Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees
This addresses the problem of reconciling probabilistic integrators with rigorous convergence analysis for researchers in numerical integration and uncertainty quantification, offering a novel theoretical framework.
The paper tackled the lack of theoretical guarantees in probabilistic integration methods like Bayesian Quadrature by introducing Frank-Wolfe Bayesian Quadrature (FWBQ), which provides exponential convergence and superexponential posterior contraction rates, and in simulations, it is competitive with state-of-the-art methods and outperforms Frank-Wolfe-based alternatives.
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.