SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications
Ilias Diakonikolas, Samuel B. Hopkins, Ankit Pensia, Stefan Tiegel
TL;DR
It is proved that there is a universal constant C>0 so that for every d ∈ ℕ, every centered subgaussian distribution D on ℝd, and every even p ∈ ℕ, the d-variate polynomial (Cp)p/2 is a sum of square polynomials, establishing that every subgaussian distribution is SoS-certifiably subgaussian.
Abstract
We prove that there is a universal constant $C>0$ so that for every $d \in \mathbb N$, every centered subgaussian distribution $\mathcal D$ on $\mathbb R^d$, and every even $p \in \mathbb N$, the $d$-variate polynomial $(Cp)^{p/2} \cdot \|v\|_{2}^p - \mathbb E_{X \sim \mathcal D} \langle v,X\rangle^p$ is a sum of square polynomials. This establishes that every subgaussian distribution is \emph{SoS-certifiably subgaussian} -- a condition that yields efficient learning algorithms for a wide variety of high-dimensional statistical tasks. As a direct corollary, we obtain computationally efficient algorithms with near-optimal guarantees for the following tasks, when given samples from an arbitrary subgaussian distribution: robust mean estimation, list-decodable mean estimation, clustering mean-separated mixture models, robust covariance-aware mean estimation, robust covariance estimation, and robust linear regression. Our proof makes essential use of Talagrand's generic chaining/majorizing measures theorem.