Theoretical and Practical Advances on Smoothing for Extensive-Form Games

Sparse iterative methods, in particular first-order methods, are known to be among the most effective in solving large-scale two-player zero-sum extensive-form games. The convergence rates of these methods depend heavily on the properties of the distance-generating function that they are based on. We investigate the acceleration of first-order methods for solving extensive-form games through better design of the dilated entropy function---a class of distance-generating functions related to the domains associated with the extensive-form games. By introducing a new weighting scheme for the dilated entropy function, we develop the first distance-generating function for the strategy spaces of sequential games that has no dependence on the branching factor of the player. This result improves the convergence rate of several first-order methods by a factor of $\Omega(b^dd)$, where $b$ is the branching factor of the player, and $d$ is the depth of the game tree. Thus far, counterfactual regret minimization methods have been faster in practice, and more popular, than first-order methods despite their theoretically inferior convergence rates. Using our new weighting scheme and practical tuning we show that, for the first time, the excessive gap technique can be made faster than the fastest counterfactual regret minimization algorithm, CFR+, in practice

Solving Large Extensive-Form Games with Strategy Constraints

Extensive-form games are a common model for multiagent interactions with imperfect information. In two-player zero-sum games, the typical solution concept is a Nash equilibrium over the unconstrained strategy set for each player. In many situations, however, we would like to constrain the set of possible strategies. For example, constraints are a natural way to model limited resources, risk mitigation, safety, consistency with past observations of behavior, or other secondary objectives for an agent. In small games, optimal strategies under linear constraints can be found by solving a linear program; however, state-of-the-art algorithms for solving large games cannot handle general constraints. In this work we introduce a generalized form of Counterfactual Regret Minimization that provably finds optimal strategies under any feasible set of convex constraints. We demonstrate the effectiveness of our algorithm for finding strategies that mitigate risk in security games, and for opponent modeling in poker games when given only partial observations of private information.Comment: Appeared in AAAI 201

Smoothing Method for Approximate Extensive-Form Perfect Equilibrium

Nash equilibrium is a popular solution concept for solving imperfect-information games in practice. However, it has a major drawback: it does not preclude suboptimal play in branches of the game tree that are not reached in equilibrium. Equilibrium refinements can mend this issue, but have experienced little practical adoption. This is largely due to a lack of scalable algorithms. Sparse iterative methods, in particular first-order methods, are known to be among the most effective algorithms for computing Nash equilibria in large-scale two-player zero-sum extensive-form games. In this paper, we provide, to our knowledge, the first extension of these methods to equilibrium refinements. We develop a smoothing approach for behavioral perturbations of the convex polytope that encompasses the strategy spaces of players in an extensive-form game. This enables one to compute an approximate variant of extensive-form perfect equilibria. Experiments show that our smoothing approach leads to solutions with dramatically stronger strategies at information sets that are reached with low probability in approximate Nash equilibria, while retaining the overall convergence rate associated with fast algorithms for Nash equilibrium. This has benefits both in approximate equilibrium finding (such approximation is necessary in practice in large games) where some probabilities are low while possibly heading toward zero in the limit, and exact equilibrium computation where the low probabilities are actually zero.Comment: Published at IJCAI 1

A Unified View of Large-scale Zero-sum Equilibrium Computation

The task of computing approximate Nash equilibria in large zero-sum extensive-form games has received a tremendous amount of attention due mainly to the Annual Computer Poker Competition. Immediately after its inception, two competing and seemingly different approaches emerged---one an application of no-regret online learning, the other a sophisticated gradient method applied to a convex-concave saddle-point formulation. Since then, both approaches have grown in relative isolation with advancements on one side not effecting the other. In this paper, we rectify this by dissecting and, in a sense, unify the two views.Comment: AAAI Workshop on Computer Poker and Imperfect Informatio

Scalable First-Order Methods for Robust MDPs

Robust Markov Decision Processes (MDPs) are a powerful framework for modeling sequential decision-making problems with model uncertainty. This paper proposes the first first-order framework for solving robust MDPs. Our algorithm interleaves primal-dual first-order updates with approximate Value Iteration updates. By carefully controlling the tradeoff between the accuracy and cost of Value Iteration updates, we achieve an ergodic convergence rate of $O \left( A^{2} S^{3}\log(S)\log(\epsilon^{-1}) \epsilon^{-1} \right)$ for the best choice of parameters on ellipsoidal and Kullback-Leibler $s$-rectangular uncertainty sets, where $S$ and $A$ is the number of states and actions, respectively. Our dependence on the number of states and actions is significantly better (by a factor of $O(A^{1.5}S^{1.5})$) than that of pure Value Iteration algorithms. In numerical experiments on ellipsoidal uncertainty sets we show that our algorithm is significantly more scalable than state-of-the-art approaches. Our framework is also the first one to solve robust MDPs with $s$-rectangular KL uncertainty sets