mdp.py 55.5 KB
 Steven Cordwell committed Jun 12, 2012 1 ``````# -*- coding: utf-8 -*- `````` Steven Cordwell committed Feb 04, 2013 2 ``````"""Markov Decision Process (MDP) Toolbox `````` Steven Cordwell committed Feb 04, 2013 3 ``````===================================== `````` Steven Cordwell committed Jan 26, 2013 4 `````` `````` Steven Cordwell committed Feb 04, 2013 5 6 ``````The MDP toolbox provides classes and functions for the resolution of descrete-time Markov Decision Processes. `````` Steven Cordwell committed Jun 12, 2012 7 `````` `````` Steven Cordwell committed Feb 04, 2013 8 9 10 11 12 ``````Available classes ----------------- MDP Base Markov decision process class FiniteHorizon `````` Steven Cordwell committed Feb 04, 2013 13 `````` Backwards induction finite horizon MDP `````` Steven Cordwell committed Feb 04, 2013 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ``````LP Linear programming MDP PolicyIteration Policy iteration MDP PolicyIterationModified Modified policy iteration MDP QLearning Q-learning MDP RelativeValueIteration Relative value iteration MDP ValueIteration Value iteration MDP ValueIterationGS Gauss-Seidel value iteration MDP `````` Steven Cordwell committed Jun 12, 2012 28 `````` `````` Steven Cordwell committed Feb 04, 2013 29 30 31 32 33 34 35 36 37 38 ``````Available functions ------------------- check Check that an MDP is properly defined checkSquareStochastic Check that a matrix is square and stochastic exampleForest A simple forest management example exampleRand A random example `````` Steven Cordwell committed Jun 12, 2012 39 `````` `````` Steven Cordwell committed Feb 04, 2013 40 41 42 43 44 ``````How to use the documentation ---------------------------- Documentation is available both as docstrings provided with the code and in html or pdf format from `The MDP toolbox homepage `_. The docstring `````` Steven Cordwell committed Aug 24, 2013 45 ``````examples assume that the `mdp` module has been imported imported like so:: `````` Steven Cordwell committed Jun 12, 2012 46 `````` `````` Steven Cordwell committed Aug 24, 2013 47 `````` >>> import mdptoolbox.mdp as mdp `````` Steven Cordwell committed Feb 04, 2013 48 49 50 51 52 `````` Code snippets are indicated by three greater-than signs:: >>> x = 17 >>> x = x + 1 `````` Steven Cordwell committed May 18, 2013 53 54 `````` >>> x 18 `````` Steven Cordwell committed Feb 04, 2013 55 56 57 58 59 `````` The documentation can be displayed with `IPython `_. For example, to view the docstring of the ValueIteration class use ``mdp.ValueIteration?``, and to view its source code use ``mdp.ValueIteration??``. `````` 60 `````` `````` 61 62 63 ``````Acknowledgments --------------- This module is modified from the MDPtoolbox (c) 2009 INRA available at `````` Steven Cordwell committed Feb 08, 2013 64 ``````http://www.inra.fr/mia/T/MDPtoolbox/. `````` 65 `````` `````` Steven Cordwell committed Jun 12, 2012 66 67 ``````""" `````` 68 69 ``````# Copyright (c) 2011-2013 Steven A. W. Cordwell # Copyright (c) 2009 INRA `````` Steven Cordwell committed Feb 04, 2013 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 ``````# # All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions are met: # # * Redistributions of source code must retain the above copyright notice, # this list of conditions and the following disclaimer. # * Redistributions in binary form must reproduce the above copyright notice, # this list of conditions and the following disclaimer in the documentation # and/or other materials provided with the distribution. # * Neither the name of the nor the names of its contributors # may be used to endorse or promote products derived from this software # without specific prior written permission. # # THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" # AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE # IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE # ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE # LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR # CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF # SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS # INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN # CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) # ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE # POSSIBILITY OF SUCH DAMAGE. `````` Steven Cordwell committed Jan 26, 2013 97 98 99 ``````from math import ceil, log, sqrt from time import time `````` Steven Cordwell committed Aug 18, 2013 100 101 ``````from numpy import absolute, array, empty, mean, mod, multiply from numpy import ndarray, ones, zeros `````` Steven Cordwell committed Aug 24, 2013 102 ``````from numpy.random import randint, random `````` Steven Cordwell committed Oct 29, 2012 103 ``````from scipy.sparse import csr_matrix as sparse `````` Steven Cordwell committed Jun 12, 2012 104 `````` `````` Steven Cordwell committed Aug 18, 2013 105 ``````from utils import check, getSpan `````` 106 `````` `````` Steven Cordwell committed Nov 04, 2012 107 ``````class MDP(object): `````` 108 `````` `````` Steven Cordwell committed Feb 04, 2013 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 `````` """A Markov Decision Problem. Parameters ---------- transitions : array transition probability matrices reward : array reward matrices discount : float or None discount factor epsilon : float or None stopping criteria max_iter : int or None maximum number of iterations Attributes ---------- P : array Transition probability matrices R : array Reward matrices V : list Value function discount : float b max_iter : int a policy : list a time : float a verbose : logical a Methods ------- iterate To be implemented in child classes, raises exception setSilent Turn the verbosity off setVerbose Turn the verbosity on """ `````` Steven Cordwell committed Oct 29, 2012 153 `````` `````` Steven Cordwell committed Jan 25, 2013 154 `````` def __init__(self, transitions, reward, discount, epsilon, max_iter): `````` Steven Cordwell committed Aug 24, 2013 155 `````` # Initialise a MDP based on the input parameters. `````` 156 `````` `````` Steven Cordwell committed Jan 24, 2013 157 158 `````` # if the discount is None then the algorithm is assumed to not use it # in its computations `````` Steven Cordwell committed Sep 10, 2013 159 160 161 162 163 164 `````` if discount is not None: self.discount = float(discount) assert 0.0 < self.discount <= 1.0, "Discount rate must be in ]0; 1]" if self.discount == 1: print("PyMDPtoolbox WARNING: check conditions of convergence. " "With no discount, convergence is not always assumed.") `````` Steven Cordwell committed Jan 24, 2013 165 166 `````` # if the max_iter is