Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games

Regret minimization is a powerful tool for solving large-scale extensive-form games. State-of-the-art methods rely on minimizing regret locally at each decision point. In this work we derive a new framework for regret minimization on sequential decision problems and extensive-form games with general compact convex sets at each decision point and general convex losses, as opposed to prior work which has been for simplex decision points and linear losses. We call our framework laminar regret decomposition. It generalizes the CFR algorithm to this more general setting. Furthermore, our framework enables a new proof of CFR even in the known setting, which is derived from a perspective of decomposing polytope regret, thereby leading to an arguably simpler interpretation of the algorithm. Our generalization to convex compact sets and convex losses allows us to develop new algorithms for several problems: regularized sequential decision making, regularized Nash equilibria in extensive-form games, and computing approximate extensive-form perfect equilibria. Our generalization also leads to the first regret-minimization algorithm for computing reduced-normal-form quantal response equilibria based on minimizing local regrets. Experiments show that our framework leads to algorithms that scale at a rate comparable to the fastest variants of counterfactual regret minimization for computing Nash equilibrium, and therefore our approach leads to the first algorithm for computing quantal response equilibria in extremely large games. Finally we show that our framework enables a new kind of scalable opponent exploitation approach

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