mdp.py

## Q-Learning Algorithm - This code is modified from the code contained in
## the MDP Toolbox Python package
## Source: https://github.com/sawcordwell/pymdptoolbox


"""Markov Decision Process (MDP) Toolbox: ``mdp`` module
=====================================================

The ``mdp`` module provides classes for the resolution of descrete-time Markov
Decision Processes.

Available classes
-----------------
:class:`~mdptoolbox.mdp.MDP`
    Base Markov decision process class
:class:`~mdptoolbox.mdp.FiniteHorizon`
    Backwards induction finite horizon MDP
:class:`~mdptoolbox.mdp.PolicyIteration`
    Policy iteration MDP
:class:`~mdptoolbox.mdp.PolicyIterationModified`
    Modified policy iteration MDP
:class:`~mdptoolbox.mdp.QLearning`
    Q-learning MDP
:class:`~mdptoolbox.mdp.RelativeValueIteration`
    Relative value iteration MDP
:class:`~mdptoolbox.mdp.ValueIteration`
    Value iteration MDP
:class:`~mdptoolbox.mdp.ValueIterationGS`
    Gauss-Seidel value iteration MDP

"""

# Copyright (c) 2011-2015 Steven A. W. Cordwell
# Copyright (c) 2009 INRA
#
# 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.

import math as _math
import time as _time

import numpy as _np
import scipy.sparse as _sp

import util as _util

_MSG_STOP_MAX_ITER = "Iterating stopped due to maximum number of iterations " \
    "condition."
_MSG_STOP_EPSILON_OPTIMAL_POLICY = "Iterating stopped, epsilon-optimal " \
    "policy found."
_MSG_STOP_EPSILON_OPTIMAL_VALUE = "Iterating stopped, epsilon-optimal value " \
    "function found."
_MSG_STOP_UNCHANGING_POLICY = "Iterating stopped, unchanging policy found."


def _computeDimensions(transition):
    A = len(transition)
    try:
        if transition.ndim == 3:
            S = transition.shape[1]
        else:
            S = transition[0].shape[0]
    except AttributeError:
        S = transition[0].shape[0]
    return S, A


def _printVerbosity(iteration, variation):
    if isinstance(variation, float):
        print("{:>10}{:>12f}".format(iteration, variation))
    elif isinstance(variation, int):
        print("{:>10}{:>12d}".format(iteration, variation))
    else:
        print("{:>10}{:>12}".format(iteration, variation))


class MDP(object):

    """A Markov Decision Problem.

    Let ``S`` = the number of states, and ``A`` = the number of acions.

    Parameters
    ----------
    transitions : array
        Transition probability matrices. These can be defined in a variety of
        ways. The simplest is a numpy array that has the shape ``(A, S, S)``,
        though there are other possibilities. It can be a tuple or list or
        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.
    reward : array
        Reward matrices or vectors. Like the transition matrices, these can
        also be defined in a variety of ways. Again the simplest is a numpy
        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``.
    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
        be displayed. Subclasses of ``MDP`` may pass ``None`` in the case where
        the algorithm does not use a discount factor.
    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
        the optimal value function. Subclasses of ``MDP`` may pass ``None`` in
        the case where the algorithm does not use an epsilon-optimal stopping
        criterion.
    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
        specified. Subclasses of ``MDP`` may pass ``None`` in the case where
        the algorithm does not use a maximum number of iterations.
    skip_check : bool
        By default we run a check on the ``transitions`` and ``rewards``
        arguments to make sure they describe a valid MDP. You can set this
        argument to True in order to skip this check.

    Attributes
    ----------
    P : array
        Transition probability matrices.
    R : array
        Reward vectors.
    V : tuple
        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.
    discount : float
        The discount rate on future rewards.
    max_iter : int
        The maximum number of iterations.
    policy : tuple
        The optimal policy.
    time : float
        The time used to converge to the optimal policy.
    verbose : boolean
        Whether verbose output should be displayed or not.

    Methods
    -------
    run
        Implemented in child classes as the main algorithm loop. Raises an
        exception if it has not been overridden.
    setSilent
        Turn the verbosity off
    setVerbose
        Turn the verbosity on

    """

    def __init__(self, transitions, reward, discount, epsilon, max_iter,
                 skip_check=False):
        # Initialise a MDP based on the input parameters.

