\"mf\">0.0</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">)]</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[(</span><span class=\"mf\">0.8</span><span class=\"p\">,</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">go</span><span class=\"p\">(</span><span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">)),</span>\n",
" <span class=\"p\">(</span><span class=\"mf\">0.1</span><span class=\"p\">,</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">go</span><span class=\"p\">(</span><span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">turn_right</span><span class=\"p\">(</span><span class=\"n\">action</span><span class=\"p\">))),</span>\n",
" <span class=\"p\">(</span><span class=\"mf\">0.1</span><span class=\"p\">,</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">go</span><span class=\"p\">(</span><span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">turn_left</span><span class=\"p\">(</span><span class=\"n\">action</span><span class=\"p\">)))]</span>\n",
" \n",
" <span class=\"k\">def</span> <span class=\"nf\">T</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">action</span><span class=\"p\">):</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">action</span> <span class=\"ow\">is</span> <span class=\"bp\">None</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[(</span><span class=\"mf\">0.0</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">)]</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">transitions</span><span class=\"p\">[</span><span class=\"n\">state</span><span class=\"p\">][</span><span class=\"n\">action</span><span class=\"p\">]</span>\n",
" \n",
" <span class=\"k\">def</span> <span class=\"nf\">go</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">direction</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return the state that results from going in this direction."""</span>\n",
" <span class=\"n\">state1</span> <span class=\"o\">=</span> <span class=\"n\">vector_add</span><span class=\"p\">(</span><span class=\"n\">state</span><span class=\"p\">,</span> <span class=\"n\">direction</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">state1</span> <span class=\"k\">if</span> <span class=\"n\">state1</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">states</span> <span class=\"k\">else</span> <span class=\"n\">state</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">to_grid</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">mapping</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Convert a mapping from (x, y) to v into a [[..., v, ...]] grid."""</span>\n",
" <span class=\"k\">return</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">reversed</span><span class=\"p\">([[</span><span class=\"n\">mapping</span><span class=\"o\">.</span><span class=\"n\">get</span><span class=\"p\">((</span><span class=\"n\">x</span><span class=\"p\">,</span> <span class=\"n\">y</span><span class=\"p\">),</span> <span class=\"bp\">None</span><span class=\"p\">)</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">x</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">cols</span><span class=\"p\">)]</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">y</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">rows</span><span class=\"p\">)]))</span>\n",
"\n",
" <span class=\"k\">def</span> <span class=\"nf\">to_arrows</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">policy</span><span class=\"p\">):</span>\n",
" <span class=\"n\">chars</span> <span class=\"o\">=</span> <span class=\"p\">{</span>\n",
" <span class=\"p\">(</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">):</span> <span class=\"s1\">'>'</span><span class=\"p\">,</span> <span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">1</span><span class=\"p\">):</span> <span class=\"s1\">'^'</span><span class=\"p\">,</span> <span class=\"p\">(</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"mi\">0</span><span class=\"p\">):</span> <span class=\"s1\">'<'</span><span class=\"p\">,</span> <span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">):</span> <span class=\"s1\">'v'</span><span class=\"p\">,</span> <span class=\"bp\">None</span><span class=\"p\">:</span> <span class=\"s1\">'.'</span><span class=\"p\">}</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">to_grid</span><span class=\"p\">({</span><span class=\"n\">s</span><span class=\"p\">:</span> <span class=\"n\">chars</span><span class=\"p\">[</span><span class=\"n\">a</span><span class=\"p\">]</span> <span class=\"k\">for</span> <span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)</span> <span class=\"ow\">in</span> <span class=\"n\">policy</span><span class=\"o\">.</span><span class=\"n\">items</span><span class=\"p\">()})</span>\n",
"</pre></div>\n",
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Aman Deep Singh
a validé
"psource(GridMDP)"
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"The **_ _init_ _** method takes **grid** as an extra parameter compared to the MDP class. The grid is a nested list of rewards in states.\n",
"\n",
"**go** method returns the state by going in particular direction by using vector_add.\n",
"\n",
"**T** method is not implemented and is somewhat different from the text. Here we return (probability, s') pairs where s' belongs to list of possible state by taking action a in state s.\n",
"\n",
"**actions** method returns list of actions possible in each state. By default it returns all actions for states other than terminal states.\n",
"\n",
"**to_arrows** are used for representing the policy in a grid like format."
