When Lower-Order Terms Dominate: Adaptive Expert Algorithms for Heavy-Tailed Losses

TL;DR

This paper introduces adaptive algorithms for prediction with expert advice under heavy-tailed losses, achieving improved regret bounds that adapt to the loss distribution without prior knowledge.

Contribution

The authors develop new adaptive algorithms that eliminate the dependence on lower-order terms in regret bounds for heavy-tailed losses, improving performance guarantees.

Findings

Achieve $ ilde{O}( oot{ heta T} ext{log}(K))$ regret bounds.

Remove dependence on maximum loss in regret guarantees.

Provide improved bounds for squared loss scenarios.

Abstract

We consider the problem setting of prediction with expert advice with possibly heavy-tailed losses, i.e. the only assumption on the losses is an upper bound on their second moments, denoted by $\theta$. We develop adaptive algorithms that do not require any prior knowledge about the range or the second moment of the losses. Existing adaptive algorithms have what is typically considered a lower-order term in their regret guarantees. We show that this lower-order term, which is often the maximum of the losses, can actually dominate the regret bound in our setting. Specifically, we show that even with small constant $\theta$, this lower-order term can scale as $\sqrt{KT}$, where $K$ is the number of experts and $T$ is the time horizon. We propose adaptive algorithms with improved regret bounds that avoid the dependence on such a lower-order term and guarantee $\mathcal{O}(\sqrt{\theta T\log(K)})$ regret in the worst case, and $\mathcal{O}(\theta \log(KT)/\Delta_{\min})$ regret when the losses are sampled i.i.d. from some fixed distribution, where $\Delta_{\min}$ is the difference between the mean losses of the second best expert and the best expert. Additionally, when the loss function is the squared loss, our algorithm also guarantees improved regret bounds over prior results.