Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, May 20, 2021, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Gleb Novikov

Consistent regression when oblivious outliers overwhelm

We consider a robust linear regression model y=Xβ+η, where an adversary oblivious to the design matrix X may choose η to be an arbitrary vector with 0 < α < 1 fraction of entries bounded by 1 in absolute value. We show that for Gaussian design matrices the Huber loss estimator for β is consistent for every sample size n= ω(d/α²) and achieves an error rate of O(d/α²n)^1/2. Both bounds are optimal (up to constant factors). Prior to our work no estimator was known to be consistent in this model except for quadratic sample size n > d² or for logarithmic inlier fraction α > 1/log n. Our results extend to designs far beyond the Gaussian case and, under some symmetry assumptions on the noise, only require the column span of X to not contain approximately sparse vectors.
This is a joint work with Tommaso d'Orsi and David Steurer.

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