Online Learning with Switching Costs and Other Adaptive Adversaries

TL;DR

This paper investigates the impact of adaptive adversaries with switching costs on online learning, revealing that bandit feedback leads to higher regret rates than full-information scenarios, and introduces new bounds and strategies.

Contribution

It characterizes the power of adaptive adversaries with switching costs and bounded memory, providing nearly complete regret bounds and a novel reduction from experts to bandits.

Findings

Bandit feedback with switching costs yields a regret rate of .67 T^{2/3}

Full-information case with switching costs achieves .5 T^{1/2} regret rate

Bounded memory adversaries can force .67 T^{2/3} regret even with full information.

Abstract

We study the power of different types of adaptive (nonoblivious) adversaries in the setting of prediction with expert advice, under both full-information and bandit feedback. We measure the player's performance using a new notion of regret, also known as policy regret, which better captures the adversary's adaptiveness to the player's behavior. In a setting where losses are allowed to drift, we characterize ---in a nearly complete manner--- the power of adaptive adversaries with bounded memories and switching costs. In particular, we show that with switching costs, the attainable rate with bandit feedback is $\widetilde{\Theta}(T^{2/3})$. Interestingly, this rate is significantly worse than the $\Theta(\sqrt{T})$ rate attainable with switching costs in the full-information case. Via a novel reduction from experts to bandits, we also show that a bounded memory adversary can force $\widetilde{\Theta}(T^{2/3})$ regret even in the full information case, proving that switching costs are easier to control than bounded memory adversaries. Our lower bounds rely on a new stochastic adversary strategy that generates loss processes with strong dependencies.