Q-Learning and Enhanced Policy Iteration in Discounted Dynamic Programming

We consider the classical nite-state discounted Markovian decision problem, and we introduce a new policy iteration-like algorithm for fi nding the optimal Q-factors. Instead of policy evaluation by solving a linear system of equations, our algorithm requires (possibly inexact) solution of a nonlinear system of equations, involving estimates of state costs as well as Q-factors. This is Bellman's equation for an optimal stopping problem that can be solved with simple Q-learning iterations, in the case where a lookup table representation is used; it can also be solved with the Q-learning algorithm of Tsitsiklis and Van Roy [TsV99], in the case where feature-based Q-factor approximations are used. In exact/lookup table representation form, our algorithm admits asynchronous and stochastic iterative implementations, in the spirit of asynchronous/modi ed policy iteration, with lower overhead and more reliable convergence advantages over existing Q-learning schemes. Furthermore, for large-scale problems, where linear basis function approximations and simulation-based temporal di erence implementations are used, our algorithm resolves e ffectively the inherent difficulties of existing schemes due to inadequate exploration

Q-Learning and Enhanced Policy Iteration in Discounted Dynamic Programming (Revised)

The revised technical report C-2010-10We consider the classical finite-state discounted Markovian decision problem, and we introduce a new policy iteration-like algorithm for finding the optimal Q-factors. Instead of policy evaluation by solving a linear system of equations, our algorithm requires (possibly inexact) solution of a nonlinear system of equations, involving estimates of state costs as well as Q-factors. This is Bellman's equation for an optimal stopping problem that can be solved with simple Q-learning iterations, in the case where a lookup table representation is used; it can also be solved with the Q-learning algorithm of Tsitsiklis and Van Roy [TsV99], in the case where feature-based Q-factor approximations are used. In exact/lookup table representation form, our algorithm admits asynchronous and stochastic iterative implementations, in the spirit of asynchronous/modified policy iteration, with lower overhead and/or more reliable convergence advantages over existing Q-learning schemes. Furthermore, for large-scale problems, where linear basis function approximations and simulation-based temporal difference implementations are used, our algorithm resolves effectively the inherent difficulties of existing schemes due to inadequate exploration

On the Convergence of Techniques that Improve Value Iteration

Prioritisation of Bellman backups or updating only a small subset of actions represent important techniques for speeding up planning in MDPs. The recent literature showed new efficient approaches which exploit these directions. Backward value iteration and backing up only the best actions were shown to lead to a significant reduction of the planning time. This paper conducts a theoretical and empirical analysis of these techniques and shows new important proofs. In particular, (1) it identifies weaker requirements for the convergence of backups based on best actions only, (2) a new method for evaluation of the Bellman error is shown for the update that updates one best action once, (3) it presents the theoretical proof of backward value iteration and establishes required initialisation, (4) and shows that the default state ordering of backups in standard value iteration can significantly influence its performance. Additionally, (5) the existing literature did not compare these methods, either empirically or analytically, against policy iteration. The rigorous empirical and novel theoretical parts of the paper reveal important associations and allow drawing guidelines on which type of value or policy iteration is suitable for a given domain. Finally, our chief message is that standard value iteration can be made far more efficient by simple modifications shown in the paper

Tight Performance Bounds for Approximate Modified Policy Iteration with Non-Stationary Policies

We consider approximate dynamic programming for the infinite-horizon stationary $\gamma$-discounted optimal control problem formalized by Markov Decision Processes. While in the exact case it is known that there always exists an optimal policy that is stationary, we show that when using value function approximation, looking for a non-stationary policy may lead to a better performance guarantee. We define a non-stationary variant of MPI that unifies a broad family of approximate DP algorithms of the literature. For this algorithm we provide an error propagation analysis in the form of a performance bound of the resulting policies that can improve the usual performance bound by a factor $O(1-\gamma)$, which is significant when the discount factor $\gamma$ is close to 1. Doing so, our approach unifies recent results for Value and Policy Iteration. Furthermore, we show, by constructing a specific deterministic MDP, that our performance guarantee is tight