Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games
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In the first-order query model for zero-sum K× K matrix games, playersobserve the expected pay-offs for all their possible actions under therandomized action played by their opponent. This is a classical model,which has received renewed interest after the discoveryby Rakhlin and Sridharan that ϵ-approximate Nash equilibria can be computedefficiently from O(ln K / ϵ) instead of O( ln K / ϵ^2) queries.Surprisingly, the optimal number of such queries, as a function of bothϵ 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 linearin K, which means that it is essentially as hard as querying the wholematrix, which can also be done with K queries. Second, for ϵ > 0, the currentquery complexity upper bound stands at O(min(ln(K) / ϵ , K)). We argue that, unfortunately, obtaining matchinglower bound is not possible with existing techniques: we prove that nolower bound can be derived by constructing hard matrices whose entriestake values in a known countable set, because such matrices can be fullyidentified by a single query. This rules out, for instance, reducing toa submodular optimization problem over the hypercube by encoding itas a binary matrix. We then introduce a new technique for lower bounds,which allows us to obtain lower bounds of orderΩ̃(log(1 / (Kϵ))) for any ϵ≤1 / cK^4, where c is a constant independent of K. We furtherdiscuss possible future directions to improve on our techniques in orderto close the gap with the upper bounds.
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