Near-Optimal No-Regret Learning for Correlated Equilibria in Multi-Player General-Sum Games

TL;DR

This paper extends recent no-regret learning results in multi-player games to internal and swap regret, achieving near-optimal convergence rates to correlated equilibria using novel techniques for analyzing fixed point operations.

Contribution

It introduces new methods to establish higher-order smoothness in learning dynamics, improving convergence rates for correlated equilibria and analyzing classic algorithms like BM.

Findings

Achieves $ ilde{O}(T^{-1})$ convergence to correlated equilibrium.

Establishes $O( extrm{polylog}(T))$ no-swap-regret bound for Blum and Mansour's algorithm.

Develops techniques for higher-order smoothness in fixed point based learning dynamics.

Abstract

Recently, Daskalakis, Fishelson, and Golowich (DFG) (NeurIPS`21) showed that if all agents in a multi-player general-sum normal-form game employ Optimistic Multiplicative Weights Update (OMWU), the external regret of every player is $O(\textrm{polylog}(T))$ after $T$ repetitions of the game. We extend their result from external regret to internal regret and swap regret, thereby establishing uncoupled learning dynamics that converge to an approximate correlated equilibrium at the rate of $\tilde{O}(T^{-1})$. This substantially improves over the prior best rate of convergence for correlated equilibria of $O(T^{-3/4})$ due to Chen and Peng (NeurIPS`20), and it is optimal -- within the no-regret framework -- up to polylogarithmic factors in $T$. To obtain these results, we develop new techniques for establishing higher-order smoothness for learning dynamics involving fixed point operations. Specifically, we establish that the no-internal-regret learning dynamics of Stoltz and Lugosi (Mach Learn`05) are equivalently simulated by no-external-regret dynamics on a combinatorial space. This allows us to trade the computation of the stationary distribution on a polynomial-sized Markov chain for a (much more well-behaved) linear transformation on an exponential-sized set, enabling us to leverage similar techniques as DFG to near-optimally bound the internal regret. Moreover, we establish an $O(\textrm{polylog}(T))$ no-swap-regret bound for the classic algorithm of Blum and Mansour (BM) (JMLR`07). We do so by introducing a technique based on the Cauchy Integral Formula that circumvents the more limited combinatorial arguments of DFG. In addition to shedding clarity on the near-optimal regret guarantees of BM, our arguments provide insights into the various ways in which the techniques by DFG can be extended and leveraged in the analysis of more involved learning algorithms.