Risk and cross validation in ridge regression with correlated samples
This addresses a gap in high-dimensional ridge regression theory for practitioners dealing with correlated data, such as in time series forecasting, though it is incremental by extending existing methods.
The authors tackled the problem of ridge regression with correlated data, showing that standard generalized cross validation fails to predict out-of-sample risk accurately, and introduced CorrGCV as an unbiased estimator that works in high-dimensional limits, validated across various datasets.
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.