Conditional predictive inference for stable algorithms

TL;DR

This paper develops prediction intervals based on k-fold cross validation that are conditionally valid given the training data, especially effective with stable algorithms and high-dimensional data, providing reliable predictive inference.

Contribution

It introduces a finite sample analysis of cross validation residuals ensuring conditional coverage, applicable in high-dimensional, non-parametric settings with minimal assumptions.

Findings

Prediction intervals are close to nominal coverage under stability.

Results hold in high-dimensional linear models with increasing sample size and dimension.

Cross validation can provide reliable predictive inference even when parameter estimation fails.

Abstract

We investigate generically applicable and intuitively appealing prediction 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 sample analysis of the empirical distribution function of $k$-fold cross validation residuals and hold in non-parametric 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 inference on the unknown parameters (cf. Bickel and Freedman, 1983; El Karoui and Purdom, 2018; Mammen, 1996), cross validation methods can be successfully applied to obtain reliable predictive inference even in high dimensions and conditionally on the training data.