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

2025-02-25

AI Summary

This paper studies $Phi$-equilibrium computation and low-$Phi$-regret online learning under a family $Phi$ of $k$-dimensional polynomial mappings, breaking the classical linear-bias restriction. We propose a nested algorithmic framework based on Ellipsoid-based Adversarial Hopes (EAH), integrating a convex-set oracle model with polynomial expectation fixed-point techniques to achieve efficient $Phi$-equilibrium approximation under polynomial-dimensional bias—first such result. Our contributions include: (i) an $varepsilon$-approximate $Phi$-equilibrium algorithm with time complexity $mathrm{poly}(n,d,k,log(1/varepsilon))$; (ii) an online learning algorithm achieving average $Phi$-regret $leq varepsilon$ within $mathrm{poly}(d,k)/varepsilon^2$ rounds; and (iii) matching upper and lower bounds establishing the first tight learnability characterization for $Phi$-equilibria under polynomial mappings.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 $ ext{poly}(n, d, k, ext{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 $ ext{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 $phi : mathcal{X} o mathcal{X}$ 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.

Problem

• Extend Φ-equilibria to k-dimensional polynomial maps.
• Develop poly-time algorithm for ε-approximate Φ-equilibria.
• Establish online algorithm for minimizing Φ-regret.

Innovation

• Extends $Φ$-equilibria to polynomial dimensions
• Uses poly-time algorithm for expected fixed points
• Implements nested ellipsoid against hope algorithm