@inproceedings{diaconis_bayesian_1988,
	title = {Bayesian Numerical Analysis},
	doi = {10.1007/978-1-4613-8768-8_20},
	abstract = {Semantic Scholar extracted view of "Bayesian Numerical Analysis" by P. Diaconis},
	author = {Diaconis, P.},
	date = {1988},
}

@article{ohagan_bayeshermite_1991,
	title = {Bayes–Hermite quadrature},
	volume = {29},
	issn = {0378-3758},
	doi = {10.1016/0378-3758(91)90002-V},
	abstract = {Bayesian quadrature treats the problem of numerical integration as one of statistical inference. A prior Gaussian process distribution is assumed for the integrand, observations arise from evaluating the integrand at selected points, and a posterior distribution is derived for the integrand and the integral. Methods are developed for quadrature in Rp. A particular application is integrating the posterior density arising from some other Bayesian analysis. Simulation results are presented, to show that the resulting Bayes–Hermite quadrature rules may perform better than the conventional Gauss–Hermite rules for this application. A key result is derived for product designs, which makes Bayesian quadrature practically useful for integrating in several dimensions. Although the method does not at present provide a solution to the more difficult problem of quadrature in high dimensions, it does seem to offer real improvements over existing methods in relatively low dimensions.},
	pages = {245--260},
	number = {3},
	journaltitle = {Journal of Statistical Planning and Inference},
	shortjournal = {Journal of Statistical Planning and Inference},
	author = {O'Hagan, A.},
	date = {1991-11-01},
	langid = {english},
	keywords = {Bayesian quadrature, Gaussian process, Gaussian quadrature, numerical integration, product rule},
}

@inproceedings{ghahramani_bayesian_2003,
	title = {Bayesian Monte Carlo},
	volume = {15},
	url = {https://papers.nips.cc/paper/2002/hash/24917db15c4e37e421866448c9ab23d8-Abstract.html},
	booktitle = {Advances in Neural Information Processing Systems},
	publisher = {{MIT} Press},
	author = {Ghahramani, Zoubin and Rasmussen, Carl},
	date = {2003},
}


@inproceedings{kandasamy_bayesian_2015,
	location = {Buenos Aires, Argentina},
	title = {Bayesian active learning for posterior estimation},
	isbn = {978-1-57735-738-4},
	series = {{IJCAI}'15},
	abstract = {This paper studies active posterior estimation in a Bayesian setting when the likelihood is expensive to evaluate. Existing techniques for posterior estimation are based on generating samples representative of the posterior. Such methods do not consider efficiency in terms of likelihood evaluations. In order to be query efficient we treat posterior estimation in an active regression framework. We propose two myopic query strategies to choose where to evaluate the likelihood and implement them using Gaussian processes. Via experiments on a series of synthetic and real examples we demonstrate that our approach is significantly more query efficient than existing techniques and other heuristics for posterior estimation.},
	pages = {3605--3611},
	booktitle = {Proceedings of the 24th International Conference on Artificial Intelligence},
	publisher = {{AAAI} Press},
	author = {Kandasamy, Kirthevasan and Schneider, Jeff and Póczos, Barnabás},
	date = {2015-07-25},
}

@article{oates_modern_2019,
	title = {A modern retrospective on probabilistic numerics},
	volume = {29},
	issn = {1573-1375},
	url = {https://doi.org/10.1007/s11222-019-09902-z},
	doi = {10.1007/s11222-019-09902-z},
	abstract = {This article attempts to place the emergence of probabilistic numerics as a mathematical–statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul\$\$'\$\$din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.},
	pages = {1335--1351},
	number = {6},
	journaltitle = {Statistics and Computing},
	shortjournal = {Stat Comput},
	author = {Oates, C. J. and Sullivan, T. J.},
	date = {2019-11-01},
	langid = {english},
}

@article{kennedy_bayesian_1998,
	title = {Bayesian quadrature with non-normal approximating functions},
	volume = {8},
	issn = {1573-1375},
	url = {https://doi.org/10.1023/A:1008832824006},
	doi = {10.1023/A:1008832824006},
	abstract = {We consider an efficient Bayesian approach to estimating integration-based posterior summaries from a separate Bayesian application. In Bayesian quadrature we model an intractable posterior density function f(·) as a Gaussian process, using an approximating function g(·), and find a posterior distribution for the integral of f(·), conditional on a few evaluations of f (·) at selected design points. Bayesian quadrature using normal g (·) is called Bayes-Hermite quadrature. We extend this theory by allowing g(·) to be chosen from two wider classes of functions. One is a family of skew densities and the other is the family of finite mixtures of normal densities. For the family of skew densities we describe an iterative updating procedure to select the most suitable approximation and apply the method to two simulated posterior density functions.},
	pages = {365--375},
	number = {4},
	journaltitle = {Statistics and Computing},
	shortjournal = {Statistics and Computing},
	author = {Kennedy, Marc},
	date = {1998-12-01},
	langid = {english},
}


