Median-of-Means as an Extremal Convex Estimator and a Nonconvex Route to the Trimmed Oracle

Abstract:We revisit median-of-means estimation from a deterministic optimization viewpoint and develop a family of block-Lp estimators for robust learning with heavy-tailed and adversarially corrupted data. In a block contamination model with at least a fraction 1 minus epsilon of good blocks, we first show that every convex block M-estimator has worst-case robustness constant at least 1 divided by 1 minus 2 epsilon. This matches the classical median-of-means bound and proves that the trimmed-block oracle constant 1 divided by 1 minus epsilon cannot be attained within the convex class. We then introduce a nonconvex block-Lp family for p between 0 and 1 and derive finite-sample deterministic robustness bounds for all global minimizers. As p decreases from 1 toward 0, these bounds continuously approach the trimmed-block oracle constant. For sufficiently small p, the global minimizers coincide with those of the oracle under a mild separation condition. We also show that the block-Lp objectives have a benign landscape, with all local minima remaining close to the truth and no bad basins. Combining these results with block-level concentration yields sub-Gaussian deviation bounds under finite 2 plus delta moments and high-dimensional extensions to robust mean estimation and sparse regression.

Machine Learning (cs.LG)

arXiv:2609.01689 [cs.LG]

(or arXiv:2609.01689v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.01689 Focus to learn more arXiv-issued DOI via DataCite

Majumdar, A. Median-of-Means as an Extremal Convex Estimator and a Nonconvex Route to the Trimmed Oracle. Mach Learn 115, 172 (2026)

https://doi.org/10.1007/s10994-026-07101-2 Focus to learn more DOI(s) linking to related resources

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