None then the algorithm is assumed to not use it # in its computations `````` Steven Cordwell committed Sep 10, 2013 167 168 169 170 `````` if max_iter is not None: self.max_iter = int(max_iter) assert self.max_iter > 0, "The maximum number of iterations " \ "must be greater than 0." `````` Steven Cordwell committed Jun 21, 2013 171 `````` # check that epsilon is something sane `````` Steven Cordwell committed Sep 10, 2013 172 173 174 `````` if epsilon is not None: self.epsilon = float(epsilon) assert self.epsilon > 0, "Epsilon must be greater than 0." `````` Steven Cordwell committed Jan 22, 2013 175 176 177 178 `````` # we run a check on P and R to make sure they are describing an MDP. If # an exception isn't raised then they are assumed to be correct. check(transitions, reward) # computePR will assign the variables self.S, self.A, self.P and self.R `````` 179 `````` self._computePR(transitions, reward) `````` Steven Cordwell committed Jan 21, 2013 180 181 182 183 `````` # the verbosity is by default turned off self.verbose = False # Initially the time taken to perform the computations is set to None self.time = None `````` Steven Cordwell committed Jan 21, 2013 184 185 `````` # set the initial iteration count to zero self.iter = 0 `````` Steven Cordwell committed Feb 10, 2013 186 `````` # V should be stored as a vector ie shape of (S,) or (1, S) `````` Steven Cordwell committed Jan 24, 2013 187 `````` self.V = None `````` Steven Cordwell committed Feb 10, 2013 188 `````` # policy can also be stored as a vector `````` Steven Cordwell committed Jan 21, 2013 189 `````` self.policy = None `````` Steven Cordwell committed Oct 29, 2012 190 `````` `````` Steven Cordwell committed Sep 10, 2013 191 192 193 194 195 196 197 198 `````` def __repr__(self): P_repr = "P: \n" R_repr = "R: \n" for aa in range(self.A): P_repr += repr(self.P[aa]) + "\n" R_repr += repr(self.R[aa]) + "\n" print(P_repr + "\n" + R_repr) `````` 199 `````` def _bellmanOperator(self, V=None): `````` Steven Cordwell committed Jun 21, 2013 200 `````` # Apply the Bellman operator on the value function. `````` Steven Cordwell committed Aug 24, 2013 201 `````` # `````` Steven Cordwell committed Jun 21, 2013 202 `````` # Updates the value function and the Vprev-improving policy. `````` Steven Cordwell committed Aug 24, 2013 203 `````` # `````` Steven Cordwell committed Jun 21, 2013 204 205 206 207 `````` # Returns: (policy, value), tuple of new policy and its value # # If V hasn't been sent into the method, then we assume to be working # on the objects V attribute `````` Steven Cordwell committed Jan 26, 2013 208 209 `````` if V is None: # this V should be a reference to the data rather than a copy `````` Steven Cordwell committed Jan 25, 2013 210 211 `````` V = self.V else: `````` Steven Cordwell committed Jun 21, 2013 212 `````` # make sure the user supplied V is of the right shape `````` Steven Cordwell committed Jan 26, 2013 213 `````` try: `````` Steven Cordwell committed Sep 10, 2013 214 215 `````` assert V.shape in ((self.S,), (1, self.S)), "V is not the " \ "right shape (Bellman operator)." `````` Steven Cordwell committed Jan 26, 2013 216 `````` except AttributeError: `````` Steven Cordwell committed Sep 10, 2013 217 `````` raise TypeError("V must be a numpy array or matrix.") `````` Steven Cordwell committed Aug 24, 2013 218 219 220 221 `````` # Looping through each action the the Q-value matrix is calculated. # P and V can be any object that supports indexing, so it is important # that you know they define a valid MDP before calling the # _bellmanOperator method. Otherwise the results will be meaningless. `````` Steven Cordwell committed May 18, 2013 222 `````` Q = empty((self.A, self.S)) `````` Steven Cordwell committed Jun 12, 2012 223 `````` for aa in range(self.A): `````` Steven Cordwell committed May 18, 2013 224 `````` Q[aa] = self.R[aa] + self.discount * self.P[aa].dot(V) `````` Steven Cordwell committed Jun 21, 2013 225 `````` # Get the policy and value, for now it is being returned but... `````` Steven Cordwell committed Jan 26, 2013 226 `````` # Which way is better? `````` 227 `````` # 1. Return, (policy, value) `````` Steven Cordwell committed May 18, 2013 228 `````` return (Q.argmax(axis=0), Q.max(axis=0)) `````` Steven Cordwell committed Jan 24, 2013 229 230 `````` # 2. update self.policy and self.V directly # self.V = Q.max(axis=1) `````` Steven Cordwell committed Jan 24, 2013 231 `````` # self.policy = Q.argmax(axis=1) `````` Steven Cordwell committed Jun 12, 2012 232 `````` `````` 233 234 235 236 237 238 239 240 241 242 243 244 245 246 `````` def _computeP(self, P): # Set self.P as a tuple of length A, with each element storing an S×S # matrix. self.A = len(P) try: if P.ndim == 3: self.S = P.shape[1] else: self.S = P[0].shape[0] except AttributeError: self.S = P[0].shape[0] # convert P to a tuple of numpy arrays self.P = tuple([P[aa] for aa in range(self.A)]) `````` 247 `````` def _computePR(self, P, R): `````` Steven Cordwell committed Jun 21, 2013 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 `````` # Compute the reward for the system in one state chosing an action. # Arguments # --------- # Let S = number of states, A = number of actions # P(SxSxA) = transition matrix # P could be an array with 3 dimensions or a cell array (1xA), # each cell containing a matrix (SxS) possibly sparse # R(SxSxA) or (SxA) = reward matrix # R could be an array with 3 dimensions (SxSxA) or a cell array # (1xA), each cell containing a sparse matrix (SxS) or a 2D # array(SxA) possibly sparse # Evaluation # ---------- # PR(SxA) = reward matrix # `````` Steven Cordwell committed Jan 26, 2013 263 `````` # We assume that P and R define a MDP i,e. assumption is that `````` Steven Cordwell committed Jan 22, 2013 264 `````` # check(P, R) has already been run and doesn't fail. `````` Steven Cordwell committed Jan 26, 2013 265 `````` # `````` 266 267 `````` # First compute store P, S, and A self._computeP(P) `````` Steven Cordwell committed Feb 10, 2013 268 269 `````` # Set self.R as a tuple of length A, with each element storing an 1×S # vector. `````` Steven Cordwell committed Feb 09, 2013 270 `````` try: `````` Steven Cordwell committed Jan 24, 2013 271 `````` if R.ndim == 2: `````` Steven Cordwell committed Sep 10, 2013 272 273 `````` self.R = tuple([array(R[:, aa]).reshape(self.S) for aa in range(self.A)]) `````` Steven Cordwell committed Jun 12, 2012 274 `````` else: `````` Steven Cordwell committed Sep 10, 2013 275 276 `````` self.R = tuple([multiply(P[aa], R[aa]).sum(1).reshape(self.S) for aa in xrange(self.A)]) `````` Steven Cordwell committed Feb 09, 2013 277 `````` except AttributeError: `````` Steven Cordwell