mdp.py 56.7 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.

60 61
import math as _math
import time as _time
Steven Cordwell's avatar
Steven Cordwell committed
62

63 64
import numpy as _np
import scipy.sparse as _sp
Steven Cordwell's avatar
Steven Cordwell committed
65

66 67 68 69 70
try:
    from .util import check, getSpan
except ValueError:
    # importing mdp as a module rather than as part of a package
    from util import check, getSpan
71

72
_MSG_STOP_MAX_ITER = "Iterating stopped due to maximum number of iterations " \
Steven Cordwell's avatar
Steven Cordwell committed
73
    "condition."
74
_MSG_STOP_EPSILON_OPTIMAL_POLICY = "Iterating stopped, epsilon-optimal " \
Steven Cordwell's avatar
Steven Cordwell committed
75
    "policy found."
76
_MSG_STOP_EPSILON_OPTIMAL_VALUE = "Iterating stopped, epsilon-optimal value " \
Steven Cordwell's avatar
Steven Cordwell committed
77
    "function found."
78
_MSG_STOP_UNCHANGING_POLICY = "Iterating stopped, unchanging policy found."
Steven Cordwell's avatar
Steven Cordwell committed
79

Steven Cordwell's avatar
Steven Cordwell committed
80
class MDP(object):
81
    
Steven Cordwell's avatar
Steven Cordwell committed
82 83
    """A Markov Decision Problem.
    
Steven Cordwell's avatar
Steven Cordwell committed
84
    Let ``S`` = the number of states, and ``A`` = the number of acions.
85
    
Steven Cordwell's avatar
Steven Cordwell committed
86 87 88
    Parameters
    ----------
    transitions : array
89
        Transition probability matrices. These can be defined in a variety of 
Steven Cordwell's avatar
Steven Cordwell committed
90
        ways. The simplest is a numpy array that has the shape ``(A, S, S)``,
91
        though there are other possibilities. It can be a tuple or list or
Steven Cordwell's avatar
Steven Cordwell committed
92 93 94 95 96 97 98
        numpy object array of length ``A``, where each element contains a numpy
        array or matrix that has the shape ``(S, S)``. This "list of matrices"
        form is useful when the transition matrices are sparse as
        ``scipy.sparse.csr_matrix`` matrices can be used. In summary, each
        action's transition matrix must be indexable like ``transitions[a]``
        where ``a`` ∈ {0, 1...A-1}, and ``transitions[a]`` returns an ``S`` ×
        ``S`` array-like object.
Steven Cordwell's avatar
Steven Cordwell committed
99
    reward : array
100 101
        Reward matrices or vectors. Like the transition matrices, these can
        also be defined in a variety of ways. Again the simplest is a numpy
Steven Cordwell's avatar
Steven Cordwell committed
102 103 104 105 106 107 108 109
        array that has the shape ``(S, A)``, ``(S,)`` or ``(A, S, S)``. A list
        of lists can be used, where each inner list has length ``S`` and the
        outer list has length ``A``. A list of numpy arrays is possible where
        each inner array can be of the shape ``(S,)``, ``(S, 1)``, ``(1, S)``
        or ``(S, S)``. Also ``scipy.sparse.csr_matrix`` can be used instead of
        numpy arrays. In addition, the outer list can be replaced by any object
        that can be indexed like ``reward[a]`` such as a tuple or numpy object
        array of length ``A``.
110 111 112 113
    discount : float
        Discount factor. The per time-step discount factor on future rewards.
        Valid values are greater than 0 upto and including 1. If the discount
        factor is 1, then convergence is cannot be assumed and a warning will
Steven Cordwell's avatar
Steven Cordwell committed
114 115
        be displayed. Subclasses of ``MDP`` may pass ``None`` in the case where
        the algorithm does not use a discount factor.
116 117 118 119
    epsilon : float
        Stopping criterion. The maximum change in the value function at each
        iteration is compared against ``epsilon``. Once the change falls below
        this value, then the value function is considered to have converged to
Steven Cordwell's avatar
Steven Cordwell committed
120 121 122
        the optimal value function. Subclasses of ``MDP`` may pass ``None`` in
        the case where the algorithm does not use an epsilon-optimal stopping
        criterion.
123 124 125
    max_iter : int
        Maximum number of iterations. The algorithm will be terminated once
        this many iterations have elapsed. This must be greater than 0 if
Steven Cordwell's avatar
Steven Cordwell committed
126 127
        specified. Subclasses of ``MDP`` may pass ``None`` in the case where
        the algorithm does not use a maximum number of iterations.
Steven Cordwell's avatar
Steven Cordwell committed
128 129 130 131
    
