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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">expected_utility</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">U</span><span class=\"p\">,</span> <span class=\"n\">mdp</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""The expected utility of doing a in state s, according to the MDP and U."""</span>\n",
" <span class=\"k\">return</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\">mdp</span><span class=\"o\">.</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",
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">policy_iteration</span><span class=\"p\">(</span><span class=\"n\">mdp</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Solve an MDP by policy iteration [Figure 17.7]"""</span>\n",
" <span class=\"n\">U</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\">pi</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">s</span><span class=\"p\">:</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">(</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> <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=\"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\">policy_evaluation</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">U</span><span class=\"p\">,</span> <span class=\"n\">mdp</span><span class=\"p\">)</span>\n",
" <span class=\"n\">unchanged</span> <span class=\"o\">=</span> <span class=\"bp\">True</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\">a</span> <span class=\"o\">=</span> <span class=\"n\">argmax</span><span class=\"p\">(</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> <span class=\"n\">key</span><span class=\"o\">=</span><span class=\"k\">lambda</span> <span class=\"n\">a</span><span class=\"p\">:</span> <span class=\"n\">expected_utility</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"n\">U</span><span class=\"p\">,</span> <span class=\"n\">mdp</span><span class=\"p\">))</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">a</span> <span class=\"o\">!=</span> <span class=\"n\">pi</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">]:</span>\n",
" <span class=\"n\">pi</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">a</span>\n",
" <span class=\"n\">unchanged</span> <span class=\"o\">=</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">unchanged</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">pi</span>\n",
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"<br>Fortunately, it is not necessary to do _exact_ policy evaluation. \n",
"The utilities can instead be reasonably approximated by performing some number of simplified value iteration steps.\n",
"The simplified Bellman update equation for the process is\n",
"\n",
"$$U_{i+1}(s) \\leftarrow R(s) + \\gamma\\sum_{s'}P(s'\\ |\\ s,\\pi_i(s))U_{i}(s')$$\n",
"\n",
"and this is repeated _k_ times to produce the next utility estimate. This is called _modified policy iteration_."
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"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">policy_evaluation</span><span class=\"p\">(</span><span class=\"n\">pi</span><span class=\"p\">,</span> <span class=\"n\">U</span><span class=\"p\">,</span> <span class=\"n\">mdp</span><span class=\"p\">,</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">20</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return an updated utility mapping U from each state in the MDP to its</span>\n",
"<span class=\"sd\"> utility, using an approximation (modified policy iteration)."""</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\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">k</span><span class=\"p\">):</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\">U</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\">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\">pi</span><span class=\"p\">[</span><span class=\"n\">s</span><span class=\"p\">])])</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">U</span>\n",
"</pre></div>\n",
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"Let us now solve **`sequential_decision_environment`** using `policy_iteration`."
]
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"### AIMA3e\n",
"__function__ POLICY-ITERATION(_mdp_) __returns__ a policy \n",
" __inputs__: _mdp_, an MDP with states _S_, actions _A_(_s_), transition model _P_(_s′_ | _s_, _a_) \n",
" __local variables__: _U_, a vector of utilities for states in _S_, initially zero \n",
"        _π_, a policy vector indexed by state, initially random \n",
"\n",
" __repeat__ \n",
"   _U_ ← POLICY\\-EVALUATION(_π_, _U_, _mdp_) \n",
"   _unchanged?_ ← true \n",
"   __for each__ state _s_ __in__ _S_ __do__ \n",
"     __if__ max<sub>_a_ ∈ _A_(_s_)</sub> Σ<sub>_s′_</sub> _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] > Σ<sub>_s′_</sub> _P_(_s′_ | _s_, _π_\\[_s_\\]) _U_\\[_s′_\\] __then do__ \n",
"       _π_\\[_s_\\] ← argmax<sub>_a_ ∈ _A_(_s_)</sub> Σ<sub>_s′_</sub> _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] \n",
"       _unchanged?_ ← false \n",
" __until__ _unchanged?_ \n",
" __return__ _π_ \n",
"\n",
"---\n",
"__Figure ??__ The policy iteration algorithm for calculating an optimal policy."
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"pseudocode('Policy-Iteration')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### AIMA3e\n",
"__function__ POLICY-ITERATION(_mdp_) __returns__ a policy \n",
" __inputs__: _mdp_, an MDP with states _S_, actions _A_(_s_), transition model _P_(_s′_ | _s_, _a_) \n",
" __local variables__: _U_, a vector of utilities for states in _S_, initially zero \n",
"        _π_, a policy vector indexed by state, initially random \n",
"\n",
" __repeat__ \n",
"   _U_ ← POLICY\\-EVALUATION(_π_, _U_, _mdp_) \n",
"   _unchanged?_ ← true \n",
"   __for each__ state _s_ __in__ _S_ __do__ \n",
"     __if__ max<sub>_a_ ∈ _A_(_s_)</sub> Σ<sub>_s′_</sub> _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] > Σ<sub>_s′_</sub> _P_(_s′_ | _s_, _π_\\[_s_\\]) _U_\\[_s′_\\] __then do__ \n",
"       _π_\\[_s_\\] ← argmax<sub>_a_ ∈ _A_(_s_)</sub> Σ<sub>_s′_</sub> _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] \n",
"       _unchanged?_ ← false \n",
" __until__ _unchanged?_ \n",
" __return__ _π_ \n",
"\n",
"---\n",
"__Figure ??__ The policy iteration algorithm for calculating an optimal policy."