committed Sep 10, 2013 278 279 `````` self.R = tuple([multiply(P[aa], R[aa]).sum(1).reshape(self.S) for aa in xrange(self.A)]) `````` Steven Cordwell committed Jan 26, 2013 280 `````` `````` 281 `````` def _iterate(self): `````` Steven Cordwell committed Jun 21, 2013 282 `````` # Raise error because child classes should implement this function. `````` 283 `````` raise NotImplementedError("You should create an _iterate() method.") `````` Steven Cordwell committed Jun 12, 2012 284 `````` `````` Steven Cordwell committed Oct 29, 2012 285 `````` def setSilent(self): `````` Steven Cordwell committed Jan 26, 2013 286 `````` """Set the MDP algorithm to silent mode.""" `````` Steven Cordwell committed Oct 29, 2012 287 288 289 `````` self.verbose = False def setVerbose(self): `````` Steven Cordwell committed Jan 26, 2013 290 `````` """Set the MDP algorithm to verbose mode.""" `````` Steven Cordwell committed Oct 29, 2012 291 `````` self.verbose = True `````` Steven Cordwell committed Oct 29, 2012 292 293 `````` class FiniteHorizon(MDP): `````` 294 `````` `````` Steven Cordwell committed Feb 04, 2013 295 `````` """A MDP solved using the finite-horizon backwards induction algorithm. `````` Steven Cordwell committed Jan 21, 2013 296 297 `````` Let S = number of states, A = number of actions `````` Steven Cordwell committed Feb 04, 2013 298 299 300 `````` Parameters ---------- `````` Steven Cordwell committed Jan 21, 2013 301 `````` P(SxSxA) = transition matrix `````` Steven Cordwell committed Jan 26, 2013 302 303 `````` P could be an array with 3 dimensions ora cell array (1xA), each cell containing a matrix (SxS) possibly sparse `````` Steven Cordwell committed Jan 21, 2013 304 305 306 307 308 309 310 `````` R(SxSxA) or (SxA) = reward matrix R could be an array with 3 dimensions (SxSxA) or a cell array (1xA), each cell containing a sparse matrix (SxS) or a 2D array(SxA) possibly sparse discount = discount factor, in ]0, 1] N = number of periods, upper than 0 h(S) = terminal reward, optional (default [0; 0; ... 0] ) `````` Steven Cordwell committed Feb 04, 2013 311 312 `````` Attributes `````` Steven Cordwell committed Jan 21, 2013 313 `````` ---------- `````` Steven Cordwell committed Feb 04, 2013 314 315 316 `````` Methods ------- `````` Steven Cordwell committed Jan 21, 2013 317 318 319 320 321 322 323 324 325 326 327 328 329 `````` V(S,N+1) = optimal value function V(:,n) = optimal value function at stage n with stage in 1, ..., N V(:,N+1) = value function for terminal stage policy(S,N) = optimal policy policy(:,n) = optimal policy at stage n with stage in 1, ...,N policy(:,N) = policy for stage N cpu_time = used CPU time Notes ----- In verbose mode, displays the current stage and policy transpose. `````` 330 `````` `````` Steven Cordwell committed Feb 04, 2013 331 332 `````` Examples -------- `````` Steven Cordwell committed Aug 24, 2013 333 334 335 `````` >>> import mdptoolbox, mdptoolbox.example >>> P, R = mdptoolbox.example.forest() >>> fh = mdptoolbox.mdp.FiniteHorizon(P, R, 0.9, 3) `````` Steven Cordwell committed Feb 04, 2013 336 337 338 339 340 341 342 343 `````` >>> fh.V array([[ 2.6973, 0.81 , 0. , 0. ], [ 5.9373, 3.24 , 1. , 0. ], [ 9.9373, 7.24 , 4. , 0. ]]) >>> fh.policy array([[0, 0, 0], [0, 0, 1], [0, 0, 0]]) `````` Steven Cordwell committed Aug 24, 2013 344 `````` `````` Steven Cordwell committed Oct 29, 2012 345 `````` """ `````` Steven Cordwell committed Jan 21, 2013 346 `````` `````` Steven Cordwell committed Jan 21, 2013 347 `````` def __init__(self, transitions, reward, discount, N, h=None): `````` Steven Cordwell committed Aug 24, 2013 348 `````` # Initialise a finite horizon MDP. `````` Steven Cordwell committed Sep 10, 2013 349 350 `````` self.N = int(N) assert self.N > 0, 'PyMDPtoolbox: N must be greater than 0.' `````` Steven Cordwell committed Feb 10, 2013 351 `````` # Initialise the base class `````` Steven Cordwell committed Jan 25, 2013 352 `````` MDP.__init__(self, transitions, reward, discount, None, None) `````` Steven Cordwell committed Feb 10, 2013 353 354 `````` # remove the iteration counter, it is not meaningful for backwards # induction `````` Steven Cordwell committed Jan 25, 2013 355 `````` del self.iter `````` Steven Cordwell committed Feb 10, 2013 356 `````` # There are value vectors for each time step up to the horizon `````` Steven Cordwell committed Jan 25, 2013 357 `````` self.V = zeros((self.S, N + 1)) `````` Steven Cordwell committed Feb 10, 2013 358 359 360 361 362 `````` # There are policy vectors for each time step before the horizon, when # we reach the horizon we don't need to make decisions anymore. self.policy = empty((self.S, N), dtype=int) # Set the reward for the final transition to h, if specified. if h is not None: `````` Steven Cordwell committed Jan 25, 2013 363 `````` self.V[:, N] = h `````` 364 365 366 367 `````` # Call the iteration method self._iterate() def _iterate(self): `````` Steven Cordwell committed Jun 21, 2013 368 `````` # Run the finite horizon algorithm. `````` Steven Cordwell committed Jan 21, 2013 369 `````` self.time = time() `````` Steven Cordwell committed Jun 21, 2013 370 `````` # loop through each time period `````` Steven Cordwell committed Jan 25, 2013 371 `````` for n in range(self.N): `````` Steven Cordwell committed Feb 10, 2013 372 373 374 `````` W, X = self._bellmanOperator(self.V[:, self.N - n]) self.V[:, self.N - n - 1] = X self.policy[:, self.N - n - 1] = W `````` Steven Cordwell committed Jan 21, 2013 375 `````` if self.verbose: `````` Steven Cordwell committed Jan 26, 2013 376 377 `````` print("stage: %s ... policy transpose : %s") % ( self.N - n, self.policy[:, self.N - n -1].tolist()) `````` Steven Cordwell committed Jun 21, 2013 378 `````` # update time spent running `````` Steven Cordwell committed Jan 21, 2013 379 `````` self.time = time() - self.time `````` Steven Cordwell committed Feb 10, 2013 380 381 382 383 384 385 386 387 388 389 `````` # After this we could create a tuple of tuples for the values and # policies. #V = [] #p = [] #for n in xrange(self.N): # V.append() # p.append() #V.append() #self.V = tuple(V) #self.policy = tuple(p) `````` Steven Cordwell committed Oct 29, 2012 390 391 `````` class LP(MDP): `````` 392 `````` `````` Steven Cordwell committed Jan 26, 2013 393 `````` """A discounted MDP soloved using linear programming. `````` Steven Cordwell committed Jan 21, 2013 394 395 396 397 398 `````` Arguments --------- Let S = number of states, A = number of actions P(SxSxA) = transition matrix `````` Steven Cordwell committed Jan 26, 2013 399 400 `````` P could be an array with 3 dimensions or a cell array (1xA), each cell containing a matrix (SxS) possibly sparse `````` Steven Cordwell committed Jan 21, 2013 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 `````` R(SxSxA) or (SxA) = reward matrix R could be an array with 3 dimensions (SxSxA) or a cell array (1xA), each cell containing a sparse matrix (SxS) or a 2D array(SxA) possibly sparse discount = discount rate, in ]0; 1[ h(S) = terminal reward, optional (default [0; 0; ... 