    Attributes
    ----------
    P : array
132
        Transition probability matrices.
Steven Cordwell's avatar
Steven Cordwell committed
133
    R : array
134 135
        Reward vectors.
    V : tuple
Steven Cordwell's avatar
Steven Cordwell committed
136 137 138
        The optimal value function. Each element is a float corresponding to
        the expected value of being in that state assuming the optimal policy
        is followed.
Steven Cordwell's avatar
Steven Cordwell committed
139
    discount : float
140
        The discount rate on future rewards.
Steven Cordwell's avatar
Steven Cordwell committed
141
    max_iter : int
142 143 144
        The maximum number of iterations.
    policy : tuple
        The optimal policy.
Steven Cordwell's avatar
Steven Cordwell committed
145
    time : float
146 147
        The time used to converge to the optimal policy.
    verbose : boolean
Steven Cordwell's avatar
Steven Cordwell committed
148
        Whether verbose output should be displayed or not.
Steven Cordwell's avatar
Steven Cordwell committed
149 150 151
    
    Methods
    -------
152
    run
Steven Cordwell's avatar
Steven Cordwell committed
153
        Implemented in child classes as the main algorithm loop. Raises an
154
        exception if it has not been overridden.
Steven Cordwell's avatar
Steven Cordwell committed
155 156 157 158 159 160
    setSilent
        Turn the verbosity off
    setVerbose
        Turn the verbosity on
    
    """
Steven Cordwell's avatar
Steven Cordwell committed
161
    
162
    def __init__(self, transitions, reward, discount, epsilon, max_iter):
163
        # Initialise a MDP based on the input parameters.
164
        
Steven Cordwell's avatar
Steven Cordwell committed
165 166
        # if the discount is None then the algorithm is assumed to not use it
        # in its computations
167 168 169 170
        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:
Steven Cordwell's avatar
Steven Cordwell committed
171
                print("WARNING: check conditions of convergence. With no "
Steven Cordwell's avatar
Steven Cordwell committed
172
                      "discount, convergence can not be assumed.")
Steven Cordwell's avatar
Steven Cordwell committed
173 174
        # if the max_iter is None then the algorithm is assumed to not use it
        # in its computations
175 176 177 178
        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
179
        # check that epsilon is something sane
180 181 182
        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
183 184 185 186
        # 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
187
        self._computePR(transitions, reward)
Steven Cordwell's avatar
Steven Cordwell committed
188 189 190 191
        # 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
192 193
        # set the initial iteration count to zero
        self.iter = 0
Steven Cordwell's avatar
Steven Cordwell committed
194
        # V should be stored as a vector ie shape of (S,) or (1, S)
Steven Cordwell's avatar
Steven Cordwell committed
195
        self.V = None
Steven Cordwell's avatar
Steven Cordwell committed
196
        # policy can also be stored as a vector
Steven Cordwell's avatar
Steven Cordwell committed
197
        self.policy = None
Steven Cordwell's avatar
Steven Cordwell committed
198
    
199 200 201 202 203 204
    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"
205
        return(P_repr + "\n" + R_repr)
206
    
207
    def _bellmanOperator(self, V=None):
Steven Cordwell's avatar
Steven Cordwell committed
208
        # Apply the Bellman operator on the value function.
209
        # 
Steven Cordwell's avatar
Steven Cordwell committed
210
        # Updates the value function and the Vprev-improving policy.
211
        # 
Steven Cordwell's avatar
Steven Cordwell committed
212 213 214 215
        # 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
216 217
        if V is None:
            # this V should be a reference to the data rather than a copy
218 219
            V = self.V
        else:
Steven Cordwell's avatar
Steven Cordwell committed
220
            # make sure the user supplied V is of the right shape
221
            try:
222 223
                assert V.shape in ((self.S,), (1, self.S)), "V is not the " \
                    "right shape (Bellman operator)."
224
            except AttributeError:
225
                raise TypeError("V must be a numpy array or matrix.")
226 227 228 229
        # 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.
230
        Q = _np.empty((self.A, self.S))
Steven Cordwell's avatar
Steven Cordwell committed
231
        for aa in range(self.A):
Steven Cordwell's avatar
Steven Cordwell committed
232
            Q[aa] = self.R[aa] + self.discount * self.P[aa].dot(V)
Steven Cordwell's avatar
Steven Cordwell committed
233
        # Get the policy and value, for now it is being returned but...
234
        # Which way is better?
235
        # 1. Return, (policy, value)
236
        return (Q.argmax(axis=0), Q.max(axis=0))
Steven Cordwell's avatar
Steven Cordwell committed
237 238
        # 2. update self.policy and self.V directly
        # self.V = Q.max(axis=1)
239
        # self.policy = Q.argmax(axis=1)
Steven Cordwell's avatar
Steven Cordwell committed
240
    
