Learning and Computation of $Φ$-Equilibria at the Frontier of Tractability

• URL: http://arxiv.org/abs/2502.18582v2
• Date: Fri, 28 Feb 2025 00:45:49 GMT
• Title: Learning and Computation of $Φ$-Equilibria at the Frontier of Tractability
• Authors: Brian Hu Zhang, Ioannis Anagnostides, Emanuel Tewolde, Ratip Emin Berker, Gabriele Farina, Vincent Conitzer, Tuomas Sandholm,
• Abstract summary: $Phi$-equilibria is a powerful and flexible framework at the heart of online learning and game theory.<n>We show that an efficient online algorithm incurs average $Phi$-regret at most $epsilon$ using $textpoly(d, k)/epsilon2$ rounds.<n>We also show nearly matching lower bounds in the online setting, thereby obtaining for the first time a family of deviations that captures the learnability of $Phi$-regret.
• Score: 85.07238533644636
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: $\Phi$-equilibria -- and the associated notion of $\Phi$-regret -- are a powerful and flexible framework at the heart of online learning and game theory, whereby enriching the set of deviations $\Phi$ begets stronger notions of rationality. Recently, Daskalakis, Farina, Fishelson, Pipis, and Schneider (STOC '24) -- abbreviated as DFFPS -- settled the existence of efficient algorithms when $\Phi$ contains only linear maps under a general, $d$-dimensional convex constraint set $\mathcal{X}$. In this paper, we significantly extend their work by resolving the case where $\Phi$ is $k$-dimensional; degree-$\ell$ polynomials constitute a canonical such example with $k = d^{O(\ell)}$. In particular, positing only oracle access to $\mathcal{X}$, we obtain two main positive results: i) a $\text{poly}(n, d, k, \text{log}(1/\epsilon))$-time algorithm for computing $\epsilon$-approximate $\Phi$-equilibria in $n$-player multilinear games, and ii) an efficient online algorithm that incurs average $\Phi$-regret at most $\epsilon$ using $\text{poly}(d, k)/\epsilon^2$ rounds. We also show nearly matching lower bounds in the online learning setting, thereby obtaining for the first time a family of deviations that captures the learnability of $\Phi$-regret. From a technical standpoint, we extend the framework of DFFPS from linear maps to the more challenging case of maps with polynomial dimension. At the heart of our approach is a polynomial-time algorithm for computing an expected fixed point of any $\phi : \mathcal{X} \to \mathcal{X}$ based on the ellipsoid against hope (EAH) algorithm of Papadimitriou and Roughgarden (JACM '08). In particular, our algorithm for computing $\Phi$-equilibria is based on executing EAH in a nested fashion -- each step of EAH itself being implemented by invoking a separate call to EAH.

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