Combining Adversarial Guarantees and Stochastic Fast Rates in Online Learning

TL;DR

This paper demonstrates that certain online learning algorithms can simultaneously guarantee worst-case regret in adversarial settings and adapt to favorable stochastic environments, achieving fast rates under the Bernstein condition.

Contribution

It shows that Squint and MetaGrad algorithms adapt automatically to stochastic environments characterized by the Bernstein condition, attaining fast rates.

Findings

Algorithms achieve fast rates in stochastic environments.

Algorithms maintain worst-case guarantees in adversarial settings.

High-probability bounds are established for the algorithms.

Abstract

We consider online learning algorithms that guarantee worst-case regret rates in adversarial environments (so they can be deployed safely and will perform robustly), yet adapt optimally to favorable stochastic environments (so they will perform well in a variety of settings of practical importance). We quantify the friendliness of stochastic environments by means of the well-known Bernstein (a.k.a. generalized Tsybakov margin) condition. For two recent algorithms (Squint for the Hedge setting and MetaGrad for online convex optimization) we show that the particular form of their data-dependent individual-sequence regret guarantees implies that they adapt automatically to the Bernstein parameters of the stochastic environment. We prove that these algorithms attain fast rates in their respective settings both in expectation and with high probability.