Competing Against Nash Equilibria in Adversarially Changing ZeroSum Games
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:921930, 2019.
Abstract
We study the problem of repeated play in a zerosum game in which the payoff matrix may change, in a possibly adversarial fashion, on each round; we call these Online Matrix Games. Finding the Nash Equilibrium (NE) of a two player zerosum game is core to many problems in statistics, optimization, and economics, and for a fixed game matrix this can be easily reduced to solving a linear program. But when the payoff matrix evolves over time our goal is to find a sequential algorithm that can compete with, in a certain sense, the NE of the longtermaveraged payoff matrix. We design an algorithm with small NE regret–that is, we ensure that the longterm payoff of both players is close to minimax optimum in hindsight. Our algorithm achieves nearoptimal dependence with respect to the number of rounds and depends polylogarithmically on the number of available actions of the players. Additionally, we show that the naive reduction, where each player simply minimizes its own regret, fails to achieve the stated objective regardless of which algorithm is used. Lastly, we consider the socalled bandit setting, where the feedback is significantly limited, and we provide an algorithm with small NE regret using onepoint estimates of each payoff matrix.
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