High-Dimensional Gaussian Mean Estimation under Realizable Contamination

TL;DR

This paper investigates the challenge of estimating the mean of a high-dimensional Gaussian distribution when data is missing in a structured adversarial manner, revealing fundamental limits and efficient algorithms within this contamination model.

Contribution

It establishes an information-computation gap for Gaussian mean estimation under realizable contamination, providing both lower bounds in the Statistical Query model and a nearly matching efficient algorithm.

Findings

Proves an SQ lower bound indicating computational hardness.

Develops an algorithm with near-optimal sample and runtime tradeoff.

Characterizes the complexity of mean estimation under structured missing data.

Abstract

We study mean estimation for a Gaussian distribution with identity covariance in $\mathbb{R}^d$ under a missing data scheme termed realizable $\epsilon$-contamination model. In this model an adversary can choose a function $r(x)$ between 0 and $\epsilon$ and each sample $x$ goes missing with probability $r(x)$. Recent work Ma et al., 2024 proposed this model as an intermediate-strength setting between Missing Completely At Random (MCAR) -- where missingness is independent of the data -- and Missing Not At Random (MNAR) -- where missingness may depend arbitrarily on the sample values and can lead to non-identifiability issues. That work established information-theoretic upper and lower bounds for mean estimation in the realizable contamination model. Their proposed estimators incur runtime exponential in the dimension, leaving open the possibility of computationally efficient algorithms in high dimensions. In this work, we establish an information-computation gap in the Statistical Query model (and, as a corollary, for Low-Degree Polynomials and PTF tests), showing that algorithms must either use substantially more samples than information-theoretically necessary or incur exponential runtime. We complement our SQ lower bound with an algorithm whose sample-time tradeoff nearly matches our lower bound. Together, these results qualitatively characterize the complexity of Gaussian mean estimation under $\epsilon$-realizable contamination.