  The goal of this paper is to show that a single robust estimator of the mean
of a multivariate Gaussian distribution can enjoy five desirable properties.
First, it is computationally tractable in the sense that it can be computed in
a time which is at most polynomial in dimension, sample size and the logarithm
of the inverse of the contamination rate. Second, it is equivariant by
translations, uniform scaling and orthogonal transformations. Third, it has a
high breakdown point equal to $0.5$, and a nearly-minimax-rate-breakdown point
approximately equal to $0.28$. Fourth, it is minimax rate optimal, up to a
logarithmic factor, when data consists of independent observations corrupted by
adversarially chosen outliers. Fifth, it is asymptotically efficient when the
rate of contamination tends to zero. The estimator is obtained by an iterative
reweighting approach. Each sample point is assigned a weight that is
iteratively updated by solving a convex optimization problem. We also establish
a dimension-free non-asymptotic risk bound for the expected error of the
proposed estimator. It is the first result of this kind in the literature and
involves only the effective rank of the covariance matrix. Finally, we show
that the obtained results can be extended to sub-Gaussian distributions, as
well as to the cases of unknown rate of contamination or unknown covariance
matrix.
