Temporal Variability in Implicit Online Learning

TL;DR

This paper analyzes implicit online learning algorithms, revealing that their regret depends on the temporal variability of loss functions, and introduces an adaptive method that achieves optimal regret bounds without prior knowledge of this variability.

Contribution

The authors provide a novel regret analysis linking variability to performance, and develop an adaptive algorithm with matching lower bounds, advancing understanding of implicit online learning.

Findings

Regret can be constant with constant temporal variability.

An adaptive algorithm matches the theoretical regret bounds.

Empirical validation on classification and regression datasets.

Abstract

In the setting of online learning, Implicit algorithms turn out to be highly successful from a practical standpoint. However, the tightest regret analyses only show marginal improvements over Online Mirror Descent. In this work, we shed light on this behavior carrying out a careful regret analysis. We prove a novel static regret bound that depends on the temporal variability of the sequence of loss functions, a quantity which is often encountered when considering dynamic competitors. We show, for example, that the regret can be constant if the temporal variability is constant and the learning rate is tuned appropriately, without the need of smooth losses. Moreover, we present an adaptive algorithm that achieves this regret bound without prior knowledge of the temporal variability and prove a matching lower bound. Finally, we validate our theoretical findings on classification and regression datasets.