On the Performance Bounds of some Policy Search Dynamic Programming Algorithms

We consider the infinite-horizon discounted optimal control problem formalized by Markov Decision Processes. We focus on Policy Search algorithms, that compute an approximately optimal policy by following the standard Policy Iteration (PI) scheme via an -approximate greedy operator (Kakade and Langford, 2002; Lazaric et al., 2010). We describe existing and a few new performance bounds for Direct Policy Iteration (DPI) (Lagoudakis and Parr, 2003; Fern et al., 2006; Lazaric et al., 2010) and Conservative Policy Iteration (CPI) (Kakade and Langford, 2002). By paying a particular attention to the concentrability constants involved in such guarantees, we notably argue that the guarantee of CPI is much better than that of DPI, but this comes at the cost of a relative--exponential in $\frac{1}{\epsilon}$-- increase of time complexity. We then describe an algorithm, Non-Stationary Direct Policy Iteration (NSDPI), that can either be seen as 1) a variation of Policy Search by Dynamic Programming by Bagnell et al. (2003) to the infinite horizon situation or 2) a simplified version of the Non-Stationary PI with growing period of Scherrer and Lesner (2012). We provide an analysis of this algorithm, that shows in particular that it enjoys the best of both worlds: its performance guarantee is similar to that of CPI, but within a time complexity similar to that of DPI

Multi-Objective Approaches to Markov Decision Processes with Uncertain Transition Parameters

Markov decision processes (MDPs) are a popular model for performance analysis and optimization of stochastic systems. The parameters of stochastic behavior of MDPs are estimates from empirical observations of a system; their values are not known precisely. Different types of MDPs with uncertain, imprecise or bounded transition rates or probabilities and rewards exist in the literature. Commonly, analysis of models with uncertainties amounts to searching for the most robust policy which means that the goal is to generate a policy with the greatest lower bound on performance (or, symmetrically, the lowest upper bound on costs). However, hedging against an unlikely worst case may lead to losses in other situations. In general, one is interested in policies that behave well in all situations which results in a multi-objective view on decision making. In this paper, we consider policies for the expected discounted reward measure of MDPs with uncertain parameters. In particular, the approach is defined for bounded-parameter MDPs (BMDPs) [8]. In this setting the worst, best and average case performances of a policy are analyzed simultaneously, which yields a multi-scenario multi-objective optimization problem. The paper presents and evaluates approaches to compute the pure Pareto optimal policies in the value vector space.Comment: 9 pages, 5 figures, preprint for VALUETOOLS 201

Nonapproximability Results for Partially Observable Markov Decision Processes

We show that for several variations of partially observable Markov decision processes, polynomial-time algorithms for finding control policies are unlikely to or simply don't have guarantees of finding policies within a constant factor or a constant summand of optimal. Here "unlikely" means "unless some complexity classes collapse," where the collapses considered are P=NP, P=PSPACE, or P=EXP. Until or unless these collapses are shown to hold, any control-policy designer must choose between such performance guarantees and efficient computation

Perseus: Randomized Point-based Value Iteration for POMDPs

Partially observable Markov decision processes (POMDPs) form an attractive and principled framework for agent planning under uncertainty. Point-based approximate techniques for POMDPs compute a policy based on a finite set of points collected in advance from the agents belief space. We present a randomized point-based value iteration algorithm called Perseus. The algorithm performs approximate value backup stages, ensuring that in each backup stage the value of each point in the belief set is improved; the key observation is that a single backup may improve the value of many belief points. Contrary to other point-based methods, Perseus backs up only a (randomly selected) subset of points in the belief set, sufficient for improving the value of each belief point in the set. We show how the same idea can be extended to dealing with continuous action spaces. Experimental results show the potential of Perseus in large scale POMDP problems

Multigrid methods for two-player zero-sum stochastic games

We present a fast numerical algorithm for large scale zero-sum stochastic games with perfect information, which combines policy iteration and algebraic multigrid methods. This algorithm can be applied either to a true finite state space zero-sum two player game or to the discretization of an Isaacs equation. We present numerical tests on discretizations of Isaacs equations or variational inequalities. We also present a full multi-level policy iteration, similar to FMG, which allows to improve substantially the computation time for solving some variational inequalities.Comment: 31 page

The Stochastic Shortest Path Problem : A polyhedral combinatorics perspective

In this paper, we give a new framework for the stochastic shortest path problem in finite state and action spaces. Our framework generalizes both the frameworks proposed by Bertsekas and Tsitsikli and by Bertsekas and Yu. We prove that the problem is well-defined and (weakly) polynomial when (i) there is a way to reach the target state from any initial state and (ii) there is no transition cycle of negative costs (a generalization of negative cost cycles). These assumptions generalize the standard assumptions for the deterministic shortest path problem and our framework encapsulates the latter problem (in contrast with prior works). In this new setting, we can show that (a) one can restrict to deterministic and stationary policies, (b) the problem is still (weakly) polynomial through linear programming, (c) Value Iteration and Policy Iteration converge, and (d) we can extend Dijkstra's algorithm