# -*- coding: utf-8 -*-
"""Markov Decision Process (MDP) Toolbox
=====================================
The MDP toolbox provides classes and functions for the resolution of
descrete-time Markov Decision Processes.
Available classes
-----------------
MDP
Base Markov decision process class
FiniteHorizon
Backwards induction finite horizon MDP
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
Available functions
-------------------
check
Check that an MDP is properly defined
checkSquareStochastic
Check that a matrix is square and stochastic
exampleForest
A simple forest management example
exampleRand
A random example
How to use the documentation
----------------------------
Documentation is available both as docstrings provided with the code and
in html or pdf format from
`The MDP toolbox homepage `_. The docstring
examples assume that the `mdp` module has been imported::
>>> import mdp
Code snippets are indicated by three greater-than signs::
>>> x = 17
>>> x = x + 1
>>> x
18
The documentation can be displayed with
`IPython `_. For example, to view the docstring of
the ValueIteration class use ``mdp.ValueIteration?``, and to view its
source code use ``mdp.ValueIteration??``.
Acknowledgments
---------------
This module is modified from the MDPtoolbox (c) 2009 INRA available at
http://www.inra.fr/mia/T/MDPtoolbox/.
"""
# Copyright (c) 2011-2013 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 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.
from math import ceil, log, sqrt
from random import random
from time import time
from numpy import absolute, array, diag, empty, mean, mod, multiply
from numpy import ndarray, ones, where, zeros
from numpy.random import rand, randint
from scipy.sparse import csr_matrix as sparse
from scipy.sparse import coo_matrix, dok_matrix
# __all__ = ["check", "checkSquareStochastic"]
# These need to be fixed so that we use classes derived from Error.
mdperr = {
"mat_nonneg" :
"PyMDPtoolbox: Probabilities must be non-negative.",
"mat_square" :
"PyMDPtoolbox: The matrix must be square.",
"mat_stoch" :
"PyMDPtoolbox: Rows of the matrix must sum to one (1).",
"mask_numpy" :
"PyMDPtoolbox: mask must be a numpy array or matrix; i.e. type(mask) is "
"ndarray or type(mask) is matrix.",
"mask_SbyS" :
"PyMDPtoolbox: The mask must have shape SxS; i.e. mask.shape = (S, S).",
"obj_shape" :
"PyMDPtoolbox: Object arrays for transition probabilities and rewards "
"must have only 1 dimension: the number of actions A. Each element of "
"the object array contains an SxS ndarray or matrix.",
"obj_square" :
"PyMDPtoolbox: Each element of an object array for transition "
"probabilities and rewards must contain an SxS ndarray or matrix; i.e. "
"P[a].shape = (S, S) or R[a].shape = (S, S).",
"P_type" :
"PyMDPtoolbox: The transition probabilities must be in a numpy array; "
"i.e. type(P) is ndarray.",
"P_shape" :
"PyMDPtoolbox: The transition probability array must have the shape "
"(A, S, S) with S : number of states greater than 0 and A : number of "
"actions greater than 0. i.e. R.shape = (A, S, S)",
"PR_incompat" :
"PyMDPtoolbox: Incompatibility between P and R dimensions.",
"prob_in01" :
"PyMDPtoolbox: Probability p must be in [0; 1].",
"R_type" :
"PyMDPtoolbox: The rewards must be in a numpy array; i.e. type(R) is "
"ndarray, or numpy matrix; i.e. type(R) is matrix.",
"R_shape" :
"PyMDPtoolbox: The reward matrix R must be an array of shape (A, S, S) or "
"(S, A) with S : number of states greater than 0 and A : number of "
"actions greater than 0. i.e. R.shape = (S, A) or (A, S, S).",
"R_gt_0" :
"PyMDPtoolbox: The rewards must be greater than 0.",
"S_gt_1" :
"PyMDPtoolbox: Number of states S must be greater than 1.",
"SA_gt_1" :
"PyMDPtoolbox: The number of states S and the number of actions A must be "
"greater than 1.",
"discount_rng" :
"PyMDPtoolbox: Discount rate must be in ]0; 1]",
"maxi_min" :
"PyMDPtoolbox: The maximum number of iterations must be greater than 0"
}
def check(P, R):
"""Check if P and R define a Markov Decision Process.
Let S = number of states, A = number of actions.
Parameters
---------
P : array
The transition matrices. It can be a three dimensional array with
a shape of (A, S, S). It can also be a one dimensional arraye with
a shape of (A, ), where each element contains a matrix of shape (S, S)
which can possibly be sparse.
R : array
The reward matrix. It can be a three dimensional array with a
shape of (S, A, A). It can also be a one dimensional array with a
shape of (A, ), where each element contains matrix with a shape of
(S, S) which can possibly be sparse. It can also be an array with
a shape of (S, A) which can possibly be sparse.
Notes
-----
Raises an error if P and R do not define a MDP.
