Tight Regret Upper and Lower Bounds for Optimistic Hedge in Two-Player Zero-Sum Games

In two-player zero-sum games, the learning dynamic based on optimistic Hedge achieves one of the best-known regret upper bounds among strongly-uncoupled learning dynamics. With an appropriately chosen learning rate, the social and individual regrets can be bounded by $O(\log(mn))$ in terms of the numbers of actions $m$ and $n$ of the two players. This study investigates the optimality of the dependence on $m$ and $n$ in the regret of optimistic Hedge. To this end, we begin by refining existing regret analysis and show that, in the strongly-uncoupled setting where the opponent's number of actions is known, both the social and individual regret bounds can be improved to $O(\sqrt{\log m \log n})$. We then show that the existing social regret bound as well as these new social and individual regret upper bounds cannot be further improved for optimistic Hedge by providing algorithm-dependent individual regret lower bounds. Importantly, these upper and lower bounds match exactly up to the constant factor in the leading term. Finally, building on these results, we improve the last-iterate convergence rate and the dynamic regret of a learning dynamic based on optimistic Hedge, and complement these results with algorithm-dependent dynamic regret lower bounds that match the improved bounds.