Online Learning and Solving Infinite Games with an ERM Oracle

TL;DR

This paper introduces online learning algorithms for infinite games that rely solely on ERM or best response oracles, achieving finite or sublinear regret and convergence to equilibrium in various game settings.

Contribution

It presents new online algorithms using only ERM or best response oracles, with regret bounds and equilibrium convergence guarantees for infinite and large games.

Findings

Finite regret in realizable online classification

Sublinear regret in agnostic setting

Convergence to approximate equilibria in large games

Abstract

While ERM suffices to attain near-optimal generalization error in the stochastic learning setting, this is not known to be the case in the online learning setting, where algorithms for general concept classes rely on computationally inefficient oracles such as the Standard Optimal Algorithm (SOA). In this work, we propose an algorithm for online binary classification setting that relies solely on ERM oracle calls, and show that it has finite regret in the realizable setting and sublinearly growing regret in the agnostic setting. We bound the regret in terms of the Littlestone and threshold dimensions of the underlying concept class. We obtain similar results for nonparametric games, where the ERM oracle can be interpreted as a best response oracle, finding the best response of a player to a given history of play of the other players. In this setting, we provide learning algorithms that only rely on best response oracles and converge to approximate-minimax equilibria in two-player zero-sum games and approximate coarse correlated equilibria in multi-player general-sum games, as long as the game has a bounded fat-threshold dimension. Our algorithms apply to both binary-valued and real-valued games and can be viewed as providing justification for the wide use of double oracle and multiple oracle algorithms in the practice of solving large games.