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 $K\times K$ 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 $\epsilon$-approximate Nash equilibria can be computedefficiently from $O(\ln K / \epsilon) $ instead of $O( \ln K / \epsilon^2)$ queries.Surprisingly, the optimal number of such queries, as a function of both$\epsilon$ 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 ($\epsilon=0$), 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 $\epsilon > 0$, the currentquery complexity upper bound stands at $O(\min(\ln(K) / \epsilon , K))$. 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 order$\tilde\Omega(\log(1 / (K\epsilon)))$ for any $\epsilon \leq1 / cK^4$, 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.