0] ) Evaluation ---------- V(S) = optimal values policy(S) = optimal policy cpu_time = used CPU time Notes ----- In verbose mode, displays the current stage and policy transpose. Examples -------- `````` Steven Cordwell committed Aug 24, 2013 420 421 422 `````` >>> import mdptoolbox, mdptoolbox.example >>> P, R = mdptoolbox.example.forest() >>> lp = mdptoolbox.mdp.LP(P, R, 0.9) `````` 423 `````` `````` Steven Cordwell committed Oct 29, 2012 424 `````` """ `````` Steven Cordwell committed Jan 21, 2013 425 `````` `````` Steven Cordwell committed Jan 21, 2013 426 `````` def __init__(self, transitions, reward, discount): `````` Steven Cordwell committed Aug 24, 2013 427 `````` # Initialise a linear programming MDP. `````` Steven Cordwell committed Jun 21, 2013 428 `````` # import some functions from cvxopt and set them as object methods `````` Steven Cordwell committed Jan 21, 2013 429 430 `````` try: from cvxopt import matrix, solvers `````` Steven Cordwell committed Jan 26, 2013 431 432 `````` self._linprog = solvers.lp self._cvxmat = matrix `````` Steven Cordwell committed Jan 21, 2013 433 `````` except ImportError: `````` Steven Cordwell committed Jan 26, 2013 434 435 `````` raise ImportError("The python module cvxopt is required to use " "linear programming functionality.") `````` Steven Cordwell committed Jun 21, 2013 436 437 `````` # we also need diagonal matrices, and using a sparse one may be more # memory efficient `````` Steven Cordwell committed Jan 21, 2013 438 `````` from scipy.sparse import eye as speye `````` Steven Cordwell committed Jan 26, 2013 439 `````` self._speye = speye `````` Steven Cordwell committed Jun 21, 2013 440 `````` # initialise the MDP. epsilon and max_iter are not needed `````` Steven Cordwell committed Jan 26, 2013 441 `````` MDP.__init__(self, transitions, reward, discount, None, None) `````` Steven Cordwell committed Jun 21, 2013 442 `````` # Set the cvxopt solver to be quiet by default, but ... `````` Steven Cordwell committed Jan 26, 2013 443 `````` # this doesn't do what I want it to do c.f. issue #3 `````` Steven Cordwell committed Jan 26, 2013 444 445 `````` if not self.verbose: solvers.options['show_progress'] = False `````` 446 447 `````` # Call the iteration method self._iterate() `````` Steven Cordwell committed Jan 26, 2013 448 `````` `````` 449 `````` def _iterate(self): `````` Steven Cordwell committed Jun 21, 2013 450 `````` #Run the linear programming algorithm. `````` Steven Cordwell committed Jan 26, 2013 451 `````` self.time = time() `````` Steven Cordwell committed Jan 21, 2013 452 `````` # The objective is to resolve : min V / V >= PR + discount*P*V `````` Steven Cordwell committed Jan 26, 2013 453 454 `````` # The function linprog of the optimisation Toolbox of Mathworks # resolves : `````` Steven Cordwell committed Jan 21, 2013 455 `````` # min f'* x / M * x <= b `````` Steven Cordwell committed Jan 26, 2013 456 457 458 459 `````` # So the objective could be expressed as : # min V / (discount*P-I) * V <= - PR # To avoid loop on states, the matrix M is structured following actions # M(A*S,S) `````` Steven Cordwell committed Jan 26, 2013 460 461 462 `````` f = self._cvxmat(ones((self.S, 1))) h = self._cvxmat(self.R.reshape(self.S * self.A, 1, order="F"), tc='d') M = zeros((self.A * self.S, self.S)) `````` Steven Cordwell committed Jan 21, 2013 463 464 `````` for aa in range(self.A): pos = (aa + 1) * self.S `````` Steven Cordwell committed Jan 26, 2013 465 466 467 `````` M[(pos - self.S):pos, :] = ( self.discount * self.P[aa] - self._speye(self.S, self.S)) M = self._cvxmat(M) `````` Steven Cordwell committed Jan 26, 2013 468 469 470 `````` # Using the glpk option will make this behave more like Octave # (Octave uses glpk) and perhaps Matlab. If solver=None (ie using the # default cvxopt solver) then V agrees with the Octave equivalent `````` Steven Cordwell committed Jun 21, 2013 471 `````` # only to 10e-8 places. This assumes glpk is installed of course. `````` Steven Cordwell committed Feb 10, 2013 472 `````` self.V = array(self._linprog(f, M, -h, solver='glpk')['x']) `````` Steven Cordwell committed Jun 21, 2013 473 `````` # apply the Bellman operator `````` 474 `````` self.policy, self.V = self._bellmanOperator() `````` Steven Cordwell committed Jun 21, 2013 475 `````` # update the time spent solving `````` Steven Cordwell committed Jan 21, 2013 476 `````` self.time = time() - self.time `````` Steven Cordwell committed Jan 26, 2013 477 `````` # store value and policy as tuples `````` 478 479 `````` self.V = tuple(self.V.tolist()) self.policy = tuple(self.policy.tolist()) `````` Steven Cordwell committed Oct 29, 2012 480 481 `````` class PolicyIteration(MDP): `````` 482 `````` `````` Steven Cordwell committed Jan 26, 2013 483 `````` """A discounted MDP solved using the policy iteration algorithm. `````` Steven Cordwell committed Nov 04, 2012 484 `````` `````` Steven Cordwell committed Jan 22, 2013 485 486 487 488 `````` Arguments --------- Let S = number of states, A = number of actions P(SxSxA) = transition matrix `````` Steven Cordwell committed Jan 26, 2013 489 490 `````` P could be an array with 3 dimensions or a cell array (1xA), each cell containing a matrix (SxS) possibly sparse `````` Steven Cordwell committed Jan 22, 2013 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 `````` R(SxSxA) or (SxA) = reward matrix R could be an array with 3 dimensions (SxSxA) or a cell array (1xA), each cell containing a sparse matrix (SxS) or a 2D array(SxA) possibly sparse discount = discount rate, in ]0, 1[ policy0(S) = starting policy, optional max_iter = maximum number of iteration to be done, upper than 0, optional (default 1000) eval_type = type of function used to evaluate policy: 0 for mdp_eval_policy_matrix, else mdp_eval_policy_iterative optional (default 0) Evaluation ---------- V(S) = value function policy(S) = optimal policy iter = number of done iterations cpu_time = used CPU time Notes ----- In verbose mode, at each iteration, displays the number of differents actions between policy n-1 and n `````` Steven Cordwell committed Nov 04, 2012 515 516 `````` Examples -------- `````` Steven Cordwell committed Aug 24, 2013 517 518 519 `````` >>> import mdptoolbox, mdptoolbox.example >>> P, R = mdptoolbox.example.rand() >>> pi = mdptoolbox.mdp.PolicyIteration(P, R, 