Online Linear Regression in Dynamic Environments via Discounting

TL;DR

This paper introduces algorithms for online linear regression in changing environments that adapt without prior knowledge, achieving optimal regret bounds and strong adaptability over all sub-intervals.

Contribution

It presents a novel discounted Vovk-Azoury-Warmuth forecaster with optimal dynamic regret guarantees and on-the-fly learned discount factors, extending to strongly adaptive settings.

Findings

Achieves optimal dynamic regret bounds in online linear regression.

Learns discount factors adaptively during online learning.

Provides strong guarantees over all sub-intervals of the data sequence.

Abstract

We develop algorithms for online linear regression which achieve optimal static and dynamic regret guarantees \emph{even in the complete absence of prior knowledge}. We present a novel analysis showing that a discounted variant of the Vovk-Azoury-Warmuth forecaster achieves dynamic regret of the form $R_{T}(\vec{u})\le O\left(d\log(T)\vee \sqrt{dP_{T}^{\gamma}(\vec{u})T}\right)$, where $P_{T}^{\gamma}(\vec{u})$ is a measure of variability of the comparator sequence, and show that the discount factor achieving this result can be learned on-the-fly. We show that this result is optimal by providing a matching lower bound. We also extend our results to \emph{strongly-adaptive} guarantees which hold over every sub-interval $[a,b]\subseteq[1,T]$ simultaneously.