Distribution-Free Robust Linear Regression

• URL: http://arxiv.org/abs/2102.12919v1
• Date: Thu, 25 Feb 2021 15:10:41 GMT
• Title: Distribution-Free Robust Linear Regression
• Authors: Jaouad Mourtada and Tomas Va\v{s}kevi\v{c}ius and Nikita Zhivotovskiy
• Abstract summary: We study random design linear regression with no assumptions on the distribution of the covariates. We construct a non-linear estimator achieving excess risk of order $d/n$ with the optimal sub-exponential tail. We prove an optimal version of the classical bound for the truncated least squares estimator due to Gy"orfi, Kohler, Krzyzak, and Walk.
• Score: 5.532477732693
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study random design linear regression with no assumptions on the distribution of the covariates and with a heavy-tailed response variable. When learning without assumptions on the covariates, we establish boundedness of the conditional second moment of the response variable as a necessary and sufficient condition for achieving deviation-optimal excess risk rate of convergence. In particular, combining the ideas of truncated least squares, median-of-means procedures and aggregation theory, we construct a non-linear estimator achieving excess risk of order $d/n$ with the optimal sub-exponential tail. While the existing approaches to learning linear classes under heavy-tailed distributions focus on proper estimators, we highlight that the improperness of our estimator is necessary for attaining non-trivial guarantees in the distribution-free setting considered in this work. Finally, as a byproduct of our analysis, we prove an optimal version of the classical bound for the truncated least squares estimator due to Gy\"{o}rfi, Kohler, Krzyzak, and Walk.

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