Finite-sample and asymptotic analysis of generalization ability with an application to penalized regression

TL;DR

This paper provides a comprehensive analysis of the generalization ability of extremum estimators, deriving bounds on out-of-sample errors, exploring hyper-parameter tuning, and establishing $L_2$-consistency for penalized regression in high-dimensional settings.

Contribution

It introduces new bounds on prediction errors, links generalization ability to hyper-parameter tuning, and proves $L_2$-consistency of penalized regression estimators in both high-dimensional and low-dimensional cases.

Findings

Derived upper bounds on out-of-sample prediction errors.

Showed how cross-validation parameter $K$ influences bias-variance trade-off.

Proved $L_2$-consistency of penalized regression estimates.

Abstract

In this paper, we study the performance of extremum estimators from the perspective of generalization ability (GA): the ability of a model to predict outcomes in new samples from the same population. By adapting the classical concentration inequalities, we derive upper bounds on the empirical out-of-sample prediction errors as a function of the in-sample errors, in-sample data size, heaviness in the tails of the error distribution, and model complexity. We show that the error bounds may be used for tuning key estimation hyper-parameters, such as the number of folds $K$ in cross-validation. We also show how $K$ affects the bias-variance trade-off for cross-validation. We demonstrate that the $\mathcal{L}_2$-norm difference between penalized and the corresponding un-penalized regression estimates is directly explained by the GA of the estimates and the GA of empirical moment conditions. Lastly, we prove that all penalized regression estimates are $L_2$-consistent for both the $n \geqslant p$ and the $n < p$ cases. Simulations are used to demonstrate key results. Keywords: generalization ability, upper bound of generalization error, penalized regression, cross-validation, bias-variance trade-off, $\mathcal{L}_2$ difference between penalized and unpenalized regression, lasso, high-dimensional data.