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Pré-Publication, Document De Travail Année : 2023

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

Résumé

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 is a classical model, which has received renewed interest after the discovery by Rakhlin and Sridharan that $\epsilon$-approximate Nash equilibria can be computed efficiently 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 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 $O(\min(\ln(K) / \epsilon , K))$. We argue that, unfortunately, obtaining 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 a submodular optimization problem over the hypercube by encoding it as 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 $\epsilon \leq 1 / cK^4$, 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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Dates et versions

hal-04079773 , version 1 (24-04-2023)
hal-04079773 , version 2 (27-10-2023)

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Hédi Hadiji, Sarah Sachs, Tim van Erven, Wouter M. Koolen. Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games. 2023. ⟨hal-04079773v1⟩
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