My PhD student Edoardo Bandoni, along with Julien Stoehr and myself, completed a paper on validating (Bayesian) kernel quadrature with unbounded domains of integration. Which connects with probabilistic numerics, since the integrand is modelled as a Gaussian process. And RKHS methods. As opposed to Monte Carlo estimators, quadrature methods approximate integrals of smooth functions with worst-case error decaying at a minimax rate α/d for smoothness α in dimension d. Existing rate-optimal quadrature methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and thus less robust in practice. This paper studies instead randomised quadrature methods, with a focus on robustness rather than on kernel-specific optimality. We construct an explicit, n-dependent, sampling distribution that achieves minimax rates for worst-case errors over smoothness classes without requiring knowledge of the kernel. This kernel-agnostic design does improve robustness while retaining optimal rates and extends Briol et al. (2019) to the unbounded case. Which cannot always be easily handled by a change of variables. Our result thus mostly covers unbounded sampling measures such as Gaussian and Student-t distributions, extending beyond compact domains. The results provide both theoretical guarantees and a practical recipe for robust, rate-optimal, randomised quadrature. [The above is mostly stated in the abstract.]
Archive for probabilistic integration
optimal sampling for kernel quadrature on unbounded domains
Posted in Books, Statistics, University life with tags Bayesian quadrature, bois de Boulogne, contraction rate, ERC Synergy Grant, Gaussian processes, minimaxity, Monte Carlo integration, numerical integration, Ocean, probabilistic integration, probabilistic numerics, RKHS, sunset, uncertainty quantification, Université Paris Dauphine on May 22, 2026 by xi'anje suis revenu de Montréal [NIPS 2015]
Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags Frank-Wolfe Bayesian quadrature, Montréal, probabilistic integration, probabilistic numerics on December 17, 2015 by xi'an
After the day trip to Montréal, a quick stop in Paris, and another one in London, I thought back on the probabilistic integration workshop of last week. First, I had a very good time discussing with people there, with no (apparent) adverse reaction to my talk on “estimating constants”. Second, I finally realised what Mark Berliner meant by saying that he was a Bayesian if not a statistician, in a discussion we had in the early 1990’s, in Cornell. Third, I became [moderately] more open to the highly structured spaces used in the approaches discussed by François-Xavier Briol, Arthur Gretton, Roman Garnett, and Francis Bach. The (RKHS) functional assumptions made in those approaches are allowing for higher and more precise convergence rates, with the question being what happens when the assumptions do not hold. A comment similar to the impact of a Gaussian process as the prior on the integrand in Bayesian quadrature.
François-Xavier presented the recently arXived probabilistic integration that Andrew discussed a week ago. (While I obviously have no relevant remark to make about the maths in this paper, I wonder at the difficulty and cost in sequentially selecting the states behind the quadrature. Which presumably is covered in the earlier Frank-Wolfe paper by the same team.) Another discussion with Arthur clarified a wee bit how RKHS can be perceived in practice, with a lingering question on the size of RKHS within the entire space of functions and more importantly the significant impact of the kernel representation on the resulting approximations. Anyway, those are exciting times, when considering that different branches of numerics and probability and statistics come together to improve upon existing techniques and I am once again glad I could took part in this workshop (although sorry I had to miss the ABC workshop that took place in parallel!)
