$\ell_1$-regression with Heavy-tailed Distributions
This work addresses robust regression for heavy-tailed distributions, offering theoretical guarantees that relax stringent assumptions, though it is incremental in extending existing ℓ1-regression methods.
The paper tackles linear regression with heavy-tailed data by using absolute loss for conditional median estimation and proposes a truncated minimization method to achieve an excess risk bound of O~(√(d/n)) without requiring exponential moment conditions on inputs and outputs.
In this paper, we consider the problem of linear regression with heavy-tailed distributions. Different from previous studies that use the squared loss to measure the performance, we choose the absolute loss, which is capable of estimating the conditional median. To address the challenge that both the input and output could be heavy-tailed, we propose a truncated minimization problem, and demonstrate that it enjoys an $\widetilde{O}(\sqrt{d/n})$ excess risk, where $d$ is the dimensionality and $n$ is the number of samples. Compared with traditional work on $\ell_1$-regression, the main advantage of our result is that we achieve a high-probability risk bound without exponential moment conditions on the input and output. Furthermore, if the input is bounded, we show that the classical empirical risk minimization is competent for $\ell_1$-regression even when the output is heavy-tailed.