241 242 243 244 245 246 247 248 249 250 251 252
    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
253
        self.P = tuple(P[aa] for aa in range(self.A))
254
    
255
    def _computePR(self, P, R):
Steven Cordwell's avatar
Steven Cordwell committed
256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
        # 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
        #
271
        # We assume that P and R define a MDP i,e. assumption is that
Steven Cordwell's avatar
Steven Cordwell committed
272
        # check(P, R) has already been run and doesn't fail.
273
        #
274 275
        # First compute store P, S, and A
        self._computeP(P)
Steven Cordwell's avatar
Steven Cordwell committed
276 277
        # Set self.R as a tuple of length A, with each element storing an 1×S
        # vector.
278
        try:
279
            if R.ndim == 1:
280
                r = _np.array(R).reshape(self.S)
281
                self.R = tuple(r for aa in range(self.A))
282
            elif R.ndim == 2:
283
                self.R = tuple(_np.array(R[:, aa]).reshape(self.S)
284
                                for aa in range(self.A))
Steven Cordwell's avatar
Steven Cordwell committed
285
            else:
286
                self.R = tuple(_np.multiply(P[aa], R[aa]).sum(1).reshape(self.S)
287
                                for aa in range(self.A))
288
        except AttributeError:
289
            if len(R) == self.A:
290
                self.R = tuple(_np.multiply(P[aa], R[aa]).sum(1).reshape(self.S)
291
                                for aa in range(self.A))
292
            else:
293
                r = _np.array(R).reshape(self.S)
294
                self.R = tuple(r for aa in range(self.A))
295
    
296
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
297
        # Raise error because child classes should implement this function.
298
        raise NotImplementedError("You should create a run() method.")
Steven Cordwell's avatar
Steven Cordwell committed
299
    
Steven Cordwell's avatar
Steven Cordwell committed
300
    def setSilent(self):
301
        """Set the MDP algorithm to silent mode."""
Steven Cordwell's avatar
Steven Cordwell committed
302 303 304
        self.verbose = False
    
    def setVerbose(self):
305
        """Set the MDP algorithm to verbose mode."""
Steven Cordwell's avatar
Steven Cordwell committed
306
        self.verbose = True
Steven Cordwell's avatar
Steven Cordwell committed
307 308

class FiniteHorizon(MDP):
309
    
Steven Cordwell's avatar
Steven Cordwell committed
310
    """A MDP solved using the finite-horizon backwards induction algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
311
    
Steven Cordwell's avatar
Steven Cordwell committed
312 313
    Parameters
    ----------
314 315 316 317 318 319 320 321 322
    transitions : array
        Transition probability matrices. See the documentation for the ``MDP``
        class for details.
    reward : array
        Reward matrices or vectors. See the documentation for the ``MDP`` class
        for details.
    discount : float
        Discount factor. See the documentation for the ``MDP`` class for
        details.
Steven Cordwell's avatar
Steven Cordwell committed
323 324 325 326
    N : int
        Number of periods. Must be greater than 0.
    h : array, optional
        Terminal reward. Default: a vector of zeros.
Steven Cordwell's avatar
Steven Cordwell committed
327
    
Steven Cordwell's avatar
Steven Cordwell committed
328 329 330 331
    Data Attributes
    ---------------
    V : array 
        Optimal value function. Shape = (S, N+1). ``V[:, n]`` = optimal value
Steven Cordwell's avatar
Steven Cordwell committed
332
        function at stage ``n`` with stage in {0, 1...N-1}. ``V[:, N]`` value
Steven Cordwell's avatar
Steven Cordwell committed
333 334 335
        function for terminal stage. 
    policy : array
        Optimal policy. ``policy[:, n]`` = optimal policy at stage ``n`` with
Steven Cordwell's avatar
Steven Cordwell committed
336
        stage in {0, 1...N}. ``policy[:, N]`` = policy for stage ``N``.
Steven Cordwell's avatar
Steven Cordwell committed
337 338
    time : float
        used CPU time
Steven Cordwell's avatar
Steven Cordwell committed
339 340 341 342
  
    Notes
    -----
    In verbose mode, displays the current stage and policy transpose.
343
    
