On Local Regret

Online learning aims to perform nearly as well as the best hypothesis in hindsight. For some hypothesis classes, though, even finding the best hypothesis offline is challenging. In such offline cases, local search techniques are often employed and only local optimality guaranteed. For online decision-making with such hypothesis classes, we introduce local regret, a generalization of regret that aims to perform nearly as well as only nearby hypotheses. We then present a general algorithm to minimize local regret with arbitrary locality graphs. We also show how the graph structure can be exploited to drastically speed learning. These algorithms are then demonstrated on a diverse set of online problems: online disjunct learning, online Max-SAT, and online decision tree learning.Comment: This is the longer version of the same-titled paper appearing in the Proceedings of the Twenty-Ninth International Conference on Machine Learning (ICML), 201

Efficient Regret Minimization in Non-Convex Games

We consider regret minimization in repeated games with non-convex loss functions. Minimizing the standard notion of regret is computationally intractable. Thus, we define a natural notion of regret which permits efficient optimization and generalizes offline guarantees for convergence to an approximate local optimum. We give gradient-based methods that achieve optimal regret, which in turn guarantee convergence to equilibrium in this framework.Comment: Published as a conference paper at ICML 201

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