Slope heuristics and V-Fold model selection in heteroscedastic regression using strongly localized bases

TL;DR

This paper analyzes the effectiveness of slope heuristics, V-fold cross-validation, and V-fold penalization for model selection in heteroscedastic regression with strongly localized bases, establishing their asymptotic optimality and comparing their practical performance.

Contribution

It introduces a new class of strongly localized bases for regression and proves the asymptotic optimality of slope heuristics and V-fold penalization in this context.

Findings

Slope heuristics are asymptotically optimal when the penalty shape is known.

V-fold cross-validation is suboptimal for fixed V, recovering an oracle with reduced data.

V-fold penalization performs comparably to V-fold cross-validation in practice.

Abstract

We investigate the optimality for model selection of the so-called slope heuristics, $V$-fold cross-validation and $V$-fold penalization in a heteroscedastic with random design regression context. We consider a new class of linear models that we call strongly localized bases and that generalize histograms, piecewise polynomials and compactly supported wavelets. We derive sharp oracle inequalities that prove the asymptotic optimality of the slope heuristics---when the optimal penalty shape is known---and $V$ -fold penalization. Furthermore, $V$-fold cross-validation seems to be suboptimal for a fixed value of $V$ since it recovers asymptotically the oracle learned from a sample size equal to $1-V^{-1}$ of the original amount of data. Our results are based on genuine concentration inequalities for the true and empirical excess risks that are of independent interest. We show in our experiments the good behavior of the slope heuristics for the selection of linear wavelet models. Furthermore, $V$-fold cross-validation and $V$-fold penalization have comparable efficiency.