Sublogarithmic Swap Regret in Multiplayer General-Sum Games via Hybrid Regularization

Swap regret governs the rate at which uncoupled learning dynamics converge to correlated equilibria in multiplayer general-sum games. Under full-information feedback, the best previous guarantee when every player follows the same dynamics grows logarithmically in the horizon T. We construct uncoupled dynamics under which every player incurs only O(nm2√log mlog T) swap regret, where n is the number of players and m bounds the number of actions per player. To our knowledge, this is the first sublogarithmic individual guarantee in this setting, and it implies that the time-averaged product distribution of play is an O(nm2√log mlog T/T)-approximate correlated equilibrium. The key algorithmic choice is to combine the Blum--Mansour reduction with optimistic follow-the-regularized-leader using a hybrid regularizer that separately weights negative Shannon entropy and the log-barrier: the entropy controls the optimistic prediction error, whereas the log-barrier controls the transition-matrix movement through its Bregman divergence. A new sensitivity theorem for stationary distributions of Markov chains, which involves neither mixing parameters nor the smallest transition probability, transfers this control to the played strategies and yields a simpler analysis without local-norm or self-concordance arguments. The guarantee is preserved by an adversarially robust variant that additionally ensures O(nm2√log mlog T+√mTlog m) swap regret against arbitrary utility sequences, and by a horizon-free variant that requires no prior knowledge of T.