@article{briol_frank-wolfe_2015,
	title = {Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees},
	url = {http://arxiv.org/abs/1506.02681},
	shorttitle = {Frank-Wolfe Bayesian Quadrature},
	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.},
	journaltitle = {{arXiv}:1506.02681 [stat]},
	author = {Briol, François-Xavier and Oates, Chris J. and Girolami, Mark and Osborne, Michael A.},
	date = {2015-12-06},
	eprinttype = {arxiv},
	eprint = {1506.02681},
	keywords = {projects/rp4-2021},
}

@inproceedings{acerbi_variational_2018,
	title = {Variational Bayesian Monte Carlo},
	volume = {31},
	url = {https://papers.nips.cc/paper/2018/hash/747c1bcceb6109a4ef936bc70cfe67de-Abstract.html},
	booktitle = {Advances in Neural Information Processing Systems},
	publisher = {Curran Associates, Inc.},
	author = {Acerbi, Luigi},
	date = {2018},
}


@article{belhadji_kernel_2019,
	title = {Kernel quadrature with {DPPs}},
	url = {http://arxiv.org/abs/1906.07832},
	abstract = {We study quadrature rules for functions from an {RKHS}, using nodes sampled from a determinantal point process ({DPP}). {DPPs} are parametrized by a kernel, and we use a truncated and saturated version of the {RKHS} kernel. This link between the two kernels, along with {DPP} machinery, leads to relatively tight bounds on the quadrature error, that depends on the spectrum of the {RKHS} kernel. Finally, we experimentally compare {DPPs} to existing kernel-based quadratures such as herding, Bayesian quadrature, or leverage score sampling. Numerical results confirm the interest of {DPPs}, and even suggest faster rates than our bounds in particular cases.},
	journaltitle = {{arXiv}:1906.07832 [cs, stat]},
	author = {Belhadji, Ayoub and Bardenet, Rémi and Chainais, Pierre},
	date = {2019-12-31},
	eprinttype = {arxiv},
	eprint = {1906.07832},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{gessner_active_2020,
	title = {Active Multi-Information Source Bayesian Quadrature},
	url = {https://proceedings.mlr.press/v115/gessner20a.html},
	abstract = {Bayesian quadrature ({BQ}) is a sample-efficient probabilistic numerical method to solve integrals of expensive-to-evaluate black-box functions, yet so far, active {BQ} learning schemes focus merely on the integrand itself as information source, and do not allow for information transfer from cheaper, related functions. Here, we set the scene for active learning in {BQ} when multiple related information sources of variable cost (in input and source) are accessible. This setting arises for example when evaluating the integrand requires a complex simulation to be run that can be approximated by simulating at lower levels of sophistication and at lesser expense. We construct meaningful cost-sensitive multi-source acquisition-rates as an extension to common utility functions from vanilla {BQ} ({VBQ}), and discuss pitfalls that arise from blindly generalizing. In proof-of-concept experiments we scrutinize the behavior of our generalized acquisition functions. On an epidemiological model, we demonstrate that active multi-source {BQ} ({AMS}-{BQ}) is more cost-efficient than {VBQ} in learning the integral to a good accuracy.},
	eventtitle = {Uncertainty in Artificial Intelligence},
	pages = {712--721},
	booktitle = {Proceedings of The 35th Uncertainty in Artificial Intelligence Conference},
	publisher = {{PMLR}},
	author = {Gessner, Alexandra and Gonzalez, Javier and Mahsereci, Maren},
	date = {2020-08-06},
	langid = {english},
	note = {{ISSN}: 2640-3498},
}

@inproceedings{osborne_bayesian_2012,
	title = {Bayesian Quadrature for Ratios},
	url = {https://proceedings.mlr.press/v22/osborne12.html},
	abstract = {We describe a novel approach to quadrature for ratios of probabilistic integrals, such as are used to compute posterior probabilities. It offers performance superior to Monte Carlo methods by exploiting a Bayesian quadrature framework. We improve upon previous Bayesian quadrature techniques by explicitly modelling the non-negativity of our integrands, and the correlations that exist between them. It offers most where the integrand is multi-modal and expensive to evaluate, as is commonplace in exoplanets research; we demonstrate the efficacy of our method on data from the Kepler spacecraft.},
	eventtitle = {Artificial Intelligence and Statistics},
	pages = {832--840},
	booktitle = {Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics},
	publisher = {{PMLR}},
	author = {Osborne, Michael and Garnett, Roman and Roberts, Stephen and Hart, Christopher and Aigrain, Suzanne and Gibson, Neale},
	date = {2012-03-21},
	langid = {english},
	note = {{ISSN}: 1938-7228},
}

