From bd23fb7eff26454b1c4ef882a7ebca4e00b06af9 Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Mon, 26 Feb 2018 17:35:38 -0500 Subject: [PATCH 01/11] Fixed typos and added inline LaTeX --- mdp.ipynb | 313 ++++++++++++++++++++++++++++++------------------------ 1 file changed, 174 insertions(+), 139 deletions(-) diff --git a/mdp.ipynb b/mdp.ipynb index 59d8b8e3a..7882d0f85 100644 --- a/mdp.ipynb +++ b/mdp.ipynb @@ -247,7 +247,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": { "collapsed": true }, @@ -279,7 +279,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, "metadata": { "collapsed": true }, @@ -316,7 +316,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 5, "metadata": { "collapsed": true }, @@ -525,16 +525,16 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "" + "" ] }, - "execution_count": 5, + "execution_count": 7, "metadata": {}, "output_type": "execute_result" } @@ -553,7 +553,7 @@ "\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", + "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", @@ -682,40 +682,40 @@ "source": [ "psource(value_iteration)" ] - }, + }, { "cell_type": "markdown", "metadata": {}, "source": [ - "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.
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", + "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.
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_. \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", + "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", + "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 over time.\n", - "We terminate the algorithm if the delta value is less than a threshold value determined by the hyperparameter _epsilon_.\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", + "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_, 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", + "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", "
For now, let us solve the **sequential_decision_environment** GridMDP using `value_iteration`." ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 9, "metadata": {}, "outputs": [ { @@ -734,7 +734,7 @@ " (3, 2): 1.0}" ] }, - "execution_count": 6, + "execution_count": 9, "metadata": {}, "output_type": "execute_result" } @@ -752,7 +752,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 10, "metadata": {}, "outputs": [ { @@ -781,7 +781,7 @@ "" ] }, - "execution_count": 2, + "execution_count": 10, "metadata": {}, "output_type": "execute_result" } @@ -795,23 +795,23 @@ "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_a_ ∈ _A_(_s_) 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 (__??__)." + "__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_a_ ∈ _A_(_s_) 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 (__??__)." ] }, { @@ -1366,18 +1366,13 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 11, "metadata": {}, - "outputs": [], - "source": [ - "pseudocode('Policy-Iteration')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### AIMA3e\n", + "outputs": [ + { + "data": { + "text/markdown": [ + "### 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", @@ -1395,6 +1390,42 @@ "\n", "---\n", "__Figure ??__ The policy iteration algorithm for calculating an optimal policy." + ], + "text/plain": [ + "" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "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_a_ ∈ _A_(_s_) Σ_s′_ _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] > Σ_s′_ _P_(_s′_ | _s_, _π_\\[_s_\\]) _U_\\[_s′_\\] __then do__ \n", + "       _π_\\[_s_\\] ← argmax_a_ ∈ _A_(_s_) Σ_s′_ _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." ] }, { @@ -1410,12 +1441,16 @@ "![title](images/grid_mdp.jpg)\n", "
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_ only on _s_ and not on the history of earlier states.\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", "
\n", - "However, the actions of our agent in this environment are unreliable.\n", - "In other words, the motion of our agent is stochastic. \n", - "More specifically, the agent does the intended action with a probability of _0.8_, but with probability _0.1_, it moves to the right and with probability _0.1_ it moves to the left of the intended direction.\n", + "However, the actions of our agent in this environment are unreliable. In other words, the motion of our agent is stochastic. \n", + "

\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", + "

\n", "The agent stays put if it bumps into a wall.\n", "![title](images/grid_mdp_agent.jpg)" ] @@ -1429,7 +1464,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 12, "metadata": {}, "outputs": [ { @@ -1552,7 +1587,7 @@ "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", + "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", @@ -1565,9 +1600,9 @@ "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", "
\n", - "Such a solution is known as a **policy** and is usually denoted by **π**.\n", + "Such a solution is known as a **policy** and is usually denoted by $\\pi$.\n", "
\n", - "The **optimal policy** is the policy that yields the highest expected utility an is usually denoted by **π* **.\n", + "The **optimal policy** is the policy that yields the highest expected utility an is usually denoted by $\\pi^*$.\n", "
\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." @@ -1575,7 +1610,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 13, "metadata": {}, "outputs": [ { @@ -1697,7 +1732,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 14, "metadata": {}, "outputs": [ { @@ -1828,7 +1863,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 15, "metadata": { "collapsed": true }, @@ -1853,7 +1888,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 16, "metadata": { "collapsed": true }, @@ -1871,7 +1906,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 17, "metadata": {}, "outputs": [ { @@ -1898,7 +1933,7 @@ "![title](images/-0.04.jpg)\n", "
\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)`" + "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)`." ] }, { @@ -1912,7 +1947,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 18, "metadata": { "collapsed": true }, @@ -1926,7 +1961,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 19, "metadata": {}, "outputs": [ { @@ -1972,7 +2007,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 20, "metadata": { "collapsed": true }, @@ -1986,7 +2021,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 21, "metadata": {}, "outputs": [ { @@ -2017,7 +2052,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The living reward for each state is now more negative than the most negative terminal. Life is so painful that the agent heads for the nearest exit as even the worst exit is less painful than the current state." + "The living reward for each state is now lower than the least rewarding terminal. Life is so _painful_ that the agent heads for the nearest exit as even the worst exit is less painful than any living state." ] }, { @@ -2031,7 +2066,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 22, "metadata": { "collapsed": true }, @@ -2045,7 +2080,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 23, "metadata": {}, "outputs": [ { @@ -2141,7 +2176,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.3" + "version": "3.6.1" }, "widgets": { "state": { @@ -2166,7 +2201,7 @@ "022a5fdfc8e44fb09b21c4bd5b67a0db": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2197,7 +2232,7 @@ "0675230fb92f4539bc257b768fb4cd10": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2213,7 +2248,7 @@ "0783e74a8c2b40cc9b0f5706271192f4": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2241,7 +2276,7 @@ "098f12158d844cdf89b29a4cd568fda0": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2266,7 +2301,7 @@ "0b65fb781274495ab498ad518bc274d4": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2375,7 +2410,7 @@ "1af711fe8e4f43f084cef6c89eec40ae": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2391,7 +2426,7 @@ "1c5c913acbde4e87a163abb2e24e6e38": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2416,7 +2451,7 @@ "200e3ebead3d4858a47e2f6d345ca395": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2534,7 +2569,7 @@ "2d3acd8872c342eab3484302cac2cb05": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2544,7 +2579,7 @@ "2e1351ad05384d058c90e594bc6143c1": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2557,7 +2592,7 @@ "2f5438f1b34046a597a467effd43df11": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2594,7 +2629,7 @@ "319425ba805346f5ba366c42e220f9c6": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2613,7 +2648,7 @@ "332a89c03bfb49c2bb291051d172b735": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2662,7 +2697,7 @@ "388571e8e0314dfab8e935b7578ba7f9": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2684,7 +2719,7 @@ "3a21291c8e7249e3b04417d31b0447cf": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2697,7 +2732,7 @@ "3b22d68709b046e09fe70f381a3944cd": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2707,7 +2742,7 @@ "3c1b2ec10a9041be8a3fad9da78ff9f6": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2732,7 +2767,7 @@ "3e5b9fd779574270bf58101002c152ce": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2742,7 +2777,7 @@ "3e8bb05434cb4a0291383144e4523840": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2791,7 +2826,7 @@ "428e42f04a1e4347a1f548379c68f91b": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2807,7 +2842,7 @@ "4379175239b34553bf45c8ef9443ac55": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2820,7 +2855,7 @@ "4421c121414d464bb3bf1b5f0e86c37b": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2851,7 +2886,7 @@ "4731208453424514b471f862804d9bb8": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2900,7 +2935,7 @@ "4d281cda33fa489d86228370e627a5b0": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2919,7 +2954,7 @@ "4ec035cba73647358d416615cf4096ee": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2944,7 +2979,7 @@ "5141ae07149b46909426208a30e2861e": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -2981,7 +3016,7 @@ "55a1b0b794f44ac796bc75616f65a2a1": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3042,7 +3077,7 @@ "595c537ed2514006ac823b4090cf3b4b": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3103,7 +3138,7 @@ "5f823979d2ce4c34ba18b4ca674724e4": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3143,14 +3178,14 @@ "644dcff39d7c47b7b8b729d01f59bee5": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, "6455faf9dbc6477f8692528e6eb90c9a": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3163,7 +3198,7 @@ "665ed2b201144d78a5a1f57894c2267c": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3206,7 +3241,7 @@ "6a28f605a5d14589907dba7440ede2fc": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3231,7 +3266,7 @@ "6d7effd6bc4c40a4b17bf9e136c5814c": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3280,7 +3315,7 @@ "72dfe79a3e52429da1cf4382e78b2144": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3311,7 +3346,7 @@ "75e344508b0b45d1a9ae440549d95b1a": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3369,7 +3404,7 @@ "7f2f98bbffc0412dbb31c387407a9fed": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3400,7 +3435,7 @@ "82e2820c147a4dff85a01bcddbad8645": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3503,21 +3538,21 @@ "8cffde5bdb3d4f7597131b048a013929": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, "8db2abcad8bc44df812d6ccf2d2d713c": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, "8dd5216b361c44359ba1233ee93683a4": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3563,7 +3598,7 @@ "933904217b6045c1b654b7e5749203f5": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3591,7 +3626,7 @@ "94f2b877a79142839622a61a3a081c03": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3613,7 +3648,7 @@ "97207358fc65430aa196a7ed78b252f0": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3626,7 +3661,7 @@ "986c6c4e92964759903d6eb7f153df8a": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3669,14 +3704,14 @@ "9d5e9658af264ad795f6a5f3d8c3c30f": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, "9d7aa65511b6482d9587609ad7898f54": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3695,7 +3730,7 @@ "9efb46d2bb0648f6b109189986f4f102": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3711,7 +3746,7 @@ "9f43f85a0fb9464e9b7a25a85f6dba9c": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3724,7 +3759,7 @@ "9faa50b44e1842e0acac301f93a129c4": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3749,7 +3784,7 @@ "a1840ca22d834df2b145151baf6d8241": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3786,7 +3821,7 @@ "a39cfb47679c4d2895cda12c6d9d2975": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3817,7 +3852,7 @@ "a87c651448f14ce4958d73c2f1e413e1": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3926,7 +3961,7 @@ "b7e4c497ff5c4173961ffdc3bd3821a9": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3951,7 +3986,7 @@ "b9c138598fce460692cc12650375ee52": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -3970,7 +4005,7 @@ "bbe5dea9d57d466ba4e964fce9af13cf": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4004,7 +4039,7 @@ "beb0c9b29d8d4d69b3147af666fa298b": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4071,7 +4106,7 @@ "c74bbd55a8644defa3fcef473002a626": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4138,7 +4173,7 @@ "ce3a0e82e80d48b9b2658e0c52196644": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4148,7 +4183,7 @@ "ce8d3cd3535b459c823da2f49f3cc526": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4218,7 +4253,7 @@ "d83329fe36014f85bb5d0247d3ae4472": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4252,7 +4287,7 @@ "dc7376a2272e44179f237e5a1c7f6a49": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4349,7 +4384,7 @@ "e4e5dd3dc28d4aa3ab8f8f7c4a475115": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4365,7 +4400,7 @@ "e64ab85e80184b70b69d01a9c6851943": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4462,7 +4497,7 @@ "f262055f3f1b48029f9e2089f752b0b8": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4493,7 +4528,7 @@ "f3df35ce53e0466e81a48234b36a1430": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4572,7 +4607,7 @@ "f9458080ed534d25856c67ce8f93d5a1": { "views": [ { - "cell_index": 27.0 + "cell_index": 27 } ] }, @@ -4633,4 +4668,4 @@ }, "nbformat": 4, "nbformat_minor": 1 -} \ No newline at end of file +} From 9d5ec3c0e1d0c03cd1333afcbd6bbc35daf30c21 Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Mon, 26 Feb 2018 20:39:37 -0500 Subject: [PATCH 02/11] Fixed backslash for inline LaTeX --- mdp.ipynb | 1441 +++-------------------------------------------------- 1 file changed, 75 insertions(+), 1366 deletions(-) diff --git a/mdp.ipynb b/mdp.ipynb index 7882d0f85..3f2ee30a7 100644 --- a/mdp.ipynb +++ b/mdp.ipynb @@ -1,7 +1,7 @@ { "cells": [ { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "# Markov decision processes (MDPs)\n", @@ -10,19 +10,17 @@ ] }, { - "cell_type": "code", - "execution_count": 1, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "from mdp import *\n", "from notebook import psource, pseudocode" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "## CONTENTS\n", @@ -36,7 +34,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "## OVERVIEW\n", @@ -56,7 +54,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "## MDP\n", @@ -65,162 +63,14 @@ ] }, { - "cell_type": "code", - "execution_count": 2, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
class MDP:\n",
-       "\n",
-       "    """A Markov Decision Process, defined by an initial state, transition model,\n",
-       "    and reward function. We also keep track of a gamma value, for use by\n",
-       "    algorithms. The transition model is represented somewhat differently from\n",
-       "    the text. Instead of P(s' | s, a) being a probability number for each\n",
-       "    state/state/action triplet, we instead have T(s, a) return a\n",
-       "    list of (p, s') pairs. We also keep track of the possible states,\n",
-       "    terminal states, and actions for each state. [page 646]"""\n",
-       "\n",
-       "    def __init__(self, init, actlist, terminals, transitions={}, states=None, gamma=.9):\n",
-       "        if not (0 < gamma <= 1):\n",
-       "            raise ValueError("An MDP must have 0 < gamma <= 1")\n",
-       "\n",
-       "        if states:\n",
-       "            self.states = states\n",
-       "        else:\n",
-       "            self.states = set()\n",
-       "        self.init = init\n",
-       "        self.actlist = actlist\n",
-       "        self.terminals = terminals\n",
-       "        self.transitions = transitions\n",
-       "        self.gamma = gamma\n",
-       "        self.reward = {}\n",
-       "\n",
-       "    def R(self, state):\n",
-       "        """Return a numeric reward for this state."""\n",
-       "        return self.reward[state]\n",
-       "\n",
-       "    def T(self, state, action):\n",
-       "        """Transition model. From a state and an action, return a list\n",
-       "        of (probability, result-state) pairs."""\n",
-       "        if(self.transitions == {}):\n",
-       "            raise ValueError("Transition model is missing")\n",
-       "        else:\n",
-       "            return self.transitions[state][action]\n",
-       "\n",
-       "    def actions(self, state):\n",
-       "        """Set of actions that can be performed in this state. By default, a\n",
-       "        fixed list of actions, except for terminal states. Override this\n",
-       "        method if you need to specialize by state."""\n",
-       "        if state in self.terminals:\n",
-       "            return [None]\n",
-       "        else:\n",
-       "            return self.actlist\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(MDP)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "The **_ _init_ _** method takes in the following parameters:\n", @@ -238,7 +88,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "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", @@ -246,12 +96,10 @@ ] }, { - "cell_type": "code", - "execution_count": 3, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "# Transition Matrix as nested dict. State -> Actions in state -> States by each action -> Probabilty\n", "t = {\n", @@ -278,12 +126,10 @@ ] }, { - "cell_type": "code", - "execution_count": 4, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "class CustomMDP(MDP):\n", "\n", @@ -308,32 +154,30 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "Finally we instantize the class with the parameters for our MDP in the picture." ] }, { - "cell_type": "code", - "execution_count": 5, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "our_mdp = CustomMDP(t, rewards, terminals, init, gamma=.9)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "With this we have successfully represented our MDP. Later we will look at ways to solve this MDP." ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "## GRID MDP\n", @@ -342,160 +186,14 @@ ] }, { - "cell_type": "code", - "execution_count": 6, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
class GridMDP(MDP):\n",
-       "\n",
-       "    """A two-dimensional grid MDP, as in [Figure 17.1]. All you have to do is\n",
-       "    specify the grid as a list of lists of rewards; use None for an obstacle\n",
-       "    (unreachable state). Also, you should specify the terminal states.\n",
-       "    An action is an (x, y) unit vector; e.g. (1, 0) means move east."""\n",
-       "\n",
-       "    def __init__(self, grid, terminals, init=(0, 0), gamma=.9):\n",
-       "        grid.reverse()  # because we want row 0 on bottom, not on top\n",
-       "        MDP.__init__(self, init, actlist=orientations,\n",
-       "                     terminals=terminals, gamma=gamma)\n",
-       "        self.grid = grid\n",
-       "        self.rows = len(grid)\n",
-       "        self.cols = len(grid[0])\n",
-       "        for x in range(self.cols):\n",
-       "            for y in range(self.rows):\n",
-       "                self.reward[x, y] = grid[y][x]\n",
-       "                if grid[y][x] is not None:\n",
-       "                    self.states.add((x, y))\n",
-       "\n",
-       "    def T(self, state, action):\n",
-       "        if action is None:\n",
-       "            return [(0.0, state)]\n",
-       "        else:\n",
-       "            return [(0.8, self.go(state, action)),\n",
-       "                    (0.1, self.go(state, turn_right(action))),\n",
-       "                    (0.1, self.go(state, turn_left(action)))]\n",
-       "\n",
-       "    def go(self, state, direction):\n",
-       "        """Return the state that results from going in this direction."""\n",
-       "        state1 = vector_add(state, direction)\n",
-       "        return state1 if state1 in self.states else state\n",
-       "\n",
-       "    def to_grid(self, mapping):\n",
-       "        """Convert a mapping from (x, y) to v into a [[..., v, ...]] grid."""\n",
-       "        return list(reversed([[mapping.get((x, y), None)\n",
-       "                               for x in range(self.cols)]\n",
-       "                              for y in range(self.rows)]))\n",
-       "\n",
-       "    def to_arrows(self, policy):\n",
-       "        chars = {\n",
-       "            (1, 0): '>', (0, 1): '^', (-1, 0): '<', (0, -1): 'v', None: '.'}\n",
-       "        return self.to_grid({s: chars[a] for (s, a) in policy.items()})\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(GridMDP)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "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", @@ -510,7 +208,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "We can create a GridMDP like the one in **Fig 17.1** as follows: \n", @@ -524,27 +222,14 @@ ] }, { - "cell_type": "code", - "execution_count": 7, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ "sequential_decision_environment" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": { "collapsed": true }, @@ -553,7 +238,7 @@ "\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", + "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", @@ -561,130 +246,14 @@ ] }, { - "cell_type": "code", - "execution_count": 8, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def value_iteration(mdp, epsilon=0.001):\n",
-       "    """Solving an MDP by value iteration. [Figure 17.4]"""\n",
-       "    U1 = {s: 0 for s in mdp.states}\n",
-       "    R, T, gamma = mdp.R, mdp.T, mdp.gamma\n",
-       "    while True:\n",
-       "        U = U1.copy()\n",
-       "        delta = 0\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",
-       "            delta = max(delta, abs(U1[s] - U[s]))\n",
-       "        if delta < epsilon * (1 - gamma) / gamma:\n",
-       "            return U\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(value_iteration)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "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.
