No-regret learning in harmonic games: Extrapolation in the face of conflicting interests

TL;DR

This paper investigates the behavior of no-regret learning algorithms in harmonic games, revealing that standard methods often fail to converge, but with extrapolation techniques, convergence to Nash equilibrium can be achieved.

Contribution

It extends understanding of no-regret learning dynamics to harmonic games, showing that augmented FTRL algorithms ensure convergence and bounded regret in these conflicting-interest settings.

Findings

FTRL dynamics are Poincaré recurrent in continuous time, failing to converge.

Standard FTRL in discrete time can trap players in cycles.

Extrapolated FTRL converges to Nash equilibrium with bounded regret.

Abstract

The long-run behavior of multi-agent learning - and, in particular, no-regret learning - is relatively well-understood in potential games, where players have aligned interests. By contrast, in harmonic games - the strategic counterpart of potential games, where players have conflicting interests - very little is known outside the narrow subclass of 2-player zero-sum games with a fully-mixed equilibrium. Our paper seeks to partially fill this gap by focusing on the full class of (generalized) harmonic games and examining the convergence properties of follow-the-regularized-leader (FTRL), the most widely studied class of no-regret learning schemes. As a first result, we show that the continuous-time dynamics of FTRL are Poincar\'e recurrent, that is, they return arbitrarily close to their starting point infinitely often, and hence fail to converge. In discrete time, the standard, "vanilla" implementation of FTRL may lead to even worse outcomes, eventually trapping the players in a perpetual cycle of best-responses. However, if FTRL is augmented with a suitable extrapolation step - which includes as special cases the optimistic and mirror-prox variants of FTRL - we show that learning converges to a Nash equilibrium from any initial condition, and all players are guaranteed at most O(1) regret. These results provide an in-depth understanding of no-regret learning in harmonic games, nesting prior work on 2-player zero-sum games, and showing at a high level that harmonic games are the canonical complement of potential games, not only from a strategic, but also from a dynamic viewpoint.