0.9) `````` Steven Cordwell committed Jun 21, 2013 520 `````` `````` Steven Cordwell committed Aug 24, 2013 521 522 `````` >>> P, R = mdptoolbox.example.forest() >>> pi = mdptoolbox.mdp.PolicyIteration(P, R, 0.9) `````` Steven Cordwell committed Jun 21, 2013 523 `````` >>> pi.V `````` Steven Cordwell committed Jun 21, 2013 524 `````` (26.244000000000018, 29.48400000000002, 33.484000000000016) `````` Steven Cordwell committed Jun 21, 2013 525 `````` >>> pi.policy `````` Steven Cordwell committed Jun 21, 2013 526 `````` (0, 0, 0) `````` Steven Cordwell committed Oct 29, 2012 527 `````` """ `````` Steven Cordwell committed Nov 04, 2012 528 `````` `````` Steven Cordwell committed Jan 26, 2013 529 530 `````` def __init__(self, transitions, reward, discount, policy0=None, max_iter=1000, eval_type=0): `````` Steven Cordwell committed Jun 21, 2013 531 532 533 `````` # Initialise a policy iteration MDP. # # Set up the MDP, but don't need to worry about epsilon values `````` Steven Cordwell committed Jan 25, 2013 534 `````` MDP.__init__(self, transitions, reward, discount, None, max_iter) `````` Steven Cordwell committed Jun 21, 2013 535 `````` # Check if the user has supplied an initial policy. If not make one. `````` Steven Cordwell committed Jan 22, 2013 536 `````` if policy0 == None: `````` Steven Cordwell committed Jun 21, 2013 537 `````` # Initialise the policy to the one which maximises the expected `````` Steven Cordwell committed Jan 22, 2013 538 `````` # immediate reward `````` Steven Cordwell committed Jun 21, 2013 539 540 `````` null = zeros(self.S) self.policy, null = self._bellmanOperator(null) `````` Steven Cordwell committed Jan 24, 2013 541 `````` del null `````` Steven Cordwell committed Jan 22, 2013 542 `````` else: `````` Steven Cordwell committed Jun 21, 2013 543 544 `````` # Use the policy that the user supplied # Make sure it is a numpy array `````` Steven Cordwell committed Jan 22, 2013 545 `````` policy0 = array(policy0) `````` Steven Cordwell committed Jun 21, 2013 546 `````` # Make sure the policy is the right size and shape `````` Steven Cordwell committed Jan 22, 2013 547 `````` if not policy0.shape in ((self.S, ), (self.S, 1), (1, self.S)): `````` Steven Cordwell committed Jan 26, 2013 548 549 `````` raise ValueError("PyMDPtolbox: policy0 must a vector with " "length S.") `````` Steven Cordwell committed Jun 21, 2013 550 `````` # reshape the policy to be a vector `````` Steven Cordwell committed Feb 10, 2013 551 `````` policy0 = policy0.reshape(self.S) `````` Steven Cordwell committed Jun 21, 2013 552 `````` # The policy can only contain integers between 1 and S `````` Steven Cordwell committed Jan 26, 2013 553 554 555 556 `````` if (mod(policy0, 1).any() or (policy0 < 0).any() or (policy0 >= self.S).any()): raise ValueError("PyMDPtoolbox: policy0 must be a vector of " "integers between 1 and S.") `````` Steven Cordwell committed Jan 22, 2013 557 558 `````` else: self.policy = policy0 `````` Steven Cordwell committed Jun 21, 2013 559 `````` # set the initial values to zero `````` Steven Cordwell committed Feb 10, 2013 560 `````` self.V = zeros(self.S) `````` Steven Cordwell committed Jun 21, 2013 561 `````` # Do some setup depending on the evaluation type `````` Steven Cordwell committed Jan 22, 2013 562 `````` if eval_type in (0, "matrix"): `````` Steven Cordwell committed Jan 24, 2013 563 `````` from numpy.linalg import solve `````` Steven Cordwell committed Jan 24, 2013 564 `````` from scipy.sparse import eye `````` 565 566 `````` self._speye = eye self._lin_eq = solve `````` Steven Cordwell committed Jan 22, 2013 567 568 569 570 `````` self.eval_type = "matrix" elif eval_type in (1, "iterative"): self.eval_type = "iterative" else: `````` Steven Cordwell committed Jan 26, 2013 571 572 573 574 `````` raise ValueError("PyMDPtoolbox: eval_type should be 0 for matrix " "evaluation or 1 for iterative evaluation. " "The strings 'matrix' and 'iterative' can also " "be used.") `````` 575 576 `````` # Call the iteration method self._iterate() `````` Steven Cordwell committed Jan 22, 2013 577 `````` `````` 578 `````` def _computePpolicyPRpolicy(self): `````` Steven Cordwell committed Jun 21, 2013 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 `````` # Compute the transition matrix and the reward matrix for a policy. # # Arguments # --------- # Let S = number of states, A = number of actions # P(SxSxA) = transition matrix # P could be an array with 3 dimensions or a cell array (1xA), # each cell containing a matrix (SxS) possibly sparse # R(SxSxA) or (SxA) = reward matrix # R could be an array with 3 dimensions (SxSxA) or # a cell array (1xA), each cell containing a sparse matrix (SxS) or # a 2D array(SxA) possibly sparse # policy(S) = a policy # # Evaluation # ---------- # Ppolicy(SxS) = transition matrix for policy # PRpolicy(S) = reward matrix for policy # `````` Steven Cordwell committed Feb 10, 2013 598 599 `````` Ppolicy = empty((self.S, self.S)) Rpolicy = zeros(self.S) `````` Steven Cordwell committed Jan 24, 2013 600 `````` for aa in range(self.A): # avoid looping over S `````` Steven Cordwell committed Feb 10, 2013 601 602 `````` # the rows that use action a. ind = (self.policy == aa).nonzero()[0] `````` Steven Cordwell committed Jan 26, 2013 603 604 `````` # if no rows use action a, then no need to assign this if ind.size > 0: `````` Steven Cordwell committed Jan 24, 2013 605 `````` Ppolicy[ind, :] = self.P[aa][ind, :] `````` 606 `````` #PR = self._computePR() # an apparently uneeded line, and `````` Steven Cordwell committed Jan 24, 2013 607 608 `````` # perhaps harmful in this implementation c.f. # mdp_computePpolicyPRpolicy.m `````` Steven Cordwell committed Feb 09, 2013 609 `````` Rpolicy[ind] = self.R[aa][ind] `````` Steven Cordwell committed Jan 24, 2013 610 611 612 613 614 615 616 617 618 619 `````` # self.R cannot be sparse with the code in its current condition, but # it should be possible in the future. Also, if R is so big that its # a good idea to use a sparse matrix for it, then converting PRpolicy # from a dense to sparse matrix doesn't seem very memory efficient if type(self.R) is sparse: Rpolicy = sparse(Rpolicy) #self.Ppolicy = Ppolicy #self.Rpolicy = Rpolicy return (Ppolicy, Rpolicy) `````` 620 `````` def _evalPolicyIterative(self, V0=0, epsilon=0.0001, max_iter=10000): `````` Steven Cordwell committed Jun 21, 2013 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 `````` # Evaluate a policy using iteration. # # Arguments # --------- # Let S = number of states, A = number of actions # P(SxSxA) = transition matrix # P could be an array with 3 dimensions or # a cell array (1xS), each cell containing a matrix possibly sparse # R(SxSxA) or (SxA) = reward