Outlier-Robust Sparse Mean Estimation for Heavy-Tailed DistributionsDownload PDF

Published: 31 Oct 2022, 18:00, Last Modified: 14 Oct 2022, 20:19NeurIPS 2022 AcceptReaders: Everyone
Keywords: sparse estimation, robust statistics, heavy-tailed estimation
TL;DR: We develop the first sample-efficient and polynomial-time algorithm for robust sparse mean estimation for heavy-tailed data.
Abstract: We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity. Specifically, given a small number of corrupted samples from a high-dimensional heavy-tailed distribution whose mean $\mu$ is guaranteed to be sparse, the goal is to efficiently compute a hypothesis that accurately approximates $\mu$ with high probability. Prior work had obtained efficient algorithms for robust sparse mean estimation of light-tailed distributions. In this work, we give the first sample-efficient and polynomial-time robust sparse mean estimator for heavy-tailed distributions under mild moment assumptions. Our algorithm achieves the optimal asymptotic error using a number of samples scaling logarithmically with the ambient dimension. Importantly, the sample complexity of our method is optimal as a function of the failure probability $\tau$, having an {\em additive} $\log(1/\tau)$ dependence. Our algorithm leverages the stability-based approach from the algorithmic robust statistics literature, with crucial (and necessary) adaptations required in our setting. Our analysis may be of independent interest, involving the delicate design of a (non-spectral) decomposition for positive semi-definite matrices satisfying certain sparsity properties.
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