On Tractable $Phi$-Equilibria in Non-Concave Games

2024-03-13

AI Summary

In non-concave games—such as those involving deep neural network-parameterized strategies or utility functions—Nash equilibria may not exist, local Nash points are computationally intractable, and mixed, correlated, or coarse-correlated equilibria may have infinite, uncomputable support. Method: This paper establishes the first tractability framework for finite and local Φ-equilibria in non-concave games, introducing a novel family of strategy modifications inspired by proximal operators and rigorously characterizing their computational boundaries. Leveraging online gradient descent (OGD) and uncoupled learning, we combine first-order stationarity analysis with computational complexity theory. Results: We prove that polynomial-time convergence to finite Φ-equilibria is achievable for bounded Φ; while local Φ-equilibria with unbounded Φ are NP-hard under higher-order approximations, OGD converges efficiently within the first-order stationary region.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to a coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when utilities are non-concave -- a common scenario in machine learning applications involving strategies parameterized by deep neural networks, or when agents' utilities are computed by neural networks, or both. Non-concave games introduce significant game-theoretic and optimization challenges: (i) Nash equilibria may not exist; (ii) local Nash equilibria, though they exist, are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria generally have infinite support and are intractable. To sidestep these challenges, we revisit the classical solution concept of $Phi$-equilibria introduced by Greenwald and Jafari [2003], which is guaranteed to exist for an arbitrary set of strategy modifications $Phi$ even in non-concave games [Stolz and Lugosi, 2007]. However, the tractability of $Phi$-equilibria in such games remains elusive. In this paper, we initiate the study of tractable $Phi$-equilibria in non-concave games and examine several natural families of strategy modifications. We show that when $Phi$ is finite, there exists an efficient uncoupled learning algorithm that converges to the corresponding $Phi$-equilibria. Additionally, we explore cases where $Phi$ is infinite but consists of local modifications. We show that approximating local $Phi$-equilibria beyond the first-order stationary regime is computationally intractable. In contrast, within this regime, we show Online Gradient Descent efficiently converges to $Phi$-equilibria for several natural infinite families of modifications, including a new structural family of modifications inspired by the well-studied proximal operator.

Problem

• Addressing intractability of equilibria in non-concave games
• Exploring efficient learning algorithms for Φ-equilibria
• Analyzing computational limits of local Φ-equilibria approximation

Innovation

• Efficient uncoupled learning for finite $Φ$-equilibria
• Online Gradient Descent for local $Φ$-equilibria
• Proximal operator-inspired structural modifications