matrix # R could be an array with 3 dimensions (SxSxA) or # a cell array (1xA), each cell containing a sparse matrix (SxS) or # a 2D array(SxA) possibly sparse # discount = discount rate in ]0; 1[ # policy(S) = a policy # V0(S) = starting value function, optional (default : zeros(S,1)) # epsilon = epsilon-optimal policy search, upper than 0, # optional (default : 0.0001) # max_iter = maximum number of iteration to be done, upper than 0, # optional (default : 10000) # # Evaluation # ---------- # Vpolicy(S) = value function, associated to a specific policy # # Notes # ----- # In verbose mode, at each iteration, displays the condition which # stopped iterations: epsilon-optimum value function found or maximum # number of iterations reached. # `````` Steven Cordwell committed Jan 25, 2013 651 `````` if (type(V0) in (int, float)) and (V0 == 0): `````` Steven Cordwell committed Feb 10, 2013 652 `````` policy_V = zeros(self.S) `````` Steven Cordwell committed Jan 22, 2013 653 `````` else: `````` Steven Cordwell committed May 18, 2013 654 `````` if (type(V0) in (ndarray)) and (V0.shape == (self.S, 1)): `````` Steven Cordwell committed Jan 25, 2013 655 656 `````` policy_V = V0 else: `````` Steven Cordwell committed Jan 26, 2013 657 658 659 `````` raise ValueError("PyMDPtoolbox: V0 vector/array type not " "supported. Use ndarray of matrix column " "vector length S.") `````` Steven Cordwell committed Jan 22, 2013 660 `````` `````` 661 `````` policy_P, policy_R = self._computePpolicyPRpolicy() `````` Steven Cordwell committed Jan 22, 2013 662 663 664 `````` if self.verbose: print(' Iteration V_variation') `````` Steven Cordwell committed Jan 24, 2013 665 `````` `````` Steven Cordwell committed Jan 22, 2013 666 667 668 `````` itr = 0 done = False while not done: `````` Steven Cordwell committed Feb 10, 2013 669 `````` itr += 1 `````` Steven Cordwell committed Jan 24, 2013 670 671 `````` Vprev = policy_V `````` 672 `````` policy_V = policy_R + self.discount * policy_P.dot(Vprev) `````` Steven Cordwell committed Jan 24, 2013 673 674 `````` variation = absolute(policy_V - Vprev).max() `````` Steven Cordwell committed Jan 22, 2013 675 676 `````` if self.verbose: print(' %s %s') % (itr, variation) `````` Steven Cordwell committed Jan 26, 2013 677 678 679 `````` # ensure |Vn - Vpolicy| < epsilon if variation < ((1 - self.discount) / self.discount) * epsilon: `````` Steven Cordwell committed Jan 22, 2013 680 681 `````` done = True if self.verbose: `````` Steven Cordwell committed Jan 26, 2013 682 683 `````` print("PyMDPtoolbox: iterations stopped, epsilon-optimal " "value function.") `````` Steven Cordwell committed Jan 22, 2013 684 685 686 `````` elif itr == max_iter: done = True if self.verbose: `````` Steven Cordwell committed Jan 26, 2013 687 688 `````` print("PyMDPtoolbox: iterations stopped by maximum number " "of iteration condition.") `````` Steven Cordwell committed Jan 22, 2013 689 `````` `````` Steven Cordwell committed Jan 24, 2013 690 `````` self.V = policy_V `````` Steven Cordwell committed Jan 21, 2013 691 `````` `````` 692 `````` def _evalPolicyMatrix(self): `````` Steven Cordwell committed Jun 21, 2013 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 `````` # Evaluate the value function of the policy using linear equations. # # Arguments # --------- # Let S = number of states, A = number of actions # P(SxSxA) = transition matrix # P could be an array with 3 dimensions or a cell array (1xA), # each cell containing a matrix (SxS) possibly sparse # R(SxSxA) or (SxA) = reward matrix # R could be an array with 3 dimensions (SxSxA) or # a cell array (1xA), each cell containing a sparse matrix (SxS) or # a 2D array(SxA) possibly sparse # discount = discount rate in ]0; 1[ # policy(S) = a policy # # Evaluation # ---------- # Vpolicy(S) = value function of the policy # `````` 712 `````` Ppolicy, Rpolicy = self._computePpolicyPRpolicy() `````` Steven Cordwell committed Jan 22, 2013 713 `````` # V = PR + gPV => (I-gP)V = PR => V = inv(I-gP)* PR `````` 714 715 `````` self.V = self._lin_eq( (self._speye(self.S, self.S) - self.discount * Ppolicy), Rpolicy) `````` Steven Cordwell committed Nov 04, 2012 716 `````` `````` 717 `````` def _iterate(self): `````` Steven Cordwell committed Jun 21, 2013 718 719 `````` # Run the policy iteration algorithm. # If verbose the print a header `````` Steven Cordwell committed Jan 21, 2013 720 721 `````` if self.verbose: print(' Iteration Number_of_different_actions') `````` Steven Cordwell committed Jun 21, 2013 722 `````` # Set up the while stopping condition and the current time `````` Steven Cordwell committed Jan 24, 2013 723 `````` done = False `````` Steven Cordwell committed Jan 21, 2013 724 `````` self.time = time() `````` Steven Cordwell committed Jun 21, 2013 725 `````` # loop until a stopping condition is reached `````` Steven Cordwell committed Nov 04, 2012 726 `````` while not done: `````` Steven Cordwell committed Feb 10, 2013 727 `````` self.iter += 1 `````` 728 `````` # these _evalPolicy* functions will update the classes value `````` Steven Cordwell committed Jan 24, 2013 729 `````` # attribute `````` Steven Cordwell committed Jan 22, 2013 730 `````` if self.eval_type == "matrix": `````` 731 `````` self._evalPolicyMatrix() `````` Steven Cordwell committed Jan 22, 2013 732 `````` elif self.eval_type == "iterative": `````` 733 `````` self._evalPolicyIterative() `````` Steven Cordwell committed Jan 24, 2013 734 735 `````` # This should update the classes policy attribute but leave the # value alone `````` 736 `````` policy_next, null = self._bellmanOperator() `````` Steven Cordwell committed Jan 24, 2013 737 `````` del null `````` Steven Cordwell committed Jun 21, 2013 738 739 `````` # calculate in how many places does the old policy disagree with # the new policy `````` Steven Cordwell committed Jan 24, 2013 740 `````` n_different = (policy_next != self.policy).sum() `````` Steven Cordwell committed Jun 21, 2013 741 `````` # if verbose then continue printing a table `````` Steven Cordwell committed Jan 21, 2013 742 `````` if self.verbose: `````` Steven Cordwell committed Jan 26, 2013 743 744 `````` print(' %s %s') % (self.iter, n_different) `````` Steven Cordwell committed Jun 21, 2013 745 746 `````` # Once the policy is unchanging of the maximum number of # of iterations has been reached then stop `````` Steven Cordwell committed Jan 24, 2013 747 `````` if n_different == 0: `````` Steven Cordwell committed Nov 04, 2012 748 `````` done = True `````` Steven Cordwell committed Jan 24, 2013 749 `````` if self.verbose: `````` Steven Cordwell committed Jan 26, 2013 750 751 `````` print("PyMDPtoolbox: iterations stopped, unchanging " "policy