发表机构
IIIT Delhi（德里印度信息技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究从确定性优化视角分析均值中位数法，提出非凸块Lp估计器，证明其鲁棒性界可趋近修剪最优估计器常数，且损失曲面良好，可扩展至高维鲁棒学习任务。
AI 中文摘要
我们从确定性优化视角重新审视均值中位数法（median-of-means）估计，并开发了一类块Lp估计器，用于对重尾分布数据及受对抗污染数据进行鲁棒学习。在块污染模型中，若至少有1-ε比例的块为良好块，我们首先证明，每个凸块M估计器的最坏情况鲁棒性常数至少为1/(1-2ε)，这与经典均值中位数法的界一致，同时证明了修剪块最优估计器的常数1/(1-ε)无法在凸类中实现。随后，我们引入p∈(0,1)的非凸块Lp家族，并推导了所有全局极小值的有限样本确定性鲁棒性界。当p从1向0减小时，这些界会连续趋近于修剪块最优估计器的常数；在适度分离条件下，对于足够小的p，全局极小值与最优估计器的全局极小值重合。我们还证明，块Lp目标具有良好的损失曲面，所有局部极小值均接近真实值且不存在不良盆地。结合这些结果与块级集中性，可在有限的2+δ矩条件下得到亚高斯偏差界，并将其扩展至高维情形下的鲁棒均值估计与稀疏回归。
英文摘要
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.
Journal refMajumdar, A. Median-of-Means as an Extremal Convex Estimator and a Nonconvex Route to the Trimmed Oracle. Mach Learn 115, 172 (2026)
DOI：10.1007/s10994-026-07101-2
