Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees

TL;DR

This paper introduces Frank-Wolfe Bayesian Quadrature, a probabilistic integration method with proven exponential convergence and superexponential posterior contraction, offering rigorous guarantees and competitive performance.

Contribution

It presents the first probabilistic integrator with theoretical guarantees, combining Bayesian Quadrature with Frank-Wolfe optimization for rigorous convergence analysis.

Findings

Exponential convergence to the true integral value.

Superexponential posterior contraction rates.

Outperforms existing methods in simulations.

Abstract

There is renewed interest in formulating integration as an inference problem, motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. An important challenge is to reconcile these probabilistic integrators with rigorous convergence guarantees. In this paper, we present the first probabilistic integrator that admits such theoretical treatment, called Frank-Wolfe Bayesian Quadrature (FWBQ). Under FWBQ, convergence to the true value of the integral is shown to be exponential and posterior contraction rates are proven to be superexponential. In simulations, FWBQ is competitive with state-of-the-art methods and out-performs alternatives based on Frank-Wolfe optimisation. Our approach is applied to successfully quantify numerical error in the solution to a challenging model choice problem in cellular biology.