High-Dimensional Robust Mean Estimation in Nearly-Linear Time

We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent error guarantees for several families of structured distributions. In this work, we give the first nearly-linear time algorithms for high-dimensional robust mean estimation. Specifically, we focus on distributions with (i) known covariance and sub-gaussian tails, and (ii) unknown bounded covariance. Given $N$ samples on $\mathbb{R}^d$, an $ε$-fraction of which may be arbitrarily corrupted, our algorithms run in time $\tilde{O}(Nd) / \mathrm{poly}(ε)$ and approximate the true mean within the information-theoretically optimal error, up to constant factors. Previous robust algorithms with comparable error guarantees have running times $\tildeΩ(N d^2)$, for $ε= Ω(1)$. Our algorithms rely on a natural family of SDPs parameterized by our current guess $ν$ for the unknown mean $μ^\star$. We give a win-win analysis establishing the following: either a near-optimal solution to the primal SDP yields a good candidate for $μ^\star$ -- independent of our current guess $ν$ -- or the dual SDP yields a new guess $ν'$ whose distance from $μ^\star$ is smaller by a constant factor. We exploit the special structure of the corresponding SDPs to show that they are approximately solvable in nearly-linear time. Our approach is quite general, and we believe it can also be applied to obtain nearly-linear time algorithms for other high-dimensional robust learning problems.