Optimistic Planning for Markov Decision Processes

International audienceThe reinforcement learning community has recently intensified its interest in online planning methods, due to their relative independence on the state space size. However, tight near-optimality guarantees are not yet available for the general case of stochastic Markov decision processes and closed-loop, state-dependent planning policies. We therefore consider an algorithm related to AO* that optimistically explores a tree representation of the space of closed-loop policies, and we analyze the near-optimality of the action it returns after n tree node expansions. While this optimistic planning requires a finite number of actions and possible next states for each transition, its asymptotic performance does not depend directly on these numbers, but only on the subset of nodes that significantly impact near-optimal policies. We characterize this set by introducing a novel measure of problem complexity, called the near-optimality exponent. Specializing the exponent and performance bound for some interesting classes of MDPs illustrates the algorithm works better when there are fewer near-optimal policies and less uniform transition probabilities

Optimistic Planning for Markov Decision Processes

International audienceThe reinforcement learning community has recently intensified its interest in online planning methods, due to their relative independence on the state space size. However, tight near-optimality guarantees are not yet available for the general case of stochastic Markov decision processes and closed-loop, state-dependent planning policies. We therefore consider an algorithm related to AO* that optimistically explores a tree representation of the space of closed-loop policies, and we analyze the near-optimality of the action it returns after n tree node expansions. While this optimistic planning requires a finite number of actions and possible next states for each transition, its asymptotic performance does not depend directly on these numbers, but only on the subset of nodes that significantly impact near-optimal policies. We characterize this set by introducing a novel measure of problem complexity, called the near-optimality exponent. Specializing the exponent and performance bound for some interesting classes of MDPs illustrates the algorithm works better when there are fewer near-optimal policies and less uniform transition probabilities

Planification Optimiste dans les Processus Décisionnels de Markov avec Croyance

Cet article décrit l'algorithme BOP (de l'anglais ``Bayesian Optimistic Planning''), un nouvel algorithme d'apprentissage par renforcement Bayésien indirect (c'est à dire fondé sur un modèle). BOP étend l'approche de l'algorithme OP-MDP (de l'anglais ``Optimistic Planning for Markov Decision Processes'', voir [Busoniu2011,Busoniu2012]) au cas où les probabilités de transitions du MDP sous-jacent sont initialement inconnues, et doivent être apprises au travers d'interactions avec l'environnement. Les connaissances sur le MDP sous-jacent sont représentées par une distribution de probabilités sur l'ensemble de tous les modèles de transitions à l'aide de distributions de Dirichlet. L'algorithme BOP planifie dans l'espace augmenté état-croyance obtenu par concaténation du vecteur d'état avec la distribution postérieure sur les modèles de transitions. On montre que BOP atteint l'optimalité Bayésienne lorsque le paramètre de budget tend vers l'infini. Quelques expériences préliminaires montrent des résultats encourageants.Peer reviewe

Beyond the One Step Greedy Approach in Reinforcement Learning

The famous Policy Iteration algorithm alternates between policy improvement and policy evaluation. Implementations of this algorithm with several variants of the latter evaluation stage, e.g, $n$-step and trace-based returns, have been analyzed in previous works. However, the case of multiple-step lookahead policy improvement, despite the recent increase in empirical evidence of its strength, has to our knowledge not been carefully analyzed yet. In this work, we introduce the first such analysis. Namely, we formulate variants of multiple-step policy improvement, derive new algorithms using these definitions and prove their convergence. Moreover, we show that recent prominent Reinforcement Learning algorithms are, in fact, instances of our framework. We thus shed light on their empirical success and give a recipe for deriving new algorithms for future study.Comment: ICML 201

