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The problem of linear predictions has been extensively studied for the pastcentury under pretty generalized frameworks. Recent advances in the robuststatistics literature allow us to analyze robust versions of classical linearmodels through the prism of Median of Means (MoM). Combining these approachesin a piecemeal way might lead to ad-hoc procedures, and the restrictedtheoretical conclusions that underpin each individual contribution may nolonger be valid. To meet these challenges coherently, in this study, we offer aunified robust framework that includes a broad variety of linear predictionproblems on a Hilbert space, coupled with a generic class of loss functions.Notably, we do not require any assumptions on the distribution of the outlyingdata points ($\mathcal{O}$) nor the compactness of the support of the inlyingones ($\mathcal{I}$). Under mild conditions on the dual norm, we show that formisspecification level $\epsilon$, these estimators achieve an error rate of$O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1}\right\}+\epsilon)$, matching the best-known rates in literature. This rate isslightly slower than the classical rates of $O(n^{-1/2})$, indicating that weneed to pay a price in terms of error rates to obtain robust estimates.Additionally, we show that this rate can be improved to achieve so-called "fastrates" under additional assumptions.

Robust Linear Predictions: Analyses of Uniform Concentration, Fast Rates and Model Misspecification