"""
# Checking P
try:
if P.ndim == 3:
aP, sP0, sP1 = P.shape
elif P.ndim == 1:
# A hack so that we can go into the next try-except statement and
# continue checking from there
raise AttributeError
else:
raise ValueError(mdperr["P_shape"])
except AttributeError:
try:
aP = len(P)
sP0, sP1 = P[0].shape
for aa in xrange(1, aP):
sP0aa, sP1aa = P[aa].shape
if (sP0aa != sP0) or (sP1aa != sP1):
raise ValueError(mdperr["obj_square"])
except AttributeError:
raise TypeError(mdperr["P_shape"])
except:
raise
# Checking R
try:
if R.ndim == 2:
sR0, aR = R.shape
sR1 = sR0
elif R.ndim == 3:
aR, sR0, sR1 = R.shape
elif R.ndim == 1:
# A hack so that we can go into the next try-except statement
raise AttributeError
else:
raise ValueError(mdperr["R_shape"])
except AttributeError:
try:
aR = len(R)
sR0, sR1 = R[0].shape
for aa in range(1, aR):
sR0aa, sR1aa = R[aa].shape
if ((sR0aa != sR0) or (sR1aa != sR1)):
raise ValueError(mdperr["obj_square"])
except AttributeError:
raise ValueError(mdperr["R_shape"])
except:
raise
# Checking dimensions
if (sP0 < 1) or (aP < 1) or (sP0 != sP1):
raise ValueError(mdperr["P_shape"])
if (sR0 < 1) or (aR < 1) or (sR0 != sR1):
raise ValueError(mdperr["R_shape"])
if (sP0 != sR0) or (aP != aR):
raise ValueError(mdperr["PR_incompat"])
# Check that the P's are square and stochastic
for aa in xrange(aP):
checkSquareStochastic(P[aa])
#checkSquareStochastic(P[aa, :, :])
# We are at the end of the checks, so if no exceptions have been raised
# then that means there are (hopefullly) no errors and we return None
return None
# These are the old code comments, which need to be converted to
# information in the docstring:
#
# tranitions must be a numpy array either an AxSxS ndarray (with any
# dtype other than "object"); or, a 1xA ndarray with a "object" dtype,
# and each element containing an SxS array. An AxSxS array will be
# be converted to an object array. A numpy object array is similar to a
# MATLAB cell array.
#
# NumPy has an array type of 'object', which is roughly equivalent to
# the MATLAB cell array. These are most useful for storing sparse
# matrices as these can only have two dimensions whereas we want to be
# able to store a transition matrix for each action. If the dytpe of
# the transition probability array is object then we store this as
# P_is_object = True.
# If it is an object array, then it should only have one dimension
# otherwise fail with a message expalining why.
# If it is a normal array then the number of dimensions must be exactly
# three, otherwise fail with a message explaining why.
#
# As above but for the reward array. A difference is that the reward
# array can have either two or 3 dimensions.
#
# We want to make sure that the transition probability array and the
# reward array are in agreement. This means that both should show that
# there are the same number of actions and the same number of states.
# Furthermore the probability of transition matrices must be SxS in
# shape, so we check for that also.
#
# If the user has put their transition matrices into a numpy array
# with dtype of 'object', then it is possible that they have made a
# mistake and not all of the matrices are of the same shape. So,
# here we record the number of actions and states that the first
# matrix in element zero of the object array says it has. After
# that we check that every other matrix also reports the same
# number of actions and states, otherwise fail with an error.
# aP: the number of actions in the transition array. This
# corresponds to the number of elements in the object array.
#
# sP0: the number of states as reported by the number of rows of
# the transition matrix
# sP1: the number of states as reported by the number of columns of
# the transition matrix
#
# Now we check to see that every element of the object array holds
# a matrix of the same shape, otherwise fail.
#
# sp0aa and sp1aa represents the number of states in each
# subsequent element of the object array. If it doesn't match
# what was found in the first element, then we need to fail
# telling the user what needs to be fixed.
#
# if we are using a normal array for this, then the first
# dimension should be the number of actions, and the second and
# third should be the number of states
#
# the first dimension of the transition matrix must report the same
# number of states as the second dimension. If not then we are not
# dealing with a square matrix and it is not a valid transition
# probability. Also, if the number of actions is less than one, or the
# number of states is less than one, then it also is not a valid
# transition probability.
#
# now we check that each transition matrix is square-stochastic. For
# object arrays this is the matrix held in each element, but for
# normal arrays this is a matrix formed by taking a slice of the array
#
# if the rewarad array has an object dtype, then we check that
# each element contains a matrix of the same shape as we did
# above with the transition array.
#
# This indicates that the reward matrices are constructed per
# transition, so that the first dimension is the actions and
# the second two dimensions are the states.
#
# then the reward matrix is per state, so the first dimension is
# the states and the second dimension is the actions.
#
# this is added just so that the next check doesn't error out
# saying that sR1 doesn't exist
#
# the number of actions must be more than zero, the number of states
# must also be more than 0, and the states must agree
#
# now we check to see that what the transition array is reporting and
# what the reward arrar is reporting agree as to the number of actions
# and states. If not then fail explaining the situation
def checkSquareStochastic(Z):
"""Check if Z is a square stochastic matrix.
Let S = number of states.
Parameters
----------
Z : matrix
This should be a two dimensional array with a shape of (S, S). It can
possibly be sparse.
Notes
----------
Returns None if no error has been detected, else it raises an error.
"""
# try to get the shape of the matrix
try:
s1, s2 = Z.shape
except AttributeError:
raise TypeError("Matrix should be a numpy type.")
except ValueError:
raise ValueError(mdperr["mat_square"])
# check that the matrix is square, and that each row sums to one
if s1 != s2:
raise ValueError(mdperr["mat_square"])
elif (absolute(Z.sum(axis=1) - ones(s2))).max() > 10e-12:
raise ValueError(mdperr["mat_stoch"])
# make sure that there are no values less than zero
try:
if (Z < 0).any():
raise ValueError(mdperr["mat_nonneg"])
except AttributeError:
try:
if (Z.data < 0).any():
raise ValueError(mdperr["mat_nonneg"])
except AttributeError:
raise TypeError("Matrix should be a numpy type.")
except:
raise
return(None)
def exampleForest(S=3, r1=4, r2=2, p=0.1, is_sparse=False):
"""Generate a MDP example based on a simple forest management scenario.
This function is used to generate a transition probability
(``A`` × ``S`` × ``S``) array ``P`` and a reward (``S`` × ``A``) matrix
``R`` that model the following problem. A forest is managed by two actions:
'Wait' and 'Cut'. An action is decided each year with first the objective
to maintain an old forest for wildlife and second to make money selling cut
wood. Each year there is a probability ``p`` that a fire burns the forest.
Here is how the problem is modelled.