@article{gunter_sampling_2014,
	title = {Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature},
	url = {http://arxiv.org/abs/1411.0439},
	abstract = {We propose a novel sampling framework for inference in probabilistic models: an active learning approach that converges more quickly (in wall-clock time) than Markov chain Monte Carlo ({MCMC}) benchmarks. The central challenge in probabilistic inference is numerical integration, to average over ensembles of models or unknown (hyper-)parameters (for example to compute the marginal likelihood or a partition function). {MCMC} has provided approaches to numerical integration that deliver state-of-the-art inference, but can suffer from sample inefficiency and poor convergence diagnostics. Bayesian quadrature techniques offer a model-based solution to such problems, but their uptake has been hindered by prohibitive computation costs. We introduce a warped model for probabilistic integrands (likelihoods) that are known to be non-negative, permitting a cheap active learning scheme to optimally select sample locations. Our algorithm is demonstrated to offer faster convergence (in seconds) relative to simple Monte Carlo and annealed importance sampling on both synthetic and real-world examples.},
	journaltitle = {{arXiv}:1411.0439 [stat]},
	author = {Gunter, Tom and Osborne, Michael A. and Garnett, Roman and Hennig, Philipp and Roberts, Stephen J.},
	date = {2014-11-03},
	eprinttype = {arxiv},
	eprint = {1411.0439},
	keywords = {Statistics - Machine Learning},
}

@inproceedings{nguyen_distributionally_2020,
	title = {Distributionally Robust Bayesian Quadrature Optimization},
	url = {https://proceedings.mlr.press/v108/nguyen20a.html},
	abstract = {Bayesian quadrature optimization ({BQO}) maximizes the expectation of an expensive black-box integrand taken over a known probability distribution. In this work, we study {BQO} under distributional uncertainty in which the underlying probability distribution is unknown except for a limited set of its i.i.d samples. A standard {BQO} approach maximizes the Monte Carlo estimate of the true expected objective given the fixed sample set. Though Monte Carlo estimate is unbiased, it has high variance given a small set of samples; thus can result in a spurious objective function. We adopt the distributionally robust optimization perspective to this problem by maximizing the expected objective under the most adversarial distribution. In particular, we propose a novel posterior sampling based algorithm, namely distributionally robust {BQO} ({DRBQO}) for this purpose. We demonstrate the empirical effectiveness of our proposed framework in synthetic and real-world problems, and characterize its theoretical convergence via Bayesian regret.},
	eventtitle = {International Conference on Artificial Intelligence and Statistics},
	pages = {1921--1931},
	booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
	publisher = {{PMLR}},
	author = {Nguyen, Thanh and Gupta, Sunil and Ha, Huong and Rana, Santu and Venkatesh, Svetha},
	date = {2020-06-03},
	langid = {english},
	note = {{ISSN}: 2640-3498},
}

@inproceedings{osborne_active_2012,
	title = {Active Learning of Model Evidence Using Bayesian Quadrature},
	volume = {25},
	url = {https://papers.nips.cc/paper/2012/hash/6364d3f0f495b6ab9dcf8d3b5c6e0b01-Abstract.html},
	booktitle = {Advances in Neural Information Processing Systems},
	publisher = {Curran Associates, Inc.},
	author = {Osborne, Michael and Garnett, Roman and Ghahramani, Zoubin and Duvenaud, David K and Roberts, Stephen J and Rasmussen, Carl},
	date = {2012},
}

@inproceedings{chai_automated_2019,
	title = {Automated Model Selection with Bayesian Quadrature},
	url = {https://proceedings.mlr.press/v97/chai19a.html},
	abstract = {We present a novel technique for tailoring Bayesian quadrature ({BQ}) to model selection. The state-of-the-art for comparing the evidence of multiple models relies on Monte Carlo methods, which converge slowly and are unreliable for computationally expensive models. Although previous research has shown that {BQ} offers sample efficiency superior to Monte Carlo in computing the evidence of an individual model, applying {BQ} directly to model comparison may waste computation producing an overly-accurate estimate for the evidence of a clearly poor model. We propose an automated and efficient algorithm for computing the most-relevant quantity for model selection: the posterior model probability. Our technique maximizes the mutual information between this quantity and observations of the models’ likelihoods, yielding efficient sample acquisition across disparate model spaces when likelihood observations are limited. Our method produces more-accurate posterior estimates using fewer likelihood evaluations than standard Bayesian quadrature and Monte Carlo estimators, as we demonstrate on synthetic and real-world examples.},
	eventtitle = {International Conference on Machine Learning},
	pages = {931--940},
	booktitle = {Proceedings of the 36th International Conference on Machine Learning},
	publisher = {{PMLR}},
	author = {Chai, Henry and Ton, Jean-Francois and Osborne, Michael A. and Garnett, Roman},
	date = {2019-05-24},
	langid = {english},
	note = {{ISSN}: 2640-3498},
}


@article{muandet_kernel_2017,
	title = {Kernel Mean Embedding of Distributions: A Review and Beyond},
	volume = {10},
	issn = {1935-8237, 1935-8245},
	url = {http://arxiv.org/abs/1605.09522},
	doi = {10.1561/2200000060},
	shorttitle = {Kernel Mean Embedding of Distributions},
	abstract = {A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space ({RKHS}) in which the whole arsenal of kernel methods can be extended to probability measures. It can be viewed as a generalization of the original "feature map" common to support vector machines ({SVMs}) and other kernel methods. While initially closely associated with the latter, it has meanwhile found application in fields ranging from kernel machines and probabilistic modeling to statistical inference, causal discovery, and deep learning. The goal of this survey is to give a comprehensive review of existing work and recent advances in this research area, and to discuss the most challenging issues and open problems that could lead to new research directions. The survey begins with a brief introduction to the {RKHS} and positive definite kernels which forms the backbone of this survey, followed by a thorough discussion of the Hilbert space embedding of marginal distributions, theoretical guarantees, and a review of its applications. The embedding of distributions enables us to apply {RKHS} methods to probability measures which prompts a wide range of applications such as kernel two-sample testing, independent testing, and learning on distributional data. Next, we discuss the Hilbert space embedding for conditional distributions, give theoretical insights, and review some applications. The conditional mean embedding enables us to perform sum, product, and Bayes' rules---which are ubiquitous in graphical model, probabilistic inference, and reinforcement learning---in a non-parametric way. We then discuss relationships between this framework and other related areas. Lastly, we give some suggestions on future research directions.},
	pages = {1--141},
	number = {1},
	journaltitle = {Foundations and Trends® in Machine Learning},
	shortjournal = {{FNT} in Machine Learning},
	author = {Muandet, Krikamol and Fukumizu, Kenji and Sriperumbudur, Bharath and Schölkopf, Bernhard},
	date = {2017},
	eprinttype = {arxiv},
	eprint = {1605.09522},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{damianou_deep_2013,
	title = {Deep Gaussian Processes},
	url = {http://proceedings.mlr.press/v31/damianou13a.html},
	abstract = {In this paper we introduce deep Gaussian process ({GP}) models. Deep {GPs} are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate {GP}. The inpu...},
	eventtitle = {Artificial Intelligence and Statistics},
	pages = {207--215},
	booktitle = {Artificial Intelligence and Statistics},
	publisher = {{PMLR}},
	author = {Damianou, Andreas and Lawrence, Neil D.},
	date = {2013-04-29},
	langid = {english},
	note = {{ISSN}: 1938-7228},
}

@article{kovachki_neural_2021,
	title = {Neural Operator: Learning Maps Between Function Spaces},
	url = {http://arxiv.org/abs/2108.08481},
	shorttitle = {Neural Operator},
	abstract = {The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators that maps between infinite dimensional function spaces. We formulate the approximation of operators by composition of a class of linear integral operators and nonlinear activation functions, so that the composed operator can approximate complex nonlinear operators. We prove a universal approximation theorem for our construction. Furthermore, we introduce four classes of operator parameterizations: graph-based operators, low-rank operators, multipole graph-based operators, and Fourier operators and describe efficient algorithms for computing with each one. The proposed neural operators are resolution-invariant: they share the same network parameters between different discretizations of the underlying function spaces and can be used for zero-shot super-resolutions. Numerically, the proposed models show superior performance compared to existing machine learning based methodologies on Burgers' equation, Darcy flow, and the Navier-Stokes equation, while being several order of magnitude faster compared to conventional {PDE} solvers.},
	journaltitle = {{arXiv}:2108.08481 [cs, math]},
	author = {Kovachki, Nikola and Li, Zongyi and Liu, Burigede and Azizzadenesheli, Kamyar and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
	date = {2021-12-16},
	eprinttype = {arxiv},
	eprint = {2108.08481},
	keywords = {Computer Science - Machine Learning, Mathematics - Numerical Analysis},
}

@article{li_fourier_2021,
	title = {Fourier Neural Operator for Parametric Partial Differential Equations},
	url = {http://arxiv.org/abs/2010.08895},
	abstract = {The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations ({PDEs}), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of {PDEs}, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first {ML}-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional {PDE} solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.},
	journaltitle = {{arXiv}:2010.08895 [cs, math]},
	author = {Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
	date = {2021-05-16},
	eprinttype = {arxiv},
	eprint = {2010.08895},
	keywords = {Computer Science - Machine Learning, Mathematics - Numerical Analysis},
}