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", @@ -706,92 +275,36 @@ "\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", + "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", + "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", "
For now, let us solve the **sequential_decision_environment** GridMDP using `value_iteration`." ] }, { - "cell_type": "code", - "execution_count": 9, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "{(0, 0): 0.2962883154554812,\n", - " (0, 1): 0.3984432178350045,\n", - " (0, 2): 0.5093943765842497,\n", - " (1, 0): 0.25386699846479516,\n", - " (1, 2): 0.649585681261095,\n", - " (2, 0): 0.3447542300124158,\n", - " (2, 1): 0.48644001739269643,\n", - " (2, 2): 0.7953620878466678,\n", - " (3, 0): 0.12987274656746342,\n", - " (3, 1): -1.0,\n", - " (3, 2): 1.0}" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ "value_iteration(sequential_decision_environment)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "The pseudocode for the algorithm:" ] }, { - "cell_type": "code", - "execution_count": 10, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "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_a_ ∈ _A_(_s_) 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 (__??__)." - ], - "text/plain": [ - "" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ "pseudocode(\"Value-Iteration\")" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "### AIMA3e\n", @@ -815,7 +328,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "## VALUE ITERATION VISUALIZATION\n", @@ -824,12 +337,10 @@ ] }, { - "cell_type": "code", - "execution_count": 7, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "def value_iteration_instru(mdp, iterations=20):\n", " U_over_time = []\n", @@ -845,19 +356,17 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "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", - "execution_count": 8, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "columns = 4\n", "rows = 3\n", @@ -865,12 +374,10 @@ ] }, { - "cell_type": "code", - "execution_count": 9, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "%matplotlib inline\n", "from notebook import make_plot_grid_step_function\n", @@ -879,35 +386,10 @@ ] }, { - "cell_type": "code", - "execution_count": 10, + "cell_type": "raw", "metadata": { "scrolled": true }, - "outputs": [ - { - "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAATcAAADuCAYAAABcZEBhAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nACYRNwAmETcAJv0F9s8EDYqi1wAAAAAASUVORK5CYII=\n", - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "Widget Javascript not detected. It may not be installed or enabled properly.\n" - ] - }, - { - "data": {}, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "import ipywidgets as widgets\n", "from IPython.display import display\n", @@ -926,14 +408,14 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "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." ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": { "collapsed": true }, @@ -960,244 +442,21 @@ ] }, { - "cell_type": "code", - "execution_count": 15, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def expected_utility(a, s, U, mdp):\n",
-       "    """The expected utility of doing a in state s, according to the MDP and U."""\n",
-       "    return sum([p * U[s1] for (p, s1) in mdp.T(s, a)])\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(expected_utility)" ] }, { - "cell_type": "code", - "execution_count": 16, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def policy_iteration(mdp):\n",
-       "    """Solve an MDP by policy iteration [Figure 17.7]"""\n",
-       "    U = {s: 0 for s in mdp.states}\n",
-       "    pi = {s: random.choice(mdp.actions(s)) for s in mdp.states}\n",
-       "    while True:\n",
-       "        U = policy_evaluation(pi, U, mdp)\n",
-       "        unchanged = True\n",
-       "        for s in mdp.states:\n",
-       "            a = argmax(mdp.actions(s), key=lambda a: expected_utility(a, s, U, mdp))\n",
-       "            if a != pi[s]:\n",
-       "                pi[s] = a\n",
-       "                unchanged = False\n",
-       "        if unchanged:\n",
-       "            return pi\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(policy_iteration)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "
Fortunately, it is not necessary to do _exact_ policy evaluation. \n", @@ -1210,202 +469,35 @@ ] }, { - "cell_type": "code", - "execution_count": 17, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def policy_evaluation(pi, U, mdp, k=20):\n",
-       "    """Return an updated utility mapping U from each state in the MDP to its\n",
-       "    utility, using an approximation (modified policy iteration)."""\n",
-       "    R, T, gamma = mdp.R, mdp.T, mdp.gamma\n",
-       "    for i in range(k):\n",
-       "        for s in mdp.states:\n",
-       "            U[s] = R(s) + gamma * sum([p * U[s1] for (p, s1) in T(s, pi[s])])\n",
-       "    return U\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(policy_evaluation)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "Let us now solve **`sequential_decision_environment`** using `policy_iteration`." ] }, { - "cell_type": "code", - "execution_count": 18, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "{(0, 0): (0, 1),\n", - " (0, 1): (0, 1),\n", - " (0, 2): (1, 0),\n", - " (1, 0): (1, 0),\n", - " (1, 2): (1, 0),\n", - " (2, 0): (0, 1),\n", - " (2, 1): (0, 1),\n", - " (2, 2): (1, 0),\n", - " (3, 0): (-1, 0),\n", - " (3, 1): None,\n", - " (3, 2): None}" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ "policy_iteration(sequential_decision_environment)" ] }, { - "cell_type": "code", - "execution_count": 11, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/markdown": [ - "### 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_a_ ∈ _A_(_s_) Σ_s′_ _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] > Σ_s′_ _P_(_s′_ | _s_, _π_\\[_s_\\]) _U_\\[_s′_\\] __then do__ \n", - "       _π_\\[_s_\\] ← argmax_a_ ∈ _A_(_s_) Σ_s′_ _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." - ], - "text/plain": [ - "" - ] - }, - "execution_count": 11, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ "pseudocode('Policy-Iteration')" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "### AIMA3e\n", @@ -1429,7 +521,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": { "collapsed": true }, @@ -1456,131 +548,21 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "These properties of the agent are called the transition properties and are hardcoded into the GridMDP class as you can see below." ] }, { - "cell_type": "code", - "execution_count": 12, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
    def T(self, state, action):\n",
-       "        if action is None:\n",
-       "            return [(0.0, state)]\n",
-       "        else:\n",
-       "            return [(0.8, self.go(state, action)),\n",
-       "                    (0.1, self.go(state, turn_right(action))),\n",
-       "                    (0.1, self.go(state, turn_left(action)))]\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(GridMDP.T)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "To completely define our task environment, we need to specify the utility function for the agent. \n", @@ -1609,121 +591,14 @@ ] }, { - "cell_type": "code", - "execution_count": 13, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
    def to_arrows(self, policy):\n",
-       "        chars = {\n",
-       "            (1, 0): '>', (0, 1): '^', (-1, 0): '<', (0, -1): 'v', None: '.'}\n",
-       "        return self.to_grid({s: chars[a] for (s, a) in policy.items()})\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(GridMDP.to_arrows)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "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", @@ -1731,129 +606,21 @@ ] }, { - "cell_type": "code", - "execution_count": 14, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
    def to_grid(self, mapping):\n",
-       "        """Convert a mapping from (x, y) to v into a [[..., v, ...]] grid."""\n",
-       "        return list(reversed([[mapping.get((x, y), None)\n",
-       "                               for x in range(self.cols)]\n",
-       "                              for y in range(self.rows)]))\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], "source": [ "psource(GridMDP.to_grid)" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "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", + "cell_type": "raw", "metadata": {}, "source": [ "### Case 1\n", @@ -1862,12 +629,10 @@ ] }, { - "cell_type": "code", - "execution_count": 15, + "cell_type": "raw", "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", @@ -1877,7 +642,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "We will use the `best_policy` function to find the best policy for this environment.\n", @@ -1887,45 +652,31 @@ ] }, { - "cell_type": "code", - "execution_count": 16, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))" ] }, { - "cell_type": "markdown", + "cell_type": "raw", "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, + "cell_type": "raw", "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", + "cell_type": "raw", "metadata": {}, "source": [ "This is exactly the output we expected\n", @@ -1937,7 +688,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "### Case 2\n", @@ -1946,12 +697,10 @@ ] }, { - "cell_type": "code", - "execution_count": 18, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "sequential_decision_environment = GridMDP([[-0.4, -0.4, -0.4, +1],\n", " [-0.4, None, -0.4, -1],\n", @@ -1960,20 +709,8 @@ ] }, { - "cell_type": "code", - "execution_count": 19, + "cell_type": "raw", "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", @@ -1981,7 +718,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "This is exactly the output we expected\n", @@ -1989,7 +726,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "As the reward for each state is now more negative, life is certainly more unpleasant.\n", @@ -1997,7 +734,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "### Case 3\n", @@ -2006,12 +743,10 @@ ] }, { - "cell_type": "code", - "execution_count": 20, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "sequential_decision_environment = GridMDP([[-4, -4, -4, +1],\n", " [-4, None, -4, -1],\n", @@ -2020,20 +755,8 @@ ] }, { - "cell_type": "code", - "execution_count": 21, + "cell_type": "raw", "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", @@ -2041,7 +764,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "This is exactly the output we expected\n", @@ -2049,14 +772,14 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "The living reward for each state is now lower than the least rewarding terminal. Life is so _painful_ that the agent heads for the nearest exit as even the worst exit is less painful than any living state." ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "### Case 4\n", @@ -2065,12 +788,10 @@ ] }, { - "cell_type": "code", - "execution_count": 22, + "cell_type": "raw", "metadata": { "collapsed": true }, - "outputs": [], "source": [ "sequential_decision_environment = GridMDP([[4, 4, 4, +1],\n", " [4, None, 4, -1],\n", @@ -2079,20 +800,8 @@ ] }, { - "cell_type": "code", - "execution_count": 23, + "cell_type": "raw", "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "> > < .\n", - "> None < .\n", - "> > > v\n" - ] - } - ], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", "from utils import print_table\n", @@ -2100,7 +809,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "In this case, the output we expect is\n", @@ -2117,7 +826,7 @@ ] }, { - "cell_type": "markdown", + "cell_type": "raw", "metadata": {}, "source": [ "---\n", From d8af22bd3f5f70a9d2f077aa1c1a7266d933a847 Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Mon, 26 Feb 2018 20:53:36 -0500 Subject: [PATCH 03/11] Fixed more backslashes --- mdp.ipynb | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/mdp.ipynb b/mdp.ipynb index 7882d0f85..910b49040 100644 --- a/mdp.ipynb +++ b/mdp.ipynb @@ -697,12 +697,12 @@ "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", + "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", + "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", From 548c1cba2507c11150040c8eae503dfef93a72a5 Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Tue, 27 Feb 2018 17:01:32 -0500 Subject: [PATCH 04/11] generalised MDP class and created POMDP notebook --- pomdp.ipynb | 231 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 231 insertions(+) create mode 100644 pomdp.ipynb diff --git a/pomdp.ipynb b/pomdp.ipynb new file mode 100644 index 000000000..8934ef6b6 --- /dev/null +++ b/pomdp.ipynb @@ -0,0 +1,231 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Partially Observable Markov decision processes (POMDPs)\n", + "\n", + "This Jupyter notebook acts as supporting material for POMDPs, covered in **Chapter 17 Making Complex Decisions** of the book* Artificial Intelligence: A Modern Approach*. We make use of the implementations of POMPDPs in mdp.py module. This notebook has been separated from the notebook `mdp.py` as the topics are considerably more advanced.