Let {1, 2 . . . ``S`` } be the states of the forest, with ``S`` being the
oldest. Let 'Wait' be action 1 and 'Cut' action 2.
After a fire, the forest is in the youngest state, that is state 1.
The transition matrix P of the problem can then be defined as follows::
| p 1-p 0.......0 |
| . 0 1-p 0....0 |
P[1,:,:] = | . . 0 . |
| . . . |
| . . 1-p |
| p 0 0....0 1-p |
| 1 0..........0 |
| . . . |
P[2,:,:] = | . . . |
| . . . |
| . . . |
| 1 0..........0 |
The reward matrix R is defined as follows::
| 0 |
| . |
R[:,1] = | . |
| . |
| 0 |
| r1 |
| 0 |
| 1 |
R[:,2] = | . |
| . |
| 1 |
| r2 |
Parameters
---------
S : int, optional
The number of states, which should be an integer greater than 0. By
default it is 3.
r1 : float, optional
The reward when the forest is in its oldest state and action 'Wait' is
performed. By default it is 4.
r2 : float, optional
The reward when the forest is in its oldest state and action 'Cut' is
performed. By default it is 2.
p : float, optional
The probability of wild fire occurence, in the range ]0, 1[. By default
it is 0.1.
Returns
-------
out : tuple
``out[1]`` contains the transition probability matrix P with a shape of
(A, S, S). ``out[2]`` contains the reward matrix R with a shape of
(S, A).
Examples
--------
>>> import mdp
>>> P, R = mdp.exampleForest()
>>> P
array([[[ 0.1, 0.9, 0. ],
[ 0.1, 0. , 0.9],
[ 0.1, 0. , 0.9]],
[[ 1. , 0. , 0. ],
[ 1. , 0. , 0. ],
[ 1. , 0. , 0. ]]])
>>> R
array([[ 0., 0.],
[ 0., 1.],
[ 4., 2.]])
"""
if S <= 1:
raise ValueError(mdperr["S_gt_1"])
if (r1 <= 0) or (r2 <= 0):
raise ValueError(mdperr["R_gt_0"])
if (p < 0) or (p > 1):
raise ValueError(mdperr["prob_in01"])
# Definition of Transition matrix P(:,:,1) associated to action Wait
# (action 1) and P(:,:,2) associated to action Cut (action 2)
# | p 1-p 0.......0 | | 1 0..........0 |
# | . 0 1-p 0....0 | | . . . |
# P(:,:,1) = | . . 0 . | and P(:,:,2) = | . . . |
# | . . . | | . . . |
# | . . 1-p | | . . . |
# | p 0 0....0 1-p | | 1 0..........0 |
if is_sparse:
P = []
rows = range(S) * 2
cols = [0] * S + range(1, S) + [S - 1]
vals = [p] * S + [1-p] * S
P.append(coo_matrix((vals, (rows, cols)), shape=(S,S)).tocsr())
rows = range(S)
cols = [0] * S
vals = [1] * S
P.append(coo_matrix((vals, (rows, cols)), shape=(S,S)).tocsr())
else:
P = zeros((2, S, S))
P[0, :, :] = (1 - p) * diag(ones(S - 1), 1)
P[0, :, 0] = p
P[0, S - 1, S - 1] = (1 - p)
P[1, :, :] = zeros((S, S))
P[1, :, 0] = 1
# Definition of Reward matrix R1 associated to action Wait and
# R2 associated to action Cut
# | 0 | | 0 |
# | . | | 1 |
# R(:,1) = | . | and R(:,2) = | . |
# | . | | . |
# | 0 | | 1 |
# | r1 | | r2 |
R = zeros((S, 2))
R[S - 1, 0] = r1
R[:, 1] = ones(S)
R[0, 1] = 0
R[S - 1, 1] = r2
# we want to return the generated transition and reward matrices
return (P, R)
def exampleRand(S, A, is_sparse=False, mask=None):
"""Generate a random Markov Decision Process.
Parameters
----------
S : int
number of states (> 0)
A : int
number of actions (> 0)
is_sparse : logical, optional
false to have matrices in dense format, true to have sparse
matrices (default false).
mask : array or None, optional
matrix with 0 and 1 (0 indicates a place for a zero
probability), (SxS) (default, random)
Returns
-------
out : tuple
``out[1]`` contains the transition probability matrix P with a shape of
(A, S, S). ``out[2]`` contains the reward matrix R with a shape of
(S, A).
Examples
--------
>>> import mdp
>>> P, R = mdp.exampleRand(5, 3)
"""
# making sure the states and actions are more than one
if (S < 1) or (A < 1):
raise ValueError(mdperr["SA_gt_1"])
# if the user hasn't specified a mask, then we will make a random one now
if mask is not None:
# the mask needs to be SxS or AxSxS
try:
if mask.shape not in ((S, S), (A, S, S)):
raise ValueError(mdperr["mask_SbyS"])
except AttributeError:
raise TypeError(mdperr["mask_numpy"])
# generate the transition and reward matrices based on S, A and mask
if is_sparse:
# definition of transition matrix : square stochastic matrix
P = [None] * A
# definition of reward matrix (values between -1 and +1)
R = [None] * A
for a in xrange(A):
# it may be more efficient to implement this by constructing lists
# of rows, columns and values then creating a coo_matrix, but this
# works for now
PP = dok_matrix((S, S))
RR = dok_matrix((S, S))
for s in xrange(S):
if mask is None:
m = rand(S)
m[m <= 2/3.0] = 0
m[m > 2/3.0] = 1
elif mask.shape == (A, S, S):
m = mask[a][s] # mask[a, s, :]
else:
m = mask[s]
n = int(m.sum()) # m[s, :]
if n == 0:
PP[s, randint(0, S)] = 1
else:
rows = s * ones(n, dtype=int)
cols = where(m)[0] # m[s, :]
vals = rand(n)
vals = vals / vals.sum()
reward = 2*rand(n) - ones(n)
