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(ln K/ε) instead of O(ln K/ε 2 ) 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( 1 Kε )) for any ε 1/(cK 4 ), 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. How many first-order queries are necessary and sufficient for a sequential learner to output an approximate saddle point for any f ∈ F ? Characterizing the query complexity of learning saddle points is of theoretical interest for understanding the hardness of computing equilibria, and for certifying the optimality of upper bounds. In this work, we restrict our attention to the special case of zero-sum matrix games, where X and Y are finite-dimensional probability simplices and f is bilinear. For this canonical setting, the optimal query complexity is, surprisingly, unresolved. Indeed, computation algorithms are known since [3], up to [37], while lower bounds remain elusive, leaving the optimal query complexity still unknown. Obtaining lower bounds is not only of fundamental interest in itself, but may also lead to interesting new techniques, since none of the existing proof strategies are applicable. New ideas provided here could prove useful to tackle other problems. Contributions We make progress towards characterizing the first-order query complexity of approximate Nash equilibria in finite-action zero-sum games, as a function of the number of actions K and approximation level ε. Our contributions are the following: