2019/03/19 by Samuel B. Hopkins, Hopkins, Samuel B., Jerry Li +1 · 8 citations
Computer Science · Mathematics · #Adversarial Robustness in Machine Learning #Algorithm #Applied mathematics #Bounded function #Combinatorics #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Computer science #Covariance #Data Structures and Algorithms (cs.DS) #Dimension (graph theory) #FOS: Computer and information sciences #FOS: Mathematics #Gaussian #Machine Learning and Algorithms #Mathematical analysis #Mathematics #Polynomial #Set (abstract data type) #Statistics #Statistics Theory (math.ST) #Time complexity #cs.CC #cs.DS #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1903.07870
published in arXiv (Cornell University) (Cornell University) · Conference on Learning Theory (COLT) 2019
openalex publication_date 2019/03/19 · arxiv created 2019/06/03 · arxiv updated 2019/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Robust mean estimation is the problem of estimating the mean μ∈ ℝd of a d-dimensional distribution D from a list of independent samples, an ε-fraction of which have been arbitrarily corrupted by a malicious adversary. Recent algorithmic progress has resulted in the first polynomial-time algorithms which achieve dimension-independent rates of error: for instance, if D has covariance I, in polynomial-time one may find μ with ‖μ- μ‖ ≤ O(√ε). However, error rates achieved by current polynomial-time algorithms, while dimension-independent, are sub-optimal in many natural settings, such as when D is sub-Gaussian, or has bounded 4-th moments. In this work we give worst-case complexity-theoretic evidence that improving on the error rates of current polynomial-time algorithms for robust mean estimation may be computationally intractable in natural settings. We show that several natural approaches to improving error rates of current polynomial-time robust mean estimation algorithms would imply efficient algorithms for the small-set expansion problem, refuting Raghavendra and Steurer's small-set expansion hypothesis (so long as P ≠ NP). We also give the first direct reduction to the robust mean estimation problem, starting from a plausible but nonstandard variant of the small-set expansion problem.