Gaussian processes are a powerful tool for modeling continuous fields, but their naive $\mathcal{O}(N^3)$ computational cost and $\mathcal{O}(N^2)$ memory requirement often limit their practical use. Vecchia's approximation is a sparse precision matrix approximation for stationary, decaying kernels that conditions each point only on its $k$ nearest neighbors. We present GraphGP, a GPU algorithm for Vecchia's approximation that scales to nearly a billion parameters with linear time and memory requirements, handling arbitrary point distributions over a large dynamic range. Our key contributions are (1) a bit-reversed k-d tree ordering that allows efficient neighbor searches while also maximizing batch parallelism, and (2) a differentiable CUDA implementation, which is substantially faster and more memory efficient than our pure JAX baseline. GraphGP provides the building blocks for inference, including forward generation, inverse application, log-determinant, and kernel parameter derivatives.

Bayesian optimization is a technique for optimizing black-box target functions. At the core of Bayesian optimization is a surrogate model that predicts the output of the target function at previously unseen inputs to facilitate the selection of promising input values. Gaussian processes (GPs) are commonly used as surrogate models but are known to scale poorly with the number of observations. We adapt the Vecchia approximation, a popular GP approximation from spatial statistics, to enable scalable high-dimensional Bayesian optimization. We develop several improvements and extensions, including training warped GPs using mini-batch gradient descent, approximate neighbor search, and selecting multiple input values in parallel. We focus on the use of our warped Vecchia GP in trust-region Bayesian optimization via Thompson sampling. On several test functions and on two reinforcement-learning problems, our methods compared favorably to the state of the art.

The Gaussian process model is an indispensible tool for the analysis of spatial and spatial-temporal datasets and has become increasingly popular as a general-purpose model for functions. Because of its high computational burden, researchers have devoted substantial effort to developing numerical approximations for Gaussian process computations. Much of the work focuses on efficient approximation of the likelihood function. Fast likelihood evaluations are crucial for optimization procedures that require many evaluations of the likelihood, such as the default Nelder-Mead algorithm (Nelder and Mead, 1965) in the R optim function. The likelihood must be repeatedly evaluated in MCMC algorithms as well. Compared to the amount of literature on efficient likelihood approximations, there has been considerably less development of techniques for numerically maximizing the likelihood (see Geoga et al. (2018) for one recent example). This article aims to address the disparity by providing: 1. Formulas for evaluating the gradient and Fisher information for Vecchia's likelihood approximation in a single pass through the data, so that the Fisher scoring algorithm can be applied. Fisher scoring is a modification of the Newton-Raphson optimization method, replacing the Hessian matrix with the Fisher information matrix.