(CREST-ENSAE, IP Paris)

**Title — **Recent advances in robust mean estimation in high dimensions

**Abstract — **Arguably the first rigorously studied question in robust statistics is the mean (or the location parameter) estimation for contaminated Gaussian distributions, dating back to 1964, pioneered by P. Huber. The natural extension of this question is to consider it in high dimensions. This problem has received renewed attention in the last few years, both from statistical and computational aspects. In this talk, I review some recent advances in the statistical performance of mean estimators under the adversarial contamination model. First, from the practical point of view, one would require a robust estimator to have a polynomial runtime. I will discuss the challenges and crucial properties of such estimators. The typical minimax rate of computationally tractable estimators is the additional log factor in the dependence on the contamination level. Secondly, from the statistical point of view, despite the recent significant interest in robust statistics, achieving dimension-free bound with optimal dependence on contamination level in the canonical Gaussian case remained open. In [3], we constructed an estimator for the mean vector that is dimension-free and has optimal dependence on the contamination level. Previously known results were either dimension-dependent and required a covariance matrix to be close to identity, or had a sub-optimal dependence on the contamination level.

**References**

The talk will be built on the following papers:

1. Dalalyan A.S. and Minasyan A. All-in-one robust estimator of the Gaussian mean. The Annals of Statistics (50) 1193–1219, 2022.

2. Bateni A.-H., Minasyan A. and Dalalyan A. Nearly minimax robust estimator of the mean vector by iterative spectral dimension reduction. arXiv preprint arXiv:2204.02323, 2022.

3. A. Minasyan and N. Zhivotovskiy. Statistically optimal robust mean and covariance estimation for anisotropic Gaussians, arXiv preprint arXiv:2301.09024, 2023.

March 08, 2023