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 $\epsilon$ swap regret over extensive-form strategy spaces of dimension $N$ in $N^{\tilde O(1/\epsilon)}$ rounds. On the other extreme, Farina and Pipis [2023] developed an efficient algorithm for minimizing the weaker notion of linear-swap regret in $\mathsf{poly}(N)/\epsilon^2$ 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 $N^{O(k)}/\epsilon^2$ 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 $N^{O(kd)^3}/\epsilon^2$ 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 $d = \mathsf{polylog} N$---the usual balancedness assumption on the game tree. The first key ingredient in our approach is a relaxation of the usual notion of a fixed point required in the framework of Gordon, Greenwald and Marks [2008]. Namely, for a given deviation $\phi$, we show that it suffices to compute what we refer to as a fixed point in expectation; that is, a distribution $\pi$ such that $\mathbb{E}_{x \sim \pi} [\phi(x) - x] \approx 0$. Unlike the problem of computing an actual (approximate) fixed point $x \approx \phi(x)$, which we show is \PPAD-hard, there is a simple and efficient algorithm for finding a solution that satisfies our relaxed notion. As a byproduct, we provide, to our knowledge, the fastest algorithm for computing $\epsilon$-correlated equilibria in normal-form games in the medium-precision regime, obviating the need to solve a linear system in every round. Our second main contribution is a characterization of the set of low-degree deviations, made possible through a connection to low-depth decisions trees from Boolean analysis.