found.") `````` Steven Cordwell committed Jan 24, 2013 752 753 754 `````` elif (self.iter == self.max_iter): done = True if self.verbose: `````` Steven Cordwell committed Jan 26, 2013 755 756 `````` print("PyMDPtoolbox: iterations stopped by maximum number " "of iteration condition.") `````` Steven Cordwell committed Jan 24, 2013 757 758 `````` else: self.policy = policy_next `````` Steven Cordwell committed Jun 21, 2013 759 `````` # update the time to return th computation time `````` Steven Cordwell committed Jan 21, 2013 760 `````` self.time = time() - self.time `````` Steven Cordwell committed Nov 04, 2012 761 `````` # store value and policy as tuples `````` 762 763 `````` self.V = tuple(self.V.tolist()) self.policy = tuple(self.policy.tolist()) `````` Steven Cordwell committed Oct 29, 2012 764 `````` `````` Steven Cordwell committed Jan 25, 2013 765 ``````class PolicyIterationModified(PolicyIteration): `````` 766 `````` `````` Steven Cordwell committed Jan 26, 2013 767 `````` """A discounted MDP solved using a modifified policy iteration algorithm. `````` Steven Cordwell committed Jan 20, 2013 768 769 770 771 772 `````` Arguments --------- Let S = number of states, A = number of actions P(SxSxA) = transition matrix `````` Steven Cordwell committed Jan 26, 2013 773 774 `````` P could be an array with 3 dimensions or a cell array (1xA), each cell containing a matrix (SxS) possibly sparse `````` Steven Cordwell committed Jan 20, 2013 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 `````` R(SxSxA) or (SxA) = reward matrix R could be an array with 3 dimensions (SxSxA) or a cell array (1xA), each cell containing a sparse matrix (SxS) or a 2D array(SxA) possibly sparse discount = discount rate, in ]0, 1[ policy0(S) = starting policy, optional max_iter = maximum number of iteration to be done, upper than 0, optional (default 1000) eval_type = type of function used to evaluate policy: 0 for mdp_eval_policy_matrix, else mdp_eval_policy_iterative optional (default 0) Data Attributes --------------- V(S) = value function policy(S) = optimal policy iter = number of done iterations cpu_time = used CPU time Notes ----- In verbose mode, at each iteration, displays the number of differents actions between policy n-1 and n Examples -------- `````` Steven Cordwell committed Aug 24, 2013 801 802 803 804 805 806 807 `````` >>> import mdptoolbox, mdptoolbox.example >>> P, R = mdptoolbox.example.forest() >>> pim = mdptoolbox.mdp.PolicyIterationModified(P, R, 0.9) >>> pim.policy FIXME >>> pim.V FIXME `````` 808 `````` `````` Steven Cordwell committed Oct 29, 2012 809 `````` """ `````` Steven Cordwell committed Jan 20, 2013 810 `````` `````` Steven Cordwell committed Jan 26, 2013 811 812 `````` def __init__(self, transitions, reward, discount, epsilon=0.01, max_iter=10): `````` Steven Cordwell committed Aug 24, 2013 813 `````` # Initialise a (modified) policy iteration MDP. `````` Steven Cordwell committed Jan 21, 2013 814 `````` `````` Steven Cordwell committed Jan 25, 2013 815 816 817 `````` # Maybe its better not to subclass from PolicyIteration, because the # initialisation of the two are quite different. eg there is policy0 # being calculated here which doesn't need to be. The only thing that `````` 818 `````` # is needed from the PolicyIteration class is the _evalPolicyIterative `````` Steven Cordwell committed Jan 25, 2013 819 `````` # function. Perhaps there is a better way to do it? `````` Steven Cordwell committed Jan 26, 2013 820 821 `````` PolicyIteration.__init__(self, transitions, reward, discount, None, max_iter, 1) `````` Steven Cordwell committed Jan 20, 2013 822 `````` `````` Steven Cordwell committed Jan 25, 2013 823 824 825 826 `````` # PolicyIteration doesn't pass epsilon to MDP.__init__() so we will # check it here if type(epsilon) in (int, float): if epsilon <= 0: `````` Steven Cordwell committed Jan 26, 2013 827 828 `````` raise ValueError("PyMDPtoolbox: epsilon must be greater than " "0.") `````` Steven Cordwell committed Jan 25, 2013 829 `````` else: `````` Steven Cordwell committed Jan 26, 2013 830 831 `````` raise ValueError("PyMDPtoolbox: epsilon must be a positive real " "number greater than zero.") `````` Steven Cordwell committed Jan 20, 2013 832 `````` `````` 833 834 `````` # computation of threshold of variation for V for an epsilon-optimal # policy `````` Steven Cordwell committed Jan 20, 2013 835 836 837 838 839 `````` if self.discount != 1: self.thresh = epsilon * (1 - self.discount) / self.discount else: self.thresh = epsilon `````` Steven Cordwell committed Jan 25, 2013 840 841 `````` self.epsilon = epsilon `````` Steven Cordwell committed Jan 20, 2013 842 `````` if discount == 1: `````` Steven Cordwell committed May 18, 2013 843 `````` self.V = zeros((self.S, 1)) `````` Steven Cordwell committed Jan 20, 2013 844 845 `````` else: # min(min()) is not right `````` Steven Cordwell committed Jan 25, 2013 846 `````` self.V = 1 / (1 - discount) * self.R.min() * ones((self.S, 1)) `````` 847 848 849 `````` # Call the iteration method self._iterate() `````` Steven Cordwell committed Jan 21, 2013 850 `````` `````` 851 `````` def _iterate(self): `````` Steven Cordwell committed Aug 24, 2013 852 `````` # Run the modified policy iteration algorithm. `````` Steven Cordwell committed Jan 20, 2013 853 854 855 856 `````` if self.verbose: print(' Iteration V_variation') `````` Steven Cordwell committed Jan 21, 2013 857 `````` self.time = time() `````` Steven Cordwell committed Jan 21, 2013 858 `````` `````` Steven Cordwell committed Jan 20, 2013 859 860 `````` done = False while not done: `````` Steven Cordwell committed Feb 10, 2013 861 `````` self.iter += 1 `````` Steven Cordwell committed Jan 20, 2013 862 `````` `````` 863 `````` self.policy, Vnext = self._bellmanOperator() `````` Steven Cordwell committed Jan 20, 2013 864 `````` #[Ppolicy, PRpolicy] = mdp_computePpolicyPRpolicy(P, PR, policy); `````` Steven Cordwell committed Jan 20, 2013 865 `````` `````` Steven Cordwell committed Jan 25, 2013 866 `````` variation = getSpan(Vnext - self.V) `````` Steven Cordwell committed Jan 20, 2013 867 868 869 `````` if self.verbose: print(" %s %s" % (self.iter, variation)) `````` Steven Cordwell committed Jan 24, 2013 870 `````` self.V = Vnext `````` Steven Cordwell committed Jan 21, 2013 871 `````` if variation < self.thresh: `````` Steven Cordwell committed Jan 20, 2013 872 873 874 875 `````` done = True else: is_verbose = False if self.verbose: `````` Steven Cordwell committed Jan 25, 2013 876 `````` self.setSilent() `````` Steven Cordwell committed Jan 20, 2013 877 878 `````` is_verbose = True `````` 879 `````` self._evalPolicyIterative(self.V, self.epsilon, self.max_iter) `````` Steven