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arXiv · 1307.1827

Loss minimization and parameter estimation with heavy tails

Daniel Hsu·Sivan Sabato
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Abstract

This work studies applications and generalizations of a simple estimation technique that provides exponential concentration under heavy-tailed distributions, assuming only bounded low-order moments. We show that the technique can be used for approximate minimization of smooth and strongly convex losses, and specifically for least squares linear regression. For instance, our $d$-dimensional estimator requires just $\tilde{O}(d\log(1/\delta))$ random samples to obtain a constant factor approximation to the optimal least squares loss with probability $1-\delta$, without requiring the covariates or noise to be bounded or subgaussian. We provide further applications to sparse linear regression and low-rank covariance matrix estimation with similar allowances on the noise and covariate distributions. The core technique is a generalization of the median-of-means estimator to arbitrary metric spaces.

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Daniel Hsu, Sivan Sabato. 2013-07-07. Loss minimization and parameter estimation with heavy tails. https://arxiv.org/abs/1307.1827

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