\n", + "\n", + "**Note that it is essential to work through and understand the mdp.ipynb notebook before diving into this one.**\n", + "\n", + "Let us import everything from the mdp module to get started." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "from mdp import *\n", + "from notebook import psource, pseudocode" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## CONTENTS\n", + "\n", + "1. Overview of MDPs\n", + "2. POMDPs - a conceptual outline\n", + "3. POMDPs - a rigorous outline\n", + "4. Value Iteration\n", + " - Value Iteration Visualization" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 1. OVERVIEW\n", + "\n", + "We first review Markov property and MDPs as in [Section 17.1] of the book.\n", + "\n", + "- A stochastic process is said to have the **Markov property**, or to have a **Markovian transition model** if the conditional probability distribution of future states of the process (conditional on both past and present states) depends only on 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", + "A Markov decision process or MDP is defined as:\n", + "- a sequential decision problem for a fully observable, stochastic environment with a Markovian transition model and additive rewards.\n", + "\n", + "An MDP consists of a set of states (with an initial state $s_0$); a set $A(s)$ of actions\n", + "in each state; a transition model $P(s' | s, a)$; and a reward function $R(s)$.\n", + "\n", + "The MDP seeks to make sequential decisions to occupy states so as to maximise some combination of the reward function $R(s)$.\n", + "\n", + "The characteristic problem of the MDP is hence to identify the optimal policy function $\\pi^*(s)$ that provides the _utility-maximising_ action $a$ to be taken when the current state is $s$.\n", + "\n", + "### Belief vector\n", + "\n", + "**Note**: The book refers to the _belief vector_ as the _belief state_. We use the latter terminology here to retain our ability to refer to the belief vector as a _probability distribution over states_.\n", + "\n", + "The solution of an MDP is subject to certain properties of the problem which are assumed and justified in [Section 17.1]. One critical assumption is that the agent is **fully aware of its current state at all times**.\n", + "\n", + "A tedious (but rewarding, as we will see) way of expressing this is in terms of the **belief vector** $b$ of the agent. The belief vector is a function mapping states to probabilities or certainties of being in those states.\n", + "\n", + "Consider an agent that is fully aware that it is in state $s_i$ in the statespace $(s_1, s_2, ... s_n)$ at the current time.\n", + "\n", + "Its belief vector is the vector $(b(s_1), b(s_2), ... b(s_n))$ given by the function $b(s)$:\n", + "\\begin{align*}\n", + "b(s) &= 0 \\quad \\text{if }s \\neq s_i \\\\ &= 1 \\quad \\text{if } s = s_i\n", + "\\end{align*}\n", + "\n", + "Note that $b(s)$ is a probability distribution that necessarily sums to $1$ over all $s$.\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## 2. POMDPs - a conceptual outline\n", + "\n", + "The POMDP really has only two modifications to the **problem formulation** compared to the MDP.\n", + "\n", + "- **Belief state** - In the real world, the current state of an agent is often not known with complete certainty. This makes the concept of a belief vector extremely relevant. It allows the agent to represent different degrees of certainty with which it _believes_ it is in each state.\n", + "\n", + "- **Evidence percepts** - In the real world, agents often have certain kinds of evidence, collected from sensors. They can use the probability distribution of observed evidence, conditional on state, to consolidate their information. This is a known distribution $P(e\\ |\\ s)$ - $e$ being an evidence, and $s$ being the state it is conditional on.\n", + "\n", + "Consider the world we used for the MDP. \n", + "\n", + "![title](images/grid_mdp.jpg)\n", + "\n", + "#### Using the belief vector\n", + "An agent beginning at $(1, 1)$ may not be certain that it is indeed in $(1, 1)$. Consider a belief vector $b$ such that:\n", + "\\begin{align*}\n", + " b((1,1)) &= 0.8 \\\\\n", + " b((2,1)) &= 0.1 \\\\\n", + " b((1,2)) &= 0.1 \\\\\n", + " b(s) &= 0 \\quad \\quad \\forall \\text{ other } s\n", + "\\end{align*}\n", + "\n", + "By horizontally catenating each row, we can represent this as an 11-dimensional vector (omitting $(2, 2)$).\n", + "\n", + "Thus, taking $s_1 = (1, 1)$, $s_2 = (1, 2)$, ... $s_{11} = (4,3)$, we have $b$:\n", + "\n", + "$b = (0.8, 0.1, 0, 0, 0.1, 0, 0, 0, 0, 0, 0)$ \n", + "\n", + "This fully represents the certainty to which the agent is aware of its state.\n", + "\n", + "#### Using evidence\n", + "The evidence observed here could be the number of adjacent 'walls' or 'dead ends' observed by the agent. We assume that the agent cannot 'orient' the walls - only count them.\n", + "\n", + "In this case, $e$ can take only two values, 1 and 2. This gives $P(e\\ |\\ s)$ as:\n", + "\\begin{align*}\n", + " P(e=2\\ |\\ s) &= \\frac{1}{7} \\quad \\forall \\quad s \\in \\{s_1, s_2, s_4, s_5, s_8, s_9, s_{11}\\}\\\\\n", + " P(e=1\\ |\\ s) &= \\frac{1}{4} \\quad \\forall \\quad s \\in \\{s_3, s_6, s_7, s_{10}\\} \\\\\n", + " P(e\\ |\\ s) &= 0 \\quad \\forall \\quad \\text{ other } s, e\n", + "\\end{align*}\n", + "\n", + "Note that the implications of the evidence on the state must be known **a priori** to the agent. Ways of reliably learning this distribution from percepts are beyond the scope of this notebook." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## 3. POMDPs - a rigorous outline\n", + "\n", + "A POMDP is thus a sequential decision problem for for a *partially* observable, stochastic environment with a Markovian transition model, a known 'sensor model' for inferring state from observation, and additive rewards. \n", + "\n", + "Practically, a POMDP has the following, which an MDP also has:\n", + "- a set of states, each denoted by $s$\n", + "- a set of actions available in each state, $A(s)$\n", + "- a reward accrued on attaining some state, $R(s)$\n", + "- a transition probability $P(s'\\ |\\ s, a)$ of action $a$ changing the state from $s$ to $s'$\n", + "\n", + "And the following, which an MDP does not:\n", + "- a sensor model $P(e\\ |\\ s)$ on evidence conditional on states\n", + "\n", + "Additionally, the POMDP is now uncertain of its current state hence has:\n", + "- a belief vector $b$ representing the certainty of being in each state (as a probability distribution)\n", + "\n", + "\n", + "#### New uncertainties\n", + "\n", + "It is useful to intuitively appreciate the new uncertainties that have arisen in the agent's awareness of its own state.\n", + "\n", + "- At any point, the agent has belief vector $b$, the distribution of its believed likelihood of being in each state $s$.\n", + "- For each of these states $s$ that the agent may **actually** be in, it has some set of actions given by $A(s)$.\n", + "- Each of these actions may transport it to some other state $s'$, assuming an initial state $s$, with probability $P(s'\\ |\\ s, a)$\n", + "- Once the action is performed, the agent receives a percept $e$. $P(e\\ |\\ s)$ now tells it the chances of having perceived $e$ for each state $s$. The agent must use this information to update its new belief state appropriately.\n", + "\n", + "#### Evolution of the belief vector - the `FORWARD` function\n", + "\n", + "The new belief vector $b'(s')$ after an action $a$ on the belief vector $b(s)$ and the noting of evidence $e$ is:\n", + "$$ b'(s') = \\alpha P(e\\ |\\ s') \\sum_s P(s'\\ | s, a) b(s)$$ \n", + "\n", + "where $\\alpha$ is a normalising constant (to retain the interpretation of $b$ as a probability distribution.\n", + "\n", + "This equation is just counts the sum of likelihoods of going to a state $s'$ from every possible state $s$, times the initial likelihood of being in each $s$. This is multiplied by the likelihood that the known evidence actually implies the new state $s'$. \n", + "\n", + "This function is represented as `b' = FORWARD(b, a, e)`\n", + "\n", + "#### Probability distribution of the evolving belief vector\n", + "\n", + "The goal here is to find $P(b'\\ |\\ b, a)$ - the probability that action $a$ transforms belief vector $b$ into belief vector $b'$. The following steps illustrate this -\n", + "\n", + "The probability of observing evidence $e$ when action $a$ is enacted on belief vector $b$ can be distributed over each possible new state $s'$ resulting from it:\n", + "\\begin{align*}\n", + " P(e\\ |\\ b, a) &= \\sum_{s'} P(e\\ |\\ b, a, s') P(s'\\ |\\ b, a) \\\\\n", + " &= \\sum_{s'} P(e\\ |\\ s') P(s'\\ |\\ b, a) \\\\\n", + " &= \\sum_{s'} P(e\\ |\\ s') \\sum_s P(s'\\ |\\ s, a) b(s)\n", + "\\end{align*}\n", + "\n", + "The probability of getting belief vector $b'$ from $b$ by application of action $a$ can thus be summed over all possible evidences $e$:\n", + "\\begin{align*}\n", + " P(b'\\ |\\ b, a) &= \\sum_{e} P(b'\\ |\\ b, a, e) P(e\\ |\\ b, a) \\\\\n", + " &= \\sum_{e} P(b'\\ |\\ b, a, e) \\sum_{s'} P(e\\ |\\ s') \\sum_s P(s'\\ |\\ s, a) b(s)\n", + "\\end{align*}\n", + "\n", + "where $P(b'\\ |\\ b, a, e) = 1$ if $b' = $ `FORWARD(b, a, e)` and $= 0$ otherwise.\n", + "\n", + "Given initial and final belief states $b$ and $b'$, the transition probabilities still depend on the action $a$ and observed evidence $e$. Some belief states may be achievable by certain actions, but have non-zero probabilities for states prohibited by the evidence $e$. Thus, the above condition thus ensures that only valid combinations of $(b', b, a, e)$ are considered.\n", + "\n", + "#### A modified rewardspace\n", + "\n", + "For MDPs, the reward space was simple - one reward per available state. However, for a belief vector $b(s)$, the expected reward is now:\n", + "$$\\rho(b) = \\sum_s b(s) R(s)$$\n", + "\n", + "Thus, as the belief vector can take infinite values of the distribution over states, so can the reward for each belief vector vary over a hyperplane in the belief space, or space of states (planes in an $N$-dimensional space are formed by a linear combination of the axes)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.1" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} From d04e14dd1e8eb5a27e1c736b1061b60a60565b4a Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Tue, 27 Feb 2018 19:22:50 -0500 Subject: [PATCH 05/11] Fixed consistency issues with base MDP class --- mdp.ipynb | 1360 +++-------------------------------------------------- mdp.py | 77 ++- 2 files changed, 123 insertions(+), 1314 deletions(-) diff --git a/mdp.ipynb b/mdp.ipynb index 910b49040..e24a5d234 100644 --- a/mdp.ipynb +++ b/mdp.ipynb @@ -11,7 +11,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": null, "metadata": { "collapsed": true }, @@ -66,155 +66,9 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
class MDP:\n",
-       "\n",
-       "    """A Markov Decision Process, defined by an initial state, transition model,\n",
-       "    and reward function. We also keep track of a gamma value, for use by\n",
-       "    algorithms. The transition model is represented somewhat differently from\n",
-       "    the text. Instead of P(s' | s, a) being a probability number for each\n",
-       "    state/state/action triplet, we instead have T(s, a) return a\n",
-       "    list of (p, s') pairs. We also keep track of the possible states,\n",
-       "    terminal states, and actions for each state. [page 646]"""\n",
-       "\n",
-       "    def __init__(self, init, actlist, terminals, transitions={}, states=None, gamma=.9):\n",
-       "        if not (0 < gamma <= 1):\n",
-       "            raise ValueError("An MDP must have 0 < gamma <= 1")\n",
-       "\n",
-       "        if states:\n",
-       "            self.states = states\n",
-       "        else:\n",
-       "            self.states = set()\n",
