Regret Matching$^+$: (In)Stability and Fast Convergence in Games
Gabriele Farina, Julien Grand-Clément, Christian Kroer, Chung-Wei Lee, Haipeng Luo
Abstract
Regret Matching$^+$ (RM$^+$) and its variants are important algorithms for solving large-scale games. However, a theoretical understanding of their success in practice is still a mystery. Moreover, recent advances on fast convergence in games are limited to no-regret algorithms such as online mirror descent, which satisfy stability. In this paper, we first give counterexamples showing that RM$^+$ and its predictive version can be unstable, which might cause other players to suffer large regret. We then provide two fixes: restarting and chopping off the positive orthant that RM$^+$ works in. We show that these fixes are sufficient to get $O(T^{1/4})$ individual regret and $O(1)$ social regret in normal-form games via RM with predictions. We also apply our stabilizing techniques to clairvoyant updates in the uncoupled learning setting for RM$^+$ and prove desirable results akin to recent works for Clairvoyant online mirror descent. Our experiments show the advantages of our algorithms over vanilla RM$^+$-based algorithms in matrix and extensive-form games.