Outlier-robust estimation of a sparse linear model using $\ell_1$-penalized Huber's $M$-estimator

Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)

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Arnak Dalalyan, Philip Thompson


We study the problem of estimating a $p$-dimensional $s$-sparse vector in a linear model with Gaussian design. In the case where the labels are contaminated by at most $o$ adversarial outliers, we prove that the $\ell_1$-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. For more general design matrices, our results highlight the importance of two properties: the transfer principle and the incoherence property. These properties with suitable constants are shown to yield the optimal rates of robust estimation with adversarial contamination.