@article{bach_equivalence_2017,
	title = {On the equivalence between kernel quadrature rules and random feature expansions},
	volume = {18},
	issn = {1532-4435},
	abstract = {We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation error, leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms. In particular, we show that the upper bound may be obtained from independent and identically distributed samples from a specific non-uniform distribution, while the lower bound if valid for any set of points. Applying our results to kernel-based quadrature, while our results are fairly general, we recover known upper and lower bounds for the special cases of Sobolev spaces. Moreover, our results extend to the more general problem of full function approximations (beyond simply computing an integral), with results in L2- and L∞-norm that match known results for special cases. Applying our results to random features, we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with Lipshitz-continuous losses.},
	pages = {714--751},
	number = {1},
	journaltitle = {The Journal of Machine Learning Research},
	shortjournal = {J. Mach. Learn. Res.},
	author = {Bach, Francis},
	date = {2017-01-01},
	keywords = {integral operators, positive-definite kernels, quadrature},
}


@inproceedings{mutny_efficient_2018,
	title = {Efficient High Dimensional Bayesian Optimization with Additivity and Quadrature Fourier Features},
	volume = {31},
	url = {https://proceedings.neurips.cc/paper/2018/hash/4e5046fc8d6a97d18a5f54beaed54dea-Abstract.html},
	booktitle = {Advances in Neural Information Processing Systems},
	publisher = {Curran Associates, Inc.},
	author = {Mutny, Mojmir and Krause, Andreas},
	date = {2018},
}

@inproceedings{wilson_gaussian_2013,
	title = {Gaussian Process Kernels for Pattern Discovery and Extrapolation},
	url = {https://proceedings.mlr.press/v28/wilson13.html},
	abstract = {Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation.  We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns and enable extrapolation.  These kernels are derived by modelling a spectral density – the Fourier transform of a kernel – with a Gaussian mixture.  The proposed kernels support a broad class of stationary covariances, but Gaussian process inference remains simple and analytic.  We demonstrate the proposed kernels by discovering patterns and performing long range extrapolation on synthetic examples, as well as atmospheric {CO}2 trends and airline passenger data.  We also show that it is possible to reconstruct several popular standard covariances within our framework.},
	eventtitle = {International Conference on Machine Learning},
	pages = {1067--1075},
	booktitle = {Proceedings of the 30th International Conference on Machine Learning},
	publisher = {{PMLR}},
	author = {Wilson, Andrew and Adams, Ryan},
	date = {2013-05-26},
	langid = {english},
	note = {{ISSN}: 1938-7228},
}

@inproceedings{oliva_bayesian_2016,
	title = {Bayesian Nonparametric Kernel-Learning},
	url = {https://proceedings.mlr.press/v51/oliva16.html},
	abstract = {Kernel methods are ubiquitous tools in machine learning. They have proven to be effective in many domains and tasks. Yet, kernel methods often require the user to select a predefined kernel to build an estimator with. However, there is often little reason for the common practice of selecting a kernel a priori. Even if a universal approximating kernel is selected, the quality of the finite sample estimator may be greatly affected by the choice of kernel. Furthermore, when directly applying kernel methods, one typically needs to compute a N {\textbackslash}times N Gram matrix of pairwise kernel evaluations to work with a dataset of N instances. The computation of this Gram matrix precludes the direct application of kernel methods on large datasets, and makes kernel learning especially difficult. In this paper we introduce Bayesian nonparmetric kernel-learning ({BaNK}), a generic, data-driven framework for scalable learning of kernels. {BaNK} places a nonparametric prior on the spectral distribution of random frequencies allowing it to both learn kernels and scale to large datasets. We show that this framework can be used for large scale regression and classification tasks. Furthermore, we show that {BaNK} outperforms several other scalable approaches for kernel learning on a variety of real world datasets.},
	eventtitle = {Artificial Intelligence and Statistics},
	pages = {1078--1086},
	booktitle = {Proceedings of the 19th International Conference on Artificial Intelligence and Statistics},
	publisher = {{PMLR}},
	author = {Oliva, Junier B. and Dubey, Avinava and Wilson, Andrew G. and Poczos, Barnabas and Schneider, Jeff and Xing, Eric P.},
	date = {2016-05-02},
	langid = {english},
	note = {{ISSN}: 1938-7228},
}

