针对非均匀污染率下的鲁棒估计，提出最优方法并给出理论极限。
Robust Estimation Under Heterogeneous Corruption Rates
- 根据各样本污染概率差异设计鲁棒估计策略
- 在高维高斯均值和线性回归中达到近最优误差界
- 适合分布式学习、众包等存在异质噪声的场景
我们研究异质污染率下的鲁棒估计问题，其中每个样本独立地以已知但不相同的概率被污染。该设置自然出现在分布式学习、联邦学习和传感器网络中，但现有鲁棒估计器通常假设统一或最坏情况下的污染，忽略了结构异质性。对于多元有界分布和一元高斯分布的均值估计，我们给出了所有异质污染模式下的紧致极小极大率。对于多元高斯均值估计和线性回归，我们建立了平方误差的极小极大率，精度为维度d的√d倍。粗略而言，我们的发现表明：超过某一污染阈值的样本会被最优估计器丢弃——该阈值由给定污染率的经验分布决定。
原文摘要 · Abstract (English)
We study the problem of robust estimation under heterogeneous corruption rates, where each sample may be independently corrupted with a known but non-identical probability. This setting arises naturally in distributed and federated learning, crowdsourcing, and sensor networks, yet existing robust estimators typically assume uniform or worst-case corruption, ignoring structural heterogeneity. For mean estimation for multivariate bounded distributions and univariate gaussian distributions, we give tight minimax rates for all heterogeneous corruption patterns. For multivariate gaussian mean estimation and linear regression, we establish the minimax rate for squared error up to a factor of $\sqrt{d}$, where $d$ is the dimension. Roughly, our findings suggest that samples beyond a certain corruption threshold may be discarded by the optimal estimators -- this threshold is determined by the empirical distribution of the corruption rates given.
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