@article{Fabozzi2017,
abstract = {Longstaff–Schwartz's least squares Monte Carlo method is one of the most applied numerical methods for pricing American-style derivatives. We examine the algorithms regression step, demonstrating that the OLS regression is not the best linear unbiased estimator because of heteroscedasticity. We prove the existence of heteroscedasticity for single-asset and multi-asset payoffs numerically and theoretically, and propose weighted-least squares MC valuation method to correct for it. An extensive numerical study shows that the proposed method produces significantly smaller pricing bias than the Longstaff–Schwartz method under several well-known price dynamics. An empirical pricing exercise using market data confirms the advantages of the improved method.},
author = {Fabozzi, F. J. and Paletta, T. and Tunaru, R.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/least-squares MC/(2017, Fabozzi) An improved least squares Monte Carlo valuation method based on heteroscedasticity.pdf:pdf},
journal = {European Journal of Operational Research},
keywords = {American options,Finance,Heteroscedasticity,Least squares Monte Carlo pricing method,Weighted least squares},
number = {2},
pages = {698--706},
publisher = {Elsevier B.V.},
title = {{An improved least squares Monte Carlo valuation method based on heteroscedasticity}},
volume = {263},
year = {2017}
}

@article{kendall2017uncertainties,
  title={What uncertainties do we need in Bayesian deep learning for computer vision?},
  author={Kendall, Alex and Gal, Yarin},
  journal={Advances in Neural Information Processing Systems},
  volume={30},
  year={2017}
}

@article{Stone1982,
author = {Stone, C. J.},
file = {:Users/fxbriol/Docs/Work/Papers/Other Statistics/nonparametric regression/(1982, Stone) Optimal global rates of convergence for nonparametric regression.pdf:pdf},
journal = {The Annals of Statistics},
number = {4},
pages = {1040--1053},
title = {{Optimal global rates of convergence for nonparametric regression}},
volume = {10},
year = {1982}
}



@article{Hu2020,
abstract = {In this paper, we propose using kernel ridge regression (KRR) to avoid the step of selecting basis functions for regression-based approaches in pricing high-dimensional American options by simulation. Our contribution is threefold. Firstly, we systematically introduce the main idea and theory of KRR and apply it to American option pricing for the first time. Secondly, we show how to use KRR with the Gaussian kernel in the regression-later method and give the computationally efficient formulas for estimating the continuation values and the Greeks. Thirdly, we propose to accelerate and improve the accuracy of KRR by performing local regression based on the bundling technique. The numerical test results show that our method is robust and has both higher accuracy and efficiency than the Least Squares Monte Carlo method in pricing high-dimensional American options.},
author = {Hu, W. and Zastawniak, T.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/least-squares MC/(2020, Hu) Pricing high dimensional American options by kernel ridge regression.pdf:pdf},
journal = {Quantitative Finance},
keywords = {High-dimensional American option,Kernel ridge regression,Machine learning,Monte Carlo,Regression-based method},
number = {5},
pages = {851--865},
publisher = {Routledge},
title = {{Pricing high-dimensional American options by kernel ridge regression}},
volume = {20},
year = {2020}
}

@article{Han2009,
abstract = {Valuation of an American option with Monte Carlo methods is one of the most important and difficult problems in pricing, since it involves the determination of optimal exercise timing in the sense that the option can be exercised at any time prior to its own maturity. Regression approaches have been widely used to price an American-style option approximately with Monte Carlo simulation. However, the conventional regression methods are very sensitive in the kind and the number of their basis functions, thereby affecting prediction accuracy. In this paper, we propose a novel kernel-based Monte Carlo simulation algorithm to overcome such shortcomings of the regression approaches and conduct a simulation on some American options with promising results on its pricing accuracy. {\textcopyright} 2008 Elsevier Ltd. All rights reserved.},
author = {Han, G. S. and Kim, B. H. and Lee, J.},
doi = {10.1016/j.eswa.2008.05.004},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/least-squares MC/(2009, Han) Kernel-based Monte Carlo simulation for American option pricing.pdf:pdf},
issn = {09574174},
journal = {Expert Systems with Applications},
keywords = {American option,Continuation value,Kernel-based regression},
number = {3},
pages = {4431--4436},
publisher = {Elsevier Ltd},
title = {{Kernel-based Monte Carlo simulation for American option pricing}},
url = {http://dx.doi.org/10.1016/j.eswa.2008.05.004},
volume = {36},
year = {2009}
}