Steven Cordwell's avatar
Steven Cordwell committed
344 345
    Examples
    --------
346 347 348
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.forest()
    >>> fh = mdptoolbox.mdp.FiniteHorizon(P, R, 0.9, 3)
349
    >>> fh.run()
Steven Cordwell's avatar
Steven Cordwell committed
350 351 352 353 354 355 356 357
    >>> 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]])
358
    
Steven Cordwell's avatar
Steven Cordwell committed
359
    """
Steven Cordwell's avatar
Steven Cordwell committed
360

Steven Cordwell's avatar
Steven Cordwell committed
361
    def __init__(self, transitions, reward, discount, N, h=None):
362
        # Initialise a finite horizon MDP.
363
        self.N = int(N)
Steven Cordwell's avatar
Steven Cordwell committed
364
        assert self.N > 0, "N must be greater than 0."
Steven Cordwell's avatar
Steven Cordwell committed
365
        # Initialise the base class
366
        MDP.__init__(self, transitions, reward, discount, None, None)
Steven Cordwell's avatar
Steven Cordwell committed
367 368
        # remove the iteration counter, it is not meaningful for backwards
        # induction
369
        del self.iter
Steven Cordwell's avatar
Steven Cordwell committed
370
        # There are value vectors for each time step up to the horizon
371
        self.V = _np.zeros((self.S, N + 1))
Steven Cordwell's avatar
Steven Cordwell committed
372 373
        # There are policy vectors for each time step before the horizon, when
        # we reach the horizon we don't need to make decisions anymore.
374
        self.policy = _np.empty((self.S, N), dtype=int)
Steven Cordwell's avatar
Steven Cordwell committed
375 376
        # Set the reward for the final transition to h, if specified.
        if h is not None:
377
            self.V[:, N] = h
378
        # Call the iteration method
379
        #self.run()
380
        
381
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
382
        # Run the finite horizon algorithm.
383
        self.time = _time.time()
Steven Cordwell's avatar
Steven Cordwell committed
384
        # loop through each time period
385
        for n in range(self.N):
Steven Cordwell's avatar
Steven Cordwell committed
386
            W, X = self._bellmanOperator(self.V[:, self.N - n])
Steven Cordwell's avatar
Steven Cordwell committed
387 388 389
            stage = self.N - n - 1
            self.V[:, stage] = X
            self.policy[:, stage] = W
Steven Cordwell's avatar
Steven Cordwell committed
390
            if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
391 392
                print(("stage: %s, policy: %s") % (
                    stage, self.policy[:, stage].tolist()))
Steven Cordwell's avatar
Steven Cordwell committed
393
        # update time spent running
394
        self.time = _time.time() - self.time
Steven Cordwell's avatar
Steven Cordwell committed
395 396
        # After this we could create a tuple of tuples for the values and 
        # policies.
Steven Cordwell's avatar
Steven Cordwell committed
397 398 399
        #self.V = tuple(tuple(self.V[:, n].tolist()) for n in range(self.N))
        #self.policy = tuple(tuple(self.policy[:, n].tolist())
        #                    for n in range(self.N))
Steven Cordwell's avatar
Steven Cordwell committed
400 401

class LP(MDP):
402
    
403
    """A discounted MDP soloved using linear programming.
Steven Cordwell's avatar
Steven Cordwell committed
404 405
    
    This class requires the Python ``cvxopt`` module to be installed.
Steven Cordwell's avatar
Steven Cordwell committed
406 407 408

    Arguments
    ---------
409 410 411 412 413 414 415 416 417
    transitions : array
        Transition probability matrices. See the documentation for the ``MDP``
        class for details.
    reward : array
        Reward matrices or vectors. See the documentation for the ``MDP`` class
        for details.
    discount : float
        Discount factor. See the documentation for the ``MDP`` class for
        details.
Steven Cordwell's avatar
Steven Cordwell committed
418 419
    h : array, optional
        Terminal reward. Default: a vector of zeros.
Steven Cordwell's avatar
Steven Cordwell committed
420
    
Steven Cordwell's avatar
Steven Cordwell committed
421 422 423 424 425 426 427 428
    Data Attributes
    ---------------
    V : tuple
        optimal values
    policy : tuple
        optimal policy
    time : float
        used CPU time
Steven Cordwell's avatar
Steven Cordwell committed
429 430 431
    
    Examples
    --------
432 433 434
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.forest()
    >>> lp = mdptoolbox.mdp.LP(P, R, 0.9)
435
    >>> lp.run()
436
    
Steven Cordwell's avatar
Steven Cordwell committed
437
    """
Steven Cordwell's avatar
Steven Cordwell committed
438