# I want to do this: PP[rows, cols] = vals, but it doesn't
# seem to work, as val[-1] is stored as the value for each
# row, column pair. Therefore the loop is needed.
for x in xrange(n):
PP[rows[x], cols[x]] = vals[x]
RR[rows[x], cols[x]] = reward[x]
# PP.tocsr() takes the same amount of time as PP.tocoo().tocsr()
# so constructing PP and RR as coo_matrix in the first place is
# probably "better"
P[a] = PP.tocsr()
R[a] = RR.tocsr()
else:
# definition of transition matrix : square stochastic matrix
P = zeros((A, S, S))
# definition of reward matrix (values between -1 and +1)
R = zeros((A, S, S))
for a in range(A):
for s in range(S):
# create our own random mask if there is no user supplied one
if mask is None:
m = rand(S)
r = random()
m[m <= r] = 0
m[m > r] = 1
elif mask.shape == (A, S, S):
m = mask[a][s] # mask[a, s, :]
else:
m = mask[s]
# Make sure that there is atleast one transition in each state
if m.sum() == 0:
P[a, s, randint(0, S)] = 1
else:
P[a][s] = m * rand(S)
P[a][s] = P[a][s] / P[a][s].sum()
R[a][s] = (m * (2*rand(S) - ones(S, dtype=int)))
# we want to return the generated transition and reward matrices
return (P, R)
def getSpan(W):
"""Return the span of W
sp(W) = max W(s) - min W(s)
"""
return (W.max() - W.min())
class MDP(object):
"""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
"""
def __init__(self, transitions, reward, discount, epsilon, max_iter):
"""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 type(discount) in (int, float):
if (discount <= 0) or (discount > 1):
raise ValueError(mdperr["discount_rng"])
else:
if discount == 1:
print("PyMDPtoolbox WARNING: check conditions of "
"convergence. With no discount, convergence is not "
"always assumed.")
self.discount = discount
elif discount is not None:
raise ValueError("PyMDPtoolbox: the discount must be a positive "
"real number less than or equal to one.")
# if the max_iter is None then the algorithm is assumed to not use it
# in its computations
if type(max_iter) in (int, float):
if max_iter <= 0:
raise ValueError(mdperr["maxi_min"])
else:
self.max_iter = max_iter
elif max_iter is not None:
raise ValueError("PyMDPtoolbox: max_iter must be a positive real "
"number greater than zero.")
if type(epsilon) in (int, float):
if epsilon <= 0:
raise ValueError("PyMDPtoolbox: epsilon must be greater than "
"0.")
elif epsilon is not None:
raise ValueError("PyMDPtoolbox: epsilon must be a positive real "
"number greater than zero.")
# 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
self._computePR(transitions, reward)
# 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 _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 is None:
# this V should be a reference to the data rather than a copy
V = self.V
else:
try:
if V.shape not in ((self.S,), (1, self.S)):
raise ValueError("bellman: V is not the right shape.")
except AttributeError:
raise TypeError("bellman: V must be a numpy array or matrix.")
Q = empty((self.A, self.S))
for aa in range(self.A):
Q[aa] = self.R[aa] + self.discount * self.P[aa].dot(V)
# 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 _computePR(self, P, R):
"""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
"""
# We assume that P and R define a MDP i,e. assumption is that
# check(P, R) has already been run and doesn't fail.
#
# 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]
except:
raise
# convert Ps to matrices
self.P = []
for aa in xrange(self.A):
self.P.append(P[aa])
self.P = tuple(self.P)
# Set self.R as a tuple of length A, with each element storing an 1×S
# vector.
try:
if R.ndim == 2:
self.R = []
for aa in xrange(self.A):
self.R.append(array(R[:, aa]).reshape(self.S))
else:
raise AttributeError
except AttributeError:
self.R = []
for aa in xrange(self.A):
try:
self.R.append(P[aa].multiply(R[aa]).sum(1).reshape(self.S))
except AttributeError:
self.R.append(multiply(P[aa],R[aa]).sum(1).reshape(self.S))
except:
raise
except:
raise
self.R = tuple(self.R)
def _iterate(self):
"""Raise error because child classes should implement this function."""
raise NotImplementedError("You should create an _iterate() 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 FiniteHorizon(MDP):
"""A MDP solved using the finite-horizon backwards induction algorithm.
Let S = number of states, A = number of actions
Parameters
----------
P(SxSxA) = transition matrix
P could be an array with 3 dimensions ora 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 factor, in ]0, 1]
N = number of periods, upper than 0
h(S) = terminal reward, optional (default [0; 0; ... 0] )
Attributes
----------
Methods
-------
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.
Examples
--------
>>> import mdp
>>> P, R = mdp.exampleForest()
>>> fh = mdp.FiniteHorizon(P, R, 0.9, 3)
>>> 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]])
"""
def __init__(self, transitions, reward, discount, N, h=None):
"""Initialise a finite horizon MDP."""
if N < 1:
raise ValueError('PyMDPtoolbox: N must be greater than 0')
else:
self.N = N
# Initialise the base class
MDP.__init__(self, transitions, reward, discount, None, None)
# remove the iteration counter, it is not meaningful for backwards
# induction
del self.iter
# There are value vectors for each time step up to the horizon
self.V = zeros((self.S, N + 1))
# 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:
self.V[:, N] = h
# Call the iteration method
self._iterate()
def _iterate(self):
"""Run the finite horizon algorithm."""
self.time = time()
for n in range(self.N):
W, X = self._bellmanOperator(self.V[:, self.N - n])
self.V[:, self.N - n - 1] = X
self.policy[:, self.N - n - 1] = W
if self.verbose:
print("stage: %s ... policy transpose : %s") % (
self.N - n, self.policy[:, self.N - n -1].tolist())
self.time = time() - self.time
# 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)
class LP(MDP):
"""A discounted MDP soloved using linear programming.