-       "        self.init = init\n",
-       "        self.actlist = actlist\n",
-       "        self.terminals = terminals\n",
-       "        self.transitions = transitions\n",
-       "        self.gamma = gamma\n",
-       "        self.reward = {}\n",
-       "\n",
-       "    def R(self, state):\n",
-       "        """Return a numeric reward for this state."""\n",
-       "        return self.reward[state]\n",
-       "\n",
-       "    def T(self, state, action):\n",
-       "        """Transition model. From a state and an action, return a list\n",
-       "        of (probability, result-state) pairs."""\n",
-       "        if(self.transitions == {}):\n",
-       "            raise ValueError("Transition model is missing")\n",
-       "        else:\n",
-       "            return self.transitions[state][action]\n",
-       "\n",
-       "    def actions(self, state):\n",
-       "        """Set of actions that can be performed in this state. By default, a\n",
-       "        fixed list of actions, except for terminal states. Override this\n",
-       "        method if you need to specialize by state."""\n",
-       "        if state in self.terminals:\n",
-       "            return [None]\n",
-       "        else:\n",
-       "            return self.actlist\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(MDP)" ] @@ -247,21 +101,21 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "# Transition Matrix as nested dict. State -> Actions in state -> States by each action -> Probabilty\n", + "# Transition Matrix as nested dict. State -> Actions in state -> List of (Probability, State) tuples\n", "t = {\n", " \"A\": {\n", - " \"X\": {\"A\":0.3, \"B\":0.7},\n", - " \"Y\": {\"A\":1.0}\n", + " \"X\": [(0.3, \"A\"), (0.7, \"B\")],\n", + " \"Y\": [(1.0, \"A\")]\n", " },\n", " \"B\": {\n", - " \"X\": {\"End\":0.8, \"B\":0.2},\n", - " \"Y\": {\"A\":1.0}\n", + " \"X\": {(0.8, \"End\"), (0.2, \"B\")},\n", + " \"Y\": {(1.0, \"A\")}\n", " },\n", " \"End\": {}\n", "}\n", @@ -279,26 +133,20 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "class CustomMDP(MDP):\n", - "\n", - " def __init__(self, transition_matrix, rewards, terminals, init, gamma=.9):\n", + " def __init__(self, init, terminals, reward, transition_matrix, gamma=.9):\n", " # All possible actions.\n", " actlist = []\n", " for state in transition_matrix.keys():\n", " actlist.extend(transition_matrix[state])\n", " actlist = list(set(actlist))\n", - "\n", - " MDP.__init__(self, init, actlist, terminals=terminals, gamma=gamma)\n", - " self.t = transition_matrix\n", - " self.reward = rewards\n", - " for state in self.t:\n", - " self.states.add(state)\n", + " MDP.__init__(self, init, actlist, terminals, reward, transition_matrix, gamma=gamma)\n", "\n", " def T(self, state, action):\n", " if action is None:\n", @@ -316,13 +164,13 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ - "our_mdp = CustomMDP(t, rewards, terminals, init, gamma=.9)" + "our_mdp = CustomMDP(init, terminals, rewards, t, gamma=.9)" ] }, { @@ -343,153 +191,9 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
class GridMDP(MDP):\n",
-       "\n",
-       "    """A two-dimensional grid MDP, as in [Figure 17.1]. All you have to do is\n",
-       "    specify the grid as a list of lists of rewards; use None for an obstacle\n",
-       "    (unreachable state). Also, you should specify the terminal states.\n",
-       "    An action is an (x, y) unit vector; e.g. (1, 0) means move east."""\n",
-       "\n",
-       "    def __init__(self, grid, terminals, init=(0, 0), gamma=.9):\n",
-       "        grid.reverse()  # because we want row 0 on bottom, not on top\n",
-       "        MDP.__init__(self, init, actlist=orientations,\n",
-       "                     terminals=terminals, gamma=gamma)\n",
-       "        self.grid = grid\n",
-       "        self.rows = len(grid)\n",
-       "        self.cols = len(grid[0])\n",
-       "        for x in range(self.cols):\n",
-       "            for y in range(self.rows):\n",
-       "                self.reward[x, y] = grid[y][x]\n",
-       "                if grid[y][x] is not None:\n",
-       "                    self.states.add((x, y))\n",
-       "\n",
-       "    def T(self, state, action):\n",
-       "        if action is None:\n",
-       "            return [(0.0, state)]\n",
-       "        else:\n",
-       "            return [(0.8, self.go(state, action)),\n",
-       "                    (0.1, self.go(state, turn_right(action))),\n",
-       "                    (0.1, self.go(state, turn_left(action)))]\n",
-       "\n",
-       "    def go(self, state, direction):\n",
-       "        """Return the state that results from going in this direction."""\n",
-       "        state1 = vector_add(state, direction)\n",
-       "        return state1 if state1 in self.states else state\n",
-       "\n",
-       "    def to_grid(self, mapping):\n",
-       "        """Convert a mapping from (x, y) to v into a [[..., v, ...]] grid."""\n",
-       "        return list(reversed([[mapping.get((x, y), None)\n",
-       "                               for x in range(self.cols)]\n",
-       "                              for y in range(self.rows)]))\n",
-       "\n",
-       "    def to_arrows(self, policy):\n",
-       "        chars = {\n",
-       "            (1, 0): '>', (0, 1): '^', (-1, 0): '<', (0, -1): 'v', None: '.'}\n",
-       "        return self.to_grid({s: chars[a] for (s, a) in policy.items()})\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(GridMDP)" ] @@ -525,20 +229,9 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "sequential_decision_environment" ] @@ -562,123 +255,9 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def value_iteration(mdp, epsilon=0.001):\n",
-       "    """Solving an MDP by value iteration. [Figure 17.4]"""\n",
-       "    U1 = {s: 0 for s in mdp.states}\n",
-       "    R, T, gamma = mdp.R, mdp.T, mdp.gamma\n",
-       "    while True:\n",
-       "        U = U1.copy()\n",
-       "        delta = 0\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",
-       "            delta = max(delta, abs(U1[s] - U[s]))\n",
-       "        if delta < epsilon * (1 - gamma) / gamma:\n",
-       "            return U\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(value_iteration)" ] @@ -715,30 +294,9 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "{(0, 0): 0.2962883154554812,\n", - " (0, 1): 0.3984432178350045,\n", - " (0, 2): 0.5093943765842497,\n", - " (1, 0): 0.25386699846479516,\n", - " (1, 2): 0.649585681261095,\n", - " (2, 0): 0.3447542300124158,\n", - " (2, 1): 0.48644001739269643,\n", - " (2, 2): 0.7953620878466678,\n", - " (3, 0): 0.12987274656746342,\n", - " (3, 1): -1.0,\n", - " (3, 2): 1.0}" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "value_iteration(sequential_decision_environment)" ] @@ -752,40 +310,9 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "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_a_ ∈ _A_(_s_) 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 (__??__)." - ], - "text/plain": [ - "" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "pseudocode(\"Value-Iteration\")" ] @@ -825,7 +352,7 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": null, "metadata": { "collapsed": true }, @@ -853,7 +380,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": null, "metadata": { "collapsed": true }, @@ -866,7 +393,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": null, "metadata": { "collapsed": true }, @@ -880,34 +407,11 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": null, "metadata": { "scrolled": true }, - "outputs": [ - { - "data": { - "image/png": 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- "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "Widget Javascript not detected. It may not be installed or enabled properly.\n" - ] - }, - { - "data": {}, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "import ipywidgets as widgets\n", "from IPython.display import display\n", @@ -961,237 +465,18 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def expected_utility(a, s, U, mdp):\n",
-       "    """The expected utility of doing a in state s, according to the MDP and U."""\n",
-       "    return sum([p * U[s1] for (p, s1) in mdp.T(s, a)])\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(expected_utility)" ] }, { "cell_type": "code", - "execution_count": 16, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def policy_iteration(mdp):\n",
-       "    """Solve an MDP by policy iteration [Figure 17.7]"""\n",
-       "    U = {s: 0 for s in mdp.states}\n",
-       "    pi = {s: random.choice(mdp.actions(s)) for s in mdp.states}\n",
-       "    while True:\n",
-       "        U = policy_evaluation(pi, U, mdp)\n",
-       "        unchanged = True\n",
-       "        for s in mdp.states:\n",
-       "            a = argmax(mdp.actions(s), key=lambda a: expected_utility(a, s, U, mdp))\n",
-       "            if a != pi[s]:\n",
-       "                pi[s] = a\n",
-       "                unchanged = False\n",
-       "        if unchanged:\n",
-       "            return pi\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(policy_iteration)" ] @@ -1211,118 +496,9 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
def policy_evaluation(pi, U, mdp, k=20):\n",
-       "    """Return an updated utility mapping U from each state in the MDP to its\n",
-       "    utility, using an approximation (modified policy iteration)."""\n",
-       "    R, T, gamma = mdp.R, mdp.T, mdp.gamma\n",
-       "    for i in range(k):\n",
-       "        for s in mdp.states:\n",
-       "            U[s] = R(s) + gamma * sum([p * U[s1] for (p, s1) in T(s, pi[s])])\n",
-       "    return U\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(policy_evaluation)" ] @@ -1336,70 +512,18 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "{(0, 0): (0, 1),\n", - " (0, 1): (0, 1),\n", - " (0, 2): (1, 0),\n", - " (1, 0): (1, 0),\n", - " (1, 2): (1, 0),\n", - " (2, 0): (0, 1),\n", - " (2, 1): (0, 1),\n", - " (2, 2): (1, 0),\n", - " (3, 0): (-1, 0),\n", - " (3, 1): None,\n", - " (3, 2): None}" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "policy_iteration(sequential_decision_environment)" ] }, { "cell_type": "code", - "execution_count": 11, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/markdown": [ - "### 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_a_ ∈ _A_(_s_) Σ_s′_ _P_(_s′_ | _s_, _a_) _U_\\[_s′_\\] > Σ_s′_ _P_(_s′_ | _s_, _π_\\[_s_\\]) _U_\\[_s′_\\] __then do__ \n", - "       _π_\\[_s_\\] ← argmax_a_ ∈ _A_(_s_) Σ_s′_ _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." - ], - "text/plain": [ - "" - ] - }, - "execution_count": 11, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "pseudocode('Policy-Iteration')" ] @@ -1464,117 +588,9 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
    def T(self, state, action):\n",
-       "        if action is None:\n",
-       "            return [(0.0, state)]\n",
-       "        else:\n",
-       "            return [(0.8, self.go(state, action)),\n",
-       "                    (0.1, self.go(state, turn_right(action))),\n",
-       "                    (0.1, self.go(state, turn_left(action)))]\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(GridMDP.T)" ] @@ -1610,114 +626,9 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
    def to_arrows(self, policy):\n",
-       "        chars = {\n",
-       "            (1, 0): '>', (0, 1): '^', (-1, 0): '<', (0, -1): 'v', None: '.'}\n",
-       "        return self.to_grid({s: chars[a] for (s, a) in policy.items()})\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(GridMDP.to_arrows)" ] @@ -1732,115 +643,9 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "\n", - "\n", - " Codestin Search App\n", - " \n", - " \n", - "\n", - "\n", - "

\n", - "\n", - "
    def to_grid(self, mapping):\n",
-       "        """Convert a mapping from (x, y) to v into a [[..., v, ...]] grid."""\n",
-       "        return list(reversed([[mapping.get((x, y), None)\n",
-       "                               for x in range(self.cols)]\n",
-       "                              for y in range(self.rows)]))\n",
-       "
