Oracle Inequalities for High-dimensional Prediction

TL;DR

This paper establishes a general oracle inequality for high-dimensional linear regression with penalized estimators, showing they can provide consistent prediction regardless of the design matrix, aiding in estimator evaluation.

Contribution

It introduces a broad oracle inequality applicable to various penalized estimators in high-dimensional prediction, independent of the design matrix.

Findings

The bound applies to a wide range of penalized estimators.

It demonstrates consistent prediction is achievable with any design matrix.

The result aids in assessing the potential and accuracy of estimators.

Abstract

The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such methods. Since the proof relies only on convexity and continuity arguments, the result holds irrespective of the design matrix and applies to a wide range of penalized estimators. Overall, the bound demonstrates that generic estimators can provide consistent prediction with any design matrix. From a practical point of view, the bound can help to identify the potential of specific estimators, and they can help to get a sense of the prediction accuracy in a given application.