Cordwell committed Jan 20, 2013 880 881 `````` if is_verbose: `````` Steven Cordwell committed Jan 25, 2013 882 `````` self.setVerbose() `````` Steven Cordwell committed Jan 20, 2013 883 `````` `````` Steven Cordwell committed Jan 20, 2013 884 `````` self.time = time() - self.time `````` Steven Cordwell committed Jan 25, 2013 885 886 `````` # store value and policy as tuples `````` 887 888 `````` self.V = tuple(self.V.tolist()) self.policy = tuple(self.policy.tolist()) `````` Steven Cordwell committed Oct 29, 2012 889 890 `````` class QLearning(MDP): `````` 891 `````` `````` Steven Cordwell committed Jan 26, 2013 892 `````` """A discounted MDP solved using the Q learning algorithm. `````` Steven Cordwell committed Oct 30, 2012 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 `````` Let S = number of states, A = number of actions Parameters ---------- P : transition matrix (SxSxA) P could be an array with 3 dimensions or a cell array (1xA), each cell containing a sparse matrix (SxS) R : reward matrix(SxSxA) or (SxA) R could be an array with 3 dimensions (SxSxA) or a cell array (1xA), each cell containing a sparse matrix (SxS) or a 2D array(SxA) possibly sparse discount : discount rate in ]0; 1[ n_iter : number of iterations to execute (optional). `````` 908 909 `````` Default value = 10000; it is an integer greater than the default value. `````` Steven Cordwell committed Oct 30, 2012 910 911 912 913 914 `````` Results ------- Q : learned Q matrix (SxA) `````` Steven Cordwell committed Jan 24, 2013 915 `````` V : learned value function (S). `````` Steven Cordwell committed Oct 30, 2012 916 917 918 919 920 921 922 `````` policy : learned optimal policy (S). mean_discrepancy : vector of V discrepancy mean over 100 iterations Then the length of this vector for the default value of N is 100 (N/100). `````` Steven Cordwell committed Jan 25, 2013 923 `````` Examples `````` Steven Cordwell committed Oct 30, 2012 924 `````` --------- `````` Steven Cordwell committed Jun 21, 2013 925 926 927 `````` >>> # These examples are reproducible only if random seed is set to 0 in >>> # both the random and numpy.random modules. >>> import numpy as np `````` Steven Cordwell committed Aug 24, 2013 928 `````` >>> import mdptoolbox, mdptoolbox.example `````` Steven Cordwell committed Jun 21, 2013 929 `````` >>> np.random.seed(0) `````` Steven Cordwell committed Aug 24, 2013 930 931 `````` >>> P, R = mdptoolbox.example.forest() >>> ql = mdptoolbox.mdp.QLearning(P, R, 0.96) `````` Steven Cordwell committed Oct 30, 2012 932 `````` >>> ql.Q `````` Steven Cordwell committed Jun 21, 2013 933 934 935 `````` array([[ 68.38037354, 43.24888454], [ 72.37777922, 42.75549145], [ 77.02892702, 64.68712932]]) `````` Steven Cordwell committed Jan 24, 2013 936 `````` >>> ql.V `````` Steven Cordwell committed Jun 21, 2013 937 `````` (68.38037354422798, 72.37777921607258, 77.02892701616531) `````` Steven Cordwell committed Oct 30, 2012 938 `````` >>> ql.policy `````` Steven Cordwell committed Jan 25, 2013 939 `````` (0, 0, 0) `````` Steven Cordwell committed Oct 30, 2012 940 `````` `````` Steven Cordwell committed Aug 24, 2013 941 `````` >>> import mdptoolbox `````` Steven Cordwell committed Oct 30, 2012 942 943 944 `````` >>> import numpy as np >>> P = np.array([[[0.5, 0.5],[0.8, 0.2]],[[0, 1],[0.1, 0.9]]]) >>> R = np.array([[5, 10], [-1, 2]]) `````` Steven Cordwell committed Jun 21, 2013 945 `````` >>> np.random.seed(0) `````` Steven Cordwell committed Aug 24, 2013 946 `````` >>> pim = mdptoolbox.mdp.QLearning(P, R, 0.9) `````` Steven Cordwell committed Oct 30, 2012 947 `````` >>> ql.Q `````` Steven Cordwell committed Jun 21, 2013 948 949 `````` array([[ 39.933691 , 43.17543338], [ 36.94394224, 35.42568056]]) `````` Steven Cordwell committed Jan 24, 2013 950 `````` >>> ql.V `````` Steven Cordwell committed Jun 21, 2013 951 `````` (43.17543338090149, 36.943942243204454) `````` Steven Cordwell committed Oct 30, 2012 952 `````` >>> ql.policy `````` Steven Cordwell committed Jan 25, 2013 953 `````` (1, 0) `````` 954 `````` `````` Steven Cordwell committed Oct 29, 2012 955 956 957 `````` """ def __init__(self, transitions, reward, discount, n_iter=10000): `````` Steven Cordwell committed Aug 24, 2013 958 `````` # Initialise a Q-learning MDP. `````` Steven Cordwell committed Oct 29, 2012 959 `````` `````` Steven Cordwell committed Jan 21, 2013 960 961 `````` # The following check won't be done in MDP()'s initialisation, so let's # do it here `````` Steven Cordwell committed Sep 10, 2013 962 963 964 `````` self.max_iter = int(n_iter) assert self.max_iter >= 10000, "PyMDPtoolbox: n_iter should be " \ "greater than 10000." `````` Steven Cordwell committed Oct 29, 2012 965 `````` `````` 966 `````` # We don't want to send this to MDP because _computePR should not be `````` Steven Cordwell committed Sep 10, 2013 967 `````` # run on it, so check that it defines an MDP `````` Steven Cordwell committed Jan 25, 2013 968 969 `````` check(transitions, reward) `````` 970 971 `````` # Store P, S, and A self._computeP(transitions) `````` Steven Cordwell committed Jan 25, 2013 972 973 974 975 976 `````` self.R = reward self.discount = discount `````` Steven Cordwell committed Oct 29, 2012 977 978 979 980 `````` # Initialisations self.Q = zeros((self.S, self.A)) self.mean_discrepancy = [] `````` 981 982 983 984 `````` # Call the iteration method self._iterate() def _iterate(self): `````` Steven Cordwell committed Aug 24, 2013 985 `````` # Run the Q-learning algoritm. `````` Steven Cordwell committed Jan 25, 2013 986 987 `````` discrepancy = [] `````` Steven Cordwell committed Oct 29, 2012 988 989 990 `````` self.time = time() # initial state choice `````` Steven Cordwell committed May 16, 2013 991 `````` s = randint(0, self.S) `````` Steven Cordwell committed Oct 29, 2012 992 `````` `````` Steven Cordwell committed Jan 25, 2013 993 `````` for n in range(1, self.max_iter + 1): `````` Steven Cordwell committed Oct 29, 2012 994 995 996 `````` # Reinitialisation of trajectories every 100 transitions if ((n % 100) == 0): `````` Steven Cordwell committed May 16, 2013 997 `````` s = randint(0, self.S) `````` Steven Cordwell committed Oct 29, 2012 998 999 1000 1001 1002 `````` # Action choice : greedy with increasing probability # probability 1-(1/log(n+2)) can be changed pn = random() if (pn < (1 - (1 / log(n + 2)))): `````` Steven Cordwell committed Oct 30, 2012 1003 1004 `````` # optimal_action = self.Q[s, :].max() a = self.Q[s, :].argmax() `````` Steven Cordwell committed Oct 29, 2012 1005 `````` else: `````` Steven Cordwell committed May 16, 2013 1006 `````` a = randint(0, self.A) `````` Steven Cordwell committed Oct 30, 2012 1007 `````` ``````