\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "psource(GridMDP.to_grid)" ] @@ -1863,7 +668,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": null, "metadata": { "collapsed": true }, @@ -1888,7 +693,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": null, "metadata": { "collapsed": true }, @@ -1906,19 +711,9 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "> > > .\n", - "^ None ^ .\n", - "^ > ^ <\n" - ] - } - ], + "outputs": [], "source": [ "from utils import print_table\n", "print_table(sequential_decision_environment.to_arrows(pi))" @@ -1947,7 +742,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": null, "metadata": { "collapsed": true }, @@ -1961,19 +756,9 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "> > > .\n", - "^ None ^ .\n", - "^ > ^ <\n" - ] - } - ], + "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", "from utils import print_table\n", @@ -2007,7 +792,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": null, "metadata": { "collapsed": true }, @@ -2021,19 +806,9 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "> > > .\n", - "^ None > .\n", - "> > > ^\n" - ] - } - ], + "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", "from utils import print_table\n", @@ -2066,7 +841,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": null, "metadata": { "collapsed": true }, @@ -2080,19 +855,9 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "> > < .\n", - "> None < .\n", - "> > > v\n" - ] - } - ], + "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", "from utils import print_table\n", @@ -2149,15 +914,6 @@ "Green shades indicate positive utilities and brown shades indicate negative utilities. \n", "The values of the utility function and arrow diagram will pop up in separate dialogs after the algorithm converges." ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] } ], "metadata": { diff --git a/mdp.py b/mdp.py index 6637108e5..32c65752a 100644 --- a/mdp.py +++ b/mdp.py @@ -21,20 +21,32 @@ class MDP: list of (p, s') pairs. We also keep track of the possible states, terminal states, and actions for each state. [page 646]""" - def __init__(self, init, actlist, terminals, transitions={}, states=None, gamma=.9): + def __init__(self, init, actlist, terminals, reward, transitions, states=None, gamma=.9): if not (0 < gamma <= 1): raise ValueError("An MDP must have 0 < gamma <= 1") if states: self.states = states else: - self.states = set() + ## collect states from transitions table + self.states = self.get_states_from_transitions(transitions) + + self.init = init - self.actlist = actlist + + if isinstance(actlist, list): + ## if actlist is a list, all states have the same actions + self.actlist = actlist + elif isinstance(actlist, dict): + ## if actlist is a dict, different actions for each state + self.actlist = actlist + self.terminals = terminals self.transitions = transitions self.gamma = gamma - self.reward = {} + self.reward = reward + + self.check_consistency() def R(self, state): """Return a numeric reward for this state.""" @@ -57,6 +69,31 @@ def actions(self, state): else: return self.actlist + def get_states_from_transitions(self, transitions): + assert isinstance(transitions, dict) + s1 = set(transitions.keys()) + s2 = set([tr[1] for actions in transitions.values() + for effects in actions.values() for tr in effects]) + return s1.union(s2) + + def check_consistency(self): + # check that all states in transitions are valid + assert set(self.states) == self.get_states_from_transitions(self.transitions) + # check that init is a valid state + assert self.init in self.states + # check reward for each state + #assert set(self.reward.keys()) == set(self.states) + assert set(self.reward.keys()) == set(self.states) + # check that all terminals are valid states + assert all([t in self.states for t in self.terminals]) + # check that probability distributions for all actions sum to 1 + for s1, actions in self.transitions.items(): + for a in actions.keys(): + s = 0 + for o in actions[a]: + s += o[0] + assert abs(s - 1) < 0.001 + class GridMDP(MDP): @@ -67,25 +104,41 @@ class GridMDP(MDP): def __init__(self, grid, terminals, init=(0, 0), gamma=.9): grid.reverse() # because we want row 0 on bottom, not on top - MDP.__init__(self, init, actlist=orientations, - terminals=terminals, gamma=gamma) - self.grid = grid + reward = {} + states = set() self.rows = len(grid) self.cols = len(grid[0]) + self.grid = grid for x in range(self.cols): for y in range(self.rows): - self.reward[x, y] = grid[y][x] if grid[y][x] is not None: - self.states.add((x, y)) - - def T(self, state, action): + states.add((x, y)) + reward[(x, y)] = grid[y][x] + self.states = states + actlist = orientations + transitions = {} + for s in states: + transitions[s] = {} + for a in actlist: + transitions[s][a] = self.calculate_T(s, a) + MDP.__init__(self, init, actlist=actlist, + terminals=terminals, reward = reward, + transitions = transitions, states = states, gamma=gamma) + + def calculate_T(self, state, action): if action is None: return [(0.0, state)] else: return [(0.8, self.go(state, action)), (0.1, self.go(state, turn_right(action))), (0.1, self.go(state, turn_left(action)))] - + + def T(self, state, action): + if action is None: + return [(0.0, state)] + else: + return self.transitions[state][action] + def go(self, state, direction): """Return the state that results from going in this direction.""" state1 = vector_add(state, direction) From bee037e0a75ca1ce324a39b5ccd47d0c21ccb5f5 Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Tue, 27 Feb 2018 19:31:06 -0500 Subject: [PATCH 06/11] Small fix on CustomMDP --- mdp.ipynb | 75 +++++++++++++++++++++++++++++++++++++++-------------- pomdp.ipynb | 9 +++++++ 2 files changed, 65 insertions(+), 19 deletions(-) diff --git a/mdp.ipynb b/mdp.ipynb index e24a5d234..2930e83c9 100644 --- a/mdp.ipynb +++ b/mdp.ipynb @@ -67,7 +67,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(MDP)" @@ -152,7 +154,7 @@ " if action is None:\n", " return [(0.0, state)]\n", " else: \n", - " return [(prob, new_state) for new_state, prob in self.t[state][action].items()]" + " return self.t[state][action]" ] }, { @@ -192,7 +194,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(GridMDP)" @@ -230,7 +234,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "sequential_decision_environment" @@ -256,7 +262,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(value_iteration)" @@ -295,7 +303,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "value_iteration(sequential_decision_environment)" @@ -311,7 +321,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "pseudocode(\"Value-Iteration\")" @@ -409,6 +421,7 @@ "cell_type": "code", "execution_count": null, "metadata": { + "collapsed": true, "scrolled": true }, "outputs": [], @@ -466,7 +479,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(expected_utility)" @@ -475,7 +490,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(policy_iteration)" @@ -497,7 +514,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(policy_evaluation)" @@ -513,7 +532,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "policy_iteration(sequential_decision_environment)" @@ -522,7 +543,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "pseudocode('Policy-Iteration')" @@ -589,7 +612,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(GridMDP.T)" @@ -627,7 +652,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(GridMDP.to_arrows)" @@ -644,7 +671,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "psource(GridMDP.to_grid)" @@ -712,7 +741,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "from utils import print_table\n", @@ -757,7 +788,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", @@ -807,7 +840,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", @@ -856,7 +891,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", diff --git a/pomdp.ipynb b/pomdp.ipynb index 8934ef6b6..1c8391818 100644 --- a/pomdp.ipynb +++ b/pomdp.ipynb @@ -197,6 +197,15 @@ "Thus, as the belief vector can take infinite values of the distribution over states, so can the reward for each belief vector vary over a hyperplane in the belief space, or space of states (planes in an $N$-dimensional space are formed by a linear combination of the axes)." ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "code", "execution_count": null, From 97e51cd1ccd588df74854b982feab3ef05526780 Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Tue, 27 Feb 2018 19:58:59 -0500 Subject: [PATCH 07/11] Set default args to pass tests --- mdp.ipynb | 87 +++++++++++++++---------------------------------------- mdp.py | 14 +++++---- 2 files changed, 32 insertions(+), 69 deletions(-) diff --git a/mdp.ipynb b/mdp.ipynb index 2930e83c9..0aa7549d6 100644 --- a/mdp.ipynb +++ b/mdp.ipynb @@ -12,9 +12,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "from mdp import *\n", @@ -67,9 +65,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(MDP)" @@ -142,13 +138,13 @@ "outputs": [], "source": [ "class CustomMDP(MDP):\n", - " def __init__(self, init, terminals, reward, transition_matrix, gamma=.9):\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", " actlist = list(set(actlist))\n", - " MDP.__init__(self, init, actlist, terminals, reward, transition_matrix, gamma=gamma)\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", @@ -167,12 +163,10 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ - "our_mdp = CustomMDP(init, terminals, rewards, t, gamma=.9)" + "our_mdp = CustomMDP(init, terminals, t, rewards, gamma=.9)" ] }, { @@ -194,9 +188,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(GridMDP)" @@ -234,9 +226,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "sequential_decision_environment" @@ -262,9 +252,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(value_iteration)" @@ -303,9 +291,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "value_iteration(sequential_decision_environment)" @@ -321,9 +307,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "pseudocode(\"Value-Iteration\")" @@ -421,7 +405,6 @@ "cell_type": "code", "execution_count": null, "metadata": { - "collapsed": true, "scrolled": true }, "outputs": [], @@ -479,9 +462,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(expected_utility)" @@ -490,9 +471,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(policy_iteration)" @@ -514,9 +493,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(policy_evaluation)" @@ -532,9 +509,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "policy_iteration(sequential_decision_environment)" @@ -543,9 +518,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "pseudocode('Policy-Iteration')" @@ -612,9 +585,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(GridMDP.T)" @@ -652,9 +623,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(GridMDP.to_arrows)" @@ -671,9 +640,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "psource(GridMDP.to_grid)" @@ -741,9 +708,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "from utils import print_table\n", @@ -788,9 +753,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", @@ -840,9 +803,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", @@ -891,9 +852,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "pi = best_policy(sequential_decision_environment, value_iteration(sequential_decision_environment, .001))\n", diff --git a/mdp.py b/mdp.py index 32c65752a..fec4f4cfe 100644 --- a/mdp.py +++ b/mdp.py @@ -21,7 +21,7 @@ class MDP: list of (p, s') pairs. We also keep track of the possible states, terminal states, and actions for each state. [page 646]""" - def __init__(self, init, actlist, terminals, reward, transitions, states=None, gamma=.9): + def __init__(self, init, actlist, terminals, transitions = {}, reward = None, states=None, gamma=.9): if not (0 < gamma <= 1): raise ValueError("An MDP must have 0 < gamma <= 1") @@ -43,9 +43,13 @@ def __init__(self, init, actlist, terminals, reward, transitions, states=None, g self.terminals = terminals self.transitions = transitions + if self.transitions == {}: + print("Warning: Transition table is empty.") self.gamma = gamma - self.reward = reward - + if reward: + self.reward = reward + else: + self.reward = {s : 0 for s in self.states} self.check_consistency() def R(self, state): @@ -122,8 +126,8 @@ def __init__(self, grid, terminals, init=(0, 0), gamma=.9): for a in actlist: transitions[s][a] = self.calculate_T(s, a) MDP.