@incollection{frazier2018bayesian,
  title={Bayesian optimization},
  author={Frazier, Peter I},
  booktitle={Recent advances in optimization and modeling of contemporary problems},
  pages={255--278},
  year={2018},
  publisher={Informs}
}

@inproceedings{Song2009,
author = {Song, L. and Huang, J. and Smola, A. and Fukumizu, K.},
booktitle = {International Conference on Machine Learning},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/Kernel Embeddings/(2009, Song) Hilbert space embeddings of conditional distributions with applications to dynamical systems.pdf:pdf},
pages = {961--968},
title = {{Hilbert space embeddings of conditional distributions with applications to dynamical systems}},
year = {2009}
}

@article{stentoft2004convergence,
  title={Convergence of the least squares {M}onte {C}arlo approach to {A}merican option valuation},
  author={Stentoft, Lars},
  journal={Management Science},
  volume={50},
  number={9},
  pages={1193--1203},
  year={2004},
  publisher={INFORMS}
}

@inproceedings{Le2005,
  title={Heteroscedastic {G}aussian process regression},
  author={Le, Quoc V and Smola, Alex J and Canu, St{\'e}phane},
  booktitle={International Conference on Machine Learning},
  pages={489--496},
  year={2005}
}


@article{Park2020,
abstract = {We present an operator-free, measure-theoretic approach to the conditional mean embedding (CME) as a random variable taking values in a reproducing kernel Hilbert space. While the kernel mean embedding of unconditional distributions has been defined rigorously, the existing operator-based approach of the conditional version depends on stringent assumptions that hinder its analysis. We overcome this limitation via a measure-theoretic treatment of CMEs. We derive a natural regression interpretation to obtain empirical estimates, and provide a thorough theoretical analysis thereof, including universal consistency. As natural by-products, we obtain the conditional analogues of the maximum mean discrepancy and Hilbert-Schmidt independence criterion, and demonstrate their behaviour via simulations.},
author = {Park, Junhyung and Muandet, Krikamol},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/Kernel Embeddings/Conditional embeddings/(2022, Park) A measure-theoretic approach to kernel conditional embeddings.pdf:pdf},
journal = {Advances in Neural Information Processing Systems},
title = {{A measure-theoretic approach to kernel conditional mean embeddings}},
year = {2020}
}

@book{Robert2004,
  title={{M}onte {C}arlo statistical methods},
  author={Robert, Christian P and Casella, George and Casella, George},
  volume={2},
  year={1999},
  publisher={Springer}
}


@article{Borogovac2008,
abstract = {The technique of control variates requires that the user identify a set of variates that are correlated with the estimation variable and whose means are known to the user. We relax the known mean requirement and instead assume the means are to be estimated. We argue that this strategy can be beneficial in parametric studies, analyze the properties of controlled estimators, and propose a class of generic and effective controls in a parametric estimation setting. We discuss the effectiveness of the estimators via analysis and simulation experiments.{\textcopyright}2008 IEEE.},
author = {Borogovac, T. and Vakili, P.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2008, Borogovac) Control variate technique - a constructive approac.pdf:pdf},
journal = {Proceedings of the 2008 Winter Simulation Conference},
pages = {320--327},
publisher = {IEEE},
title = {{Control variate technique: A constructive approach}},
year = {2008}
}

@article{Oates2019Modern,
  title={A modern retrospective on probabilistic numerics},
  author={Oates, Chris J and Sullivan, Timothy John},
  journal={Statistics and Computing},
  volume={29},
  number={6},
  pages={1335--1351},
  year={2019},
  publisher={Springer}
}


@article{wang2020generalizing,
  title={Generalizing from a few examples: A survey on few-shot learning},
  author={Wang, Yaqing and Yao, Quanming and Kwok, James T and Ni, Lionel M},
  journal={ACM Computing Surveys (CSUR)},
  volume={53},
  number={3},
  pages={1--34},
  year={2020},
  publisher={ACM New York, NY, USA}
}


@inproceedings{LapNTKSimilarity,
 author = {Geifman, Amnon and Yadav, Abhay and Kasten, Yoni and Galun, Meirav and Jacobs, David and Ronen, Basri},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Larochelle and M. Ranzato and R. Hadsell and M. F. Balcan and H. Lin},
 pages = {1451--1461},
 publisher = {Curran Associates, Inc.},
 title = {On the Similarity between the Laplace and Neural Tangent Kernels},
 url = {https://proceedings.neurips.cc/paper/2020/file/1006ff12c465532f8c574aeaa4461b16-Paper.pdf},
 volume = {33},
 year = {2020}
}