Steven Cordwell's avatar
Steven Cordwell committed
439
    def __init__(self, transitions, reward, discount):
440
        # Initialise a linear programming MDP.
Steven Cordwell's avatar
Steven Cordwell committed
441
        # import some functions from cvxopt and set them as object methods
Steven Cordwell's avatar
Steven Cordwell committed
442 443
        try:
            from cvxopt import matrix, solvers
444 445
            self._linprog = solvers.lp
            self._cvxmat = matrix
Steven Cordwell's avatar
Steven Cordwell committed
446
        except ImportError:
447 448
            raise ImportError("The python module cvxopt is required to use "
                              "linear programming functionality.")
Steven Cordwell's avatar
Steven Cordwell committed
449
        # initialise the MDP. epsilon and max_iter are not needed
450
        MDP.__init__(self, transitions, reward, discount, None, None)
Steven Cordwell's avatar
Steven Cordwell committed
451
        # Set the cvxopt solver to be quiet by default, but ...
452
        # this doesn't do what I want it to do c.f. issue #3
453 454
        if not self.verbose:
            solvers.options['show_progress'] = False
455
        # Call the iteration method
456
        #self.run()
457
    
458
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
459
        #Run the linear programming algorithm.
460
        self.time = _time.time()
Steven Cordwell's avatar
Steven Cordwell committed
461
        # The objective is to resolve : min V / V >= PR + discount*P*V
462 463
        # The function linprog of the optimisation Toolbox of Mathworks
        # resolves :
Steven Cordwell's avatar
Steven Cordwell committed
464
        # min f'* x / M * x <= b
465 466 467 468
        # 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)
469
        f = self._cvxmat(_np.ones((self.S, 1)))
470
        h = self._cvxmat(self.R.reshape(self.S * self.A, 1, order="F"), tc='d')
471
        M = _np.zeros((self.A * self.S, self.S))
Steven Cordwell's avatar
Steven Cordwell committed
472 473
        for aa in range(self.A):
            pos = (aa + 1) * self.S
474
            M[(pos - self.S):pos, :] = (
475
                self.discount * self.P[aa] - _sp.eye(self.S, self.S))
476
        M = self._cvxmat(M)
477 478 479
        # 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
480
        # only to 10e-8 places. This assumes glpk is installed of course.
481
        self.V = _np.array(self._linprog(f, M, -h, solver='glpk')['x'])
Steven Cordwell's avatar
Steven Cordwell committed
482
        # apply the Bellman operator
483
        self.policy, self.V =  self._bellmanOperator()
Steven Cordwell's avatar
Steven Cordwell committed
484
        # update the time spent solving
485
        self.time = _time.time() - self.time
486
        # store value and policy as tuples
487 488
        self.V = tuple(self.V.tolist())
        self.policy = tuple(self.policy.tolist())
Steven Cordwell's avatar
Steven Cordwell committed
489 490

class PolicyIteration(MDP):
491
    
492
    """A discounted MDP solved using the policy iteration algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
493
    
Steven Cordwell's avatar
Steven Cordwell committed
494 495
    Arguments
    ---------
496 497 498 499 500 501 502 503 504
    transitions : array
        Transition probability matrices. See the documentation for the ``MDP``
        class for details.
    reward : array
        Reward matrices or vectors. See the documentation for the ``MDP`` class
        for details. 
    discount : float
        Discount factor. See the documentation for the ``MDP`` class for
        details.
Steven Cordwell's avatar
Steven Cordwell committed
505 506 507
    policy0 : array, optional
        Starting policy.
    max_iter : int, optional
508 509
        Maximum number of iterations. See the documentation for the ``MDP``
        class for details. Default is 1000.
Steven Cordwell's avatar
Steven Cordwell committed
510 511 512 513
    eval_type : int or string, optional
        Type of function used to evaluate policy. 0 or "matrix" to solve as a
        set of linear equations. 1 or "iterative" to solve iteratively.
        Default: 0.
Steven Cordwell's avatar
Steven Cordwell committed
514
             
Steven Cordwell's avatar
Steven Cordwell committed
515 516 517 518 519 520 521 522 523 524
    Data Attributes
    ---------------
    V : tuple
        value function 
    policy : tuple
        optimal policy
    iter : int
        number of done iterations
    time : float
        used CPU time
Steven Cordwell's avatar
Steven Cordwell committed
525 526 527 528 529 530
    
    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
531 532
    Examples
    --------
533 534 535
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.rand()
    >>> pi = mdptoolbox.mdp.PolicyIteration(P, R, 0.9)
536
    >>> pi.run()
537
    
538 539
    >>> P, R = mdptoolbox.example.forest()
    >>> pi = mdptoolbox.mdp.PolicyIteration(P, R, 0.9)
540
    >>> pi.run()
Steven Cordwell's avatar
Steven Cordwell committed
541
    >>> pi.V
542
    (26.244000000000018, 29.48400000000002, 33.484000000000016)
Steven Cordwell's avatar
Steven Cordwell committed
543
    >>> pi.policy
544
    (0, 0, 0)
Steven Cordwell's avatar
Steven Cordwell committed
545
    """
Steven Cordwell's avatar
Steven Cordwell committed
546
    
547 548
    def __init__(self, transitions, reward, discount, policy0=None,
                 max_iter=1000, eval_type=0):
Steven Cordwell's avatar
Steven Cordwell committed
549 550 551
        # Initialise a policy iteration MDP.
        #
        # Set up the MDP, but don't need to worry about epsilon values
552
        MDP.__init__(self, transitions, reward, discount, None, max_iter)
Steven Cordwell's avatar
Steven Cordwell committed
553
        # Check if the user has supplied an initial policy. If not make one.
Steven Cordwell's avatar
Steven Cordwell committed
554
        if policy0 == None:
Steven Cordwell's avatar
Steven Cordwell committed
555
            # Initialise the policy to the one which maximises the expected
Steven Cordwell's avatar
Steven Cordwell committed
556
            # immediate reward
557
            null = _np.zeros(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
558
            self.policy, null = self._bellmanOperator(null)
559
            del null
Steven Cordwell's avatar
Steven Cordwell committed
560
        else:
Steven Cordwell's avatar
Steven Cordwell committed
561 562
            # Use the policy that the user supplied
            # Make sure it is a numpy array
563
            policy0 = _np.array(policy0)
Steven Cordwell's avatar
Steven Cordwell committed
564
            # Make sure the policy is the right size and shape
565 566
            assert policy0.shape in ((self.S, ), (self.S, 1), (1, self.S)), \
                "'policy0' must a vector with length S."
Steven Cordwell's avatar
Steven Cordwell committed
567
            # reshape the policy to be a vector
Steven Cordwell's avatar
Steven Cordwell committed
568
            policy0 = policy0.reshape(self.S)
569 570
            # The policy can only contain integers between 0 and S-1
            msg = "'policy0' must be a vector of integers between 0 and S-1."
571
            assert not _np.mod(policy0, 1).any(), msg
572 573 574
            assert (policy0 >= 0).all(), msg
            assert (policy0 < self.S).all(), msg
            self.policy = policy0
Steven Cordwell's avatar
Steven Cordwell committed
575
        # set the initial values to zero
576
        self.V = _np.zeros(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
577
        # Do some setup depending on the evaluation type
Steven Cordwell's avatar
Steven Cordwell committed
578 579 580 581 582
        if eval_type in (0, "matrix"):
            self.eval_type = "matrix"
        elif eval_type in (1, "iterative"):
            self.eval_type = "iterative"
        else:
Steven Cordwell's avatar
Steven Cordwell committed
583 584 585
            raise ValueError("'eval_type' should be '0' for matrix evaluation "
                             "or '1' for iterative evaluation. The strings "
                             "'matrix' and 'iterative' can also be used.")
586
        # Call the iteration method
587
        #self.run()
Steven Cordwell's avatar
Steven Cordwell committed
588
    
589
    def _computePpolicyPRpolicy(self):
Steven Cordwell's avatar
Steven Cordwell committed
590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608
        # 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
        #
609 610
        Ppolicy = _np.empty((self.S, self.S))
        Rpolicy = _np.zeros(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
611
        for aa in range(self.A): # avoid looping over S
Steven Cordwell's avatar
Steven Cordwell committed
612 613
            # the rows that use action a.
            ind = (self.policy == aa).nonzero()[0]
614 615
            # if no rows use action a, then no need to assign this
            if ind.size > 0:
616 617 618 619
                try:
                    Ppolicy[ind, :] = self.P[aa][ind, :]
                except ValueError:
                    Ppolicy[ind, :] = self.P[aa][ind, :].todense()
620
                #PR = self._computePR() # an apparently uneeded line, and
Steven Cordwell's avatar
Steven Cordwell committed
621 622
                # perhaps harmful in this implementation c.f.
                # mdp_computePpolicyPRpolicy.m
623
                Rpolicy[ind] = self.R[aa][ind]
Steven Cordwell's avatar
Steven Cordwell committed
624 625 626 627
        # 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
628 629
        if type(self.R) is _sp.csr_matrix:
            Rpolicy = _sp.csr_matrix(Rpolicy)
Steven Cordwell's avatar
Steven Cordwell committed
630 631 632 633
        #self.Ppolicy = Ppolicy
        #self.Rpolicy = Rpolicy
        return (Ppolicy, Rpolicy)
    