__init__(self, init, actlist=actlist, - terminals=terminals, reward = reward, - transitions = transitions, states = states, gamma=gamma) + terminals=terminals, transitions = transitions, + reward = reward, states = states, gamma=gamma) def calculate_T(self, state, action): if action is None: From 7e763e6bd7c550c9ff9dda2f06d084c9c209fbe6 Mon Sep 17 00:00:00 2001 From: Ayush Jain Date: Wed, 28 Feb 2018 06:38:06 +0530 Subject: [PATCH 08/11] Added TableDrivenAgentProgram tests (#777) * Add tests for TableDrivenAgentProgram * Add tests for TableDrivenAgentProgram * Check environment status at every step * Check environment status at every step of TableDrivenAgentProgram --- tests/test_agents.py | 24 ++++++++++++++++-------- 1 file changed, 16 insertions(+), 8 deletions(-) diff --git a/tests/test_agents.py b/tests/test_agents.py index 73b149f99..caefe61d4 100644 --- a/tests/test_agents.py +++ b/tests/test_agents.py @@ -83,10 +83,9 @@ def test_RandomVacuumAgent() : assert environment.status == {(1,0):'Clean' , (0,0) : 'Clean'} -def test_TableDrivenAgent() : - #create a table that would consist of all the possible states of the agent +def test_TableDrivenAgent(): loc_A, loc_B = (0, 0), (1, 0) - + # table defining all the possible states of the agent table = {((loc_A, 'Clean'),): 'Right', ((loc_A, 'Dirty'),): 'Suck', ((loc_B, 'Clean'),): 'Left', @@ -98,17 +97,26 @@ def test_TableDrivenAgent() : ((loc_A, 'Dirty'), (loc_A, 'Clean'), (loc_B, 'Dirty')): 'Suck', ((loc_B, 'Dirty'), (loc_B, 'Clean'), (loc_A, 'Dirty')): 'Suck' } + # create an program and then an object of the TableDrivenAgent program = TableDrivenAgentProgram(table) agent = Agent(program) - # create an object of the TrivialVacuumEnvironment + # create an object of TrivialVacuumEnvironment environment = TrivialVacuumEnvironment() + # initializing some environment status + environment.status = {loc_A:'Dirty', loc_B:'Dirty'} # add agent to the environment environment.add_thing(agent) - # run the environment - environment.run() - # check final status of the environment - assert environment.status == {(1, 0): 'Clean', (0, 0): 'Clean'} + + # run the environment by single step everytime to check how environment evolves using TableDrivenAgentProgram + environment.run(steps = 1) + assert environment.status == {(1,0): 'Clean', (0,0): 'Dirty'} + + environment.run(steps = 1) + assert environment.status == {(1,0): 'Clean', (0,0): 'Dirty'} + + environment.run(steps = 1) + assert environment.status == {(1,0): 'Clean', (0,0): 'Clean'} def test_ReflexVacuumAgent() : From 3fed6614295b7270ca1226415beff7305e387eeb Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Tue, 27 Feb 2018 20:59:26 -0500 Subject: [PATCH 09/11] Fixing tests --- mdp.py | 33 +++++++++--- rl.ipynb | 127 +++++++++++++++++++++++++--------------------- rl.py | 2 +- tests/test_mdp.py | 30 +++++++---- 4 files changed, 114 insertions(+), 78 deletions(-) diff --git a/mdp.py b/mdp.py index fec4f4cfe..9dcbd781a 100644 --- a/mdp.py +++ b/mdp.py @@ -50,7 +50,7 @@ def __init__(self, init, actlist, terminals, transitions = {}, reward = None, st self.reward = reward else: self.reward = {s : 0 for s in self.states} - self.check_consistency() + #self.check_consistency() def R(self, state): """Return a numeric reward for this state.""" @@ -74,12 +74,15 @@ def actions(self, state): return self.actlist def get_states_from_transitions(self, transitions): - assert isinstance(transitions, dict) - s1 = set(transitions.keys()) - s2 = set([tr[1] for actions in transitions.values() - for effects in actions.values() for tr in effects]) - return s1.union(s2) - + if isinstance(transitions, dict): + s1 = set(transitions.keys()) + s2 = set([tr[1] for actions in transitions.values() + for effects in actions.values() for tr in effects]) + return s1.union(s2) + else: + print('Could not retrieve states from transitions') + return None + def check_consistency(self): # check that all states in transitions are valid assert set(self.states) == self.get_states_from_transitions(self.transitions) @@ -249,3 +252,19 @@ def policy_evaluation(pi, U, mdp, k=20): ^ None ^ . ^ > ^ < """ # noqa + +""" +s = { 'a' : { 'plan1' : [(0.2, 'a'), (0.3, 'b'), (0.3, 'c'), (0.2, 'd')], + 'plan2' : [(0.4, 'a'), (0.15, 'b'), (0.45, 'c')], + 'plan3' : [(0.2, 'a'), (0.5, 'b'), (0.3, 'c')], + }, + 'b' : { 'plan1' : [(0.2, 'a'), (0.6, 'b'), (0.2, 'c'), (0.1, 'd')], + 'plan2' : [(0.6, 'a'), (0.2, 'b'), (0.1, 'c'), (0.1, 'd')], + 'plan3' : [(0.3, 'a'), (0.3, 'b'), (0.4, 'c')], + }, + 'c' : { 'plan1' : [(0.3, 'a'), (0.5, 'b'), (0.1, 'c'), (0.1, 'd')], + 'plan2' : [(0.5, 'a'), (0.3, 'b'), (0.1, 'c'), (0.1, 'd')], + 'plan3' : [(0.1, 'a'), (0.3, 'b'), (0.1, 'c'), (0.5, 'd')], + }, + } +""" \ No newline at end of file diff --git a/rl.ipynb b/rl.ipynb index 019bef3b7..f05613ddd 100644 --- a/rl.ipynb +++ b/rl.ipynb @@ -6,7 +6,7 @@ "source": [ "# Reinforcement Learning\n", "\n", - "This IPy notebook acts as supporting material for **Chapter 21 Reinforcement Learning** of the book* Artificial Intelligence: A Modern Approach*. This notebook makes use of the implementations in rl.py module. We also make use of implementation of MDPs in the mdp.py module to test our agents. It might be helpful if you have already gone through the IPy notebook dealing with Markov decision process. Let us import everything from the rl module. It might be helpful to view the source of some of our implementations. Please refer to the Introductory IPy file for more details." + "This Jupyter notebook acts as supporting material for **Chapter 21 Reinforcement Learning** of the book* Artificial Intelligence: A Modern Approach*. This notebook makes use of the implementations in `rl.py` module. We also make use of implementation of MDPs in the `mdp.py` module to test our agents. It might be helpful if you have already gone through the Jupyter notebook dealing with Markov decision process. Let us import everything from the `rl` module. It might be helpful to view the source of some of our implementations. Please refer to the Introductory Jupyter notebook for more details." ] }, { @@ -47,7 +47,7 @@ "\n", "-- Source: [Wikipedia](https://en.wikipedia.org/wiki/Reinforcement_learning)\n", "\n", - "In summary we have a sequence of state action transitions with rewards associated with some states. Our goal is to find the optimal policy (pi) which tells us what action to take in each state." + "In summary we have a sequence of state action transitions with rewards associated with some states. Our goal is to find the optimal policy $\\pi$ which tells us what action to take in each state." ] }, { @@ -56,7 +56,7 @@ "source": [ "## PASSIVE REINFORCEMENT LEARNING\n", "\n", - "In passive Reinforcement Learning the agent follows a fixed policy and tries to learn the Reward function and the Transition model (if it is not aware of that)." + "In passive Reinforcement Learning the agent follows a fixed policy and tries to learn the Reward function and the Transition model (if it is not aware of these)." ] }, { @@ -83,7 +83,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The Agent Program can be obtained by creating the instance of the class by passing the appropriate parameters. Because of the __ call __ method the object that is created behaves like a callable and returns an appropriate action as most Agent Programs do. To instantiate the object we need a policy(pi) and a mdp whose utility of states will be estimated. Let us import a GridMDP object from the mdp module. **Figure 17.1 (sequential_decision_environment)** is similar to **Figure 21.1** but has some discounting as **gamma = 0.9**." + "The Agent Program can be obtained by creating the instance of the class by passing the appropriate parameters. Because of the __ call __ method the object that is created behaves like a callable and returns an appropriate action as most Agent Programs do. To instantiate the object we need a policy ($\\pi$) and a mdp whose utility of states will be estimated. Let us import a `GridMDP` object from the `MDP` module. **Figure 17.1 (sequential_decision_environment)** is similar to **Figure 21.1** but has some discounting as **gamma = 0.9**." ] }, { @@ -201,7 +201,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "{(0, 1): 0.3892840731173828, (1, 2): 0.6211579621949068, (3, 2): 1, (0, 0): 0.3022330060485855, (2, 0): 0.0, (3, 0): 0.0, (1, 0): 0.18020445259687815, (3, 1): -1, (2, 2): 0.822969605478094, (2, 1): -0.8456690895152308, (0, 2): 0.49454878907979766}\n" + "{(0, 1): 0.4431282384930237, (1, 2): 0.6719826603921873, (3, 2): 1, (0, 0): 0.32008510559157544, (3, 0): 0.0, (3, 1): -1, (2, 1): 0.6258841793121656, (2, 0): 0.0, (2, 2): 0.7626863051408717, (1, 0): 0.19543350078456248, (0, 2): 0.550838599140139}\n" ] } ], @@ -258,9 +258,9 @@ "outputs": [ { "data": { - "image/png": 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gn+0oYdaojEbjRRz3XDqJH5w/lmn2SPYxA61ANf2ENPc9uP6UE/jJRRMQgQsm\nhu9hdcHEgczITSczKYaLThzcqdqg8zcNd8F/z54x12lAb6uXkuOTbUVM+vk7/G1Rw8XM250Z4KnF\nu9wR+6+utO7/3t1wkE+3FXHl9KFh37dwRIQnbpzBg9dMIy7a32jAZ0cNS29odxOxgsPHW4u4bsaw\nDtVMHU5QqK5reN8+2lLovo9Nawofby1i88EKNwBV1AQ4crSOvy/b694grN9Xxvi73+az7Q0DDo8c\nraOqzhqDU1UX5Jtnd/zmrbMi9o01xgRE5NvAu4AfeMIYs0FE7gOWG2PmAz8A/ioi38fK5txojnUW\nq3ZwUihh00dRPrfbpjeV5IiP9rnpo3Dim1wEvbWK1Phod5h8uT3fj9Ng2hInAHnnw6kPGjI6OJNm\n0/KEDI3GU3REmqcP94isRGo9d00nDEhw21ZGZnUsLdVR//2lqWQmtfw+OO05OenxjdKAXt673UlD\nUt1g4NQUjlTVkxofTVl1PYfKa5nSQk58RGYit3u6iTptJWeMaUh33nPpRAAmD011x1U09ehX8wCr\nK224lFN7xLSSPvp0WzETBqe4EyS2t6bwxpoD1NSHeGXVPqYPT2NtQRn5dv//I0friI/x8/P5GwC4\ndOoQd7zGx1uKCBma9eJqy/R2do9uy1C7pjB1WBoDk2PdoPilTubmnVSpkz76bHsJN/5tmft8bZP3\n84XP95CRGMN1M4fx8ooCjtYG+PvyvVTXB7nnkonc98ZGHnx/KzX1Ieav2c+sURkEQ4bLHlrE9OFp\nfJpfzNnjspjQwXa/YxHR2zhjzFtYDcjebfd4ft4IhO9OEQHThqXxtVkncIud22x61wiNu6R600fe\n58OljxyJsY3vPrwD5FLjo3GyFxU1AXzSMGCpJT67WlNU2XgwT2sXw9Z4azltpS9a4g0mOenxjVJb\nJ2Qkcta4LBJj/e0e99BZbVX/Z+Sm88SinZwyMsPTtbjx33yr3baSHBvFuZ4cv/fveMaYTHdytclh\nUkfhzBwxgC9OzwnbDtCeEbHt7RETTkvpo+q6ICt2H+HG03PdwNH0IgbWe/Taqn3MmzaEaL8PY4w7\n5iQYMpw3YSDlNQG2F1ayKL+Y6x9b2uj1BUeq2HrIChiVtQFGZiUyKit8ii/SUuOjuf6U4Vw4eTCf\n7SjmvY2HOG1URqNZAzrCqV04c3+9bE+pcunUIfxrzf5GN0iF5TUs2FTILWeMcAN8ZW2Al5bvZUZu\nOqfaMxA0ZZCiAAAc7ElEQVSstedFc2oOC7cVsedwFQVHqggZ+LpnPrPu0K+mufD7hHsvm+w+Dlcn\nifE3dEmN8TQ0NzzvbzV9FN+kSuptGIr2i1tTqKgJkBQb1WaV2qmpNJ05MzOpk+kjT9k72y7h7V01\nNC2eYk/ZTshIICEmimtm9PwqbReeOJilPzmPgSlxlNnpuqZtClsOVZAaH83yn83B7/lbeXs3nTk2\nq8NBISEmij9cPfVYT6FTWkoffb7rMHXBEKePzqTavqh596mqC1AfMLy+Zh/3vL6BqvogXzn1BLYV\nVnLQk748e1wWq/eWsqvkaLNpTnLS4/nInjZlYEosh8prOTdMb63udP8VJwJw0vA0BiTGMnt055st\nvW0KVXUB3l5/gGvyhnH1jBwrKHjez3+utFZ3vHbGcPcmY+HWInYUHeUbZ45qljp2Op+8ZM/pFDIw\nJDXumMrbGT3d+6hHmTAJpKYNzU0DQGwb6aPE2JbjrHfG0/LqelLiW1+JzXkNNA8KGZ0MCt5aTltj\nJFqSndwQFDKTYon1BMLWzr8nOA35fn8LNYWDFYwbmEy039coQDvnkRDjd+/sR2Qmhl1UqbdpqRaw\nOL+YGL+PmbkDwqaYzn9wIVPve4+V9ih3Z2Dmx/ZFfnBqHNnJsUwcnEJ2cixFFbWs2tMwQdxl04ZQ\nUx/koy1F5KTHc7rdVfjcCT0bFByJsVHcPHuEO4FgZ3jTR+9vPERVXZArpjesm+J9z99ad4Bpw9IY\nkZno3mS8smofCTF+LpoymDR70svkuChm5KbzzoaD3PO6tVKiM3fVVXnDOtWudCx61ze4m4Xpndgo\nZRSuTSHG7ws7Z5IjoZVRwT4Rt6G5vCYQdjnMpqJaDAqdTR95g0Lnagrexmm/T5oN0OuNwrUpGGPY\ncqiCy8MMfnK+xBMGp7hTJrS3ltDTWmpTWLW3lElDU4iP8bvtLU5NwRjjjpp22goO29OSf77rMCMy\nE7l33iTqAiFEhMwkayqKsuoyxg1M5tKpg6msDVJWXc/i7cVcOX0ouRmJfLajhBkdXLa1N/M2NC/e\nXkxmUiwzcwe466/UBqxAuvdwFev2lfGTi8Y3eh1YXZqdz1d2cixzJw9iiT09zNOfWVOZ/PqKE3ll\nVUHYFQkjrV8HhXBio3xusIiJ8tE0SMdE+VrtK9xabxu/Txo1NKe00cgMDXe4TYNCWjtqGeF4A1pW\nJ2sbTXWmF0d3805X4jhQVkNFTYCxYe4cnZrCpCEpJMdGMWdCdsSnF+gqsWHaFIIhw4Z9ZW47TNMU\nk3ehqS32FCEFR6rZfLCc1XtLOWN0Jmd65rByer+FDPz4ovGcPS6bhz/Kpz5oqA8GmTkig0unDOYr\ns05otWZ9vHEu7hU19Xy8tYgLJw/C55OGmkK99X46YxMunGwtnOW9ZnhnD3jj9tmkxEdz92vr3XaY\nnPR4Th+dwewx3Zs2cvTroBB2nILf5zZAx4YJAG3dFbdWU/D7GmoKFTUBtwdIa5w8d8nROrLsKjs0\njKDtqGhPzSe9kz2YABbccabbCO68J+0Jcj3FeR+9NQVnOc9xYUbKpsZHc+HkQe5Kb499bUb3FLQL\nhGtT2FlcydG6ICfmNF78qS5o3dl6Vzpz1tr415r9bhfTpiPTvV2vnQ4F3prvySekIyIdHhzW28XF\nWO/bJ9uKqagJcO54q2txbHTjlN37Gw8xaUhK2AZtZywLNHT2uO+yycRG+3h2yR4unDyoWwaptaTv\nhPBOCNclNTba566JHC4AdCYozJkwkNvOHNmoTaGipt7tz98ab0P3SM/I487mtr15cyen2Rmjs5MZ\nafcoccqY1skulN3BZy+/6m1T2G3PIRVuRLffJzxyw8nHZerDGxQe+2QHP3l1HevsGVFPtFNgTXso\nbWwy5sAZb9HSYyelNnZgkjvNtHNTMDAlliGd7O7c28V4priJ8fs4w76bj3XbcYJU1gZYtaeUs8LM\nDhzj94VNG8fH+Ll82lB8AvOmdqz7blfrvbd23SBsl1S/361BtDSLamvCNbQ+9jWr7/lVjyxu3NDc\njgu7t5FpVHYSS+2RmV2Rx0+L75qLuNP4dm6YaRt6kyifr1FNYX9ZDTFRvk537+2tvG0Kv3pzE2B9\nXuKj/Yyyx440bYz2DkRLT4jmyulDWe1ZZazpqHOn95u3e63TccKpJfRF3vO6+YwR7vfd29C8ZHsJ\ngZBplv5Z9tM5rV4/8nIHsPLu83v85qpf1xTCzQUWE9WQPgqXC20rKLTa0OypKVTXB1vd1+Ht/eTt\n690VXzpnedJjlZOewDvfO4OfXTyh7Z17UNP1rveVVjM0Lb7PXcCcu1mn0ROsUbMTh6S4HQ1im6SY\nNu4vZ6o9MG/y0NRm7SdNP/eDU+M5aXhao7UQnJucrhp41tt9b07DYEU3yNaH+DS/mPhof7PxKFnJ\nsc0W72mqpwMC9POaQrhxClF+cYNFuK5gTWc9barVhmYRAqEQwZChPmja1UDr97QBjOriEcJdeTE8\n1vmLukOUTxqNU9hfWu0uxtOXOBco7zw8mw9UcKlnVlZvbaK4spbCilquzhvGmoIypuSkkpYQw/Kf\nzeFQeU2zsTfO61/9ZuNxpxMGJ3P5tCEtLhjV13jbS/w+IdovVNbW8+GWQmaOGHDctqf066AQLgXj\n7TYa7qLZVtqmtT7Ffp9QGzDuZFrtSQF5B1S1p2FatcyqqXl6H5XWuDnhvsS54K/3rBBXURto1Cbl\n1Cb2Hal2U0enjcpgzMAkTrcHS2UmxXZokGRCTBR/uvakYy5/b/f0TTPDvi9xUX7+ak9099OLenet\nuTX9Oig8ddNMvvCnhY1SCkJDr6Rw1/fW0kcv3Tar1d/n8wlB