CONDITIONAL PREDICTIVE INFERENCE FOR STABLE ALGORITHMS

Lukas Steinberger

doi:10.1214/22-AOS2250

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CONDITIONAL PREDICTIVE INFERENCE FOR STABLE ALGORITHMS

2023-02-01

Lukas Steinberger *

Hannes Leeb

DOI：10.1214/22-AOS2250

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Abstract

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We investigate generically applicable and intuitively appealing predic-tion intervals based on k-fold cross-validation. We focus on the conditional coverage probability of the proposed intervals, given the observations in the training sample (hence, training conditional validity), and show that it is close to the nominal level, in an appropriate sense, provided that the underlying algorithm used for computing point predictions is sufficiently stable when feature-response pairs are omitted. Our results are based on a finite sam-ple analysis of the empirical distribution function of k-fold cross-validation residuals and hold in nonparametric settings with only minimal assumptions on the error distribution. To illustrate our results, we also apply them to high -dimensional linear predictors, where we obtain uniform asymptotic training conditional validity as both sample size and dimension tend to infinity at the same rate and consistent parameter estimation typically fails. These results show that despite the serious problems of resampling procedures for infer-ence on the unknown parameters (cf. in A Festschrift for Erich L. Lehmann (1983) 28???48 Wadsworth; Ann. Statist. 24 (1996) 307???335; J. Mach. Learn. Res. 19 (2018) 5), cross-validation methods can be successfully applied to ob-tain reliable predictive inference even in high dimensions and conditionally on the training data.

Keywords:

Prediction intervals

high-dimensional regression

algorithmic stability

cross-validation