]
},
{
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"source": [
"We can create a GridMDP like the one in **Fig 17.1** as follows: \n",
"\n",
" GridMDP([[-0.04, -0.04, -0.04, +1],\n",
" [-0.04, None, -0.04, -1],\n",
" [-0.04, -0.04, -0.04, -0.04]],\n",
" terminals=[(3, 2), (3, 1)])\n",
" \n",
"In fact the **sequential_decision_environment** in mdp module has been instantized using the exact same code."
]
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"\n",
"Now that we have looked how to represent MDPs. Let's aim at solving them. Our ultimate goal is to obtain an optimal policy. We start with looking at Value Iteration and a visualisation that should help us understanding it better.\n",
"\n",
"We start by calculating Value/Utility for each of the states. The Value of each state is the expected sum of discounted future rewards given we start in that state and follow a particular policy $\\pi$. The value or the utility of a state is given by\n",
"\n",
"$$U(s)=R(s)+\\gamma\\max_{a\\epsilon A(s)}\\sum_{s'} P(s'\\ |\\ s,a)U(s')$$\n",
"\n",
"This is called the Bellman equation. The algorithm Value Iteration (**Fig. 17.4** in the book) relies on finding solutions of this Equation. The intuition Value Iteration works is because values propagate through the state space by means of local updates. This point will we more clear after we encounter the visualisation. For more information you can refer to **Section 17.2** of the book. \n"
]
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">value_iteration</span><span class=\"p\">(</span><span class=\"n\">mdp</span><span class=\"p\">,</span> <span class=\"n\">epsilon</span><span class=\"o\">=</span><span class=\"mf\">0.001</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Solving an MDP by value iteration. [Figure 17.4]"""</span>\n",
" <span class=\"n\">U1</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">s</span><span class=\"p\">:</span> <span class=\"mi\">0</span> <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">mdp</span><span class=\"o\">.</span><span class=\"n\">states</span><span class=\"p\">}</span>\n",
" <span class=\"n\">R</span><span class=\"p\">,</span> <span class=\"n\">T</span><span class=\"p\">,</span> <span class=\"n\">gamma</span> <span class=\"o\">=</span> <span class=\"n\">mdp</span><span class=\"o\">.</span><span class=\"n\">R</span><span class=\"p\">,</span> <span class=\"n\">mdp</span><span class=\"o\">.</span><span class=\"n\">T</span><span class=\"p\">,</span> <span class=\"n\">mdp</span><span class=\"o\">.</span><span class=\"n\">gamma</span>\n",
" <span class=\"k\">while</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
" <span class=\"n\">U</span> <span class=\"o\">=</span> <span class=\"n\">U1</span><span class=\"o\">.</span><span class=\"n\">copy</span><span class=\"p\">()</span>\n",
" <span class=\"n\">delta</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">mdp</span><span class=\"o\">.</span><span class=\"n\">states</span><span class=\"p\">:</span>\n",
" <span class=\"n\">U1</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">R</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"o\">+</span> <span class=\"n\">gamma</span> <span class=\"o\">*</span> <span class=\"nb\">max</span><span class=\"p\">([</span><span class=\"nb\">sum</span><span class=\"p\">([</span><span class=\"n\">p</span> <span class=\"o\">*</span> <span class=\"n\">U</span><span class=\"p\">[</span><span class=\"n\">s1</span><span class=\"p\">]</span> <span class=\"k\">for</span> <span class=\"p\">(</span><span class=\"n\">p</span><span class=\"p\">,</span> <span class=\"n\">s1</span><span class=\"p\">)</span> <span class=\"ow\">in</span> <span class=\"n\">T</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"p\">)])</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">a</span> <span class=\"ow\">in</span> <span class=\"n\">mdp</span><span class=\"o\">.</span><span class=\"n\">actions</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)])</span>\n",
