Convergence Guarantees for Adaptive Bayesian Quadrature Methods
This work provides foundational theoretical support for adaptive Bayesian quadrature, which is important for practitioners in numerical integration and machine learning who rely on these methods for medium-dimensional problems.
The paper addresses the lack of theoretical guarantees for adaptive Bayesian quadrature methods by proving consistency and deriving convergence rates for a broad class of these methods, introducing the concept of weak adaptivity to facilitate the analysis.
Adaptive Bayesian quadrature (ABQ) is a powerful approach to numerical integration that empirically compares favorably with Monte Carlo integration on problems of medium dimensionality (where non-adaptive quadrature is not competitive). Its key ingredient is an acquisition function that changes as a function of previously collected values of the integrand. While this adaptivity appears to be empirically powerful, it complicates analysis. Consequently, there are no theoretical guarantees so far for this class of methods. In this work, for a broad class of adaptive Bayesian quadrature methods, we prove consistency, deriving non-tight but informative convergence rates. To do so we introduce a new concept we call weak adaptivity. Our results identify a large and flexible class of adaptive Bayesian quadrature rules as consistent, within which practitioners can develop empirically efficient methods.