Online Linear Regression in Dynamic Environments via Discounting

TL;DR

This work addresses online linear regression in dynamic, nonstationary environments without any prior knowledge. It introduces a discounted Vovk-Azoury-Warmuth forecaster that forgets old data with factor $\gamma$ and optionally uses hints $\tilde{y}_t$, achieving dynamic regret bounds of the form $R_T(\boldsymbol{u})=O\big(d\log T \;\vee\; \sqrt{d P_T^\gamma(\boldsymbol{u})T}\big)$ and small-loss variants. The authors prove a matching dimension-dependent lower bound, show that the optimal discount factor can be learned on the fly via a grid of experts and a clipping-based meta-algorithm, and extend the guarantees to strongly adaptive bounds that hold on every sub-interval. These contributions establish optimal, prior-knowledge-free guarantees for online regression in dynamic environments and pave the way for robust, nonstationary learning in unbounded domains. The results have potential impact on streaming analytics and adaptive decision-making where data distributions drift over time.

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 $R_{T}(\vec{u})\le O\left(d\log(T)\vee \sqrt{dP_{T}^γ(\vec{u})T}\right)$, where $P_{T}^γ(\vec{u})$ 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.

Theorems & Definitions (60)

• Theorem 2.1
• Theorem 3.1
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Proposition 1.0