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The goal of this paper is to show that a single robust estimator of the meanof a multivariate Gaussian distribution can enjoy five desirable properties.First, it is computationally tractable in the sense that it can be computed ina time which is at most polynomial in dimension, sample size and the logarithmof the inverse of the contamination rate. Second, it is equivariant bytranslations, uniform scaling and orthogonal transformations. Third, it has ahigh breakdown point equal to $0.5$, and a nearly-minimax-rate-breakdown pointapproximately equal to $0.28$. Fourth, it is minimax rate optimal, up to alogarithmic factor, when data consists of independent observations corrupted byadversarially chosen outliers. Fifth, it is asymptotically efficient when therate of contamination tends to zero. The estimator is obtained by an iterativereweighting approach. Each sample point is assigned a weight that isiteratively updated by solving a convex optimization problem. We also establisha dimension-free non-asymptotic risk bound for the expected error of theproposed estimator. It is the first result of this kind in the literature andinvolves only the effective rank of the covariance matrix. Finally, we showthat the obtained results can be extended to sub-Gaussian distributions, aswell as to the cases of unknown rate of contamination or unknown covariancematrix.

All-In-One Robust Estimator of the Gaussian Mean