Quasi-Likelihood and/or Robust Estimation in High Dimensions

TL;DR

This paper extends theoretical results for high-dimensional generalized linear models with Lasso to include quasi-likelihood and robust loss functions, providing bounds on prediction and estimation errors under certain conditions.

Contribution

It introduces new theoretical bounds for quasi-likelihood estimators in high dimensions, including robustness considerations and false positive control.

Findings

Bounds for prediction error and ℓ₁-error established

Robust loss functions analyzed under fourth moment conditions

No false positives under irrepresentable condition

Abstract

We consider the theory for the high-dimensional generalized linear model with the Lasso. After a short review on theoretical results in literature, we present an extension of the oracle results to the case of quasi-likelihood loss. We prove bounds for the prediction error and $\ell_1$-error. The results are derived under fourth moment conditions on the error distribution. The case of robust loss is also given. We moreover show that under an irrepresentable condition, the $\ell_1$-penalized quasi-likelihood estimator has no false positives.