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[
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
--------
"""
def __init__(self, transitions, reward, discount):
"""Initialise a linear programming MDP."""
try:
from cvxopt import matrix, solvers
self._linprog = solvers.lp
self._cvxmat = matrix
except ImportError:
raise ImportError("The python module cvxopt is required to use "
"linear programming functionality.")
from scipy.sparse import eye as speye
self._speye = speye
MDP.__init__(self, transitions, reward, discount, None, None)
# this doesn't do what I want it to do c.f. issue #3
if not self.verbose:
solvers.options['show_progress'] = False
# Call the iteration method
self._iterate()
def _iterate(self):
"""Run the linear programming algorithm."""
self.time = time()
# The objective is to resolve : min V / V >= PR + discount*P*V
# The function linprog of the optimisation Toolbox of Mathworks
# resolves :
# min f'* x / M * x <= b
# 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)
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))
for aa in range(self.A):
pos = (aa + 1) * self.S
M[(pos - self.S):pos, :] = (
self.discount * self.P[aa] - self._speye(self.S, self.S))
M = self._cvxmat(M)
# 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
# only to 10e-8 places.
self.V = array(self._linprog(f, M, -h, solver='glpk')['x'])
self.policy, self.V = self._bellmanOperator()
self.time = time() - self.time
# store value and policy as tuples
self.V = tuple(self.V.tolist())
self.policy = tuple(self.policy.tolist())
class PolicyIteration(MDP):
"""A discounted MDP solved using the policy iteration algorithm.
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[
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
Examples
--------
>>> import mdp
>>> P, R = mdp.exampleRand(5, 3)
>>> pi = mdp.PolicyIteration(P, R, 0.9)
"""
def __init__(self, transitions, reward, discount, policy0=None,
max_iter=1000, eval_type=0):
"""Initialise a policy iteration MDP."""
MDP.__init__(self, transitions, reward, discount, None, max_iter)
if policy0 == None:
# initialise the policy to the one which maximises the expected
# immediate reward
self.V = zeros(self.S)
self.policy, null = self._bellmanOperator()
del null
else:
policy0 = array(policy0)
if not policy0.shape in ((self.S, ), (self.S, 1), (1, self.S)):
raise ValueError("PyMDPtolbox: policy0 must a vector with "
"length S.")
policy0 = policy0.reshape(self.S)
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.")
else:
self.policy = policy0
# set or reset the initial values to zero
self.V = zeros(self.S)
if eval_type in (0, "matrix"):
from numpy.linalg import solve
from scipy.sparse import eye
self._speye = eye
self._lin_eq = solve
self.eval_type = "matrix"
elif eval_type in (1, "iterative"):
self.eval_type = "iterative"
else:
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.")
# Call the iteration method
self._iterate()
def _computePpolicyPRpolicy(self):
"""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
"""
Ppolicy = empty((self.S, self.S))
Rpolicy = zeros(self.S)
for aa in range(self.A): # avoid looping over S
# the rows that use action a.
ind = (self.policy == aa).nonzero()[0]
# if no rows use action a, then no need to assign this
if ind.size > 0:
Ppolicy[ind, :] = self.P[aa][ind, :]
#PR = self._computePR() # an apparently uneeded line, and
# perhaps harmful in this implementation c.f.
# mdp_computePpolicyPRpolicy.m
Rpolicy[ind] = self.R[aa][ind]
# 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)
def _evalPolicyIterative(self, V0=0, epsilon=0.0001, max_iter=10000):
"""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.
"""
if (type(V0) in (int, float)) and (V0 == 0):
policy_V = zeros(self.S)
else:
if (type(V0) in (ndarray)) and (V0.shape == (self.S, 1)):
policy_V = V0
else:
raise ValueError("PyMDPtoolbox: V0 vector/array type not "
"supported. Use ndarray of matrix column "
"vector length S.")
policy_P, policy_R = self._computePpolicyPRpolicy()
if self.verbose:
print(' Iteration V_variation')
itr = 0
done = False
while not done:
itr += 1
Vprev = policy_V
policy_V = policy_R + self.discount * policy_P.dot(Vprev)
variation = absolute(policy_V - Vprev).max()
if self.verbose:
print(' %s %s') % (itr, variation)
# ensure |Vn - Vpolicy| < epsilon
if variation < ((1 - self.discount) / self.discount) * epsilon:
done = True
if self.verbose:
print("PyMDPtoolbox: iterations stopped, epsilon-optimal "
"value function.")
elif itr == max_iter:
done = True
if self.verbose:
print("PyMDPtoolbox: iterations stopped by maximum number "
"of iteration condition.")
self.V = policy_V
def _evalPolicyMatrix(self):
"""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
"""
Ppolicy, Rpolicy = self._computePpolicyPRpolicy()
# V = PR + gPV => (I-gP)V = PR => V = inv(I-gP)* PR
self.V = self._lin_eq(
(self._speye(self.S, self.S) - self.discount * Ppolicy), Rpolicy)
def _iterate(self):
"""Run the policy iteration algorithm."""
if self.verbose:
print(' Iteration Number_of_different_actions')
done = False
self.time = time()
while not done:
self.iter += 1
# these _evalPolicy* functions will update the classes value
# attribute
if self.eval_type == "matrix":
self._evalPolicyMatrix()
elif self.eval_type == "iterative":
self._evalPolicyIterative()
# This should update the classes policy attribute but leave the
# value alone
policy_next, null = self._bellmanOperator()
del null
n_different = (policy_next != self.policy).sum()
if self.verbose:
print(' %s %s') % (self.iter,
n_different)
if n_different == 0:
done = True
if self.verbose:
print("PyMDPtoolbox: iterations stopped, unchanging "
"policy found.")
elif (self.iter == self.max_iter):
done = True
if self.verbose:
print("PyMDPtoolbox: iterations stopped by maximum number "
"of iteration condition.")
else:
self.policy = policy_next
self.time = time() - self.time
# store value and policy as tuples
self.V = tuple(self.V.tolist())
self.policy = tuple(self.policy.tolist())
class PolicyIterationModified(PolicyIteration):
"""A discounted MDP solved using a modifified policy iteration algorithm.
