Training with (Swap) Regret Loss in a Single-Layer Self-Attention Model: A Case Study on the Probability Simplex

We give deterministic and uncoupled learning dynamics for finite multiplayer general-sum games under full-information feedback that achieve constant individual swap regret, independent of the horizon $T$. With $n$ players and at most $m$ actions each, the individual swap regret of every player is $O(\sqrt{n} m \log m \log^{5/2}(nm))$ at every finite horizon. Each player predicts the deviation gains, then uses these predictions to update a row-stochastic transition matrix, and plays its stationary distribution. The proof combines a potential argument exploiting stationarity with a two-scale higher-order prediction analysis, using rooted-tree representations to handle the nonlinear dependence of deviation gains on the stationary distributions. An adversarially robust variant, obtained through a generic common-prefix switching wrapper, preserves the self-play bound up to a universal constant and guarantees individual swap regret at most $7\sqrt{m T \log m}$ in the adversarial setting.

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 $O(nm^2\sqrt{\log m\log T})$ 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 $O(nm^2\sqrt{\log m\log T}/T)$-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 $O(nm^2\sqrt{\log m\log T}+\sqrt{mT\log m})$ swap regret against arbitrary utility sequences, and by a horizon-free variant that requires no prior knowledge of $T$.

In this paper, we study regret minimization in repeated games with \emph{adaptive} opponents who can respond based on histories of play. The standard metric of \emph{external regret} in online learning is known to fail to capture such adaptivity. To account for players' counterfactual reasoning, we introduce {\tt Repeated Policy Regret (RP-Regret)}, a game-theoretic metric that measures the difference between the \emph{realized} and the \emph{best-in-hindsight} accumulated utility when all players can \emph{respond} to the history of play. Compared to existing regret notions in this setting, ours is native to repeated game playing, enabling stronger comparators and opponents with fewer constraints, while maintaining the possibility of finding better equilibria when all players minimize it. We first identify necessary conditions for obtaining {\tt RP-Regret} sublinear in time, on the variation of the player's comparator strategies in the regret definition and on the memories of both the comparator and opponents' strategies. We then study additional conditions and provable algorithms to minimize {\tt RP-Regret}, which is by definition \emph{non-convex} in the strategy space. To address this challenge, we propose three algorithms: (i) one based on an optimization oracle, as assumed in some prior work in online non-convex learning; (ii) one that minimizes a convex and \emph{linearized} surrogate of {\tt RP-Regret} at each iteration; (iii) one that directly minimizes {\tt RP-Regret} when opponents change strategies slowly. Furthermore, when all players can run algorithms to minimize the {\tt RP-Regret} (or its linearized variant), certain subgame perfect equilibria of the repeated game can be learned. We also provide experiments showing that minimizing our regret notions can lead to more cooperative solutions with higher utility in games such as Stag-Hunt.