634
    def _evalPolicyIterative(self, V0=0, epsilon=0.0001, max_iter=10000):
Steven Cordwell's avatar
Steven Cordwell committed
635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664
        # 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.
        #
665 666 667
        try:
            assert V0.shape in ((self.S, ), (self.S, 1), (1, self.S)), \
                "'V0' must be a vector of length S."
668
            policy_V = _np.array(V0).reshape(self.S)
669
        except AttributeError:
Steven Cordwell's avatar
Steven Cordwell committed
670
            if V0 == 0:
671
                policy_V = _np.zeros(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
672
            else:
673
                policy_V = _np.array(V0).reshape(self.S)
Steven Cordwell's avatar
Steven Cordwell committed
674
        
675
        policy_P, policy_R = self._computePpolicyPRpolicy()
Steven Cordwell's avatar
Steven Cordwell committed
676 677
        
        if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
678
            print('    Iteration\t\t    V variation')
679
        
Steven Cordwell's avatar
Steven Cordwell committed
680 681 682
        itr = 0
        done = False
        while not done:
683
            itr += 1
684 685
            
            Vprev = policy_V
686
            policy_V = policy_R + self.discount * policy_P.dot(Vprev)
687
            
688
            variation = _np.absolute(policy_V - Vprev).max()
Steven Cordwell's avatar
Steven Cordwell committed
689
            if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
690
                print(('      %s\t\t      %s') % (itr, variation))
691 692 693
            
            # ensure |Vn - Vpolicy| < epsilon
            if variation < ((1 - self.discount) / self.discount) * epsilon:
Steven Cordwell's avatar
Steven Cordwell committed
694 695
                done = True
                if self.verbose:
696
                    print(_MSG_STOP_EPSILON_OPTIMAL_VALUE)
Steven Cordwell's avatar
Steven Cordwell committed
697 698 699
            elif itr == max_iter:
                done = True
                if self.verbose:
700
                    print(_MSG_STOP_MAX_ITER)
Steven Cordwell's avatar
Steven Cordwell committed
701
        
Steven Cordwell's avatar
Steven Cordwell committed
702
        self.V = policy_V
703
    
704
    def _evalPolicyMatrix(self):
Steven Cordwell's avatar
Steven Cordwell committed
705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723
        # 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
        #
724
        Ppolicy, Rpolicy = self._computePpolicyPRpolicy()
Steven Cordwell's avatar
Steven Cordwell committed
725
        # V = PR + gPV  => (I-gP)V = PR  => V = inv(I-gP)* PR
726 727
        self.V = _np.linalg.solve(
            (_sp.eye(self.S, self.S) - self.discount * Ppolicy), Rpolicy)
Steven Cordwell's avatar
Steven Cordwell committed
728
    
729
    def run(self):
Steven Cordwell's avatar
Steven Cordwell committed
730 731
        # Run the policy iteration algorithm.
        # If verbose the print a header
732
        if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
733
            print('  Iteration\t\tNumber of different actions')
Steven Cordwell's avatar
Steven Cordwell committed
734
        # Set up the while stopping condition and the current time
Steven Cordwell's avatar
Steven Cordwell committed
735
        done = False
736
        self.time = _time.time()
Steven Cordwell's avatar
Steven Cordwell committed
737
        # loop until a stopping condition is reached
Steven Cordwell's avatar
Steven Cordwell committed
738
        while not done:
739
            self.iter += 1
740
            # these _evalPolicy* functions will update the classes value
Steven Cordwell's avatar
Steven Cordwell committed
741
            # attribute
Steven Cordwell's avatar
Steven Cordwell committed
742
            if self.eval_type == "matrix":
743
                self._evalPolicyMatrix()
Steven Cordwell's avatar
Steven Cordwell committed
744
            elif self.eval_type == "iterative":
745
                self._evalPolicyIterative()
Steven Cordwell's avatar
Steven Cordwell committed
746 747
            # This should update the classes policy attribute but leave the
            # value alone
748
            policy_next, null = self._bellmanOperator()
749
            del null
Steven Cordwell's avatar
Steven Cordwell committed
750 751
            # calculate in how many places does the old policy disagree with
            # the new policy
752
            n_different = (policy_next != self.policy).sum()
Steven Cordwell's avatar
Steven Cordwell committed
753
            # if verbose then continue printing a table
754
            if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
755
                print(('    %s\t\t  %s') % (self.iter, n_different))
Steven Cordwell's avatar
Steven Cordwell committed
756 757
            # Once the policy is unchanging of the maximum number of 
            # of iterations has been reached then stop
758
            if n_different == 0:
Steven Cordwell's avatar
Steven Cordwell committed
759
                done = True
760
                if self.verbose:
761
                    print(_MSG_STOP_UNCHANGING_POLICY)
762 763 764
            elif (self.iter == self.max_iter):
                done = True 
                if self.verbose:
765
                    print(_MSG_STOP_MAX_ITER)
766 767
            else:
                self.policy = policy_next
Steven Cordwell's avatar
Steven Cordwell committed
768
        # update the time to return th computation time
769
        self.time = _time.time() - self.time
Steven Cordwell's avatar
Steven Cordwell committed
770
        # store value and policy as tuples
771 772
        self.V = tuple(self.V.tolist())
        self.policy = tuple(self.policy.tolist())
Steven Cordwell's avatar
Steven Cordwell committed
773