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[
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
--------
>>> import mdp
"""
def __init__(self, transitions, reward, discount, epsilon=0.01,
max_iter=10):
"""Initialise a (modified) policy iteration MDP."""
# 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
# is needed from the PolicyIteration class is the _evalPolicyIterative
# function. Perhaps there is a better way to do it?
PolicyIteration.__init__(self, transitions, reward, discount, None,
max_iter, 1)
# PolicyIteration doesn't pass epsilon to MDP.__init__() so we will
# check it here
if type(epsilon) in (int, float):
if epsilon <= 0:
raise ValueError("PyMDPtoolbox: epsilon must be greater than "
"0.")
else:
raise ValueError("PyMDPtoolbox: epsilon must be a positive real "
"number greater than zero.")
# computation of threshold of variation for V for an epsilon-optimal
# policy
if self.discount != 1:
self.thresh = epsilon * (1 - self.discount) / self.discount
else:
self.thresh = epsilon
self.epsilon = epsilon
if discount == 1:
self.V = zeros((self.S, 1))
else:
# min(min()) is not right
self.V = 1 / (1 - discount) * self.R.min() * ones((self.S, 1))
# Call the iteration method
self._iterate()
def _iterate(self):
"""Run the modified policy iteration algorithm."""
if self.verbose:
print(' Iteration V_variation')
self.time = time()
done = False
while not done:
self.iter += 1
self.policy, Vnext = self._bellmanOperator()
#[Ppolicy, PRpolicy] = mdp_computePpolicyPRpolicy(P, PR, policy);
variation = getSpan(Vnext - self.V)
if self.verbose:
print(" %s %s" % (self.iter, variation))
self.V = Vnext
if variation < self.thresh:
done = True
else:
is_verbose = False
if self.verbose:
self.setSilent()
is_verbose = True
self._evalPolicyIterative(self.V, self.epsilon, self.max_iter)
if is_verbose:
self.setVerbose()
self.time = time() - self.time
# store value and policy as tuples
self.V = tuple(self.V.tolist())
self.policy = tuple(self.policy.tolist())
class QLearning(MDP):
"""A discounted MDP solved using the Q learning algorithm.
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).
Default value = 10000; it is an integer greater than the default
value.
Results
-------
Q : learned Q matrix (SxA)
V : learned value function (S).
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).
Examples
---------
>>> import random # this example is reproducible only if random seed is set
>>> import mdp
>>> random.seed(0)
>>> P, R = mdp.exampleForest()
>>> ql = mdp.QLearning(P, R, 0.96)
>>> ql.Q
array([[ 68.80977389, 46.62560314],
[ 72.58265749, 43.1170545 ],
[ 77.1332834 , 65.01737419]])
>>> ql.V
(68.80977388561172, 72.5826574913828, 77.13328339600116)
>>> ql.policy
(0, 0, 0)
>>> import random # this example is reproducible only if random seed is set
>>> import mdp
>>> 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]])
>>> random.seed(0)
>>> ql = mdp.QLearning(P, R, 0.9)
>>> ql.Q
array([[ 36.63245946, 42.24434307],
[ 35.96582807, 32.70456417]])
>>> ql.V
(42.24434307022128, 35.96582807367007)
>>> ql.policy
(1, 0)
"""
def __init__(self, transitions, reward, discount, n_iter=10000):
"""Initialise a Q-learning MDP."""
# The following check won't be done in MDP()'s initialisation, so let's
# do it here
if (n_iter < 10000):
raise ValueError("PyMDPtoolbox: n_iter should be greater than "
"10000.")
# We don't want to send this to MDP because _computePR should not be
# run on it
# MDP.__init__(self, transitions, reward, discount, None, n_iter)
check(transitions, reward)
if (transitions.dtype is object):
self.P = transitions
self.A = self.P.shape[0]
self.S = self.P[0].shape[0]
else: # convert to an object array
self.A = transitions.shape[0]
self.S = transitions.shape[1]
self.P = zeros(self.A, dtype=object)
for aa in range(self.A):
self.P[aa] = transitions[aa, :, :]
self.R = reward
self.discount = discount
self.max_iter = n_iter
# Initialisations
self.Q = zeros((self.S, self.A))
self.mean_discrepancy = []
# Call the iteration method
self._iterate()
def _iterate(self):
"""Run the Q-learning algoritm."""
discrepancy = []
self.time = time()
# initial state choice
s = randint(0, self.S)
for n in range(1, self.max_iter + 1):
# Reinitialisation of trajectories every 100 transitions
if ((n % 100) == 0):
s = randint(0, self.S)
# Action choice : greedy with increasing probability
# probability 1-(1/log(n+2)) can be changed
pn = random()
if (pn < (1 - (1 / log(n + 2)))):
# optimal_action = self.Q[s, :].max()
a = self.Q[s, :].argmax()
else:
a = randint(0, self.A)
# Simulating next state s_new and reward associated to ~~
p_s_new = 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]
if (self.R.dtype == object):
r = self.R[a][s, s_new]
elif (self.R.ndim == 3):
r = self.R[a, s, s_new]
else:
r = self.R[s, a]
# 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 = (1 / sqrt(n + 2)) * 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(absolute(dQ))
# Computing means all over maximal Q variations values
if len(discrepancy) == 100:
self.mean_discrepancy.append(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.time = time() - self.time
# convert V and policy to tuples
self.V = tuple(self.V.tolist())
self.policy = tuple(self.policy.tolist())
class RelativeValueIteration(MDP):
"""A MDP solved using the relative value iteration algorithm.
