Corrupted Learning Dynamics in Games

Taira Tsuchiya, Shinji Ito, Haipeng Luo

Abstract: Learning in games refers to scenarios where multiple players interact in a shared environment, each aiming to minimize their regret. It is well known that an equilibrium can be computed at a fast rate of $O(1/T)$ when all players follow the optimistic follow-the-regularized-leader (OFTRL). However, this acceleration is limited to the \textit{honest regime}, in which all players fully adhere to a prescribed algorithm—a situation that may not be realistic in practice. To address this issue, we present \textit{corrupted learning dynamics} that adaptively find an equilibrium at a rate that depends on the extent to which each player deviates from the strategy suggested by the prescribed algorithm. First, in two-player zero-sum corrupted games, we provide learning dynamics for which the external regret of the $x$-player (and similarly for the $y$-player) is roughly bounded by $O(\log (m_x m_y) + \sqrt{\hat{C}_y} + \hat{C}_x)$, where $m_x$ and $m_y$ denote the number of actions of the $x$- and $y$-players, respectively, and $\hat{C}_x$ and $\hat{C}_y$ represent their cumulative deviations. We then extend our approach to multiplayer general-sum corrupted games, providing learning dynamics for which the swap regret of player $i$ is bounded by $O(\log T + \sqrt{\sum_{k} \hat{C}_k \log T} + \hat{C}_i)$ ignoring dependence on the number of players and actions, where $\hat{C}_i$ is the cumulative deviation of player $i$ from the prescribed algorithm. Our learning dynamics are agnostic to the levels of corruption. A key technical contribution is a new analysis that ensures the stability of a stationary distribution of a Markov chain under a new adaptive learning rate, thereby allowing us to achieve the desired bound in the corrupted regime while matching the best existing bound in the honest regime. Notably, our framework can be extended to address not only corruption in strategies but also corruption in the observed expected utilities, and we provide several matching lower bounds.

External IDs:dblp:conf/colt/TsuchiyaIL25