Agnostic Smoothed Online Regression with Adversarial Responses

Xuanyu Chen, Yue Yu

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…