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
epsilon = epsilon-optimal policy search, upper than 0,
optional (default: 0.01)
max_iter = maximum number of iteration to be done, upper than 0,
optional (default 1000)
Evaluation
----------
policy(S) = epsilon-optimal policy
average_reward = average reward of the optimal policy
cpu_time = used CPU time
Notes
-----
In verbose mode, at each iteration, displays the span of U variation
and the condition which stopped iterations : epsilon-optimum policy found
or maximum number of iterations reached.
Examples
--------
>>> import mdp
>>> P, R = exampleForest()
>>> rvi = mdp.RelativeValueIteration(P, R)
>>> rvi.average_reward
2.4300000000000002
>>> rvi.policy
(0, 0, 0)
>>> rvi.iter
4
>>> import mdp
>>> 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]])
>>> rvi = mdp.RelativeValueIteration(P, R)
>>> rvi.V
(10.0, 3.885235246411831)
>>> rvi.average_reward
3.8852352464118312
>>> rvi.policy
(1, 0)
>>> rvi.iter
29
"""
def __init__(self, transitions, reward, epsilon=0.01, max_iter=1000):
"""Initialise a relative value iteration MDP."""
MDP.__init__(self, transitions, reward, None, epsilon, max_iter)
self.epsilon = epsilon
self.discount = 1
self.V = zeros(self.S)
self.gain = 0 # self.U[self.S]
self.average_reward = None
# Call the iteration method
self._iterate()
def _iterate(self):
"""Run the relative value iteration algorithm."""
done = False
if self.verbose:
print(' Iteration U_variation')
self.time = time()
while not done:
self.iter += 1;
self.policy, Vnext = self._bellmanOperator()
Vnext = Vnext - self.gain
variation = getSpan(Vnext - self.V)
if self.verbose:
print(" %s %s" % (self.iter, variation))
if variation < self.epsilon:
done = True
self.average_reward = self.gain + (Vnext - self.V).min()
if self.verbose:
print("MDP Toolbox : iterations stopped, epsilon-optimal "
"policy found.")
elif self.iter == self.max_iter:
done = True
self.average_reward = self.gain + (Vnext - self.V).min()
if self.verbose:
print("MDP Toolbox : iterations stopped by maximum "
"number of iteration condition.")
self.V = Vnext
self.gain = float(self.V[self.S - 1])
self.time = time() - self.time
# store value and policy as tuples
self.V = tuple(self.V.tolist())
self.policy = tuple(self.policy.tolist())
class ValueIteration(MDP):
"""A discounted MDP solved using the value iteration algorithm.
Description
-----------
mdp.ValueIteration applies the value iteration algorithm to solve
discounted MDP. The algorithm consists in solving Bellman's equation
iteratively.
Iterating is stopped when an epsilon-optimal policy is found or after a
specified number (max_iter) of iterations.
This function uses verbose and silent modes. In verbose mode, the function
displays the variation of V (value function) for each iteration and the
condition which stopped iterations: epsilon-policy found or maximum number
of iterations reached.
Let S = number of states, A = number of actions.
Parameters
----------
P : array
transition matrix
P could be a numpy ndarray with 3 dimensions (AxSxS) or a
numpy ndarray of dytpe=object with 1 dimenion (1xA), each
element containing a numpy ndarray (SxS) or scipy sparse matrix.
R : array
reward matrix
R could be a numpy ndarray with 3 dimensions (AxSxS) or numpy
ndarray of dtype=object with 1 dimension (1xA), each element
containing a sparse matrix (SxS). R also could be a numpy
ndarray with 2 dimensions (SxA) possibly sparse.
discount : float
discount rate
Greater than 0, less than or equal to 1. Beware to check conditions of
convergence for discount = 1.
epsilon : float, optional
epsilon-optimal policy search
Greater than 0, optional (default: 0.01).
max_iter : int, optional
maximum number of iterations to be done
Greater than 0, optional (default: computed)
initial_value : array, optional
starting value function
optional (default: zeros(S,)).
Data Attributes
---------------
V : value function
A vector which stores the optimal value function. Prior to calling the
_iterate() method it has a value of None. Shape is (S, ).
policy : epsilon-optimal policy
A vector which stores the optimal policy. Prior to calling the
_iterate() method it has a value of None. Shape is (S, ).
iter : number of iterations taken to complete the computation
An integer
time : used CPU time
A float
Methods
-------
setSilent()
Sets the instance to silent mode.
setVerbose()
Sets the instance to verbose mode.
Notes
-----
In verbose mode, at each iteration, displays the variation of V
and the condition which stopped iterations: epsilon-optimum policy found
or maximum number of iterations reached.
Examples
--------
>>> import mdp
>>> P, R = mdp.exampleForest()
>>> vi = mdp.ValueIteration(P, R, 0.96)
>>> vi.verbose
False
>>> vi.V
(5.93215488, 9.38815488, 13.38815488)
>>> vi.policy
(0, 0, 0)
>>> vi.iter
4
>>> vi.time
0.0009911060333251953
>>> import mdp
>>> 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]])
>>> vi = mdp.ValueIteration(P, R, 0.9)
>>> vi.V
(40.04862539271682, 33.65371175967546)
>>> vi.policy
(1, 0)
>>> vi.iter
26
>>> vi.time
0.0066509246826171875
>>> import mdp
>>> import numpy as np
>>> from scipy.sparse import csr_matrix as sparse
>>> P = [None] * 2
>>> P[0] = sparse([[0.5, 0.5],[0.8, 0.2]])
>>> P[1] = sparse([[0, 1],[0.1, 0.9]])
>>> R = np.array([[5, 10], [-1, 2]])
>>> vi = mdp.ValueIteration(P, R, 0.9)
>>> vi.V
(40.04862539271682, 33.65371175967546)
>>> vi.policy
(1, 0)
"""
def __init__(self, transitions, reward, discount, epsilon=0.01,
max_iter=1000, initial_value=0):
"""Initialise a value iteration MDP."""
