Scale-Invariant Fast Convergence in Games

TL;DR

This paper introduces scale-invariant learning dynamics that guarantee fast convergence to equilibrium in games without prior knowledge of utility scales, applicable to both two-player zero-sum and multiplayer general-sum games.

Contribution

It develops scale-free, scale-invariant algorithms with provable convergence rates for various game types, eliminating the need for utility scale prior knowledge.

Findings

Bounded external regret by ten A_{ ext{diff}}

Achieved ten A_{ ext{diff}} / T convergence rate in zero-sum games

Bounded swap regret by O(U_{ ext{max}} \u2217 \u007elog T) in general-sum games

Abstract

Scale-invariance in games has recently emerged as a widely valued desirable property. Yet, almost all fast convergence guarantees in learning in games require prior knowledge of the utility scale. To address this, we develop learning dynamics that achieve fast convergence while being both scale-free, requiring no prior information about utilities, and scale-invariant, remaining unchanged under positive rescaling of utilities. For two-player zero-sum games, we obtain scale-free and scale-invariant dynamics with external regret bounded by $\tilde{O}(A_{\mathrm{diff}})$, where $A_{\mathrm{diff}}$ is the payoff range, which implies an $\tilde{O}(A_{\mathrm{diff}} / T)$ convergence rate to Nash equilibrium after $T$ rounds. For multiplayer general-sum games with $n$ players and $m$ actions, we obtain scale-free and scale-invariant dynamics with swap regret bounded by $O(U_{\mathrm{max}} \log T)$, where $U_{\mathrm{max}}$ is the range of the utilities, ignoring the dependence on the number of players and actions. This yields an $O(U_{\mathrm{max}} \log T / T)$ convergence rate to correlated equilibrium. Our learning dynamics are based on optimistic follow-the-regularized-leader with an adaptive learning rate that incorporates the squared path length of the opponents' gradient vectors, together with a new stopping-time analysis that exploits negative terms in regret bounds without scale-dependent tuning. For general-sum games, scale-free learning is enabled also by a technique called doubling clipping, which clips observed gradients based on past observations.