Online Optimization in Games via Control Theory: Connecting Regret, Passivity and Poincar\'e Recurrence

TL;DR

This paper introduces a control-theoretic framework for online optimization in games, showing that passivity and energy conservation in dynamics lead to bounded regret and recurrent behaviors, extending stability analysis to Poincaré recurrence.

Contribution

It establishes a novel connection between passivity in control theory and bounded regret in online game learning, introducing lossless dynamics and energy-preserving behaviors.

Findings

All continuous-time FTRL dynamics are lossless and passive.

Convex combinations of FTRL dynamics are also lossless with bounded regret.

Lossless game dynamics can exhibit Poincaré recurrence, indicating recurrent behaviors.

Abstract

We present a novel control-theoretic understanding of online optimization and learning in games, via the notion of passivity. Passivity is a fundamental concept in control theory, which abstracts energy conservation and dissipation in physical systems. It has become a standard tool in analysis of general feedback systems, to which game dynamics belong. Our starting point is to show that all continuous-time Follow-the-Regularized-Leader (FTRL) dynamics, which include the well-known Replicator Dynamic, are lossless, i.e. it is passive with no energy dissipation. Interestingly, we prove that passivity implies bounded regret, connecting two fundamental primitives of control theory and online optimization. The observation of energy conservation in FTRL inspires us to present a family of lossless learning dynamics, each of which has an underlying energy function with a simple gradient structure. This family is closed under convex combination; as an immediate corollary, any convex combination of FTRL dynamics is lossless and thus has bounded regret. This allows us to extend the framework of Fox and Shamma [Games, 2013] to prove not just global asymptotic stability results for game dynamics, but Poincar\'e recurrence results as well. Intuitively, when a lossless game (e.g. graphical constant-sum game) is coupled with lossless learning dynamics, their feedback interconnection is also lossless, which results in a pendulum-like energy-preserving recurrent behavior, generalizing the results of Piliouras and Shamma [SODA, 2014] and Mertikopoulos, Papadimitriou and Piliouras [SODA, 2018].