Variable Selection Consistency of Gaussian Process Regression

TL;DR

This paper proves that hierarchical Gaussian process models with stochastic variable selection can consistently identify true important variables in high-dimensional nonparametric regression, under certain smoothness conditions.

Contribution

It establishes variable selection consistency for Gaussian process regression models with hierarchical priors in high-dimensional settings, a previously unresolved question.

Findings

Variable selection consistency is achievable with Gaussian process models.

The results hold in high-dimensional asymptotic regimes.

Sharp bounds on small ball probabilities are developed and may be of independent interest.

Abstract

Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped with stochastic variable selection, are known to also adapt to the unknown intrinsic dimension of a sparse true regression function. But it remains unclear if such extensions offer variable selection consistency, i.e., if the true subset of important variables could be consistently learned from the data. It is shown here that variable consistency may indeed be achieved with such models at least when the true regression function has finite smoothness to induce a polynomially larger penalty on inclusion of false positive predictors. Our result covers the high dimensional asymptotic setting where the predictor dimension is allowed to grow with the sample size. The proof utilizes Schwartz theory to establish that the posterior probability of wrong selection vanishes asymptotically. A necessary and challenging technical development involves providing sharp upper and lower bounds to small ball probabilities at all rescaling levels of the Gaussian process prior, a result that could be of independent interest.