        # if the discount is None then the algorithm is assumed to not use it
        # in its computations
        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("WARNING: check conditions of convergence. With no "
                      "discount, convergence can not be assumed.")

        # if the max_iter is None then the algorithm is assumed to not use it
        # in its computations
        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."
            )

        # check that epsilon is something sane
        if epsilon is not None:
            self.epsilon = float(epsilon)
            assert self.epsilon > 0, "Epsilon must be greater than 0."

        if not skip_check:
            # 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.
            _util.check(transitions, reward)

        self.S, self.A = _computeDimensions(transitions)
        self.P = self._computeTransition(transitions)
        self.R = self._computeReward(reward, transitions)

        # 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
        # set the initial iteration count to zero
        self.iter = 0
        # V should be stored as a vector ie shape of (S,) or (1, S)
        self.V = None
        # policy can also be stored as a vector
        self.policy = None

    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"
        return(P_repr + "\n" + R_repr)

    def _bellmanOperator(self, V=None):
        # Apply the Bellman operator on the value function.
        #
        # Updates the value function and the Vprev-improving policy.
        #
        # 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
        if V is None:
            # this V should be a reference to the data rather than a copy
            V = self.V
        else:
            # make sure the user supplied V is of the right shape
            try:
                assert V.shape in ((self.S,), (1, self.S)), "V is not the " \
                    "right shape (Bellman operator)."
            except AttributeError:
                raise TypeError("V must be a numpy array or matrix.")
        # 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.
        Q = _np.empty((self.A, self.S))
        for aa in range(self.A):
            Q[aa] = self.R[aa] + self.discount * self.P[aa].dot(V)
        # Get the policy and value, for now it is being returned but...
        # Which way is better?
        # 1. Return, (policy, value)
        return (Q.argmax(axis=0), Q.max(axis=0))
        # 2. update self.policy and self.V directly
        # self.V = Q.max(axis=1)
        # self.policy = Q.argmax(axis=1)

    def _computeTransition(self, transition):
        return tuple(transition[a] for a in range(self.A))

    def _computeReward(self, reward, transition):
        # Compute the reward for the system in one state chosing an action.
        # Arguments
        # Let S = number of states, A = number of actions
        # P could be an array with 3 dimensions or  a cell array (1xA),
        # each cell containing a matrix (SxS) possibly sparse
        # 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
        try:
            if reward.ndim == 1:
                return self._computeVectorReward(reward)
            elif reward.ndim == 2:
                return self._computeArrayReward(reward)
            else:
                r = tuple(map(self._computeMatrixReward, reward, transition))
                return r
        except (AttributeError, ValueError):
            if len(reward) == self.A:
                r = tuple(map(self._computeMatrixReward, reward, transition))
                return r
            else:
                return self._computeVectorReward(reward)

    def _computeVectorReward(self, reward):
        if _sp.issparse(reward):
            raise NotImplementedError
        else:
            r = _np.array(reward).reshape(self.S)
            return tuple(r for a in range(self.A))

    def _computeArrayReward(self, reward):
        if _sp.issparse(reward):
            raise NotImplementedError
        else:
            def func(x):
                return _np.array(x).reshape(self.S)

            return tuple(func(reward[:, a]) for a in range(self.A))

    def _computeMatrixReward(self, reward, transition):
        if _sp.issparse(reward):
            # An approach like this might be more memory efficeint
            # reward.data = reward.data * transition[reward.nonzero()]
            # return reward.sum(1).A.reshape(self.S)
            # but doesn't work as it is.
            return reward.multiply(transition).sum(1).A.reshape(self.S)
        elif _sp.issparse(transition):
            return transition.multiply(reward).sum(1).A.reshape(self.S)
        else:
            return _np.multiply(transition, reward).sum(1).reshape(self.S)

    def _startRun(self):
        if self.verbose:
            _printVerbosity('Iteration', 'Variation')

        self.time = _time.time()

    def _endRun(self):
        # store value and policy as tuples
        self.V = tuple(self.V.tolist())

        try:
            self.policy = tuple(self.policy.tolist())
        except AttributeError:
            self.policy = tuple(self.policy)

        self.time = _time.time() - self.time

    def run(self):
        """Raises error because child classes should implement this function.
        """
        raise NotImplementedError("You should create a run() method.")

    def setSilent(self):
        """Set the MDP algorithm to silent mode."""
        self.verbose = False

    def setVerbose(self):
        """Set the MDP algorithm to verbose mode."""
        self.verbose = True



class QLearning(MDP):

    """A discounted MDP solved using the Q learning algorithm.