Simple Regret Optimization in Online Planning for Markov Decision Processes

We consider online planning in Markov decision processes (MDPs). In online planning, the agent focuses on its current state only, deliberates about the set of possible policies from that state onwards and, when interrupted, uses the outcome of that exploratory deliberation to choose what action to perform next. The performance of algorithms for online planning is assessed in terms of simple regret, which is the agent's expected performance loss when the chosen action, rather than an optimal one, is followed. To date, state-of-the-art algorithms for online planning in general MDPs are either best effort, or guarantee only polynomial-rate reduction of simple regret over time. Here we introduce a new Monte-Carlo tree search algorithm, BRUE, that guarantees exponential-rate reduction of simple regret and error probability. This algorithm is based on a simple yet non-standard state-space sampling scheme, MCTS2e, in which different parts of each sample are dedicated to different exploratory objectives. Our empirical evaluation shows that BRUE not only provides superior performance guarantees, but is also very effective in practice and favorably compares to state-of-the-art. We then extend BRUE with a variant of "learning by forgetting." The resulting set of algorithms, BRUE(alpha), generalizes BRUE, improves the exponential factor in the upper bound on its reduction rate, and exhibits even more attractive empirical performance

Optimistic minimax search for noncooperative switched control with or without dwell time

International audienceWe consider adversarial problems in which two agents control two switching signals, the first agent aiming to maximize a discounted sum of rewards, and the second aiming to minimize it. Both signals may be subject to constraints on the dwell time after a switch. We search the tree of possible mode sequences with an algorithm called optimistic minimax search with dwell time (OMSd), showing that it obtains a solution close to the minimax-optimal one, and we characterize the rate at which the suboptimality goes to zero. The analysis is driven by a novel measure of problem complexity, and it is first given in the general dwell-time case, after which it is specialized to the unconstrained case. We exemplify the framework for networked control systems where the minimizer signal is a discrete time delay on the control channel, and we provide extensive simulations and a real-time experiment for nonlinear systems of this type

Planning in Markov Decision Processes with Gap-Dependent Sample Complexity

We propose MDP-GapE, a new trajectory-based Monte-Carlo Tree Search algorithm for planning in a Markov Decision Process in which transitions have a finite support. We prove an upper bound on the number of calls to the generative models needed for MDP-GapE to identify a near-optimal action with high probability. This problem-dependent sample complexity result is expressed in terms of the sub-optimality gaps of the state-action pairs that are visited during exploration. Our experiments reveal that MDP-GapE is also effective in practice, in contrast with other algorithms with sample complexity guarantees in the fixed-confidence setting, that are mostly theoretical

Optimistic planning for continuous–action deterministic systems.

Abstract : We consider the optimal control of systems with deterministic dynamics, continuous, possibly large-scale state spaces, and continuous, low-dimensional action spaces. We describe an online planning algorithm called SOOP, which like other algorithms in its class has no direct dependence on the state space structure. Unlike previous algorithms, SOOP explores the true solution space, consisting of infinite sequences of continuous actions, without requiring knowledge about the smoothness of the system. To this end, it borrows the principle of the simultaneous optimistic optimization method, and develops a nontrivial adaptation of this principle to the planning problem. Experiments on four problems show SOOP reliably ranks among the best algorithms, fully dominating competing methods when the problem requires both long horizons and fine discretization

Monte-Carlo Graph Search: the Value of Merging Similar States

International audienceWe consider the problem of planning in a Markov Decision Process (MDP) with a generative model and limited computational budget. Despite the underlying MDP transitions having a graph structure, the popular Monte-Carlo Tree Search algorithms such as UCT rely on a tree structure to represent their value estimates. That is, they do not identify together two similar states reached via different trajectories and represented in separate branches of the tree. In this work, we propose a graph-based planning algorithm, which takes into account this state similarity. In our analysis, we provide a regret bound that depends on a novel problem-dependent measure of difficulty, which improves on the original tree-based bound in MDPs where the trajectories overlap, and recovers it otherwise. Then, we show that this methodology can be adapted to existing planning algorithms that deal with stochastic systems. Finally, numerical simulations illustrate the benefits of our approach