Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees

By François‐Xavier Briol, Chris J. Oates, Mark A. Girolami, Michael A. Osborne. 2015

97 citations recorded by the source corpus.

There is renewed interest in formulating integration as a statistical 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 therefore 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 up to exponential and posterior contraction rates are proven to be up to super-exponential. 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 Bayesian model choice problem in cellular biology.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Statistical and Nonlinear Physics

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Citations in the corpus

Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, background): “Because of this nice property, kernel quadratures have been studied extensively in recent years [7, 3, 5, 2, 17] and have started to find applications in machine learning and statistics [23, 14, 12, 6].”
Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences (2018, methodology): “The former are called kernel means, and the latter the maximum mean discrepancy (MMD). These have been widely used in machine learning, and interested readers may have a look at the recent survey by Muandet et al. [2017]. While the MMD is characterized as the worst-case error of integrals with respe”
Monte Carlo with determinantal point processes (2016, methodology): “There are theoretical results for hybrid methods between Monte Carlo and Bayesian quadrature [Briol et al., 2015], see Section 4 for further discussion.”
Black-box Importance Sampling (2016, methodology): “We note that the idea of using reference weights was used in Briol et al. (2015a) to establish the convergence rate of Bayesian Monte Carlo.”