Computational Lower Bounds for Regret Minimization in Normal-Form Games

TL;DR

This paper establishes fundamental lower bounds on the number of iterations needed for regret minimization algorithms to approximate correlated equilibria in normal-form games, showing existing algorithms are nearly optimal.

Contribution

It introduces new computational lower bounds for approximating correlated equilibria, using the sum-of-squares framework, demonstrating near-optimality of current algorithms.

Findings

Lower bounds for uniform T-sparse CE with specific approximation guarantees.

Existing algorithms like multiplicative weights are close to these lower bounds.

Hardness results apply to both log-sparse and near-complete sparse correlated equilibria.

Abstract

A celebrated connection in the interface of online learning and game theory establishes that players minimizing swap regret converge to correlated equilibria (CE) -- a seminal game-theoretic solution concept. Despite the long history of this problem and the renewed interest it has received in recent years, a basic question remains open: how many iterations are needed to approximate an equilibrium under the usual normal-form representation? In this paper, we provide evidence that existing learning algorithms, such as multiplicative weights update, are close to optimal. In particular, we prove lower bounds for the problem of computing a CE that can be expressed as a uniform mixture of $T$ product distributions -- namely, a uniform $T$-sparse CE; such lower bounds immediately circumscribe (computationally bounded) regret minimization algorithms in games. Our results are obtained in the algorithmic framework put forward by Kothari and Mehta (STOC 2018) in the context of computing Nash equilibria, which consists of the sum-of-squares (SoS) relaxation in conjunction with oracle access to a verification oracle; the goal in that framework is to lower bound either the degree of the SoS relaxation or the number of queries to the verification oracle. Here, we obtain two such hardness results, precluding computing i) uniform $\text{log }n$-sparse CE when $\epsilon =\text{poly}(1/\text{log }n)$ and ii) uniform $n^{1 - o(1)}$-sparse CE when $\epsilon = \text{poly}(1/n)$.