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Adaptive Bayesian quadrature (ABQ) is a powerful approach to numericalintegration that empirically compares favorably with Monte Carlo integration onproblems of medium dimensionality (where non-adaptive quadrature is notcompetitive). Its key ingredient is an acquisition function that changes as afunction of previously collected values of the integrand. While this adaptivityappears to be empirically powerful, it complicates analysis. Consequently,there are no theoretical guarantees so far for this class of methods. In thiswork, for a broad class of adaptive Bayesian quadrature methods, we proveconsistency, deriving non-tight but informative convergence rates. To do so weintroduce a new concept we call weak adaptivity. Our results identify a largeand flexible class of adaptive Bayesian quadrature rules as consistent, withinwhich practitioners can develop empirically efficient methods.

Convergence Guarantees for Adaptive Bayesian Quadrature Methods