Towards characterizing the first-order query complexity of learning (approximate) Nash equilibria in zero-sum matrix games

In the first-order query model for zero-sum K×K matrix games, players observe the expected pay-offs for all their possible actions under the randomized action played by their opponent. This classical model has received renewed interest after the discovery by Rakhlin and Sridharan that ϵ-approximate Nash equilibria can be computed efficiently from O(lnKϵ) instead of O(lnKϵ2) queries. Surprisingly, the optimal number of such queries, as a function of both ϵ and K, is not known. We make progress on this question on two fronts. First, we fully characterise the query complexity of learning exact equilibria (ϵ=0), by showing that they require a number of queries that is linear in K, which means that it is essentially as hard as querying the whole matrix, which can also be done with K queries. Second, for ϵ>0, the current query complexity upper bound stands at O(min(ln(K)ϵ,K)). We argue that, unfortunately, obtaining a matching lower bound is not possible with existing techniques: we prove that no lower bound can be derived by constructing hard matrices whose entries take values in a known countable set, because such matrices can be fully identified by a single query. This rules out, for instance, reducing to an optimization problem over the hypercube by encoding it as a binary payoff matrix. We then introduce a new technique for lower bounds, which allows us to obtain lower bounds of order Ω~(log(1Kϵ) for any ϵ≤1/(cK4), where c is a constant independent of K. We further discuss possible future directions to improve on our techniques in order to close the gap with the upper bounds.

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inProceedings Towards characterizing the first-order query complexity of learning (approximate) Nash Equilibria in zero-sum matrix games H. Hadiji (Hédi), S. Sachs (Sarah), T. van Even (Tim) and W.M. Koolen-Wijkstra (Wouter)