Efficient Φ-Regret Minimization with Low-Degree Swap Deviations in Extensive-Form Games

Recent breakthrough results by Dagan, Daskalakis, Fishelson and Golowich [2023] and Peng and Rubinstein [2023] established an efficient algorithm attaining at most ϵ swap regret over extensive-form strategy spaces of dimension N in N^Õ(1/ϵ) rounds. On the other extreme, Farina and Pipis [2023] developed an efficient algorithm for minimizing the weaker notion of linear-swap regret in 𝗉𝗈𝗅𝗒(N)/ϵ^2 rounds. In this paper, we take a step toward bridging the gap between those two results. We introduce the set of k-mediator deviations, which generalize the untimed communication deviations recently introduced by Zhang, Farina and Sandholm [2024] to the case of having multiple mediators. We develop parameterized algorithms for minimizing the regret with respect to this set of deviations in N^O(k)/ϵ^2 rounds. This closes the gap in the sense that k=1 recovers linear swap regret, while k=N recovers swap regret. Moreover, by relating k-mediator deviations to low-degree polynomials, we show that regret minimization against degree-k polynomial swap deviations is achievable in N^O(kd)^3/ϵ^2 rounds, where d is the depth of the game, assuming constant branching factor. For a fixed degree k, this is polynomial for Bayesian games and quasipolynomial more broadly when d = 𝗉𝗈𝗅𝗒𝗅𝗈𝗀 N – the usual balancedness assumption on the game tree.