List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussians

TL;DR

This paper introduces new algorithms for list-decodable Gaussian mean estimation and learning mixtures of spherical Gaussians, achieving improved guarantees and weaker separation conditions, with theoretical bounds and practical implications.

Contribution

It develops efficient algorithms with better error bounds for list-decodable mean estimation and Gaussian mixture learning, using novel polynomial techniques and weaker separation assumptions.

Findings

Algorithm achieves error $O( ext{poly}(rac{1}{eta}))$ for mean estimation.

Learns Gaussian mixtures with separation $\Omega(k^{\epsilon})$, improving over previous bounds.

Provides lower bounds indicating near-optimality of the algorithms.

Abstract

We study the problem of list-decodable Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for these problems. {\bf List-Decodable Mean Estimation.} Fix any $d \in \mathbb{Z}_+$ and $0< \alpha <1/2$. We design an algorithm with runtime $O (\mathrm{poly}(n/\alpha)^{d})$ that outputs a list of $O(1/\alpha)$ many candidate vectors such that with high probability one of the candidates is within $\ell_2$-distance $O(\alpha^{-1/(2d)})$ from the true mean. The only previous algorithm for this problem achieved error $\tilde O(\alpha^{-1/2})$ under second moment conditions. For $d = O(1/\epsilon)$, our algorithm runs in polynomial time and achieves error $O(\alpha^{\epsilon})$. We also give a Statistical Query lower bound suggesting that the complexity of our algorithm is qualitatively close to best possible. {\bf Learning Mixtures of Spherical Gaussians.} We give a learning algorithm for mixtures of spherical Gaussians that succeeds under significantly weaker separation assumptions compared to prior work. For the prototypical case of a uniform mixture of $k$ identity covariance Gaussians we obtain: For any $\epsilon>0$, if the pairwise separation between the means is at least $\Omega(k^{\epsilon}+\sqrt{\log(1/\delta)})$, our algorithm learns the unknown parameters within accuracy $\delta$ with sample complexity and running time $\mathrm{poly} (n, 1/\delta, (k/\epsilon)^{1/\epsilon})$. The previously best known polynomial time algorithm required separation at least $k^{1/4} \mathrm{polylog}(k/\delta)$. Our main technical contribution is a new technique, using degree-$d$ multivariate polynomials, to remove outliers from high-dimensional datasets where the majority of the points are corrupted.