Beyond Time-Average Convergence: Near-Optimal Uncoupled Online Learning via Clairvoyant Multiplicative Weights Update

TL;DR

This paper introduces the Clairvoyant Multiplicative Weights Update (CMWU), an algorithm for regret minimization in games that achieves near-optimal convergence rates by leveraging a clairvoyant approach and efficient computation.

Contribution

The paper presents a novel CMWU algorithm that attains constant regret and fast convergence to coarse correlated equilibrium in general games, improving upon existing rates.

Findings

CMWU achieves constant regret of (rac{\u2212 ext{ln}(m)}{\u03b7}) in all normal-form games.

The updates can be computed linearly fast via a contraction map under certain step-size conditions.

The dynamics converge at a rate of O(nV ( ext{log} m ext{log} T / T)) to a coarse correlated equilibrium.

Abstract

In this paper, we provide a novel and simple algorithm, Clairvoyant Multiplicative Weights Updates (CMWU) for regret minimization in general games. CMWU effectively corresponds to the standard MWU algorithm but where all agents, when updating their mixed strategies, use the payoff profiles based on tomorrow's behavior, i.e. the agents are clairvoyant. CMWU achieves constant regret of $\ln(m)/\eta$ in all normal-form games with m actions and fixed step-sizes $\eta$. Although CMWU encodes in its definition a fixed point computation, which in principle could result in dynamics that are neither computationally efficient nor uncoupled, we show that both of these issues can be largely circumvented. Specifically, as long as the step-size $\eta$ is upper bounded by $\frac{1}{(n-1)V}$, where $n$ is the number of agents and $[0,V]$ is the payoff range, then the CMWU updates can be computed linearly fast via a contraction map. This implementation results in an uncoupled online learning dynamic that admits a $O (\log T)$-sparse sub-sequence where each agent experiences at most $O(nV\log m)$ regret. This implies that the CMWU dynamics converge with rate $O(nV \log m \log T / T)$ to a \textit{Coarse Correlated Equilibrium}. The latter improves on the current state-of-the-art convergence rate of \textit{uncoupled online learning dynamics} \cite{daskalakis2021near,anagnostides2021near}.