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Outlier-robust estimation of a sparse linear model using $\ell_1$-penalized Huber's $M$-estimator
Arnak Dalalyan · Philip Thompson

Thu Dec 12 10:45 AM -- 12:45 PM (PST) @ East Exhibition Hall B + C #20
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.

Author Information

Arnak Dalalyan (ENSAE ParisTech)
Philip Thompson (University of Cambridge, Statistical Laboratory)

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