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Gaussian process (GP) regression is a flexible, nonparametric approach toregression that naturally quantifies uncertainty. In many applications, thenumber of responses and covariates are both large, and a goal is to selectcovariates that are related to the response. For this setting, we propose anovel, scalable algorithm, coined VGPR, which optimizes a penalized GPlog-likelihood based on the Vecchia GP approximation, an ordered conditionalapproximation from spatial statistics that implies a sparse Cholesky factor ofthe precision matrix. We traverse the regularization path from strong to weakpenalization, sequentially adding candidate covariates based on the gradient ofthe log-likelihood and deselecting irrelevant covariates via a new quadraticconstrained coordinate descent algorithm. We propose Vecchia-based mini-batchsubsampling, which provides unbiased gradient estimators. The resultingprocedure is scalable to millions of responses and thousands of covariates.Theoretical analysis and numerical studies demonstrate the improved scalabilityand accuracy relative to existing methods.

Scalable Gaussian-process regression and variable selection using Vecchia approximations