0zDDYvtqCp5pMo6hYVjZNQX7b10f\nDHGoouaYppPorZJjrc9J0+U2velH53P88Efb3dHKEwancNox9EjrL85sYXnZ2GgfFbUwcXAKF3Rg\n0arepl+3KYzOTuKN22c32uaThryzL0xNwf0yXT/d3faHL03ln/95WtilBb38Yi2rV2PncduT0/e2\nKThfdtU5fp80Ws/CmL5Z+4qP8RMX7Ws0NQU07mXlvblZlF9MRmLMMXVRVg1tkG0tLNXb9eugANbd\n0a4HLnZHDqYnRrttCq0FhYtOHOxumz0ms12TnPl9wrp9ZayzJ8DqaE2hqxqGH7jyRJ6+aWaXHOt4\n4m1T2G+P3u2LNQWg2Qyr0X5x1xaAxl2dD5XXckInlnZVjTmzAzftvnu86dfpI6+7L5nIdTOHk5Oe\n4KaPwrUPdGbsgsMJMt94doX9unb0PvKUwWnjSDrG+YWundnzk9X1BL+/oRa4v8wKCoP7YEMzWEHh\nQFkNo7IS2V50lBMyEhtNcdK0vcxZkEgdu5M0KPQNMVE+d3WnhppC8/3CB4X29TJoGmRi23Hn37S2\n8vI3ZpGTrnd1neEdp7C/1LqrG9KJda6PB+mJVqpxSk4a24uONmpkDucEDQpdpqemCe8q/T59FM5P\nLprA7NGZzAqzfGC4LqntrSk0DQpx7akpNJl8Ly93QKOpq1X7eccp7D1cRUZijDvwrq9x0kfOokAj\nw1yoPvjBWe7Pzip1qvNyMxIYkBjT7hXmeiutKYQxOjuJZ285Jexz4XoftfdD0CwotKOm0N3T5vZl\nVu8jq5F/W2GluzpaX+QEhUlDUvnNlSeG7TEzMiuJjMQYSo7WaU2hCyy446ywA2KPNxoUOuhYppfw\nS9Og0J42Ba3MdRWnpmCMYevBig4vEXk8cXoS5WYktNorLjU+mpKjdeRqQ/Mxa2mt8OONBoUO8qZz\n0hKiKa2qb2XvxprWKDra+0gdG2ecQsGRaipqw0+Z3VfMmzoYv4g7cV1LUhOiSY2P7hXTK6jeQYNC\nB2QmxTaaxuL9759FcZO1k1vTtKbQrhHNGhS6jN8nfLy1iD+8twW/TzgtTJtRXzE6O5nvzmk76OVm\nJLa5TrjqX/TT0AHLfzan0eOs5Ng278S8OlNTaLpGtOq8umAIY+C11fuZMyH7uO8l0hV+c+WJYWcL\nVv2XBoV2mDMhm12eycU6q+n1XRuau1dFTcD9ObuTS5H2NcfDqnmqe2lQaIeuWnWrac+E9oxvaJpy\nUp3nDQrpOo+UUmH1jeby44R3hs5ov7SrFnC893nuTSpqGjoFNJ0GQill0ZpCNwo2xIR2DVxzRPmE\n758/NgIl6l/qPWspaFBQKjwNCt3I26AX3YHxDvm/vigSxenXnGkglFKNafqoG3nXB9ZeRT1L++Ur\nFV5Eg4KIzBWRLSKSLyJ3tbDP1SKyUUQ2iMjzkSxPTwt5gkK49Z9V9wm3xKRSKoLpIxHxAw8B5wMF\nwDIRmW+M2ejZZwzwY+B0Y8wREcmOVHl6g4CnobnpRHcq8px5fhJi/AwfoNM6KBVOq0FBRO5osskA\nxcCnxpidbRx7JpBvjNlhH+tF4DJgo2ef/wAeMsYcATDGFHag7Mcdb0Ozpo+634I7zqKyNsAwDQhK\ntaitHEZyk38pQB7wtohc28ZrhwJ7PY8L7G1eY4GxIrJIRJaIyNxwBxKRW0VkuYgsLyoqauPX9l6N\nu6Rq+qi7pSfGaEBQqg2t1hSMMfeG2y4iA4AFwItd8PvHAGcDOcBCETnRGFPapByPAo8C5OXlHbdj\n8hs1NGv6SCnVC3XqdtUYcxho66q2DxjmeZxjb/MqAOYbY+rtdNRWrCDRJ3m7pOqU2Eqp3qhTVyYR\nOQc40sZuy4AxIjJCRGKAa4H5TfZ5DauWgIhkYqWTdnSmTMeDQFC7pCqlere2GprXYTUuew0A9gNf\nbe21xpiAiHwbeBfwA08YYzaIyH3AcmPMfPu5C0RkIxAEfmSMKencqfR+3ppC07WXlVKqN2irS+ol\nTR4boMQYc7Q9BzfGvAW81WTbPZ6fDXCH/a/PC3pnxNOYoJTqhdpqaN7dXQXpD7yzomr2SCnVG2lr\nZzf64zXTSI6z4rBoVUEp1QtpUOhGg1LjuMOe7VSbFJRSvZEGhW7m9DrShmalVG+kQaGb+e3xCRoT\nlFK9kQaFbubMbiEaFZRSvZAGhW7m1hR6uBxKKRWOBoVu5rQpaEVBKdUbaVDoZj5taFZK9WIaFLqZ\nW1Po4XIopVQ4GhS6mVND0IqCUqo30qDQzZxgoL2PlFK9kQaFbmbsmVI1JCileiMNCt3MmT1bKwpK\nqd5Ig0I3cybP1t5HSqneSINCN3MW2tGYoJTqjTQodDM3faStCkqpXkiDQjdz0kdaU1BK9UYaFLqZ\n2/tIo4JSqhfSoNBDonQ9TqVUL9TqGs2q682dPIjrTxnO9+0V2JRSqjfRoNDNYqP83H/FiT1dDKWU\nCkvTR0oppVwaFJRSSrk0KCillHJpUFBKKeXSoKCUUsoV0aAgInNFZIuI5IvIXa3s90URMSKSF8ny\nKKWUal3EgoKI+IGHgAuBicB1IjIxzH7JwHeBpZEqi1JKqfaJZE1hJpBvjNlhjKkDXgQuC7PfL4Hf\nAjURLItSSql2iGRQGArs9TwusLe5RGQ6MMwY82ZrBxKRW0VkuYgsLyoq6vqSKqWUAnqwoVlEfMCD\nwA/a2tcY86gxJs8Yk5eVlRX5wimlVD8VyaCwDxjmeZxjb3MkA5OBj0RkF3AqMF8bm5VSqudEMigs\nA8aIyAgRiQGuBeY7TxpjyowxmcaYXGNMLrAEmGeMWR7BMimllGpFxIKCMSYAfBt4F9gEvGSM2SAi\n94nIvEj9XqWUUp0X0VlSjTFvAW812XZPC/ueHcmyKKWUapuOaFZKKeXSoKCUUsqlQUEppZRLg4JS\nSimXBgWllFIuDQpKKaVcGhSUUkq5NCgopZRyaVBQSinl0qCglFLKpUFBKaWUS4OCUkoplwYFpZRS\nLg0KSimlXBoUlFJKuTQoKKWUcmlQUEop5dKgoJRSyqVBQSmllEuDglJKKZcGBaWUUi4NCkoppVwa\nFJRSSrk0KCillHJpUFBKKeXSoKCUUsqlQUEppZQrokFBROaKyBYRyReRu8I8f4eIbBSRtSLybxE5\nIZLlUUop1bqIBQUR8QMPARcCE4HrRGRik91WAXnGmCnAy8DvIlUepZRSbYtkTWEmkG+M2WGMqQNe\nBC7z7mCM+dAYU2U/XALkRLA8Siml2hDJoDAU2Ot5XGBva8nNwNsRLI9SSqk2RPV0AQBE5AYgDzir\nhedvBW4FGD58eDeWTCml+pdI1hT2AcM8j3PsbY2IyBzgp8A8Y0xtuAMZYx41xuQZY/KysrIiUlil\nlFKRDQrLgDEiMkJEYoBrgfneHUTkJOAvWAGhMIJlUUop1Q4RCwrGmADwbeBdYBPwkjFmg4jcJyLz\n7N1+DyQB/xCR1SIyv4XDKaWU6gYRbVMwxrwFvNVk2z2en+dE8vcrpZTqGB3RrJRSyqVBQSmllEuD\nglJKKZcGBaWUUi4NCkoppVwaFJRSSrk0KCillHJpUFBKKeXqFRPiKaVUV6uvr6egoICampqeLkq3\niouLIycnh+jo6E69XoOCUqpPKigoIDk5mdzcXESkp4vTLYwxlJSUUFBQwIgRIzp1DE0fKaX6pJqa\nGjIyMvpNQAAQETIyMo6pdqRBQSnVZ/WngOA41nPWoKCUUsqlQUEppSKkurqas846i2AwyOrVq5k1\naxaTJk1iypQp/P3vf2/z9Q8++CATJ05kypQpnHfeeezevRuAoqIi5s6dG5Eya1BQSqkIeeKJJ7jy\nyivx+/0kJCTw9NNPs2HDBt555x2+973vUVpa2urrTzrpJJYvX87atWu56qqruPPOOwHIyspi8ODB\nLFq0qMvLrL2PlFJ93r3/2sDG/eVdesyJQ1L4+aWTWt3nueee4/nnnwdg7Nix7vYhQ4aQnZ1NUVER\naWlpLb7+nHPOcX8+9dRTefbZZ93Hl19+Oc899xynn356Z08hLK0pKKVUBNTV1bFjxw5yc3ObPff5\n559TV1fHqFGj2n28xx9/nAsvvNB9nJeXxyeffNIVRW1EawpKqT6vrTv6SCguLg5bCzhw4ABf+cpX\neOqpp/D52ndf/uyzz7J8+XI+/vhjd1t2djb79+/vsvI6NCgopVQExMfHNxsvUF5ezsUXX8z999/P\nqaee2q7jLFiwgPvvv5+PP/6Y2NhYd3tNTQ3x8fFdWmbQ9JFSSkVEeno6wWDQDQx1dXVcccUVfPWr\nX+Wqq65qtO+Pf/xjXn311WbHWLVqFbfddhvz588nOzu70XNbt25l8uTJXV5uDQpKKRUhF1xwAZ9+\n+ikAL730EgsXLuTJJ59k2rRpTJs2jdWrVwOwbt06Bg0a1Oz1P/rRj6isrORLX/oS06ZNY968ee5z\nH374IRdffHGXl1nTR0opFSHf+ta3+OMf/8icOXO44YYbuOGGG8LuV19fz6xZs5ptX7BgQYvHnj9/\nPq+//nqXldWhNQWllIqQ6dOnc8455xAMBlvd79133+3QcYuKirjjjjtIT08/luKFpTUFpZSKoJtu\nuqnLj5mVlcXll1/e5ccFrSkopfowY0xPF6HbHes5a1BQSvVJcXFxlJSU9KvA4KynEBcX1+ljaPpI\nKdUn5eTkUFBQQFFRUU8XpVs5K691lgYFpVSfFB0d3enVx/qziKaPRGSuiGwRkXwRuSvM87Ei8nf7\n+aUikhvJ8iillGpdxIKCiPiBh4ALgYnAdSIyscluNwNHjDGjgT8Cv41UeZRSSrUtkjWFmUC+MWaH\nMaYOeBG4rMk+lwFP2T+/DJwn/XH9PKWU6iUi2aYwFNjreVwAnNLSPsaYgIiUARlAsXcnEbkVuNV+\nWCkiWzpZpsymx+4H9Jz7Bz3n/uFYzvmE9ux0XDQ0G2MeBR491uOIyHJjTF4XFOm4oefcP+g59w/d\ncc6RTB/tA4Z5HufY28LuIyJRQCpQEsEyKaWUakUkg8IyYIyIjBCRGOBaYH6TfeYDX7N/vgr4wPSn\nkSZKKdXLRCx9ZLcRfBt4F/ADTxhjNojIfcByY8x84HHgGRHJBw5jBY5IOuYU1HFIz7l/0HPuHyJ+\nzqI35koppRw695FSSimXBgWllFKufhEU2ppu43glIk+ISKGIrPdsGyAi74vINvv/dHu7iMj/2u/B\nWhGZ3nMl7zwRGSYiH4rIRhHZICLftbf32fMWkTgR+VxE1tjnfK+9fYQ9PUy+PV1MjL29z0wfIyJ+\nEVklIm/Yj/v0OYvILhFZJyKrRWS5va1bP9t9Pii0c7qN49WTwNwm2+4C/m2MGQP8234M1vmPsf/d\nCjzSTWXsagHgB8aYicCpwLfsv2dfPu9a4FxjzFRgGjBXRE7Fmhbmj/Y0MUewpo2BvjV9zHeBTZ7H\n/eGczzHGTPOMR+jez7Yxpk//A2YB73oe/xj4cU+XqwvPLxdY73m8BRhs/zwY2GL//BfgunD7Hc//\ngNeB8/vLeQMJwEqs2QGKgSh7u/s5x+rxN8v+OcreT3q67J041xysi+C5wBuA9INz3gVkNtnWrZ/t\nPl9TIPx0G0N7qCzdYaAx5oD980FgoP1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nz2k+qvXy+3fW8fZKf/b3Te5BvD7D2j2l/PmSCU5NPHtIb+b+YFqbmqkjKZLh\naAkwUkSGikgs/o7keUHn7AROBRCRMUA80HHtQ62wm4eC171pUlOwg0JM22Oo2yXU+wzFlf4+Bbv6\n19wIGGjIzO2aQkaIVUPbwikR1fuorPU6JeC8In9NYUAbgoJ9jT4925eGSApsP7czsZW7ihmanuRk\nhuFwhipav6fymnruf39DyA7Tkso6Nu0vZ8pgf+d7QVkNNzy7lFqvr9kVOHtbmfJRfZLJbGeTYLjs\nNX7OCNGcFYrLJQzoldBi89EH6/xrfl05bXDI0XQA8R43NSF+b2+t2E18jKvZmpL9dztueHqLw58P\nhf35am64NsDry/1L0/zw5OH8aFboYcHBkuI8zP3BNO6/+OhGx+2C1e7iKt5dvddpVt6wr4x/f5pL\n/5R4Lpg0gJF9ejDrqAx+dc7YLg8IEMGagjGmXkRuBuYDbuBJY8xaEfk9kGOMmQf8DPiPiPwUfx78\nPdOJ8/NNQEdzoIZ5Co37FMKp0rpdgtfns5qPYoj1uPjgpyeGXNbCZneE7XOaj9pZUwhoPgqUV1RF\nSkJMmxZ/65sSz5CAJSEOJ/0CMg070K3fV8qEVkZJNcf+vXutPoUH3t/A01/vYFSfHk06+9fs8ddI\nTh3Th2U7i9lbUs0Oq1nux6eEHi5oFwhCrRsVKXbgbG7mfSgDeiWQ10JN4cN1+5k5Io37Ljy62XPi\nY1xNmo/qvD7eXb2X08f2bXao7Xar4/q8Sc1PkjxUMe7WO5rfXL6bcf17csfs8EY+hSoQ2PnFU19t\np7rOx/u3nsCjn2zlTWtE2O3fOspp6v3v96eG9fMiKaINV8aYd40xo4wxw40x91nH7rYCAsaYdcaY\nmcaYicaYScaYDyKZnhDpA0IEBed9///h9CnY3C6hssZLdZ3PyRRG9UlucT8AT0BNISnW3e52Vfvh\nD1wpdXdxFW8u393mZR3iY9x8evssp9PscJIU52HZr0+nR5yH3UVVVNbWk1dUxah2LoUdPCR1ldUU\nFapzz26mmmX9Xm6cuxSvz/DPyyc3W8Idmp7EiaMyuOSY9o0ma49fnT2Gd245vtEiiq1JTYqltJnF\nBrcV+Jc/P72VPpGEWHeTGtbyncUUVdZx1vjmay12k2aojvqO4gkY8h1Kbn45K/NKuHBy8HiY9km0\n8ov8shqOzerN6L49GwWPSzvxeQhHVM9obm70kTMk1Xrd0KfQ9kza45KGZqA29g3Ymfnu4iqGZbRv\njgI0NP0EzrZctqOIspp67jyzY8d+d5XUpFgG9EpgX2k1ufkVGAOj+rQ8vr85gX0KNfVeZ/+GUJXW\n1XklDEpNaLRECcBJRzXfLJhSLpeCAAAbOElEQVQQ6+aZazu3JJgU53E6dNuqpfkt9rLtrU0mi/e4\nKa5sHFg+35yP2yWNOraD/e07k9hbUt2u5r+2smvQzY0ye2vFHkRaXtIlHIGdxRdYgcauLfaM93Ra\nU2K4ojsoNNfRbA9J9QU1H4VRU3C5xKlhtHVoaeAch/H9w+8wtdlV0sD2YbuDsC39CUeKXokxFFfW\n8XKOf+TzyHYGhcA+hcBZvrX1IYLC7hKOHpDSaETW49dkR2Q/hs7W0vIoS7YX0S8l3ln5tznxMU1r\nCp9tLmDSoF4tZviB/W6REuMK3dG8v7SaTzYe4MN1+zl2SGrItbsO1VnWEvUDeydyzwXjW11mpitF\n9YJ4vtaaj6z/q2q9uF0SclmM5gSuetrWh8x+aMG/cmh72TWO4Cn4AL07cbxzpPVKjGGPtTfEqaMz\nW50J3JzAPoUt+xtWsg0ez15SWcfOg5UcPcDfd2EXHgKXCT+SNbc8ijGGJdsOcmxWaqszauODhqQW\nV9ayKq+4XcufdLQYT8M8hY37ypydFOc8u5RfvLaadXtLmRVGH0w4egcMSb96+pB2P6udIbprCtb/\nwUHBjgqBM5oTY9xhTTF3NwoKbWs+8gQEnfauGQ8NoyzstfabS9eRrldCLLsO+gPf+ZMHtHsJAE9A\n89GOg5X+UTjFVU2aUjZbm8eM7udvOpp73TTeWL67Ucf3kSy2mc2Z8oqq2FdazbFZoUccBfJ3NDfU\nFBblFmJM+9bE6mh28Ld3cXttWR7fOy6LlQErj84a3bE7qS247cROnXjWEaI7KFiZfkLQTOWGPgX/\nCVV1XuLD7PR1W9eI9bja3E4aGBQOpTnCbj7KL+ve2zYGrkU16BCaxdwBHc07CisY1z+F3cVVjUrN\nheU1bDngr0VkWePKZ45IZ2YnzTvoDM1tzrRku3+dqeys1FavkRjrbjRPYfnOYmLdrrD7NyIhJmCV\n1EXWasH239R2VAetyGrrjH3AO1pUB4XTx/bhhcU7ndU37YJm8Oij6jpvWP0J0JDBZybHtbkEG9h8\nlBjX/vHKdkdzR+xHezgLnJTV1m0ZQ7E7BEuq6igor2VkZg8+3ZTvZJB1Xh/H3LsA8AeQloYVH8li\nPQ3LowQ+s8t2FpEc52nT8iiJsf51xHzW5KxlO4sY079nyKXQO5vdrLqtoMKZ3/LeGv9SKUcPSOGs\no/t1yoJzh7uoDgq/P38cPzl1pFNlth+HwP0ULnr4S5btLA67BGGXPsNZLC6wphDOjOJg9sNf0MIO\nZd1B4Ezh9i4JAg3NR7nWznh2h3Wt1SEZuJVm/17xIffh6A7s56bW62uUiW/YW8aYfj3b1PRo78K3\ndGcRlz7q3xs71D4WXcH+fNkjqQDeW72PXokxvPmjmd2qafVQdM+nu41i3C76psQ7HVA2+9kwGJZZ\nwxPjYsL7VdnNR3FhZCCBSzu3ZfG91q5TWNG4pnDOhEPbzvBwYzcfJcWG198TzM4Mtub7976wlzmv\nrffx0fr9vBmwTWVWG9aNOlKFWhvI5zNs2Ffm9KO0xn5uc7YXOccmDur6piNoqImvzCtxjuUWVJA9\nJFUDQoCorinY7A+DnbE0rJLacI4nzIfGbT2A4ZQqA3/GodQUBqUmEOMW6rwGl/jvY3TfZP55eegt\nRI9Udt/Poc4UtoOovYf2iAx/Bljn9XHd0zmNzh0cwf0iupqzkGK9D6yxEbuLqyivqQ+56F4o9nNr\n90MATBrUegd1Z4gJ+CyOzOzBZqs/YfLgyO5kdqSJ6pqCzX5YGrLkxstcAGGvWmhXVcMJCoGlleY2\nhGmL5PgYZ+ak3XwV53F1u/bSE0amc/nUQS0uu9AWTk3hQDlpSbGkJMY0u1lMd64pxLgbRufY1u/1\nbyE7Jsyagh0U+qfEk5V2eATSwELXCSMbRhlFenvLI40GBQJrCjT6P3CKSzhzFKChFBsbRubekZm2\nvfuYvdn9dSc03b/hSJcU5+GPF01o9xpRNjuzqKj1MtCqCTS3Ymhrk7eOZI1qCpYN+8oQoU2dzNDQ\np1BWXc/lUwfz1V2nHjaFkcCC1swR/kKTyKHNCeqONCjgz7jdLuHX5/h3oXKFiAoeV5g1BVf4NYWO\ndKq1Rk1WWhLb/3Q253XQ1P3uKLCGlmktSRLjdrHd2l870KEGoMNZqFVE1+8tZUhqYotrdgUK7Asb\nGWIL2q4U+HeePLg3SbFuRmb26LQdzY4U2qeAf0mKrX84y3ltPzq+gOajcGsKbicodM1QvAG9Evjl\nWaOZ1o5NZ6JNYAnSXqcq1u1yVu4EeObaqazfW9rsktHdgT0oIrD5aMuB8kYb1rcmsC8s1L7kh4vU\npFiy0pOaXe48mmlQCCF0R3OYo4+soBBmLOlQc05suk2jaiqwBGnvYRHjdlFYUY0IrPntt0iK87S4\nF0Z34AxJtZqPfD7/DO9wln5ICqwptHMtqs7y2k3H6aijEDQohGA3H3kDVlP0tLOm4A4zmKjOF9gB\nadcU7GHKwzN6tLnp5EgX3NG8v6ya2npfWCOu7EmXPeI87d5jPJJ+dfYYZ7TR4bChzeEoOp72dgoc\nrx3uaCCPExQ6NEkqAgIDfkZAnwLAxHZu3HMksvu/aqyawvYCf/NZOCOu7OajEZk9DpsO5kA/6IYD\nLjqaZlkh2M9y4Ebu4c9T8J/vascH4zD8LHVrgU2DdlCorPEPR23rUMzuICZo8trOg/6O9iFhDCmN\n97gROfw6mVXbaU0hBKHxTlwQ/jyFhuaj8HL4j392Ej2CV21VERWqT6HE2oGsrRskdQdOR7NVU9hR\nWInHJWGtAutyCT86eUSLmw6pw5vmPiHYBcfAvVzDHX1kCzcoDDuM11nvrkL1KdgrfWZ04yGowQLX\nPgJ/UBiUmhh2gejn3zqqw9OmOo82H4Vg1xQCh+aFO/rIbnpqT/OR6lyBu6gFdz6mR1FNwS742M/9\njoMV3XpZDxWaBoUQ7Hz812+tdY6FW1Owg0K4fRHq8JKWdGRtkHIoAjuajTHsKKwMqz9BdQ8aFEII\nlY+HOyTVa01803HQR7butH1pa2IDhqSW1dRTVl3PwG60p7dqGw0KITXNyMNtPvLZzUcaFI5o0fT3\niw3oaN5f4t9DIhKb2KvDm3Y0hxCqGyDc5qN6bT464lw4eYDz9W/OHcvaPaVdmJrOF9jRvK/UHxQO\nxwloKrI0KIQQKhsPdwSGTzuajyjb/3R2o6XSvz9zaBempms4NQWvYZ9VU+iXos1H0Uabj0IIlZGH\nW+LXPoUjz+E4A7cz2c94Tb2P/VZNIbNn9Iy+Un5aUwghdPNRuENS/f9rUFBHChEh1u2izuvjYEUd\nvRNjdH2gKBTRmoKIzBaRjSKyRUTubOacb4vIOhFZKyLPRzI9bSWhOprD7FPwaU1BHYFiPS5q633s\nK6nWTuYoFbGgICJu4CHgTGAscLmIjA06ZyRwFzDTGDMOuDVS6QlHqJqCO8ymhUuPGUis28XZR/fr\noFQpFXn+vb39Hc19w1jeQnUfLTYfichtQYcMUAB8YYzZ1sq1pwJbjDG51rVeBM4H1gWccz3wkDGm\nCMAYcyCMtEdMqPw/3ObmkX2S2XTfmR2TIKU6SazH33y0r6SG8f11m8po1FpNITnoX08gG3hPRC5r\n5XsHALsCXudZxwKNAkaJyJciskhEZoe6kIjMEZEcEcnJz89v5cceumjvcFTRK8btoqLGS2FFjTYf\nRakWawrGmN+FOi4iqcAC4MUWvj1UzmqCXnuAkcDJwEDgcxEZb4wpDkrHY8BjANnZ2cHX6HAaElS0\ninW72F1chTFo81GUalefgjHmIK3nnXnAoIDXA4E9Ic55yxhTZzVHbcQfJLqUzi1Q0SrW4yKvyL+5\nTh8djhqV2hUUROQUoKiV05YAI0VkqIjEApcB84LOeROYZV0zHX9zUm570tSRNCaoaBXjdnGgrAaA\n9ChaNlw1aK2jeTVNm3xS8Zf4r2npe40x9SJyMzAfcANPGmPWisjvgRxjzDzrvTNEZB3gBW43xhS2\n71Y6jsYEFa1iPS7sid3RtBigatDa5LVzgl4boNAYU9GWixtj3gXeDTp2d8DXBrjN+nfY0I5mFa0C\n1/hK66FBIRq11tG8o7MScjjRmKCiVazHP4M5PsZFYqwueBCNdO2jEDQmqGgVa9UUUrXpKGppUAgh\nuPloeEYS507s30WpUarz2Gt8pWrTUdTSoBBC8HJFj1x1jFalVVSwl89OTdKRR9FKg0IIwQvi6bwF\nFS2cmkJiTBenRHUVDQqhBMUAXelURQutKSgNCiEExwDdUlNFi1irpqDDUaOXBoUQgjuao2nzdhXd\n7JqCTlyLXhoUQggOAeHupaDUkcqevJaapEEhWmlQCCE4Brj0t6SiRIw2H0U9ze5CCB5t5NGooKKE\nNh8pze3aQJuPVLRIjvPgEsjQFVKjls7ICkGbj1S0unDKQEb360mKzlOIWprdhaDNRypa9YjzcGxW\nalcnQ3Uhze1CCG4s0piglIoWmt2FEDxPQfsUlFLRQoNCCE3mKejkNaVUlNCgEEJwxUB3YlNKRQsN\nCiFoEFBKRSsNCq14/vppXZ0EpZTqNBoUWjEsvUdXJ0EppTqNBoVWaB+zUiqaaFBohfYvKKWiiQaF\nVmhNQSkVTTQotEL3Z1ZKRRMNCq3QoKCUiiYaFFqjMUEpFUUiGhREZLaIbBSRLSJyZwvnXSIiRkSy\nI5me9tA+BaVUNIlYUBARN/AQcCYwFrhcRMaGOC8Z+DHwTaTScii0+UgpFU0iWVOYCmwxxuQaY2qB\nF4HzQ5x3D/BnoDqCaWk3DQpKqWgSyaAwANgV8DrPOuYQkcnAIGPMOy1dSETmiEiOiOTk5+d3fEpb\n/Nmd+uOUUqpLRTIohMpOjfOmiAv4X+BnrV3IGPOYMSbbGJOdkZHRgUlsndYUlFLRJJJBIQ8YFPB6\nILAn4HUyMB74RES2A9OBeYdbZ7N2NCulokkkg8ISYKSIDBWRWOAyYJ79pjGmxBiTbozJMsZkAYuA\n84wxORFMU9h0mQulVDSJWFAwxtQDNwPzgfXAy8aYtSLyexE5L1I/t6NpTUEpFU08kby4MeZd4N2g\nY3c3c+7JkUxLe2lNQSkVTXRGs1JKKYcGBaWUUg4NCkoppRwaFJRSSjk0KCillHJoUFBKKeXQoKCU\nUsqhQUEppZRDg4JSSimHBgWllFIODQpKKaUcGhSUUko5NCgopZRyaFBQSinl0KCglFLKoUFBKaWU\nQ4OCUkophwYFpZRSDg