@article{bezanson_julia_2015,
	title = {Julia: A Fresh Approach to Numerical Computing},
	url = {http://arxiv.org/abs/1411.1607},
	shorttitle = {Julia},
	abstract = {Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast. Julia questions notions generally held as "laws of nature" by practitioners of numerical computing: 1. High-level dynamic programs have to be slow. 2. One must prototype in one language and then rewrite in another language for speed or deployment, and 3. There are parts of a system for the programmer, and other parts best left untouched as they are built by the experts. We introduce the Julia programming language and its design --- a dance between specialization and abstraction. Specialization allows for custom treatment. Multiple dispatch, a technique from computer science, picks the right algorithm for the right circumstance. Abstraction, what good computation is really about, recognizes what remains the same after differences are stripped away. Abstractions in mathematics are captured as code through another technique from computer science, generic programming. Julia shows that one can have machine performance without sacrificing human convenience.},
	journaltitle = {{arXiv}:1411.1607 [cs]},
	author = {Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B.},
	date = {2015-07-19},
	eprinttype = {arxiv},
	eprint = {1411.1607},
	keywords = {Computer Science - Mathematical Software},
}

@misc{hamid_marginalising_2022,
	title = {Marginalising over Stationary Kernels with Bayesian Quadrature},
	url = {http://arxiv.org/abs/2106.07452},
	abstract = {Marginalising over families of Gaussian Process kernels produces flexible model classes with well-calibrated uncertainty estimates. Existing approaches require likelihood evaluations of many kernels, rendering them prohibitively expensive for larger datasets. We propose a Bayesian Quadrature scheme to make this marginalisation more efficient and thereby more practical. Through use of the maximum mean discrepancies between distributions, we define a kernel over kernels that captures invariances between Spectral Mixture ({SM}) Kernels. Kernel samples are selected by generalising an information-theoretic acquisition function for warped Bayesian Quadrature. We show that our framework achieves more accurate predictions with better calibrated uncertainty than state-of-the-art baselines, especially when given limited (wall-clock) time budgets.},
	author = {Hamid, Saad and Schulze, Sebastian and Osborne, Michael A. and Roberts, Stephen J.},
	date = {2022-03-27},
	eprinttype = {arxiv},
	eprint = {2106.07452 [cs, stat]},
	note = {Number: {arXiv}:2106.07452},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}


@article{halton_efficiency_1960,
	title = {On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals},
	volume = {2},
	issn = {0945-3245},
	url = {https://doi.org/10.1007/BF01386213},
	doi = {10.1007/BF01386213},
	pages = {84--90},
	number = {1},
	journaltitle = {Numerische Mathematik},
	shortjournal = {Numer. Math.},
	author = {Halton, J. H.},
	date = {1960-12-01},
	langid = {english},
}

@book{rasmussen_gaussian_2006,
	location = {Cambridge, Mass},
	title = {Gaussian processes for machine learning},
	isbn = {978-0-262-18253-9},
	series = {Adaptive computation and machine learning},
	pagetotal = {248},
	publisher = {{MIT} Press},
	author = {Rasmussen, Carl Edward and Williams, Christopher K. I.},
	date = {2006},
	note = {{OCLC}: ocm61285753},
	keywords = {Data processing, Gaussian processes, Machine learning, Mathematical models},
}

@inproceedings{zhen_learning_2020,
	title = {Learning to Learn Kernels with Variational Random Features},
	url = {http://proceedings.mlr.press/v119/zhen20a.html},
	abstract = {We introduce kernels with random Fourier features in the meta-learning framework for few-shot learning. We propose meta variational random features ({MetaVRF}) to learn adaptive kernels for the base-...},
	eventtitle = {International Conference on Machine Learning},
	pages = {11409--11419},
	booktitle = {International Conference on Machine Learning},
	publisher = {{PMLR}},
	author = {Zhen, Xiantong and Sun, Haoliang and Du, Yingjun and Xu, Jun and Yin, Yilong and Shao, Ling and Snoek, Cees},
	date = {2020-11-21},
	langid = {english},
	note = {{ISSN}: 2640-3498},
}

@article{briol_probabilistic_2019,
	title = {Probabilistic Integration: A Role in Statistical Computation?},
	volume = {34},
	issn = {0883-4237, 2168-8745},
	url = {https://projecteuclid.org/journals/statistical-science/volume-34/issue-1/Probabilistic-Integration-A-Role-in-Statistical-Computation/10.1214/18-STS660.full},
	doi = {10.1214/18-STS660},
	shorttitle = {Probabilistic Integration},
	abstract = {A research frontier has emerged in scientific computation, wherein discretisation error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational work-flow, in order to assess the impact of discretisation error on the computer output. This paper examines the case for probabilistic numerical methods in routine statistical computation. Our focus is on numerical integration, where a probabilistic integrator is equipped with a full distribution over its output that reflects the fact that the integrand has been discretised. Our main technical contribution is to establish, for the first time, rates of posterior contraction for one such method. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and a computer model for an oil reservoir.},
	pages = {1--22},
	number = {1},
	journaltitle = {Statistical Science},
	author = {Briol, François-Xavier and Oates, Chris J. and Girolami, Mark and Osborne, Michael A. and Sejdinovic, Dino},
	date = {2019-02},
	note = {Publisher: Institute of Mathematical Statistics},
	keywords = {computational statistics, nonparametric statistics, probabilistic numerics, uncertainty quantification},
}