MDP.__init__(self, transitions, reward, discount, epsilon, max_iter)
# initialization of optional arguments
if initial_value == 0:
self.V = zeros(self.S)
else:
if len(initial_value) != self.S:
raise ValueError("PyMDPtoolbox: The initial value must be "
"a vector of length S.")
else:
try:
self.V = initial_value.reshape(self.S)
except AttributeError:
self.V = array(initial_value)
except:
raise
if self.discount < 1:
# compute a bound for the number of iterations and update the
# stored value of self.max_iter
self._boundIter(epsilon)
# computation of threshold of variation for V for an epsilon-
# optimal policy
self.thresh = epsilon * (1 - self.discount) / self.discount
else: # discount == 1
# threshold of variation for V for an epsilon-optimal policy
self.thresh = epsilon
# Call the iteration method
self._iterate()
def _boundIter(self, epsilon):
"""Compute a bound for the number of iterations.
for the value iteration
algorithm to find an epsilon-optimal policy with use of span for the
stopping criterion
Arguments -------------------------------------------------------------
Let S = number of states, A = number of actions
epsilon = |V - V*| < epsilon, upper than 0,
optional (default : 0.01)
Evaluation ------------------------------------------------------------
max_iter = bound of the number of iterations for the value
iteration algorithm to find an epsilon-optimal policy with use of
span for the stopping criterion
cpu_time = used CPU time
"""
# See Markov Decision Processes, M. L. Puterman,
# Wiley-Interscience Publication, 1994
# p 202, Theorem 6.6.6
# k = max [1 - S min[ P(j|s,a), p(j|s',a')] ]
# s,a,s',a' j
k = 0
h = zeros(self.S)
for ss in range(self.S):
PP = zeros((self.A, self.S))
for aa in range(self.A):
try:
PP[aa] = self.P[aa][:, ss]
except ValueError:
PP[aa] = self.P[aa][:, ss].todense().A1
except:
raise
# the function "min()" without any arguments finds the
# minimum of the entire array.
h[ss] = PP.min()
k = 1 - h.sum()
Vprev = self.V
null, value = self._bellmanOperator()
# p 201, Proposition 6.6.5
max_iter = (log((epsilon * (1 - self.discount) / self.discount) /
getSpan(value - Vprev) ) / log(self.discount * k))
#self.V = Vprev
self.max_iter = int(ceil(max_iter))
def _iterate(self):
"""Run the value iteration algorithm."""
if self.verbose:
print(' Iteration V_variation')
self.time = time()
done = False
while not done:
self.iter += 1
Vprev = self.V.copy()
# Bellman Operator: compute policy and value functions
self.policy, self.V = self._bellmanOperator()
# The values, based on Q. For the function "max()": the option
# "axis" means the axis along which to operate. In this case it
# finds the maximum of the the rows. (Operates along the columns?)
variation = getSpan(self.V - Vprev)
if self.verbose:
print(" %s %s" % (self.iter, variation))
if variation < self.thresh:
done = True
if self.verbose:
print("...iterations stopped, epsilon-optimal policy "
"found.")
elif (self.iter == self.max_iter):
done = True
if self.verbose:
print("...iterations stopped by maximum number of "
"iteration condition.")
# store value and policy as tuples
self.V = tuple(self.V.tolist())
self.policy = tuple(self.policy.tolist())
self.time = time() - self.time
class ValueIterationGS(ValueIteration):
"""
A discounted MDP solved using the value iteration Gauss-Seidel algorithm.
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]
beware to check conditions of convergence for discount = 1.
epsilon = epsilon-optimal policy search, upper than 0,
optional (default : 0.01)
max_iter = maximum number of iteration to be done, upper than 0,
optional (default : computed)
V0(S) = starting value function, optional (default : zeros(S,1))
Evaluation
----------
policy(S) = epsilon-optimal policy
iter = number of done iterations
cpu_time = used CPU time
Notes
-----
In verbose mode, at each iteration, displays the variation of V
and the condition which stopped iterations: epsilon-optimum policy found
or maximum number of iterations reached.
Examples
--------
"""
def __init__(self, transitions, reward, discount, epsilon=0.01,
max_iter=10, initial_value=0):
"""Initialise a value iteration Gauss-Seidel MDP."""
ValueIteration.__init__(self, transitions, reward, discount, epsilon,
max_iter, initial_value)
# Call the iteration method
self._iterate()
def _iterate(self):
"""Run the value iteration Gauss-Seidel algorithm."""
done = False
if self.verbose:
print(' Iteration V_variation')
self.time = time()
while not done:
self.iter += 1
Vprev = self.V.copy()
for s in range(self.S):
Q = [float(self.R[a][s]+
self.discount * self.P[a][s, :].dot(self.V))
for a in range(self.A)]
self.V[s] = max(Q)
variation = getSpan(self.V - Vprev)
if self.verbose:
print(" %s %s" % (self.iter, variation))
if variation < self.thresh:
done = True
if self.verbose:
print("MDP Toolbox : iterations stopped, epsilon-optimal "
"policy found.")
elif self.iter == self.max_iter:
done = True
if self.verbose:
print("MDP Toolbox : iterations stopped by maximum number "
"of iteration condition.")
self.policy = []
for s in range(self.S):
Q = zeros(self.A)
for a in range(self.A):
Q[a] = self.R[a][s] + self.discount * self.P[a][s,:].dot(self.V)
self.V[s] = Q.max()
self.policy.append(int(Q.argmax()))
self.time = time() - self.time
self.V = tuple(self.V.tolist())
self.policy = tuple(self.policy)
if __name__ == "__main__":
import doctest
doctest.testmod()
~~