Loss Minimization and Parameter Estimation with Heavy Tails

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 O(d log(1/d)) random samples to obtain a constant factor approximation to the optimal least squares loss with probability 1-d, 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.