A Lower Bound on Swap Regret in Extensive-Form Games

TL;DR

This paper establishes a fundamental lower bound on the number of rounds needed for algorithms to achieve low swap regret in extensive-form games, showing that polynomial-time solutions are impossible in general.

Contribution

It proves a lower bound on swap regret algorithms, demonstrating that achieving low swap regret requires exponential rounds in general extensive-form games.

Findings

No polynomial-round algorithm exists for low swap regret in general extensive-form games.

Achieving $ ext{average swap regret} \, ext{$ extless$} \, ext{$ ext{epsilon}$}$ requires exponential rounds.

The lower bound depends on the number of nodes and the desired regret level.

Abstract

Recent simultaneous works by Peng and Rubinstein [2024] and Dagan et al. [2024] have demonstrated the existence of a no-swap-regret learning algorithm that can reach $\epsilon$ average swap regret against an adversary in any extensive-form game within $m^{\tilde{\mathcal O}(1/\epsilon)}$ rounds, where $m$ is the number of nodes in the game tree. However, the question of whether a $\mathrm{poly}(m, 1/\epsilon)$-round algorithm could exist remained open. In this paper, we show a lower bound that precludes the existence of such an algorithm. In particular, we show that achieving average swap regret $\epsilon$ against an oblivious adversary in general extensive-form games requires at least $\mathrm{exp}\left(\Omega\left(\min\left\{m^{1/14}, \epsilon^{-1/6}\right\}\right)\right)$ rounds.