Online Linear Regression in Dynamic Environments via Discounting

• URL: http://arxiv.org/abs/2405.19175v1
• Date: Wed, 29 May 2024 15:17:53 GMT
• Title: Online Linear Regression in Dynamic Environments via Discounting
• Authors: Andrew Jacobsen, Ashok Cutkosky,
• Abstract summary: We develop algorithms for online linear regression which achieve optimal static and dynamic regret guarantees. We show that a discounted variant of the Vovk-Azoury-Warmuth forecaster achieves dynamic regret of the form $R_T(vecu)le Oleft(dlog(T)vee sqrtdP_Tgamma(vecu)$, where $P_Tgamma(vecu)$ is a measure of variability of the comparator sequence, and show that the
• Score: 38.15622843661145
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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