Risk and cross validation in ridge regression with correlated samples

TL;DR

This paper develops asymptotic risk formulas for ridge regression with correlated samples, showing limitations of standard GCV and proposing CorrGCV for better risk estimation, especially in time series contexts.

Contribution

It extends high-dimensional ridge regression theory to correlated data, introduces CorrGCV, and analyzes risks in time series forecasting scenarios.

Findings

GCV fails with correlated samples

CorrGCV provides unbiased risk estimates

Time series test points can lead to overly optimistic risk predictions

Abstract

Recent years have seen substantial advances in our understanding of high-dimensional ridge regression, but existing theories assume that training examples are independent. By leveraging techniques from random matrix theory and free probability, we provide sharp asymptotics for the in- and out-of-sample risks of ridge regression when the data points have arbitrary correlations. We demonstrate that in this setting, the generalized cross validation estimator (GCV) fails to correctly predict the out-of-sample risk. However, in the case where the noise residuals have the same correlations as the data points, one can modify the GCV to yield an efficiently-computable unbiased estimator that concentrates in the high-dimensional limit, which we dub CorrGCV. We further extend our asymptotic analysis to the case where the test point has nontrivial correlations with the training set, a setting often encountered in time series forecasting. Assuming knowledge of the correlation structure of the time series, this again yields an extension of the GCV estimator, and sharply characterizes the degree to which such test points yield an overly optimistic prediction of long-time risk. We validate the predictions of our theory across a variety of high dimensional data.