774
class PolicyIterationModified(PolicyIteration):
775
    
776
    """A discounted MDP  solved using a modifified policy iteration algorithm.
Steven Cordwell's avatar
Steven Cordwell committed
777 778 779
    
    Arguments
    ---------
780 781 782 783 784 785 786 787 788
    transitions : array
        Transition probability matrices. See the documentation for the ``MDP``
        class for details.
    reward : array
        Reward matrices or vectors. See the documentation for the ``MDP`` class
        for details.
    discount : float
        Discount factor. See the documentation for the ``MDP`` class for
        details.
Steven Cordwell's avatar
Steven Cordwell committed
789 790 791 792
    epsilon : float, optional
        Stopping criterion. See the documentation for the ``MDP`` class for
        details. Default: 0.01.
    max_iter : int, optional
793
        Maximum number of iterations. See the documentation for the ``MDP``
Steven Cordwell's avatar
Steven Cordwell committed
794
        class for details. Default is 10.
Steven Cordwell's avatar
Steven Cordwell committed
795 796 797
    
    Data Attributes
    ---------------
Steven Cordwell's avatar
Steven Cordwell committed
798 799 800 801 802 803 804 805
    V : tuple
        value function 
    policy : tuple
        optimal policy
    iter : int
        number of done iterations
    time : float
        used CPU time
Steven Cordwell's avatar
Steven Cordwell committed
806 807 808
    
    Examples
    --------
809 810 811
    >>> import mdptoolbox, mdptoolbox.example
    >>> P, R = mdptoolbox.example.forest()
    >>> pim = mdptoolbox.mdp.PolicyIterationModified(P, R, 0.9)
812
    >>> pim.run()
813
    >>> pim.policy
814
    (0, 0, 0)
815
    >>> pim.V
816
    (21.81408652334702, 25.054086523347017, 29.054086523347017)
817
    
Steven Cordwell's avatar
Steven Cordwell committed
818
    """
819
    
820 821
    def __init__(self, transitions, reward, discount, epsilon=0.01,
                 max_iter=10):
822
        # Initialise a (modified) policy iteration MDP.
Steven Cordwell's avatar
Steven Cordwell committed
823
        
824 825 826
        # 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
827
        # is needed from the PolicyIteration class is the _evalPolicyIterative
828
        # function. Perhaps there is a better way to do it?
829 830
        PolicyIteration.__init__(self, transitions, reward, discount, None,
                                 max_iter, 1)
831
        
832 833
        # PolicyIteration doesn't pass epsilon to MDP.__init__() so we will
        # check it here
834 835
        self.epsilon = float(epsilon)
        assert epsilon > 0, "'epsilon' must be greater than 0."
836
        
837 838
        # computation of threshold of variation for V for an epsilon-optimal
        # policy
839
        if self.discount != 1:
840
            self.thresh = self.epsilon * (1 - self.discount) / self.discount
841
        else:
842
            self.thresh = self.epsilon
843
        
844
        if self.discount == 1:
845
            self.V = _np.zeros((self.S, 1))
846
        else:
847
            Rmin = min(R.min() for R in self.R)
848
            self.V = 1 / (1 - self.discount) * Rmin * _np.ones((self.S,))
849 850
        
        # Call the iteration method
851
        #self.run()
Steven Cordwell's avatar
Steven Cordwell committed
852
    
853
    def run(self):
854
        # Run the modified policy iteration algorithm.
855 856
        
        if self.verbose:
Steven Cordwell's avatar
Steven Cordwell committed
857
            print('  \tIteration