Barriers to Welfare Maximization with No-Regret Learning

TL;DR

This paper proves fundamental computational lower bounds on the number of iterations needed for no-regret learning algorithms to approximate welfare-maximizing equilibria in two-player general-sum games, highlighting inherent complexity barriers.

Contribution

It establishes the first known computational lower bounds for welfare approximation via no-regret learning in two-player games, using reductions from clique problems.

Findings

Computational lower bounds match maximum clique hardness.

Hardness extends to low-precision regimes via planted clique conjecture.

No efficient algorithms can achieve non-trivial sparsity in polynomial time.

Abstract

A celebrated result in the interface of online learning and game theory guarantees that the repeated interaction of no-regret players leads to a coarse correlated equilibrium (CCE) -- a natural game-theoretic solution concept. Despite the rich history of this foundational problem and the tremendous interest it has received in recent years, a basic question still remains open: how many iterations are needed for no-regret players to approximate an equilibrium? In this paper, we establish the first computational lower bounds for that problem in two-player (general-sum) games under the constraint that the CCE reached approximates the optimal social welfare (or some other natural objective). From a technical standpoint, our approach revolves around proving lower bounds for computing a near-optimal $T$-sparse CCE -- a mixture of $T$ product distributions, thereby circumscribing the iteration complexity of no-regret learning even in the centralized model of computation. Our proof proceeds by extending a classical reduction of Gilboa and Zemel [1989] for optimal Nash to sparse (approximate) CCE. In particular, we show that the inapproximability of maximum clique precludes attaining any non-trivial sparsity in polynomial time. Moreover, we strengthen our hardness results to apply in the low-precision regime as well via the planted clique conjecture.