]
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"source": [
"## Sequential Decision Problems\n",
"\n",
"Now that we have the tools required to solve MDPs, let us see how Sequential Decision Problems can be solved step by step and how a few built-in tools in the GridMDP class help us better analyse the problem at hand. \n",
"As always, we will work with the grid world from **Figure 17.1** from the book.\n",
"\n",
"<br>This is the environment for our agent.\n",
"We assume for now that the environment is _fully observable_, so that the agent always knows where it is.\n",
"We also assume that the transitions are **Markovian**, that is, the probability of reaching state $s'$ from state $s$ depends only on $s$ and not on the history of earlier states.\n",
"Almost all stochastic decision problems can be reframed as a Markov Decision Process just by tweaking the definition of a _state_ for that particular problem.\n",
"<br>\n",
"However, the actions of our agent in this environment are unreliable. In other words, the motion of our agent is stochastic. \n",
"<br><br>\n",
"More specifically, the agent may - \n",
"* move correctly in the intended direction with a probability of _0.8_, \n",
"* move $90^\\circ$ to the right of the intended direction with a probability 0.1\n",
"* move $90^\\circ$ to the left of the intended direction with a probability 0.1\n",
"<br><br>\n",
"The agent stays put if it bumps into a wall.\n",
""
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"These properties of the agent are called the transition properties and are hardcoded into the GridMDP class as you can see below."
]
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"cell_type": "code",
"execution_count": 12,
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"<div class=\"highlight\"><pre><span></span> <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",
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" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
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"To completely define our task environment, we need to specify the utility function for the agent. \n",
"This is the function that gives the agent a rough estimate of how good being in a particular state is, or how much _reward_ an agent receives by being in that state.\n",
"The agent then tries to maximize the reward it gets.\n",
"As the decision problem is sequential, the utility function will depend on a sequence of states rather than on a single state.\n",
"For now, we simply stipulate that in each state $s$, the agent receives a finite reward $R(s)$.\n",
"\n",
"For any given state, the actions the agent can take are encoded as given below:\n",
"- Move Up: (0, 1)\n",
"- Move Down: (0, -1)\n",
"- Move Left: (-1, 0)\n",
"- Move Right: (1, 0)\n",
"- Do nothing: `None`\n",
"\n",
"We now wonder what a valid solution to the problem might look like. \n",
"We cannot have fixed action sequences as the environment is stochastic and we can eventually end up in an undesirable state.\n",
"Therefore, a solution must specify what the agent shoulddo for _any_ state the agent might reach.\n",
"<br>\n",
"Such a solution is known as a **policy** and is usually denoted by $\\pi$.\n",
"The **optimal policy** is the policy that yields the highest expected utility an is usually denoted by $\\pi^*$.\n",
"<br>\n",
"The `GridMDP` class has a useful method `to_arrows` that outputs a grid showing the direction the agent should move, given a policy.\n",
"We will use this later to better understand the properties of the environment."
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"\n",
"<div class=\"highlight\"><pre><span></span> <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",
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]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This method directly encodes the actions that the agent can take (described above) to characters representing arrows and shows it in a grid format for human visalization purposes. \n",
"It converts the received policy from a `dictionary` to a grid using the `to_grid` method."
]
},
{
"cell_type": "code",
"execution_count": 14,
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"metadata": {},
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{
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"\n",
"<div class=\"highlight\"><pre><span></span> <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",
"</pre></div>\n",
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"source": [
"psource(GridMDP.to_grid)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now that we have all the tools required and a good understanding of the agent and the environment, we consider some cases and see how the agent should behave for each case."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Case 1\n",
"---\n",
"R(s) = -0.04 in all states except terminal states"
]
},
{
"cell_type": "code",
"execution_count": 15,
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"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# Note that this environment is also initialized in mdp.py by default\n",
"sequential_decision_environment = 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)])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will use the `best_policy` function to find the best policy for this environment.\n",
"But, as you can see, `best_policy` requires a utility function as well.\n",
"We already know that the utility function can be found by `value_iteration`.\n",
"Hence, our best policy is:"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now use the `to_arrows` method to see how our agent should pick its actions in the environment."
]
},
{
"cell_type": "code",
"execution_count": 17,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"> > > .\n",
"^ None ^ .\n",
"^ > ^ <\n"
]
}
],
"source": [
"from utils import print_table\n",
"print_table(sequential_decision_environment.to_arrows(pi))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is exactly the output we expected\n",
"<br>\n",
"\n",
"<br>\n",
"Notice that, because the cost of taking a step is fairly small compared with the penalty for ending up in `(4, 2)` by accident, the optimal policy is conservative. \n",
"In state `(3, 1)` it recommends taking the long way round, rather than taking the shorter way and risking getting a large negative reward of -1 in `(4, 2)`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Case 2\n",
"---\n",
"R(s) = -0.4 in all states except in terminal states"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"sequential_decision_environment = GridMDP([[-0.4, -0.4, -0.4, +1],\n",
" [-0.4, None, -0.4, -1],\n",
" [-0.4, -0.4, -0.4, -0.4]],\n",
" terminals=[(3, 2), (3, 1)])"
]
},
{
"cell_type": "code",
"execution_count": 19,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"> > > .\n",
"^ None ^ .\n",
"^ > ^ <\n"
]
}
],
"source": [
"pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n",
"from utils import print_table\n",
"print_table(sequential_decision_environment.to_arrows(pi))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is exactly the output we expected\n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As the reward for each state is now more negative, life is certainly more unpleasant.\n",
"The agent takes the shortest route to the +1 state and is willing to risk falling into the -1 state by accident."
]
},
{
"cell_type": "markdown",