# LEARNING ALGORITHMS FOR MARKOV DECISION PROCESSES

##### Abstract

We propose various computational schemes for solving Partially Observable
Markov Decision Processes with the finite stage additive cost and infinite
horizon discounted cost criterion. Error bounds for the corresponding algorithms
are given and it is further shown that at the expense of more computational
effort the Partially Observable Markov Decision Problem (POMDP) can be solved
as closely to the optimal as desired.
It is well known that a sufficient statistic for taking the best action at any time for
the POMDP is the aposteriori probability distribution on the underlying states, given
all the past history, and that this can be updated recursively. We prove that the finite
stage optimal costs as well as the optimal cost for the infinite horizon discounted
cost problem are both Lipschitz continuous (with domain the unit simplex of probability
distributions over the underlying states) and gives bounds for the Lipschitz constant.
We use these bounds to provide error bounds for computational algorithms for solving
POMDPs.
We extend the almost sure convergence result of a very general stochastic approximation
algorithm to the case when the underlying Markov process exhibits periodicity. This result
is used to extend the proof of convergence of Temporal Difference (TD) reinforcement learning
schemes with linear function approximation for Markov Cost processes in order to estimate the
cost to go function for the discounted cost criterion, and the differential cost function for the
average cost criterion, respectively.
Adaptive control of Markov Decision Problems (MDPs) is a problem in which a full knowledge
of the system parameters, namely transition probabilities as well as the distribution of the
immediate costs, are not available apriori. We give direct adaptive control schemes for
infinite horizon discounted cost and average cost MDPs. Approximate Policy Iteration
using on-line TD schemes for policy evaluation is detailed for the discounted cost and
average cost criteria.
Possible extensions of direct adaptive control schemes to the POMDP framework are
discussed.
Auxiliary results relevant to the core results of the dissertation are stated
and proved in the appendices. In particular an efficient discretization scheme
for the finite dimensional unit simplex is given. Some general error bounds for
MDPs are also given. Also TD schemes for learning in Stochastic Shortest Path
problems (SSP) are discussed.

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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