Outlier-robust estimation of a sparse linear model using ℓ_1-penalized Huber's M-estimator

04/12/2019

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by   Arnak S. Dalalyan, et al.

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We study the problem of estimating a p-dimensional s-sparse vector in a linear model with Gaussian design and additive noise. In the case where the labels are contaminated by at most o adversarial outliers, we prove that the ℓ_1-penalized Huber's M-estimator based on n samples attains the optimal rate of convergence (s/n)^1/2 + (o/n), up to a logarithmic factor. This is proved when the proportion of contaminated samples goes to zero at least as fast as 1/(n), but we argue that constant fraction of outliers can be achieved by slightly more involved techniques.