Poster
Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games
Hedi Hadiji · Sarah Sachs · Tim van Erven · Wouter Koolen
Great Hall & Hall B1+B2 (level 1) #1707
Abstract:
In the first-order query model for zero-sum 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 -approximate Nash equilibria can be computed efficiently from instead of queries. Surprisingly, the optimal number of such queries, as a function of both and , is not known. We make progress on this question on two fronts. First, we fully characterise the query complexity of learning exact equilibria (), by showing that they require a number of queries that is linear in , which means that it is essentially as hard as querying the whole matrix, which can also be done with queries. Second, for , the current query complexity upper bound stands at . 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 for any , where is a constant independent of . 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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