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arXiv · 2609.32478

Agnostic Smoothed Online Regression with Adversarial Responses

Xuanyu Chen·Yue Yu
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Abstract

We study smoothed online prediction with bounded adversarial responses. This widely studied framework bridges i.i.d. sampling and adversarial covariate selection through a smoothness parameter $\textsf{C}_{\textsf{cov}}$, which bounds conditional covariate densities relative to a fixed, unknown base measure. We propose \textsc{Hedge-Cover}, an information-theoretic algorithm that achieves sublinear regret $\widetilde{O}(\sqrt{\text{Pdim}(\mathcal{F}) \textsf{C}_{\textsf{cov}} T})$ for function classes with bounded pseudo-dimension. The algorithm aggregates a carefully constructed family of experts using \textsc{Hedge}, with a prior that links regret to the number of disagreements between a consistent selector and a target function. We bound this number by exploiting covariate smoothness. This answers an open problem posed in \cite{blanchard2025agnostic} on the minimax optimal adaptive regret of the smoothed online regression problem. We establish a matching lower bound for the class of linear predictors. The main intricacy of the lower bound lies in explicitly constructing a challenging sequential covariate distribution supported on mutually orthogonal hyperplanes. This construction may be of independent technical interest. Finally, we revisit the well-specified setting and quantify the effect of response noise. For conditionally $ν^2$-subGaussian responses, we extend the existing lower bound under realizable responses by showing that the minimax expected regret is $Ω((1\vee ν)\sqrt{(\textsf{C}_{\textsf{cov}}-1)dT})$ for a function class of VC dimension $d$. A corresponding upper bound for ERM matches this dependence on $ν$.

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BibTeXRIS

Xuanyu Chen, Yue Yu. 2026-09-26. Agnostic Smoothed Online Regression with Adversarial Responses. https://arxiv.org/abs/2609.32478

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