Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games

• URL: http://arxiv.org/abs/2304.12768v2
• Date: Thu, 2 Nov 2023 09:02:30 GMT
• Title: Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games
• Authors: H\'edi Hadiji (L2S), Sarah Sachs (UvA), Tim van Erven (UvA), Wouter M. Koolen (CWI)
• Abstract summary: We show that exact equilibria can be computed efficiently from $O(fracln Kepsilon)$ instead of $O(fracln Kepsilon2)$ queries. We introduce a new technique for lower bounds, which allows us to obtain lower bounds of order $tildeOmega(frac1Kepsilon)$ for any $epsilon leq 1 / (cK4)$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the first-order query model for zero-sum $K\times K$ 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 $\epsilon$-approximate Nash equilibria can be computed efficiently from $O(\frac{\ln K}{\epsilon})$ instead of $O(\frac{\ln K}{\epsilon^2})$ 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 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 $\epsilon > 0$, the current query complexity upper bound stands at $O(\min(\frac{\ln(K)}{\epsilon} , K))$. 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 $\tilde\Omega(\log(\frac{1}{K\epsilon})$ for any $\epsilon \leq 1 / (cK^4)$, 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.

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