List-Decodable Mean Estimation in Nearly-PCA Time

Part of Advances in Neural Information Processing Systems 34 (NeurIPS 2021)

Authors

Ilias Diakonikolas, Daniel Kane, Daniel Kongsgaard, Jerry Li, Kevin Tian

Abstract

Robust statistics has traditionally focused on designing estimators tolerant to a minority of contaminated data. {\em List-decodable learning}~\cite{CharikarSV17} studies the more challenging regime where only a minority $\tfrac 1 k$ fraction of the dataset, $k \geq 2$, is drawn from the distribution of interest, and no assumptions are made on the remaining data. We study the fundamental task of list-decodable mean estimation in high dimensions. Our main result is a new algorithm for bounded covariance distributions with optimal sample complexity and near-optimal error guarantee, running in {\em nearly-PCA time}. Assuming the ground truth distribution on $\mathbb{R}^d$ has identity-bounded covariance, our algorithm outputs $O(k)$ candidate means, one of which is within distance $O(\sqrt{k\log k})$ from the truth. Our algorithm runs in time $\widetilde{O}(ndk)$, where $n$ is the dataset size. This runtime nearly matches the cost of performing $k$-PCA on the data, a natural bottleneck of known algorithms for (very) special cases of our problem, such as clustering well-separated mixtures. Prior to our work, the fastest runtimes were $\widetilde{O}(n^2 d k^2)$~\cite{DiakonikolasKK20}, and $\widetilde{O}(nd k^C)$ \cite{CherapanamjeriMY20} for an unspecified constant $C \geq 6$. Our approach builds on a novel soft downweighting method we term SIFT, arguably the simplest known polynomial-time mean estimator in the list-decodable setting. To develop our fast algorithms, we boost the computational cost of SIFT via a careful win-win-win'' analysis of an approximate Ky Fan matrix multiplicative weights procedure we develop, which may be of independent interest.