How Hard Is Robust Mean Estimation?

Robust mean estimation is the problem of estimating the mean $\mu \in \mathbb{R}^d$ of a $d$-dimensional distribution $D$ from a list of independent samples, an $\epsilon$-fraction of which have been arbitrarily corrupted by a malicious adversary. Recent algorithmic progress has resulted in the first polynomial-time algorithms which achieve \emph{dimension-independent} rates of error: for instance, if $D$ has covariance $I$, in polynomial-time one may find $\hat{\mu}$ with $\|\mu - \hat{\mu}\| \leq O(\sqrt{\epsilon})$. 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 $P \neq NP$). 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.