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# Online Linear Regression in Dynamic Environments via Discounting

*Proceedings of the 41st International Conference on Machine Learning*, PMLR 235:21083-21120, 2024.

#### Abstract

We develop algorithms for online linear regression which achieve optimal static and dynamic regret guarantees

*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 $R_{T}(\vec{u})\le O\Big(d\log(T)\vee \sqrt{dP_{T}^{\gamma}(\vec{u})T}\Big)$, 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*strongly-adaptive*guarantees which hold over every sub-interval $[a,b]\subseteq[1,T]$ simultaneously.