@inproceedings{briol_sampling_2017,
	location = {Sydney, {NSW}, Australia},
	title = {On the sampling problem for Kernel Quadrature},
	series = {{ICML}'17},
	abstract = {The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio s/d, where s and d encode the smoothness and dimension of the integrand. However, an empirical investigation reveals that the rate constant C is highly sensitive to the distribution of the random points. In contrast to standard Monte Carlo integration, for which optimal importance sampling is well-understood, the sampling distribution that minimises C for Kernel Quadrature does not admit a closed form. This paper argues that the practical choice of sampling distribution is an important open problem. One solution is considered; a novel automatic approach based on adaptive tempering and sequential Monte Carlo. Empirical results demonstrate a dramatic reduction in integration error of up to 4 orders of magnitude can be achieved with the proposed method.},
	pages = {586--595},
	booktitle = {Proceedings of the 34th International Conference on Machine Learning - Volume 70},
	publisher = {{JMLR}.org},
	author = {Briol, François-Xavier and Oates, Chris J. and Cockayne, Jon and Chen, Wilson Ye and Girolami, Mark},
	date = {2017-08-06},
}


@book{rudin_fourier_2011,
	location = {Hoboken},
	title = {Fourier Analysis on Groups.},
	isbn = {978-1-118-16564-5},
	url = {http://www.123library.org/book_details/?id=30165},
	abstract = {In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact abelian ({LCA}) groups. Rudin's book, published in 1962, was the first to give a systematic account of these developments and has come to be regarded as a classic in the field. The basic facts concerning Fourier analysis and the structure of {LCA} groups are proved in the opening chapters, in order to make the treatment relatively self-contained.},
	publisher = {John Wiley \& Sons},
	author = {Rudin, Walter},
	date = {2011},
	note = {{OCLC}: 772844593},
}

@inproceedings{rahimi_random_2008,
	title = {Random Features for Large-Scale Kernel Machines},
	volume = {20},
	booktitle = {Advances in Neural Information Processing Systems},
	publisher = {Curran Associates, Inc.},
	author = {Rahimi, Ali and Recht, Benjamin},
	date = {2008},
}

@article{chang_data-driven_2017,
	title = {Data-driven Random Fourier Features using Stein Effect},
	url = {http://arxiv.org/abs/1705.08525},
	abstract = {Large-scale kernel approximation is an important problem in machine learning research. Approaches using random Fourier features have become increasingly popular [Rahimi and Recht, 2007], where kernel approximation is treated as empirical mean estimation via Monte Carlo ({MC}) or Quasi-Monte Carlo ({QMC}) integration [Yang et al., 2014]. A limitation of the current approaches is that all the features receive an equal weight summing to 1. In this paper, we propose a novel shrinkage estimator from "Stein effect", which provides a data-driven weighting strategy for random features and enjoys theoretical justifications in terms of lowering the empirical risk. We further present an efficient randomized algorithm for large-scale applications of the proposed method. Our empirical results on six benchmark data sets demonstrate the advantageous performance of this approach over representative baselines in both kernel approximation and supervised learning tasks.},
	journaltitle = {{arXiv}:1705.08525 [cs, stat]},
	author = {Chang, Wei-Cheng and Li, Chun-Liang and Yang, Yiming and Poczos, Barnabas},
	date = {2017-05-23},
	eprinttype = {arxiv},
	eprint = {1705.08525},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}


@inproceedings{tompkins_black_2019,
	title = {Black Box Quantiles for Kernel Learning},
	url = {http://proceedings.mlr.press/v89/tompkins19a.html},
	abstract = {Kernel methods have been successfully used in various domains to model nonlinear patterns. However, the structure of the kernels is typically handcrafted for each dataset based on the experience of...},
	eventtitle = {The 22nd International Conference on Artificial Intelligence and Statistics},
	pages = {1427--1437},
	booktitle = {The 22nd International Conference on Artificial Intelligence and Statistics},
	publisher = {{PMLR}},
	author = {Tompkins, Anthony and Senanayake, Ransalu and Morere, Philippe and Ramos, Fabio},
	date = {2019-04-11},
	langid = {english},
	note = {{ISSN}: 2640-3498},
}

@article{rossberg_positive_1995,
	title = {Positive definite probability densities and probability distributions},
	volume = {76},
	issn = {1573-8795},
	url = {https://doi.org/10.1007/BF02363232},
	doi = {10.1007/BF02363232},
	abstract = {In the first short survey article [H.-J. Rossberg,Wiss. Z. Karl Marx Univ.,37, 366–374 (1988)] the author indicated that positive definite probability densities deserve attention and that the investigation of this class leads to serious new mathematical problems. Later [R. Riedel and H.-J. Rossberg,Metrika, (submitted for publication)] it turned out that it is worthwhile to consider an even larger class of distribution functions, namely those having characteristic function f≥0; they were first briefly considered in [P. Lévy,C. R. Acad. Sci.,265, 249–252 (1967)]. We survey the basic ideas and the present state of the developing new theory; moreover, we formulate open problems. Some new results are proved. For the sake of brevity we do not enter into a discussion of continuation problems and limit theorems; for these subjects see [H.-J. Rossberg,Appendix to Gnedenko B. W., Einführung in die Wahrscheinlichkeitstheorie, 9th ed., Akademie-Verlag, Berlin (1991)], [H.-J. Rossberg,Math. Nachr.,141, 227–232 (1989)], and [H.-J. Rossberg,Lect. Notes Math.,1412, 296–308 (1989)].},
	pages = {2181--2197},
	number = {1},
	journaltitle = {Journal of Mathematical Sciences},
	shortjournal = {J Math Sci},
	author = {Rossberg, H. -J.},
	date = {1995-08-01},
	langid = {english},
}

@article{durand_streaming_2017,
	title = {Streaming kernel regression with provably adaptive mean, variance, and regularization},
	url = {http://arxiv.org/abs/1708.00768},
	abstract = {We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an {RKHS}. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finite-dimensional linear regression with fixed regularization and known variance to the kernel setup with a regularization parameter allowed to be a measurable function of past observations. Then, using appropriate self-normalized inequalities we build upper and lower bound estimates for the variance, leading to Bersntein-like concentration bounds. The later is used in order to define the adaptive regularization. The bounds resulting from our technique are valid uniformly over all observation points and all time steps, and are compared against the literature with numerical experiments. Finally, the potential of these tools is illustrated by an application to kernelized bandits, where we revisit the Kernel {UCB} and Kernel Thompson Sampling procedures, and show the benefits of the novel adaptive kernel tuning strategy.},
	journaltitle = {{arXiv}:1708.00768 [cs, stat]},
	author = {Durand, Audrey and Maillard, Odalric-Ambrym and Pineau, Joelle},
	date = {2017-08-02},
	eprinttype = {arxiv},
	eprint = {1708.00768},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{sutherland_error_2015,
	location = {Arlington, Virginia, {USA}},
	title = {On the error of random fourier features},
	isbn = {978-0-9966431-0-8},
	series = {{UAI}'15},
	abstract = {Kernel methods give powerful, flexible, and theoretically grounded approaches to solving many problems in machine learning. The standard approach, however, requires pairwise evaluations of a kernel function, which can lead to scalability issues for very large datasets. Rahimi and Recht (2007) suggested a popular approach to handling this problem, known as random Fourier features. The quality of this approximation, however, is not well understood. We improve the uniform error bound of that paper, as well as giving novel understandings of the embedding's variance, approximation error, and use in some machine learning methods. We also point out that surprisingly, of the two main variants of those features, the more widely used is strictly higher-variance for the Gaussian kernel and has worse bounds.},
	pages = {862--871},
	booktitle = {Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence},
	publisher = {{AUAI} Press},
	author = {Sutherland, Danica J. and Schneider, Jeff},
	date = {2015-07-12},
}

@inproceedings{Mutny2018,
address = {Montreal, Canada},
author = {Mutn{\'{y}}, Mojm{\'{i}}r and Krause, Andreas},
booktitle = {Conference on Neural Information Processing Systems (NIPS)},
title = {{Efficient High Dimensional Bayesian Optimization with Additivity and Quadrature Fourier Features}},
year = {2018}
}

@inproceedings{Sutherland2015,
author = {Sutherland, Dougal J. and Schneider, Jeff},
booktitle = {Uncertainty in Artificial Intelligence - Proceedings of the 31st Conference, UAI 2015},
pages = {862--871},
title = {{On the error of random Fourier features}},
year = {2015}
}

@article{Durand2018,
author = {Durand, Audrey and Maillard, Odalric-Ambrym and Pineau, Joelle},
editor = {Szepesvari, Csaba},
journal = {Journal of Machine Learning Research},
title = {{Streaming kernel regression with provably adaptive mean, variance, and regularization}},
volume = {19},
year = {2018}
}

@incollection{Rudin1990,
author = {Rudin, Walter},
booktitle = {Fourier Analysis on Groups},
chapter = {1},
number = {2},
pages = {1--34},
publisher = {John Wiley {\&} Sons, Ltd},
title = {{The Basic Theorems of Fourier Analysis}},
year = {1990}
}

@article{Rossberg1995,
author = {Rossberg, H.-J.},
journal = {Journal of Mathematical Sciences},
number = {1},
pages = {2181--2197},
title = {{Positive definite probability densities and probability distributions}},
volume = {76},
year = {1995}
}