Fast and Furious Symmetric Learning in Zero-Sum Games: Gradient Descent as Fictitious Play

TL;DR

This paper proves that both Fictitious Play and Gradient Descent algorithms achieve sublinear regret in symmetric zero-sum games, extending their effectiveness beyond small games and confirming conjectures about Fictitious Play.

Contribution

It establishes $O( oot{2}{T})$ regret bounds for Fictitious Play and Gradient Descent in generalized symmetric zero-sum games, demonstrating fast convergence without diminishing stepsizes.

Findings

Fictitious Play has $O( oot{2}{T})$ regret in symmetric zero-sum games.

Gradient Descent also achieves $O( oot{2}{T})$ regret with large constant stepsizes.

First demonstration of fast, non-vanishing stepsize behavior in larger zero-sum games.

Abstract

This paper investigates the sublinear regret guarantees of two non-no-regret algorithms in zero-sum games: Fictitious Play, and Online Gradient Descent with constant stepsizes. In general adversarial online learning settings, both algorithms may exhibit instability and linear regret due to no regularization (Fictitious Play) or small amounts of regularization (Gradient Descent). However, their ability to obtain tighter regret bounds in two-player zero-sum games is less understood. In this work, we obtain strong new regret guarantees for both algorithms on a class of symmetric zero-sum games that generalize the classic three-strategy Rock-Paper-Scissors to a weighted, n-dimensional regime. Under symmetric initializations of the players' strategies, we prove that Fictitious Play with any tiebreaking rule has $O(\sqrt{T})$ regret, establishing a new class of games for which Karlin's Fictitious Play conjecture holds. Moreover, by leveraging a connection between the geometry of the iterates of Fictitious Play and Gradient Descent in the dual space of payoff vectors, we prove that Gradient Descent, for almost all symmetric initializations, obtains a similar $O(\sqrt{T})$ regret bound when its stepsize is a sufficiently large constant. For Gradient Descent, this establishes the first "fast and furious" behavior (i.e., sublinear regret without time-vanishing stepsizes) for zero-sum games larger than 2x2.