Clairvoyant Regret Minimization: Equivalence with Nemirovski's Conceptual Prox Method and Extension to General Convex Games

TL;DR

This paper establishes the equivalence between Clairvoyant MWU and Nemirovski's conceptual prox method, extending the algorithm to general convex games and improving regret bounds to scale with the square root of players.

Contribution

It demonstrates the equivalence of CMWU with the conceptual prox method and extends it to convex games, providing improved regret bounds.

Findings

CMWU is equivalent to the conceptual prox method.

The extended Clairvoyant OMD algorithm applies to convex games.

Regret bounds scale with the square root of the number of players.

Abstract

A recent paper by Piliouras et al. [2021, 2022] introduces an uncoupled learning algorithm for normal-form games -- called Clairvoyant MWU (CMWU). In this note we show that CMWU is equivalent to the conceptual prox method described by Nemirovski [2004]. This connection immediately shows that it is possible to extend the CMWU algorithm to any convex game, a question left open by Piliouras et al. We call the resulting algorithm -- again equivalent to the conceptual prox method -- Clairvoyant OMD. At the same time, we show that our analysis yields an improved regret bound compared to the original bound by Piliouras et al., in that the regret of CMWU scales only with the square root of the number of players, rather than the number of players themselves.