" <span class=\"n\">delta</span> <span class=\"o\">=</span> <span class=\"nb\">max</span><span class=\"p\">(</span><span class=\"n\">delta</span><span class=\"p\">,</span> <span class=\"nb\">abs</span><span class=\"p\">(</span><span class=\"n\">U1</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">]</span> <span class=\"o\">-</span> <span class=\"n\">U</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">]))</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">delta</span> <span class=\"o\"><</span> <span class=\"n\">epsilon</span> <span class=\"o\">*</span> <span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">-</span> <span class=\"n\">gamma</span><span class=\"p\">)</span> <span class=\"o\">/</span> <span class=\"n\">gamma</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">U</span>\n",
"</pre></div>\n",
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"It takes as inputs two parameters, an MDP to solve and epsilon, the maximum error allowed in the utility of any state. It returns a dictionary containing utilities where the keys are the states and values represent utilities. <br> Value Iteration starts with arbitrary initial values for the utilities, calculates the right side of the Bellman equation and plugs it into the left hand side, thereby updating the utility of each state from the utilities of its neighbors. \n",
"This is repeated until equilibrium is reached. \n",
"It works on the principle of _Dynamic Programming_ - using precomputed information to simplify the subsequent computation. \n",
"If $U_i(s)$ is the utility value for state $s$ at the $i$ th iteration, the iteration step, called Bellman update, looks like this:\n",
"\n",
"$$ U_{i+1}(s) \\leftarrow R(s) + \\gamma \\max_{a \\epsilon A(s)} \\sum_{s'} P(s'\\ |\\ s,a)U_{i}(s') $$\n",
"\n",
"As you might have noticed, `value_iteration` has an infinite loop. How do we decide when to stop iterating? \n",
"The concept of _contraction_ successfully explains the convergence of value iteration. \n",
"Refer to **Section 17.2.3** of the book for a detailed explanation. \n",
"In the algorithm, we calculate a value $delta$ that measures the difference in the utilities of the current time step and the previous time step. \n",
"\n",
"$$\\delta = \\max{(\\delta, \\begin{vmatrix}U_{i + 1}(s) - U_i(s)\\end{vmatrix})}$$\n",
"\n",
"This value of delta decreases as the values of $U_i$ converge.\n",
"We terminate the algorithm if the $\\delta$ value is less than a threshold value determined by the hyperparameter _epsilon_.\n",
"\n",
"$$\\delta \\lt \\epsilon \\frac{(1 - \\gamma)}{\\gamma}$$\n",
"\n",
"To summarize, the Bellman update is a _contraction_ by a factor of $gamma$ on the space of utility vectors. \n",
"Hence, from the properties of contractions in general, it follows that `value_iteration` always converges to a unique solution of the Bellman equations whenever $gamma$ is less than 1.\n",
"We then terminate the algorithm when a reasonable approximation is achieved.\n",
"In practice, it often occurs that the policy $pi$ becomes optimal long before the utility function converges. For the given 4 x 3 environment with $gamma = 0.9$, the policy $pi$ is optimal when $i = 4$ (at the 4th iteration), even though the maximum error in the utility function is stil 0.46. This can be clarified from **figure 17.6** in the book. Hence, to increase computational efficiency, we often use another method to solve MDPs called Policy Iteration which we will see in the later part of this notebook. \n",
"<br>For now, let us solve the **sequential_decision_environment** GridMDP using `value_iteration`."
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"The pseudocode for the algorithm:"
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"text/markdown": [
"### AIMA3e\n",
"__function__ VALUE-ITERATION(_mdp_, _ε_) __returns__ a utility function \n",
" __inputs__: _mdp_, an MDP with states _S_, actions _A_(_s_), transition model _P_(_s′_ | _s_, _a_), \n",
"      rewards _R_(_s_), discount _γ_ \n",
"   _ε_, the maximum error allowed in the utility of any state \n",
" __local variables__: _U_, _U′_, vectors of utilities for states in _S_, initially zero \n",
"        _δ_, the maximum change in the utility of any state in an iteration \n",
"\n",
" __repeat__ \n",
"   _U_ ← _U′_; _δ_ ← 0 \n",
"   __for each__ state _s_ in _S_ __do__ \n",
"     _U′_\\[_s_\\] ← _R_(_s_) + _γ_ max<sub>_a_ ∈ _A_(_s_)</sub> Σ _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] \n",
"     __if__ | _U′_\\[_s_\\] − _U_\\[_s_\\] | > _δ_ __then__ _δ_ ← | _U′_\\[_s_\\] − _U_\\[_s_\\] | \n",
" __until__ _δ_ < _ε_(1 − _γ_)/_γ_ \n",
" __return__ _U_ \n",
"\n",
"---\n",
"__Figure ??__ The value iteration algorithm for calculating utilities of states. The termination condition is from Equation (__??__)."
