Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent

TL;DR

This paper introduces stochastic gradient algorithms for sampling from Gaussian process posteriors, offering a computationally efficient alternative to traditional methods, with accurate predictions and uncertainty estimates in large-scale or ill-conditioned problems.

Contribution

It develops low-variance stochastic gradient objectives for Gaussian process sampling and extends these to inducing points, providing a scalable and effective approach.

Findings

SGD produces accurate predictions even without quick convergence.

Predictive distributions are close to the true posterior in data-rich and data-sparse regions.

Achieves state-of-the-art performance on large-scale regression and Bayesian optimization tasks.

Abstract

Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian optimization task.