Refined Lower Bounds for Adversarial Bandits

TL;DR

This paper establishes new lower bounds on adversarial bandit regret, showing that recent upper bounds are nearly tight and proving fundamental impossibility results that differentiate bandit from full-information settings.

Contribution

It introduces refined lower bounds for adversarial bandit regret and proves key impossibility results that highlight limitations of bandit algorithms.

Findings

Recent upper bounds are close to tight.

Existence of a universally optimal arm does not improve regret.

Regret cannot scale with the effective range of losses.

Abstract

We provide new lower bounds on the regret that must be suffered by adversarial bandit algorithms. The new results show that recent upper bounds that either (a) hold with high-probability or (b) depend on the total lossof the best arm or (c) depend on the quadratic variation of the losses, are close to tight. Besides this we prove two impossibility results. First, the existence of a single arm that is optimal in every round cannot improve the regret in the worst case. Second, the regret cannot scale with the effective range of the losses. In contrast, both results are possible in the full-information setting.