Best-Case Lower Bounds in Online Learning

TL;DR

This paper explores the limits of how much an online learning algorithm can improve in the best case, providing bounds that match known upper bounds and analyzing their implications for fairness and adaptivity.

Contribution

It introduces a general method for establishing best-case lower bounds in FTRL algorithms with time-varying regularizers, aligning these bounds with existing upper regret bounds.

Findings

Best-case lower bounds are of the same order as upper regret bounds in various settings.

Linearized FTRL can achieve negative linear regret, contrasting with standard FTRL.

Complete characterization of best-case sequences in binary DTOL with two experts.

Abstract

Much of the work in online learning focuses on the study of sublinear upper bounds on the regret. In this work, we initiate the study of best-case lower bounds in online convex optimization, wherein we bound the largest improvement an algorithm can obtain relative to the single best action in hindsight. This problem is motivated by the goal of better understanding the adaptivity of a learning algorithm. Another motivation comes from fairness: it is known that best-case lower bounds are instrumental in obtaining algorithms for decision-theoretic online learning (DTOL) that satisfy a notion of group fairness. Our contributions are a general method to provide best-case lower bounds in Follow The Regularized Leader (FTRL) algorithms with time-varying regularizers, which we use to show that best-case lower bounds are of the same order as existing upper regret bounds: this includes situations with a fixed learning rate, decreasing learning rates, timeless methods, and adaptive gradient methods. In stark contrast, we show that the linearized version of FTRL can attain negative linear regret. Finally, in DTOL with two experts and binary predictions, we fully characterize the best-case sequences, which provides a finer understanding of the best-case lower bounds.