    Parameters
    ----------
    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.
    alpha : float
        Learning rate.
    epsilon : float
        Probability of selecting a random action.
    decay : float
        Epsilon decay rate.     
    n_iter : int, optional
        Number of iterations to execute. This is ignored unless it is an
        integer greater than the default value. Defaut: 10,000.
    skip_check : bool
        By default we run a check on the ``transitions`` and ``rewards``
        arguments to make sure they describe a valid MDP. You can set this
        argument to True in order to skip this check.

    Data Attributes
    ---------------
    Q : array
        learned Q matrix (SxA)
    V : tuple
        learned value function (S).
    policy : tuple
        learned optimal policy (S).
    mean_discrepancy : array
        Vector of V discrepancy mean over 100 iterations. Then the length of
        this vector for the default value of N is 100 (N/100).
    """

    def __init__(self, transitions, reward, discount, alpha, epsilon = 0.5, 
                 decay = 1.0, n_iter=10000, skip_check=False):
        # Initialise a Q-learning MDP.

        # The following check won't be done in MDP()'s initialisation, so let's
        # do it here
        self.max_iter = int(n_iter)
        #assert self.max_iter >= 10000, "'n_iter' should be greater than 10000."

        if not skip_check:
            # We don't want to send this to MDP because _computePR should not
            #  be run on it, so check that it defines an MDP
            _util.check(transitions, reward)

        # Store P, S, and A
        self.S, self.A = _computeDimensions(transitions)
        self.P = self._computeTransition(transitions)

        self.R = reward
        self.discount = discount
        
        self.alpha = alpha
        self.epsilon = epsilon
        self.decay = decay

        # Initialisations
        self.Q = _np.zeros((self.S, self.A))
        self.mean_discrepancy = []

    def run(self):
        # Run the Q-learning algoritm.
        discrepancy = []

        self.time = _time.time()

        # initial state choice
        s = _np.random.randint(0, self.S)

        for n in range(1, self.max_iter + 1):

            # Reinitialisation of trajectories every 100 transitions
            if (n % 100) == 0:
                s = _np.random.randint(0, self.S)

            # Action choice 
            pn = _np.random.random()
            if pn < (1 - self.epsilon):
                # optimal_action = self.Q[s, :].max()
                a = self.Q[s, :].argmax()
            else:
                a = _np.random.randint(0, self.A)
            
            self.epsilon *= self.decay
            
            # Simulating next state s_new and reward associated to <s,s_new,a>
            p_s_new = _np.random.random()
            p = 0
            s_new = -1
            while (p < p_s_new) and (s_new < (self.S - 1)):
                s_new = s_new + 1
                p = p + self.P[a][s, s_new]

            try:
                r = self.R[a][s, s_new]
            except IndexError:
                try:
                    r = self.R[s, a]
                except IndexError:
                    r = self.R[s]

            # Updating the value of Q
            # Decaying update coefficient (1/sqrt(n+2)) can be changed
            delta = r + self.discount * self.Q[s_new, :].max() - self.Q[s, a]

            dQ = self.alpha * delta
            self.Q[s, a] = self.Q[s, a] + dQ

            # current state is updated
            s = s_new
            
            # Computing and saving maximal values of the Q variation
            discrepancy.append(_np.absolute(dQ))

            # Computing means all over maximal Q variations values
            if len(discrepancy) == 1000:
                self.mean_discrepancy.append(_np.mean(discrepancy))
                discrepancy = []

            # compute the value function and the policy
            self.V = self.Q.max(axis=1)
            self.policy = self.Q.argmax(axis=1)
            
        self._endRun()