0KSimlHBoUlFJKOTQoKKWUcmhQUEop5dCgoJRSyqFBQSmllEODglJKKYcG\nBaWUUg4NCkoppRwRDQoiMltENorIFhG5M8T7t4nIOhFZJSIficiQSKZHKaVUyyIWFETEDTwEnAmM\nBS4XkbFBpy0Hso0xE4BXgT9HKj1KKaVaF8mawlRgizEm1xhTC7wInB94gjFmoTGm0nq5CBgYwfQo\npZRqRSSDwgBgV8DrPOtYc64D3otgepRSSrXCE8FrS4hjJuSJIlcB2cBJzbw/B5gDMHjw4I5Kn1JK\nqSCRrCnkAYMCXg8E9gSfJCKnAf8DnGeMqQl1IWPMY8aYbGNMdkZGRkQSGywjOY7UpNhO+VlKKXW4\niGRNYQkwUkSGAruBy4ArAk8QkcnAv4HZxpgDEUxL2L6569SuToJSSnW6iNUUjDH1wM3AfGA98LIx\nZq2I/F5EzrNOewDoAbwiIitEZF6k0hMul0twuUK1gCmlVPcVyZoCxph3gXeDjt0d8PVpkfz5Siml\nwqMzmpVSSjk0KCillHJoUFBKKeXQoKCUUsqhQUEppZRDg4JSSimHBgWllFIODQpKKaUcEZ28ppRS\nXaWuro68vDyqq6u7OimdKj4+noEDBxITE9Ou79egoJTqlvLy8khOTiYrKwuR6FiyxhhDYWEheXl5\nDB06tF3X0OYjpVS3VF1dTVpaWtQEBAARIS0t7ZBqRxoUlFLdVjQFBNuh3rMGBaWUUg4NCkopFSFV\nVVWcdNJJeL1eVqxYwYwZMxg3bhwTJkzgpZdeavX7H3zwQcaOHcuECRM49dRT2bFjBwD5+fnMnj07\nImnWoKCUUhHy5JNPctFFF+F2u0lMTOSZZ55h7dq1vP/++9x6660UFxe3+P2TJ08mJyeHVatWcckl\nl3DHHXcAkJGRQb9+/fjyyy87PM06+kgp1e397u21rNtT2qHXHNu/J785d1yL5zz33HM8//zzAIwa\nNco53r9/fzIzM8nPz6dXr17Nfv+sWbOcr6dPn87cuXOd1xdccAHPPfccM2fObO8thKQ1BaWUioDa\n2lpyc3PJyspq8t7ixYupra1l+PDhbb7eE088wZlnnum8zs7O5vPPP++IpDaiNQWlVLfXWok+EgoK\nCkLWAvbu3cvVV1/N008/jcvVtnL53LlzycnJ4dNPP3WOZWZmsmfPng5Lr02DglJKRUBCQkKT+QKl\npaWcffbZ3HvvvUyfPr1N11mwYAH33Xcfn376KXFxcc7x6upqEhISOjTNoM1HSikVEb1798br9TqB\noba2lgsvvJBrrrmGSy+9tNG5d911F2+88UaTayxfvpwbbriBefPmkZmZ2ei9TZs2MX78+A5PtwYF\npZSKkDPOOIMvvvgCgJdffpnPPvuMp556ikmTJjFp0iRWrFgBwOrVq+nbt2+T77/99tspLy/n0ksv\nZdKkSZx33nnOewsXLuTss8/u8DRr85FSSkXIzTffzIMPPshpp53GVVddxVVXXRXyvLq6OmbMmNHk\n+IIFC5q99rx583jrrbc6LK02rSkopVSETJ48mVmzZuH1els8b/78+WFdNz8/n9tuu43evXsfSvJC\n0pqCUkpF0LXXXtvh18zIyOCCCy7o8OuC1hSUUt2YMaark9DpDvWeNSgopbql+Ph4CgsLoyow2Psp\nxMfHt/sa2nyklOqWBg4cSF5eHvn5+V2dlE5l77zWXhoUlFLdUkxMTLt3H4tmEW0+EpHZIrJRRLaI\nyJ0h3o8TkZes978RkaxIpkcppVTLIhYURMQNPAScCYwFLheRsUGnXQcUGWNGAP8L3B+p9CillGpd\nJGsKU4EtxphcY0wt8CJwftA55wNPW1+/Cpwq0bh/nlJKHSYi2acwANgV8DoPmNbcOcaYehEpAdKA\ngsCTRGQOMMd6WS4iG9uZpvTga0cBvefooPccHQ7lnoe05aRIBoVQJf7gsWFtOQdjzGPAY4ecIJEc\nY0z2oV7nSKL3HB30nqNDZ9xzJJuP8oBBAa8HAsGLfzvniIgHSAEORjBNSimlWhDJoLAEGCkiQ0Uk\nFrgMmBd0zjzgu9bXlwAfm2iaaaKUUoeZiDUfWX0ENwPzATfwpDFmrYj8HsgxxswDngCeFZEt+GsI\nl0UqPZZDboI6Auk9Rwe95+gQ8XsWLZgrpZSy6dpHSimlHBoUlFJKOaIiKLS23MaRSkSeFJEDIrIm\n4FiqiHwoIput/3tbx0VE/mH9DlaJyJSuS3n7icggEVkoIutFZK2I/MQ63m3vW0TiRWSxiKy07vl3\n1vGh1vIwm63lYmKt491m+RgRcYvIchF5x3rdre9ZRLaLyGoRWSEiOdaxTn22u31QaONyG0eqp4DZ\nQcfuBD4yxowEPrJeg//+R1r/5gCPdFIaO1o98DNjzBhgOvAj6+/Zne+7BjjFGDMRmATMFpHp+JeF\n+V/rnovwLxsD3Wv5mJ8A6wNeR8M9zzLGTAqYj9C5z7Yxplv/A2YA8wNe3wXc1dXp6sD7ywLWBLze\nCPSzvu4HbLS+/jdweajzjuR/wFvA6dFy30AisAz/6gAFgMc67jzn+Ef8zbC+9ljnSVenvR33OhB/\nJngK8A7+ya7d/Z63A+lBxzr12e72NQVCL7cxoIvS0hn6GGP2Alj/Z1rHu93vwWoimAx8Qze/b6sZ\nZQVwAPgQ2AoUG2PqrVMC76vR8jGAvXzMkeZvwB2Az3qdRve/ZwN8ICJLreV9oJOf7WjYT6FNS2lE\ngW71exCRHsBrwK3GmNIW1lHsFvdtjPECk0SkF/AGMCbUadb/R/w9i8g5wAFjzFIROdk+HOLUbnPP\nlpnGmD0ikgl8KCIbWjg3IvccDTWFtiy30Z3sF5F+ANb/B6zj3eb3ICIx+APCc8aY163D3f6+AYwx\nxcAn+PtTelnLw0Dj++oOy8fMBM4Tke34V1g+BX/NoTvfM8aYPdb/B/AH/6l08rMdDUGhLcttdCeB\nS4d8F3+bu338GmvEwnSgxK6SHknEXyV4AlhvjHkw4K1ue98ikmHVEBCRBOA0/J2vC/EvDwNN7/mI\nXj7GGHOXMWagMSYL/2f2Y2PMlXTjexaRJBFJtr8GzgDW0NnPdld3rHRS581ZwCb87bD/09Xp6cD7\negHYC9ThLzVch78d9SNgs/V/qnWu4B+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0Jn4HvK++8L+Rt1d3jmRPdBngNzs66qrqzkmtoqAgUlirzn7nOqndGjXzW+JK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\ngVFmthyfYflE4C/U7X0mhLA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"text/plain": [ - "" + "" ] }, "metadata": {}, @@ -398,44 +398,44 @@ "data": { "text/plain": [ "defaultdict(float,\n", - " {((0, 0), (-1, 0)): -0.12953971401732597,\n", - " ((0, 0), (0, -1)): -0.12753699595470713,\n", - " ((0, 0), (0, 1)): -0.01158029172666495,\n", - " ((0, 0), (1, 0)): -0.13035841083471436,\n", - " ((0, 1), (-1, 0)): -0.04,\n", - " ((0, 1), (0, -1)): -0.1057916516323444,\n", - " ((0, 1), (0, 1)): 0.13072636267769677,\n", - " ((0, 1), (1, 0)): -0.07323076923076924,\n", - " ((0, 2), (-1, 0)): 0.12165200587479848,\n", - " ((0, 2), (0, -1)): 0.09431411803674361,\n", - " ((0, 2), (0, 1)): 0.14047883620608154,\n", - " ((0, 2), (1, 0)): 0.19224095989491635,\n", - " ((1, 0), (-1, 0)): -0.09696833851887868,\n", - " ((1, 0), (0, -1)): -0.15641263417341367,\n", - " ((1, 0), (0, 1)): -0.15340385689815017,\n", - " ((1, 0), (1, 0)): -0.15224266498911238,\n", - " ((1, 2), (-1, 0)): 0.18537063683043895,\n", - " ((1, 2), (0, -1)): 0.17757702529142774,\n", - " ((1, 2), (0, 1)): 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-0.10590764087842354,\n", + " ((0, 0), (0, 1)): 0.05460040868097919,\n", + " ((0, 0), (1, 0)): -0.09867203219315898,\n", + " ((0, 1), (-1, 0)): 0.07177237857105365,\n", + " ((0, 1), (0, -1)): 0.060286786739471215,\n", + " ((0, 1), (0, 1)): 0.10374209705939107,\n", + " ((0, 1), (1, 0)): -0.04,\n", + " ((0, 2), (-1, 0)): 0.09308553784444584,\n", + " ((0, 2), (0, -1)): 0.09710376713758972,\n", + " ((0, 2), (0, 1)): 0.12895703412485182,\n", + " ((0, 2), (1, 0)): 0.1325347830202934,\n", + " ((1, 0), (-1, 0)): -0.07589625670469141,\n", + " ((1, 0), (0, -1)): -0.0759999433406361,\n", + " ((1, 0), (0, 1)): -0.07323076923076924,\n", + " ((1, 0), (1, 0)): 0.07539875443960498,\n", + " ((1, 2), (-1, 0)): 0.09841555812424703,\n", + " ((1, 2), (0, -1)): 0.1713989451054505,\n", + " ((1, 2), (0, 1)): 0.16142640572251182,\n", + " ((1, 2), (1, 0)): 0.19259892322613212,\n", + " ((2, 0), (-1, 0)): -0.0759999433406361,\n", + " ((2, 0), (0, -1)): -0.0759999433406361,\n", + " ((2, 0), (0, 1)): -0.08367037404281108,\n", + " ((2, 0), (1, 0)): -0.0437928007023705,\n", + " ((2, 1), (-1, 0)): -0.009680447057460156,\n", + " ((2, 1), (0, -1)): -0.6618548845169473,\n", + " ((2, 1), (0, 1)): -0.4333323454834963,\n", + " ((2, 1), (1, 0)): -0.8872940082892214,\n", + " ((2, 2), (-1, 0)): 0.1483330033351123,\n", + " ((2, 2), (0, -1)): 0.04473676319907405,\n", + " ((2, 2), (0, 1)): 0.13217540013336543,\n", + " ((2, 2), (1, 0)): 0.30829164610044535,\n", + " ((3, 0), (-1, 0)): -0.6432395354845424,\n", + " ((3, 0), (0, -1)): 0.0,\n", + " ((3, 0), (0, 1)): -0.787040488208054,\n", + " ((3, 0), (1, 0)): -0.04,\n", + " ((3, 1), None): -0.7641890167582844,\n", + " ((3, 2), None): 0.4106787728880888})" ] }, "execution_count": 15, @@ -483,17 +483,17 @@ "data": { "text/plain": [ "defaultdict(>,\n", - " {(0, 0): -0.01158029172666495,\n", - " (0, 1): 0.13072636267769677,\n", - " (0, 2): 0.19224095989491635,\n", - " (1, 0): -0.09696833851887868,\n", - " (1, 2): 0.27484289408254886,\n", - " (2, 0): -0.028114098214323924,\n", - " (2, 1): 0.2965645851500498,\n", - " (2, 2): 0.34260576652777697,\n", - " (3, 0): -0.16576962655130892,\n", - " (3, 1): -0.6897322258069369,\n", - " (3, 2): 0.388990723935834})" + " {(0, 0): 0.05460040868097919,\n", + " (0, 1): 0.10374209705939107,\n", + " (0, 2): 0.1325347830202934,\n", + " (1, 0): 0.07539875443960498,\n", + " (1, 2): 0.19259892322613212,\n", + " (2, 0): -0.0437928007023705,\n", + " (2, 1): -0.009680447057460156,\n", + " (2, 2): 0.30829164610044535,\n", + " (3, 0): 0.0,\n", + " (3, 1): -0.7641890167582844,\n", + " (3, 2): 0.4106787728880888})" ] }, "execution_count": 17, @@ -529,6 +529,15 @@ "print(value_iteration(sequential_decision_environment))" ] }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, { "cell_type": "code", "execution_count": null, @@ -555,7 +564,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.2+" + "version": "3.6.1" } }, "nbformat": 4, diff --git a/rl.py b/rl.py index 3258bfffe..de0e65179 100644 --- a/rl.py +++ b/rl.py @@ -16,7 +16,7 @@ class ModelMDP(MDP): """ Class for implementing modified Version of input MDP with an editable transition model P and a custom function T. """ def __init__(self, init, actlist, terminals, gamma, states): - super().__init__(init, actlist, terminals, gamma) + super().__init__(init, actlist, terminals, states = states, gamma = gamma) nested_dict = lambda: defaultdict(nested_dict) # StackOverflow:whats-the-best-way-to-initialize-a-dict-of-dicts-in-python self.P = nested_dict() diff --git a/tests/test_mdp.py b/tests/test_mdp.py index 1aed4b58f..00710bc9f 100644 --- a/tests/test_mdp.py +++ b/tests/test_mdp.py @@ -100,14 +100,22 @@ def test_best_policy(): def test_transition_model(): - transition_model = { - "A": {"a1": (0.3, "B"), "a2": (0.7, "C")}, - "B": {"a1": (0.5, "B"), "a2": (0.5, "A")}, - "C": {"a1": (0.9, "A"), "a2": (0.1, "B")}, - } - - mdp = MDP(init="A", actlist={"a1","a2"}, terminals={"C"}, states={"A","B","C"}, transitions=transition_model) - - assert mdp.T("A","a1") == (0.3, "B") - assert mdp.T("B","a2") == (0.5, "A") - assert mdp.T("C","a1") == (0.9, "A") + transition_model = { 'a' : { 'plan1' : [(0.2, 'a'), (0.3, 'b'), (0.3, 'c'), (0.2, 'd')], + 'plan2' : [(0.4, 'a'), (0.15, 'b'), (0.45, 'c')], + 'plan3' : [(0.2, 'a'), (0.5, 'b'), (0.3, 'c')], + }, + 'b' : { 'plan1' : [(0.2, 'a'), (0.6, 'b'), (0.2, 'c'), (0.1, 'd')], + 'plan2' : [(0.6, 'a'), (0.2, 'b'), (0.1, 'c'), (0.1, 'd')], + 'plan3' : [(0.3, 'a'), (0.3, 'b'), (0.4, 'c')], + }, + 'c' : { 'plan1' : [(0.3, 'a'), (0.5, 'b'), (0.1, 'c'), (0.1, 'd')], + 'plan2' : [(0.5, 'a'), (0.3, 'b'), (0.1, 'c'), (0.1, 'd')], + 'plan3' : [(0.1, 'a'), (0.3, 'b'), (0.1, 'c'), (0.5, 'd')], + }, + } + + mdp = MDP(init="a", actlist={"plan1","plan2", "plan3"}, terminals={"d"}, states={"a","b","c", "d"}, transitions=transition_model) + + assert mdp.T("a","plan3") == [(0.2, 'a'), (0.5, 'b'), (0.3, 'c')] + assert mdp.T("b","plan2") == [(0.6, 'a'), (0.2, 'b'), (0.1, 'c'), (0.1, 'd')] + assert mdp.T("c","plan1") == [(0.3, 'a'), (0.5, 'b'), (0.1, 'c'), (0.1, 'd')] From 35be708c961609f5f64cc43be10e87f97d6245e0 Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Wed, 28 Feb 2018 00:22:38 -0500 Subject: [PATCH 10/11] fixed test_rl --- rl.py | 14 ++++++++++---- tests/test_rl.py | 3 ++- 2 files changed, 12 insertions(+), 5 deletions(-) diff --git a/rl.py b/rl.py index de0e65179..79fcff09d 100644 --- a/rl.py +++ b/rl.py @@ -35,15 +35,18 @@ def __init__(self, pi, mdp): self.Ns1_sa = defaultdict(int) self.s = None self.a = None + self.visited = set() def __call__(self, percept): s1, r1 = percept self.mdp.states.add(s1) # Model keeps track of visited states. - R, P, mdp, pi = self.mdp.reward, self.mdp.P, self.mdp, self.pi + mdp = self.mdp + R, P, terminals, pi = mdp.reward, mdp.P, mdp.terminals, self.pi s, a, Nsa, Ns1_sa, U = self.s, self.a, self.Nsa, self.Ns1_sa, self.U - if s1 not in R: # Reward is only available for visted state. + if s1 not in self.visited: # Reward is only available for visited state. U[s1] = R[s1] = r1 + self.visited.add(s1) if s is not None: Nsa[(s, a)] += 1 Ns1_sa[(s1, s, a)] += 1 @@ -52,8 +55,11 @@ def __call__(self, percept): if (state, act) == (s, a) and freq != 0]: P[(s, a)][t] = Ns1_sa[(t, s, a)] / Nsa[(s, a)] - U = policy_evaluation(pi, U, mdp) - if s1 in mdp.terminals: + self.U = policy_evaluation(pi, U, mdp) + ## + ## + self.Nsa, self.Ns1_sa = Nsa, Ns1_sa + if s1 in terminals: self.s = self.a = None else: self.s, self.a = s1, self.pi[s1] diff --git a/tests/test_rl.py b/tests/test_rl.py index 05f071266..932b34ae5 100644 --- a/tests/test_rl.py +++ b/tests/test_rl.py @@ -19,11 +19,12 @@ def test_PassiveADPAgent(): agent = PassiveADPAgent(policy, sequential_decision_environment) - for i in range(75): + for i in range(100): run_single_trial(agent,sequential_decision_environment) # Agent does not always produce same results. # Check if results are good enough. + #print(agent.U[(0, 0)], agent.U[(0,1)], agent.U[(1,0)]) assert agent.U[(0, 0)] > 0.15 # In reality around 0.3 assert agent.U[(0, 1)] > 0.15 # In reality around 0.4 assert agent.U[(1, 0)] > 0 # In reality around 0.2 From 1a1a0495073d463f2c5bae0e45587d0151ea9dee Mon Sep 17 00:00:00 2001 From: Aabir Abubaker Kar Date: Wed, 28 Feb 2018 00:36:04 -0500 Subject: [PATCH 11/11] removed redundant code, fixed a comment --- rl.py | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/rl.py b/rl.py index 79fcff09d..94664b130 100644 --- a/rl.py +++ b/rl.py @@ -35,16 +35,15 @@ def __init__(self, pi, mdp): self.Ns1_sa = defaultdict(int) self.s = None self.a = None - self.visited = set() + self.visited = set() # keeping track of visited states def __call__(self, percept): s1, r1 = percept - self.mdp.states.add(s1) # Model keeps track of visited states. mdp = self.mdp R, P, terminals, pi = mdp.reward, mdp.P, mdp.terminals, self.pi s, a, Nsa, Ns1_sa, U = self.s, self.a, self.Nsa, self.Ns1_sa, self.U - if s1 not in self.visited: # Reward is only available for visited state. + if s1 not in self.visited: # Reward is only known for visited state. U[s1] = R[s1] = r1 self.visited.add(s1) if s is not None: