mdp.py 56 KB
Newer Older
Steven Cordwell's avatar
Steven Cordwell committed
1
# -*- coding: utf-8 -*-
2
3
"""Markov Decision Process (MDP) Toolbox: ``mdp`` module
=====================================================
4

5
6
The ``mdp`` module provides classes for the resolution of descrete-time Markov
Decision Processes.
Steven Cordwell's avatar
Steven Cordwell committed
7

Steven Cordwell's avatar
Steven Cordwell committed
8
9
10
11
12
Available classes
-----------------
MDP
    Base Markov decision process class
FiniteHorizon
Steven Cordwell's avatar
Steven Cordwell committed
13
    Backwards induction finite horizon MDP
Steven Cordwell's avatar
Steven Cordwell committed
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's avatar
Steven Cordwell committed
28
29
30

"""

31
32
# Copyright (c) 2011-2013 Steven A. W. Cordwell
# Copyright (c) 2009 INRA
Steven Cordwell's avatar
Steven Cordwell committed
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
# 
# 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 <ORGANIZATION> 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's avatar
Steven Cordwell committed
60
61
62
from math import ceil, log, sqrt
from time import time

63
64
from numpy import absolute, array, empty, mean, mod, multiply
from numpy import ndarray, ones, zeros
65
from numpy.random import randint, random
Steven Cordwell's avatar
Steven Cordwell committed
66
from scipy.sparse import csr_matrix as sparse
Steven Cordwell's avatar
Steven Cordwell committed
67

Steven Cordwell's avatar
Steven Cordwell committed
68
from .utils import check, getSpan
69

Steven Cordwell's avatar
Steven Cordwell committed
70
class MDP(object):
71
    
Steven Cordwell's avatar
Steven Cordwell committed
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
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
    """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's avatar
Steven Cordwell committed
116
    
117
    def __init__(self, transitions, reward, discount, epsilon, max_iter):
118
        # Initialise a MDP based on the input parameters.
119
        
Steven Cordwell's avatar
Steven Cordwell committed
120
121
        # if the discount is None then the algorithm is assumed to not use it
        # in its computations
122
123
124
125
126
127
        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's avatar
Steven Cordwell committed
128
129
        # if the max_iter is None then the algorithm is assumed to not use it
        # in its computations
130
131
132
133
        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's avatar
Steven Cordwell committed
134
        # check that epsilon is something sane
135
136
137
        if epsilon is not None:
            self.epsilon = float(epsilon)
            assert self.epsilon > 0, "Epsilon must be greater than 0."
Steven Cordwell's avatar
Steven Cordwell committed
138
139
140
141
        # 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
142
        self._computePR(transitions, reward)
Steven Cordwell's avatar
Steven Cordwell committed
143
144
145
146
        # 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
147
148
        # set the initial iteration count to zero
        self.iter = 0
Steven Cordwell's avatar
Steven Cordwell committed
149
        # V should be stored as a vector ie shape of (S,) or (1, S)
Steven Cordwell's avatar
Steven Cordwell committed
150
        self.V = None
Steven Cordwell's avatar
Steven Cordwell committed
151
        # policy can also be stored as a vector
Steven Cordwell's avatar
Steven Cordwell committed
152
        self.policy = None
Steven Cordwell's avatar
Steven Cordwell committed
153
    
154
155
156
157
158
159
    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"
160
        return(P_repr + "\n" + R_repr)
161
    
162
    def _bellmanOperator(self, V=None):
Steven Cordwell's avatar
Steven Cordwell committed
163
        # Apply the Bellman operator on the value function.
164
        # 
Steven Cordwell's avatar
Steven Cordwell committed
165
        # Updates the value function and the Vprev-improving policy.
166
        # 
Steven Cordwell's avatar
Steven Cordwell committed
167
168
169
170
        # 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
171
172
        if V is None:
            # this V should be a reference to the data rather than a copy
173
174
            V = self.V
        else:
Steven Cordwell's avatar
Steven Cordwell committed
175
            # make sure the user supplied V is of the right shape
176
            try:
177
178
                assert V.shape in ((self.S,), (1, self.S)), "V is not the " \
                    "right shape (Bellman operator)."
179
            except AttributeError:
180
                raise TypeError("V must be a numpy array or matrix.")
181
182
183
184
        # 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's avatar
Steven Cordwell committed
185
        Q = empty((self.A, self.S))
Steven Cordwell's avatar
Steven Cordwell committed
186
        for aa in range(self.A):
Steven Cordwell's avatar
Steven Cordwell committed
187
            Q[aa] = self.R[aa] + self.discount * self.P[aa].dot(V)
Steven Cordwell's avatar
Steven Cordwell committed
188
        # Get the policy and value, for now it is being returned but...
189
        # Which way is better?
190
        # 1. Return, (policy, value)
191
        return (Q.argmax(axis=0), Q.max(axis=0))
Steven Cordwell's avatar
Steven Cordwell committed
192
193
        # 2. update self.policy and self.V directly
        # self.V = Q.max(axis=1)
194
        # self.policy = Q.argmax(axis=1)
Steven Cordwell's avatar
Steven Cordwell committed
195
    
196
197
198
199
200
201
202
203
204
205
206
207
208
209
    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)])
    
210
    def _computePR(self, P, R):
Steven Cordwell's avatar
Steven Cordwell committed
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
        # 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
        #
226
        # We assume that P and R define a MDP i,e. assumption is that
Steven Cordwell's avatar
Steven Cordwell committed
227
        # check(P, R) has already been run and doesn't fail.
228
        #
229
230
        # First compute store P, S, and A
        self._computeP(P)
Steven Cordwell's avatar
Steven Cordwell committed
231
232
        # Set self.R as a tuple of length A, with each element storing an 1×S
        # vector.
233
        try:
234
            if R.ndim == 2:
235
236
                self.R = tuple([array(R[:, aa]).reshape(self.S)
                                for aa in range(self.A)])
Steven Cordwell's avatar
Steven Cordwell committed
237
            else:
238
                self.R = tuple([multiply(P[aa], R[aa]).sum(1).reshape(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
239
                                for aa in range(self.A)])
240
        except AttributeError:
241
            self.R = tuple([multiply(P[aa], R[aa]).sum(1).reshape(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
242
                            for aa in range(self.A)])
243
    
244
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
245
        # Raise error because child classes should implement this function.
246
        raise NotImplementedError("You should create a run() method.")
Steven Cordwell's avatar
Steven Cordwell committed
247
    
Steven Cordwell's avatar
Steven Cordwell committed
248
    def setSilent(self):
249
        """Set the MDP algorithm to silent mode."""
Steven Cordwell's avatar
Steven Cordwell committed
250
251
252
        self.verbose = False
    
    def setVerbose(self):
253
        """Set the MDP algorithm to verbose mode."""
Steven Cordwell's avatar
Steven Cordwell committed
254
        self.verbose = True
Steven Cordwell's avatar
Steven Cordwell committed
255
256

class FiniteHorizon(MDP):
257
    
Steven Cordwell's avatar
Steven Cordwell committed
258
    """A MDP solved using the finite-horizon backwards induction algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
259
260
    
    Let S = number of states, A = number of actions
Steven Cordwell's avatar
Steven Cordwell committed
261
262
263
    
    Parameters
    ----------
Steven Cordwell's avatar
Steven Cordwell committed
264
    P(SxSxA) = transition matrix 
265
266
             P could be an array with 3 dimensions ora cell array (1xA),
             each cell containing a matrix (SxS) possibly sparse
Steven Cordwell's avatar
Steven Cordwell committed
267
268
269
270
271
272
273
    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's avatar
Steven Cordwell committed
274
275
    
    Attributes
Steven Cordwell's avatar
Steven Cordwell committed
276
    ----------
Steven Cordwell's avatar
Steven Cordwell committed
277
278
279
    
    Methods
    -------
Steven Cordwell's avatar
Steven Cordwell committed
280
281
282
283
284
285
286
287
288
289
290
291
292
    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.
293
    
Steven Cordwell's avatar
Steven Cordwell committed
294
295
    Examples
    --------
296
297
298
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.forest()
    >>> fh = mdptoolbox.mdp.FiniteHorizon(P, R, 0.9, 3)
299
    >>> fh.run()
Steven Cordwell's avatar
Steven Cordwell committed
300
301
302
303
304
305
306
307
    >>> 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]])
308
    
Steven Cordwell's avatar
Steven Cordwell committed
309
    """
Steven Cordwell's avatar
Steven Cordwell committed
310

Steven Cordwell's avatar
Steven Cordwell committed
311
    def __init__(self, transitions, reward, discount, N, h=None):
312
        # Initialise a finite horizon MDP.
313
314
        self.N = int(N)
        assert self.N > 0, 'PyMDPtoolbox: N must be greater than 0.'
Steven Cordwell's avatar
Steven Cordwell committed
315
        # Initialise the base class
316
        MDP.__init__(self, transitions, reward, discount, None, None)
Steven Cordwell's avatar
Steven Cordwell committed
317
318
        # remove the iteration counter, it is not meaningful for backwards
        # induction
319
        del self.iter
Steven Cordwell's avatar
Steven Cordwell committed
320
        # There are value vectors for each time step up to the horizon
321
        self.V = zeros((self.S, N + 1))
Steven Cordwell's avatar
Steven Cordwell committed
322
323
324
325
326
        # 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:
327
            self.V[:, N] = h
328
        # Call the iteration method
329
        #self.run()
330
        
331
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
332
        # Run the finite horizon algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
333
        self.time = time()
Steven Cordwell's avatar
Steven Cordwell committed
334
        # loop through each time period
335
        for n in range(self.N):
Steven Cordwell's avatar
Steven Cordwell committed
336
337
338
            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's avatar
Steven Cordwell committed
339
            if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
340
341
                print(("stage: %s ... policy transpose : %s") % (
                    self.N - n, self.policy[:, self.N - n -1].tolist()))
Steven Cordwell's avatar
Steven Cordwell committed
342
        # update time spent running
Steven Cordwell's avatar
Steven Cordwell committed
343
        self.time = time() - self.time
Steven Cordwell's avatar
Steven Cordwell committed
344
345
346
347
348
349
350
351
352
353
        # 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's avatar
Steven Cordwell committed
354
355

class LP(MDP):
356
    
357
    """A discounted MDP soloved using linear programming.
Steven Cordwell's avatar
Steven Cordwell committed
358
359
360
361
362

    Arguments
    ---------
    Let S = number of states, A = number of actions
    P(SxSxA) = transition matrix 
363
364
             P could be an array with 3 dimensions or a cell array (1xA),
             each cell containing a matrix (SxS) possibly sparse
Steven Cordwell's avatar
Steven Cordwell committed
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
    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
    --------
384
385
386
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.forest()
    >>> lp = mdptoolbox.mdp.LP(P, R, 0.9)
387
    >>> lp.run()
388
    
Steven Cordwell's avatar
Steven Cordwell committed
389
    """
Steven Cordwell's avatar
Steven Cordwell committed
390

Steven Cordwell's avatar
Steven Cordwell committed
391
    def __init__(self, transitions, reward, discount):
392
        # Initialise a linear programming MDP.
Steven Cordwell's avatar
Steven Cordwell committed
393
        # import some functions from cvxopt and set them as object methods
Steven Cordwell's avatar
Steven Cordwell committed
394
395
        try:
            from cvxopt import matrix, solvers
396
397
            self._linprog = solvers.lp
            self._cvxmat = matrix
Steven Cordwell's avatar
Steven Cordwell committed
398
        except ImportError:
399
400
            raise ImportError("The python module cvxopt is required to use "
                              "linear programming functionality.")
Steven Cordwell's avatar
Steven Cordwell committed
401
402
        # we also need diagonal matrices, and using a sparse one may be more
        # memory efficient
Steven Cordwell's avatar
Steven Cordwell committed
403
        from scipy.sparse import eye as speye
404
        self._speye = speye
Steven Cordwell's avatar
Steven Cordwell committed
405
        # initialise the MDP. epsilon and max_iter are not needed
406
        MDP.__init__(self, transitions, reward, discount, None, None)
Steven Cordwell's avatar
Steven Cordwell committed
407
        # Set the cvxopt solver to be quiet by default, but ...
408
        # this doesn't do what I want it to do c.f. issue #3
409
410
        if not self.verbose:
            solvers.options['show_progress'] = False
411
        # Call the iteration method
412
        #self.run()
413
    
414
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
415
        #Run the linear programming algorithm.
416
        self.time = time()
Steven Cordwell's avatar
Steven Cordwell committed
417
        # The objective is to resolve : min V / V >= PR + discount*P*V
418
419
        # The function linprog of the optimisation Toolbox of Mathworks
        # resolves :
Steven Cordwell's avatar
Steven Cordwell committed
420
        # min f'* x / M * x <= b
421
422
423
424
        # 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)
425
426
427
        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's avatar
Steven Cordwell committed
428
429
        for aa in range(self.A):
            pos = (aa + 1) * self.S
430
431
432
            M[(pos - self.S):pos, :] = (
                self.discount * self.P[aa] - self._speye(self.S, self.S))
        M = self._cvxmat(M)
433
434
435
        # 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's avatar
Steven Cordwell committed
436
        # only to 10e-8 places. This assumes glpk is installed of course.
Steven Cordwell's avatar
Steven Cordwell committed
437
        self.V = array(self._linprog(f, M, -h, solver='glpk')['x'])
Steven Cordwell's avatar
Steven Cordwell committed
438
        # apply the Bellman operator
439
        self.policy, self.V =  self._bellmanOperator()
Steven Cordwell's avatar
Steven Cordwell committed
440
        # update the time spent solving
Steven Cordwell's avatar
Steven Cordwell committed
441
        self.time = time() - self.time
442
        # store value and policy as tuples
443
444
        self.V = tuple(self.V.tolist())
        self.policy = tuple(self.policy.tolist())
Steven Cordwell's avatar
Steven Cordwell committed
445
446

class PolicyIteration(MDP):
447
    
448
    """A discounted MDP solved using the policy iteration algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
449
    
Steven Cordwell's avatar
Steven Cordwell committed
450
451
452
453
    Arguments
    ---------
    Let S = number of states, A = number of actions
    P(SxSxA) = transition matrix 
454
455
             P could be an array with 3 dimensions or a cell array (1xA),
             each cell containing a matrix (SxS) possibly sparse
Steven Cordwell's avatar
Steven Cordwell committed
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
    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's avatar
Steven Cordwell committed
480
481
    Examples
    --------
482
483
484
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.rand()
    >>> pi = mdptoolbox.mdp.PolicyIteration(P, R, 0.9)
485
    >>> pi.run()
486
    
487
488
    >>> P, R = mdptoolbox.example.forest()
    >>> pi = mdptoolbox.mdp.PolicyIteration(P, R, 0.9)
489
    >>> pi.run()
Steven Cordwell's avatar
Steven Cordwell committed
490
    >>> pi.V
491
    (26.244000000000018, 29.48400000000002, 33.484000000000016)
Steven Cordwell's avatar
Steven Cordwell committed
492
    >>> pi.policy
493
    (0, 0, 0)
Steven Cordwell's avatar
Steven Cordwell committed
494
    """
Steven Cordwell's avatar
Steven Cordwell committed
495
    
496
497
    def __init__(self, transitions, reward, discount, policy0=None,
                 max_iter=1000, eval_type=0):
Steven Cordwell's avatar
Steven Cordwell committed
498
499
500
        # Initialise a policy iteration MDP.
        #
        # Set up the MDP, but don't need to worry about epsilon values
501
        MDP.__init__(self, transitions, reward, discount, None, max_iter)
Steven Cordwell's avatar
Steven Cordwell committed
502
        # Check if the user has supplied an initial policy. If not make one.
Steven Cordwell's avatar
Steven Cordwell committed
503
        if policy0 == None:
Steven Cordwell's avatar
Steven Cordwell committed
504
            # Initialise the policy to the one which maximises the expected
Steven Cordwell's avatar
Steven Cordwell committed
505
            # immediate reward
Steven Cordwell's avatar
Steven Cordwell committed
506
507
            null = zeros(self.S)
            self.policy, null = self._bellmanOperator(null)
508
            del null
Steven Cordwell's avatar
Steven Cordwell committed
509
        else:
Steven Cordwell's avatar
Steven Cordwell committed
510
511
            # Use the policy that the user supplied
            # Make sure it is a numpy array
Steven Cordwell's avatar
Steven Cordwell committed
512
            policy0 = array(policy0)
Steven Cordwell's avatar
Steven Cordwell committed
513
            # Make sure the policy is the right size and shape
Steven Cordwell's avatar
Steven Cordwell committed
514
            if not policy0.shape in ((self.S, ), (self.S, 1), (1, self.S)):
515
516
                raise ValueError("PyMDPtolbox: policy0 must a vector with "
                                 "length S.")
Steven Cordwell's avatar
Steven Cordwell committed
517
            # reshape the policy to be a vector
Steven Cordwell's avatar
Steven Cordwell committed
518
            policy0 = policy0.reshape(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
519
            # The policy can only contain integers between 1 and S
520
521
522
523
            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's avatar
Steven Cordwell committed
524
525
            else:
                self.policy = policy0
Steven Cordwell's avatar
Steven Cordwell committed
526
        # set the initial values to zero
Steven Cordwell's avatar
Steven Cordwell committed
527
        self.V = zeros(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
528
        # Do some setup depending on the evaluation type
Steven Cordwell's avatar
Steven Cordwell committed
529
        if eval_type in (0, "matrix"):
530
            from numpy.linalg import solve
531
            from scipy.sparse import eye
532
533
            self._speye = eye
            self._lin_eq = solve
Steven Cordwell's avatar
Steven Cordwell committed
534
535
536
537
            self.eval_type = "matrix"
        elif eval_type in (1, "iterative"):
            self.eval_type = "iterative"
        else:
538
539
540
541
            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.")
542
        # Call the iteration method
543
        #self.run()
Steven Cordwell's avatar
Steven Cordwell committed
544
    
545
    def _computePpolicyPRpolicy(self):
Steven Cordwell's avatar
Steven Cordwell committed
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
        # 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's avatar
Steven Cordwell committed
565
566
        Ppolicy = empty((self.S, self.S))
        Rpolicy = zeros(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
567
        for aa in range(self.A): # avoid looping over S
Steven Cordwell's avatar
Steven Cordwell committed
568
569
            # the rows that use action a.
            ind = (self.policy == aa).nonzero()[0]
570
571
            # if no rows use action a, then no need to assign this
            if ind.size > 0:
572
573
574
575
                try:
                    Ppolicy[ind, :] = self.P[aa][ind, :]
                except ValueError:
                    Ppolicy[ind, :] = self.P[aa][ind, :].todense()
576
                #PR = self._computePR() # an apparently uneeded line, and
Steven Cordwell's avatar
Steven Cordwell committed
577
578
                # perhaps harmful in this implementation c.f.
                # mdp_computePpolicyPRpolicy.m
579
                Rpolicy[ind] = self.R[aa][ind]
Steven Cordwell's avatar
Steven Cordwell committed
580
581
582
583
584
585
586
587
588
589
        # 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)
    
590
    def _evalPolicyIterative(self, V0=0, epsilon=0.0001, max_iter=10000):
Steven Cordwell's avatar
Steven Cordwell committed
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
        # 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.
        #
621
        if (type(V0) in (int, float)) and (V0 == 0):
Steven Cordwell's avatar
Steven Cordwell committed
622
            policy_V = zeros(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
623
        else:
Steven Cordwell's avatar
Steven Cordwell committed
624
            if (type(V0) in (ndarray)) and (V0.shape == (self.S, 1)):
625
626
                policy_V = V0
            else:
627
628
629
                raise ValueError("PyMDPtoolbox: V0 vector/array type not "
                                 "supported. Use ndarray of matrix column "
                                 "vector length S.")
Steven Cordwell's avatar
Steven Cordwell committed
630
        
631
        policy_P, policy_R = self._computePpolicyPRpolicy()
Steven Cordwell's avatar
Steven Cordwell committed
632
633
634
        
        if self.verbose:
            print('  Iteration    V_variation')
635
        
Steven Cordwell's avatar
Steven Cordwell committed
636
637
638
        itr = 0
        done = False
        while not done:
639
            itr += 1
640
641
            
            Vprev = policy_V
642
            policy_V = policy_R + self.discount * policy_P.dot(Vprev)
643
644
            
            variation = absolute(policy_V - Vprev).max()
Steven Cordwell's avatar
Steven Cordwell committed
645
            if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
646
                print(('      %s         %s') % (itr, variation))
647
648
649
            
            # ensure |Vn - Vpolicy| < epsilon
            if variation < ((1 - self.discount) / self.discount) * epsilon:
Steven Cordwell's avatar
Steven Cordwell committed
650
651
                done = True
                if self.verbose:
652
653
                    print("PyMDPtoolbox: iterations stopped, epsilon-optimal "
                          "value function.")
Steven Cordwell's avatar
Steven Cordwell committed
654
655
656
            elif itr == max_iter:
                done = True
                if self.verbose:
657
658
                    print("PyMDPtoolbox: iterations stopped by maximum number "
                          "of iteration condition.")
Steven Cordwell's avatar
Steven Cordwell committed
659
        
Steven Cordwell's avatar
Steven Cordwell committed
660
        self.V = policy_V
661
    
662
    def _evalPolicyMatrix(self):
Steven Cordwell's avatar
Steven Cordwell committed
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
        # 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
        #
682
        Ppolicy, Rpolicy = self._computePpolicyPRpolicy()
Steven Cordwell's avatar
Steven Cordwell committed
683
        # V = PR + gPV  => (I-gP)V = PR  => V = inv(I-gP)* PR
684
685
        self.V = self._lin_eq(
            (self._speye(self.S, self.S) - self.discount * Ppolicy), Rpolicy)
Steven Cordwell's avatar
Steven Cordwell committed
686
    
687
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
688
689
        # Run the policy iteration algorithm.
        # If verbose the print a header
690
691
        if self.verbose:
            print('  Iteration  Number_of_different_actions')
Steven Cordwell's avatar
Steven Cordwell committed
692
        # Set up the while stopping condition and the current time
Steven Cordwell's avatar
Steven Cordwell committed
693
        done = False
694
        self.time = time()
Steven Cordwell's avatar
Steven Cordwell committed
695
        # loop until a stopping condition is reached
Steven Cordwell's avatar
Steven Cordwell committed
696
        while not done:
697
            self.iter += 1
698
            # these _evalPolicy* functions will update the classes value
Steven Cordwell's avatar
Steven Cordwell committed
699
            # attribute
Steven Cordwell's avatar
Steven Cordwell committed
700
            if self.eval_type == "matrix":
701
                self._evalPolicyMatrix()
Steven Cordwell's avatar
Steven Cordwell committed
702
            elif self.eval_type == "iterative":
703
                self._evalPolicyIterative()
Steven Cordwell's avatar
Steven Cordwell committed
704
705
            # This should update the classes policy attribute but leave the
            # value alone
706
            policy_next, null = self._bellmanOperator()
707
            del null
Steven Cordwell's avatar
Steven Cordwell committed
708
709
            # calculate in how many places does the old policy disagree with
            # the new policy
710
            n_different = (policy_next != self.policy).sum()
Steven Cordwell's avatar
Steven Cordwell committed
711
            # if verbose then continue printing a table
712
            if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
713
714
                print(('       %s                 %s') % (self.iter,
                                                         n_different))
Steven Cordwell's avatar
Steven Cordwell committed
715
716
            # Once the policy is unchanging of the maximum number of 
            # of iterations has been reached then stop
717
            if n_different == 0:
Steven Cordwell's avatar
Steven Cordwell committed
718
                done = True
719
                if self.verbose:
720
721
                    print("PyMDPtoolbox: iterations stopped, unchanging "
                          "policy found.")
722
723
724
            elif (self.iter == self.max_iter):
                done = True 
                if self.verbose:
725
726
                    print("PyMDPtoolbox: iterations stopped by maximum number "
                          "of iteration condition.")
727
728
            else:
                self.policy = policy_next
Steven Cordwell's avatar
Steven Cordwell committed
729
        # update the time to return th computation time
730
        self.time = time() - self.time
Steven Cordwell's avatar
Steven Cordwell committed
731
        # store value and policy as tuples
732
733
        self.V = tuple(self.V.tolist())
        self.policy = tuple(self.policy.tolist())
Steven Cordwell's avatar
Steven Cordwell committed
734

735
class PolicyIterationModified(PolicyIteration):
736
    
737
    """A discounted MDP  solved using a modifified policy iteration algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
738
739
740
741
742
    
    Arguments
    ---------
    Let S = number of states, A = number of actions
    P(SxSxA) = transition matrix 
743
744
             P could be an array with 3 dimensions or a cell array (1xA),
             each cell containing a matrix (SxS) possibly sparse
Steven Cordwell's avatar
Steven Cordwell committed
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
    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
    --------
771
772
773
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.forest()
    >>> pim = mdptoolbox.mdp.PolicyIterationModified(P, R, 0.9)
774
    >>> pim.run()
775
776
777
778
    >>> pim.policy
    FIXME
    >>> pim.V
    FIXME
779
    
Steven Cordwell's avatar
Steven Cordwell committed
780
    """
781
    
782
783
    def __init__(self, transitions, reward, discount, epsilon=0.01,
                 max_iter=10):
784
        # Initialise a (modified) policy iteration MDP.
Steven Cordwell's avatar
Steven Cordwell committed
785
        
786
787
788
        # 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
789
        # is needed from the PolicyIteration class is the _evalPolicyIterative
790
        # function. Perhaps there is a better way to do it?
791
792
        PolicyIteration.__init__(self, transitions, reward, discount, None,
                                 max_iter, 1)
793
        
794
795
796
797
        # PolicyIteration doesn't pass epsilon to MDP.__init__() so we will
        # check it here
        if type(epsilon) in (int, float):
            if epsilon <= 0:
798
799
                raise ValueError("PyMDPtoolbox: epsilon must be greater than "
                                 "0.")
800
        else:
801
802
            raise ValueError("PyMDPtoolbox: epsilon must be a positive real "
                             "number greater than zero.")
803
        
804
805
        # computation of threshold of variation for V for an epsilon-optimal
        # policy
806
807
808
809
810
        if self.discount != 1:
            self.thresh = epsilon * (1 - self.discount) / self.discount
        else:
            self.thresh = epsilon
        
811
812
        self.epsilon = epsilon
        
813
        if discount == 1:
Steven Cordwell's avatar
Steven Cordwell committed
814
            self.V = zeros((self.S, 1))
815
816
        else:
            # min(min()) is not right
817
            self.V = 1 / (1 - discount) * self.R.min() * ones((self.S, 1))
818
819
        
        # Call the iteration method
820
        #self.run()
Steven Cordwell's avatar
Steven Cordwell committed
821
    
822
    def run(self):
823
        # Run the modified policy iteration algorithm.
824
825
        
        if self.verbose:
826
            print('\tIteration\tV-variation')
827
        
Steven Cordwell's avatar
Steven Cordwell committed
828
        self.time = time()
Steven Cordwell's avatar
Steven Cordwell committed
829
        
830
831
        done = False
        while not done:
832
            self.iter += 1
833
            
834
            self.policy, Vnext = self._bellmanOperator()
835
            #[Ppolicy, PRpolicy] = mdp_computePpolicyPRpolicy(P, PR, policy);
836
            
837
            variation = getSpan(Vnext - self.V)
838
            if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
839
                print(("\t%s\t%s" % (self.iter, variation)))
840
            
Steven Cordwell's avatar
Steven Cordwell committed
841
            self.V = Vnext
Steven Cordwell's avatar
Steven Cordwell committed
842
            if variation < self.thresh:
843
844
845
846
                done = True
            else:
                is_verbose = False
                if self.verbose:
847
                    self.setSilent()
848
849
                    is_verbose = True
                
850
                self._evalPolicyIterative(self.V, self.epsilon, self.max_iter)
851
852
                
                if is_verbose:
853
                    self.setVerbose()
854
        
855
        self.time = time() - self.time
856
857
        
        # store value and policy as tuples
858
859
        self.V = tuple(self.V.tolist())
        self.policy = tuple(self.policy.tolist())
Steven Cordwell's avatar
Steven Cordwell committed
860
861

class QLearning(MDP):
862
    
863
    """A discounted MDP solved using the Q learning algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
    
    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).
879
880
        Default value = 10000; it is an integer greater than the default
        value.
Steven Cordwell's avatar
Steven Cordwell committed
881
882
883
884
885
    
    Results
    -------
    Q : learned Q matrix (SxA) 
    
Steven Cordwell's avatar
Steven Cordwell committed
886
    V : learned value function (S).
Steven Cordwell's avatar
Steven Cordwell committed
887
888
889
890
891
892
893
    
    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).

894
    Examples
Steven Cordwell's avatar
Steven Cordwell committed
895
    ---------
896
897
898
    >>> # These examples are reproducible only if random seed is set to 0 in
    >>> # both the random and numpy.random modules.
    >>> import numpy as np
899
    >>> import mdptoolbox, mdptoolbox.example
900
    >>> np.random.seed(0)
901
902
    >>> P, R = mdptoolbox.example.forest()
    >>> ql = mdptoolbox.mdp.QLearning(P, R, 0.96)
903
    >>> ql.run()
Steven Cordwell's avatar
Steven Cordwell committed
904
    >>> ql.Q
905
906
907
    array([[ 68.38037354,  43.24888454],
           [ 72.37777922,  42.75549145],
           [ 77.02892702,  64.68712932]])
Steven Cordwell's avatar
Steven Cordwell committed
908
    >>> ql.V
909
    (68.38037354422798, 72.37777921607258, 77.02892701616531)
Steven Cordwell's avatar
Steven Cordwell committed
910
    >>> ql.policy
911
    (0, 0, 0)
Steven Cordwell's avatar
Steven Cordwell committed
912
    
913
    >>> import mdptoolbox
Steven Cordwell's avatar
Steven Cordwell committed
914
915
916
    >>> 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]])
917
    >>> np.random.seed(0)
918
919
    >>> ql = mdptoolbox.mdp.QLearning(P, R, 0.9)
    >>> ql.run()
Steven Cordwell's avatar
Steven Cordwell committed
920
    >>> ql.Q
921
922
    array([[ 39.933691  ,  43.17543338],
           [ 36.94394224,  35.42568056]])
Steven Cordwell's avatar
Steven Cordwell committed
923
    >>> ql.V
924
    (43.17543338090149, 36.943942243204454)
Steven Cordwell's avatar
Steven Cordwell committed
925
    >>> ql.policy
926
    (1, 0)
927
    
Steven Cordwell's avatar
Steven Cordwell committed
928
929
930
    """
    
    def __init__(self, transitions, reward, discount, n_iter=10000):
931
        # Initialise a Q-learning MDP.
Steven Cordwell's avatar
Steven Cordwell committed
932
        
933
934
        # The following check won't be done in MDP()'s initialisation, so let's
        # do it here
935
936
937
        self.max_iter = int(n_iter)
        assert self.max_iter >= 10000, "PyMDPtoolbox: n_iter should be " \
                                        "greater than 10000."
Steven Cordwell's avatar
Steven Cordwell committed
938
        
939
        # We don't want to send this to MDP because _computePR should not be
940
        # run on it, so check that it defines an MDP
941
942
        check(transitions, reward)
        
943
944
        # Store P, S, and A
        self._computeP(transitions)
945
946
947
948
949
        
        self.R = reward
        
        self.discount = discount
        
Steven Cordwell's avatar
Steven Cordwell committed
950
951
952
953
        # Initialisations
        self.Q = zeros((self.S, self.A))
        self.mean_discrepancy = []
        
954
        # Call the iteration method
955
        #self.run()
956
        
957
    def run(self):
958
        # Run the Q-learning algoritm.
959
960
        discrepancy = []
        
Steven Cordwell's avatar
Steven Cordwell committed
961
962
963
        self.time = time()
        
        # initial state choice
964
        s = randint(0, self.S)
Steven Cordwell's avatar
Steven Cordwell committed
965
        
966
        for n in range(1, self.max_iter + 1):
Steven Cordwell's avatar
Steven Cordwell committed
967
968
969
            
            # Reinitialisation of trajectories every 100 transitions
            if ((n % 100) == 0):
970
                s = randint(0, self.S)
Steven Cordwell's avatar
Steven Cordwell committed
971
972
973
974
975
            
            # 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's avatar
Steven Cordwell committed
976
977
                # optimal_action = self.Q[s, :].max()
                a = self.Q[s, :].argmax()
Steven Cordwell's avatar
Steven Cordwell committed
978
            else:
979
                a = randint(0, self.A)
Steven Cordwell's avatar
Steven Cordwell committed
980
            
Steven Cordwell's avatar
Steven Cordwell committed
981
982
983
            # Simulating next state s_new and reward associated to <s,s_new,a>
            p_s_new = random()
            p = 0
Steven Cordwell's avatar
Steven Cordwell committed
984
            s_new = -1
985
            while ((p < p_s_new) and (s_new < (self.S - 1))):
Steven Cordwell's avatar
Steven Cordwell committed
986
                s_new = s_new + 1
Steven Cordwell's avatar
Steven Cordwell committed
987
988
                p = p + self.P[a][s, s_new]
            
989
            try:
Steven Cordwell's avatar
Steven Cordwell committed
990
                r = self.R[a][s, s_new]
991
            except IndexError:
Steven Cordwell's avatar
Steven Cordwell committed
992
993
                r = self.R[s, a]