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"pseudocode(\"Value-Iteration\")"
]
},
"metadata": {},
"source": [
"### AIMA3e\n",
"__function__ VALUE-ITERATION(_mdp_, _ε_) __returns__ a utility function \n",
" __inputs__: _mdp_, an MDP with states _S_, actions _A_(_s_), transition model _P_(_s′_ | _s_, _a_), \n",
"      rewards _R_(_s_), discount _γ_ \n",
"   _ε_, the maximum error allowed in the utility of any state \n",
" __local variables__: _U_, _U′_, vectors of utilities for states in _S_, initially zero \n",
"        _δ_, the maximum change in the utility of any state in an iteration \n",
"\n",
" __repeat__ \n",
"   _U_ ← _U′_; _δ_ ← 0 \n",
"   __for each__ state _s_ in _S_ __do__ \n",
"     _U′_\\[_s_\\] ← _R_(_s_) + _γ_ max<sub>_a_ ∈ _A_(_s_)</sub> Σ _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] \n",
"     __if__ | _U′_\\[_s_\\] − _U_\\[_s_\\] | > _δ_ __then__ _δ_ ← | _U′_\\[_s_\\] − _U_\\[_s_\\] | \n",
" __until__ _δ_ < _ε_(1 − _γ_)/_γ_ \n",
" __return__ _U_ \n",
"\n",
"---\n",
"__Figure ??__ The value iteration algorithm for calculating utilities of states. The termination condition is from Equation (__??__)."
"metadata": {},
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"## VALUE ITERATION VISUALIZATION\n",
"\n",
"To illustrate that values propagate out of states let us create a simple visualisation. We will be using a modified version of the value_iteration function which will store U over time. We will also remove the parameter epsilon and instead add the number of iterations we want."
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"def value_iteration_instru(mdp, iterations=20):\n",
" U_over_time = []\n",
" U1 = {s: 0 for s in mdp.states}\n",
" R, T, gamma = mdp.R, mdp.T, mdp.gamma\n",
" for _ in range(iterations):\n",
" U = U1.copy()\n",
" for s in mdp.states:\n",
" U1[s] = R(s) + gamma * max([sum([p * U[s1] for (p, s1) in T(s, a)])\n",
" for a in mdp.actions(s)])\n",
" U_over_time.append(U)\n",
" return U_over_time"
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"metadata": {},
"source": [
"Next, we define a function to create the visualisation from the utilities returned by **value_iteration_instru**. The reader need not concern himself with the code that immediately follows as it is the usage of Matplotib with IPython Widgets. If you are interested in reading more about these visit [ipywidgets.readthedocs.io](http://ipywidgets.readthedocs.io)"
]
},
{
"cell_type": "code",
"source": [
"columns = 4\n",
"rows = 3\n",
"U_over_time = value_iteration_instru(sequential_decision_environment)"
]
},
{
"cell_type": "code",
"%matplotlib inline\n",
"from notebook import make_plot_grid_step_function\n",
"\n",
"plot_grid_step = make_plot_grid_step_function(columns, rows, U_over_time)"
]
},
{
"cell_type": "code",
"metadata": {
"scrolled": true
},
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"outputs": [
{
"data": {
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"text/plain": [
"<matplotlib.figure.Figure at 0x1d38c9c67f0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"The installed widget Javascript is the wrong version. It must satisfy the semver range ~2.1.4.\n"
]
},
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "77e9849e074841e49d8b0ebc8191507c"
}
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import ipywidgets as widgets\n",
"from IPython.display import display\n",
"iteration_slider = widgets.IntSlider(min=1, max=15, step=1, value=0)\n",
"w=widgets.interactive(plot_grid_step,iteration=iteration_slider)\n",
"\n",
"visualize_callback = make_visualize(iteration_slider)\n",
"\n",
"visualize_button = widgets.ToggleButton(description = \"Visualize\", value = False)\n",
"time_select = widgets.ToggleButtons(description='Extra Delay:',options=['0', '0.1', '0.2', '0.5', '0.7', '1.0'])\n",
"a = widgets.interactive(visualize_callback, Visualize = visualize_button, time_step=time_select)\n",
"display(a)"
"metadata": {},
"source": [
Aman Deep Singh
a validé
"Move the slider above to observe how the utility changes across iterations. It is also possible to move the slider using arrow keys or to jump to the value by directly editing the number with a double click. The **Visualize Button** will automatically animate the slider for you. The **Extra Delay Box** allows you to set time delay in seconds upto one second for each time step. There is also an interactive editor for grid-world problems `grid_mdp.py` in the gui folder for you to play around with."
"metadata": {
"collapsed": true
},
"source": [