Efficient Phi-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 Ntilde O(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 mathsfpoly(N)/ε2 rounds. In this paper, we develop efficient parameterized algorithms for regimes between these two extremes. 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, and we develop algorithms for minimizing the regret with respect to this set of deviations in NO(k)/ε2 rounds. Moreover, by relating k-mediator deviations to low-degree polynomials, we show that regret minimization against degree-k polynomial swap deviations is achievable in NO(kd)3/ε2 rounds, where d is the depth of the game, assuming a constant branching factor. For a fixed degree k, this is polynomial for Bayesian games and quasipolynomial more broadly when d = mathsfpolylog N -- the usual balancedness assumption on the game tree.