Scaling Gaussian Process Regression with Derivatives (1810.12283v1)

Abstract: Gaussian processes (GPs) with derivatives are useful in many applications, including Bayesian optimization, implicit surface reconstruction, and terrain reconstruction. Fitting a GP to function values and derivatives at $n$ points in $d$ dimensions requires linear solves and log determinants with an ${n(d+1) \times n(d+1)}$ positive definite matrix -- leading to prohibitive $\mathcal{O}(n^3d^3)$ computations for standard direct methods. We propose iterative solvers using fast $\mathcal{O}(nd)$ matrix-vector multiplications (MVMs), together with pivoted Cholesky preconditioning that cuts the iterations to convergence by several orders of magnitude, allowing for fast kernel learning and prediction. Our approaches, together with dimensionality reduction, enables Bayesian optimization with derivatives to scale to high-dimensional problems and large evaluation budgets.

• The paper introduces D-SKI and D-SKIP methods that incorporate derivative information to enhance GP prediction efficiency.
• It employs preconditioning and active subspace techniques to overcome cubic scaling challenges in large, high-dimensional datasets.
• Empirical evaluations demonstrate improved accuracy in terrain reconstruction and Bayesian optimization compared to traditional GP models.

An Overview of "Scaling Gaussian Process Regression with Derivatives"

This paper focuses on advancing Gaussian Process (GP) regression to integrate derivative information, providing scalability to high-dimensional problems. Gaussian Processes are established as powerful tools in probabilistic modeling, particularly in Bayesian optimization, surface reconstruction, and terrain reconstruction. Incorporating derivatives within GPs enhances the model's prediction efficacy, leveraging both function values and gradient information.

Background on Gaussian Processes with Derivatives

GPs offer a probabilistic framework where any collection of points follows a joint Gaussian distribution, defined by a mean and a covariance kernel. Predictive modeling with GPs typically requires solving linear systems involving the kernel matrix, which becomes computationally prohibitive as the dataset size scales, especially when it includes derivatives.

The computational challenges discussed in the paper arise primarily from the size of the kernel matrices. When derivatives are included, the matrix size increases by a factor related to the dimensionality of the input data. For standard methods, this results in complexity growing cubically with both the number of data points and the dimensionality, posing a barrier to model scalability.

Methodological Advancements

Two notable techniques, structured kernel interpolation (SKI) and its derivative-based extension, SKIP, are pivotal in the proposed approach. These methods approximate the kernel matrix using a structured interpolation scheme, allowing for efficient matrix-vector multiplications. SKIP further exploits kernel product structures, decomposing high-dimensional problems through Hadamard products and employing Lanczos decompositions.

• D-SKI and D-SKIP: The authors propose D-SKI, extending SKI to incorporate derivative information, and D-SKIP, a variant of SKIP to handle derivatives in scenarios with separable kernels. These methods enable scalable GP regression by reducing computation complexity substantially—from cubic to linear in terms of data points and dimensions.
• Preconditioning Techniques: The paper emphasizes the importance of preconditioning for efficient convergence of iterative solvers when working with large kernel matrices augmented with derivatives. Applying pivoted Cholesky preconditioners, they achieve substantial reduction in the iterations needed for convergence.
• Dimensionality Reduction: Leveraging active subspace techniques for dimensionality reduction, the authors showcase improvements in model performance by focusing on the more informative subspace of the data, reducing unnecessary computational overhead in high-dimensional spaces.

Empirical Validation

The research includes extensive empirical evaluation across several tasks:

• Terrain and Surface Reconstruction: Demonstration of effective terrain reconstruction using real-world datasets shows that incorporating gradient information yields significantly better fidelity in predictions compared to models without derivatives.
• Implicit Surface Fitting: The capability of GPs with derivatives to reconstruct complex surfaces like the Stanford bunny under varying noise conditions further underlines the practical applicability of the proposed methods.
• Bayesian Optimization: When coupled with active subspace methods, Bayesian optimization using this enhanced GP framework successfully tackles high-dimensional problems, demonstrating superior performance over traditional optimization methods.

Practical and Theoretical Implications

The research extends the utility of GPs, making them feasible for large-scale applications. The scaling improvements allow for tackling demanding problems in optimization and spatial modeling domains more effectively. Moreover, the theoretical contributions include introducing a novel integration of kernel interpolation with derivatives, setting a foundation for further research into scalable probabilistic learning methodologies.

Future Directions

Further refinements could enhance computational efficiency and broaden application scope. Potential research avenues include exploring alternative kernel approximations and integrating dimensionality reduction techniques into broader classes of problems, paving the way for unifying local and global optimization paradigms in the context of AI advancements.

In summary, this paper presents significant advancements in scaling Gaussian process regression with derivatives, emphasizing computational efficiency and practical utility in high-dimensional spaces. The methodologies proposed could reshape probabilistic modeling approaches, driving enhanced performance across diversified applications in AI research.