@article{aronszajn50reproducing,
  added-at = {2008-03-10T11:21:00.000+0100},
  author = {Aronszajn, N.},
  biburl = {https://www.bibsonomy.org/bibtex/2024c71f807cbf95a8fb6b934c01f4919/sb3000},
  description = {CiteULike: Theory of reproducing kernels},
  interhash = {5f0e5e40a1512aa0b21f287a39b81b31},
  intrahash = {024c71f807cbf95a8fb6b934c01f4919},
  journal = {Transactions of the American Mathematical Society},
  keywords = {kernel},
  number = 3,
  pages = {337--404},
  timestamp = {2010-10-07T14:13:58.000+0200},
  title = {Theory of Reproducing Kernels},
  url = {http://dx.doi.org/10.2307/1990404},
  volume = 68,
  year = 1950
}

@inproceedings{huang2021fl,
  title={FL-NTK: A Neural Tangent Kernel-based Framework for Federated Learning Analysis},
  author={Huang, Baihe and Li, Xiaoxiao and Song, Zhao and Yang, Xin},
  booktitle={International Conference on Machine Learning},
  pages={4423--4434},
  year={2021},
  
}

@inproceedings{liu2020finding,
  title={Finding trainable sparse networks through neural tangent transfer},
  author={Liu, Tianlin and Zenke, Friedemann},
  booktitle={International Conference on Machine Learning},
  pages={6336--6347},
  year={2020},
  
}

@article{cortes2010generalization,
  title={Generalization bounds for learning kernels},
  author={Cortes, Corinna and Mohri, Mehryar and Rostamizadeh, Afshin},
  year={2010}
}

@article{cortes1995support,
  title={Support-vector networks},
  author={Cortes, Corinna and Vapnik, Vladimir},
  journal={Machine learning},
  volume={20},
  number={3},
  pages={273--297},
  year={1995},
  publisher={Springer}
}

@misc{poole2016exponential,
      title={Exponential expressivity in deep neural networks through transient chaos}, 
      author={Ben Poole and Subhaneil Lahiri and Maithra Raghu and Jascha Sohl-Dickstein and Surya Ganguli},
      year={2016},
      eprint={1606.05340},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@misc{schoenholz2017deep,
      title={Deep Information Propagation}, 
      author={Samuel S. Schoenholz and Justin Gilmer and Surya Ganguli and Jascha Sohl-Dickstein},
      year={2017},
      eprint={1611.01232},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}
@misc{xiao2018dynamical,
      title={Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networks}, 
      author={Lechao Xiao and Yasaman Bahri and Jascha Sohl-Dickstein and Samuel S. Schoenholz and Jeffrey Pennington},
      year={2018},
      eprint={1806.05393},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}
@misc{yang2017mean,
      title={Mean Field Residual Networks: On the Edge of Chaos}, 
      author={Greg Yang and Samuel S. Schoenholz},
      year={2017},
      eprint={1712.08969},
      archivePrefix={arXiv},
      primaryClass={cs.NE}
}
@misc{matthews2018gaussian,
      title={{G}aussian Process Behaviour in Wide Deep Neural Networks}, 
      author={Alexander G. de G. Matthews and Mark Rowland and Jiri Hron and Richard E. Turner and Zoubin Ghahramani},
      year={2018},
      eprint={1804.11271},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}
@misc{novak2020bayesian,
      title={{B}ayesian Deep Convolutional Networks with Many Channels are {G}aussian Processes}, 
      author={Roman Novak and Lechao Xiao and Jaehoon Lee and Yasaman Bahri and Greg Yang and Jiri Hron and Daniel A. Abolafia and Jeffrey Pennington and Jascha Sohl-Dickstein},
      year={2020},
      eprint={1810.05148},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}
@misc{yu2021s2mlp,
      title={S$^2$-MLP: Spatial-Shift MLP Architecture for Vision}, 
      author={Tan Yu and Xu Li and Yunfeng Cai and Mingming Sun and Ping Li},
      year={2021},
      eprint={2106.07477},
      archivePrefix={arXiv},
      primaryClass={cs.CV}
}
@article{Rasmussen2003,
  title={{B}ayesian {M}onte {C}arlo},
  author={Rasmussen, Carl E. and Ghahramani, Zoubin},
  journal={Advances in Neural Information Processing Systems},
  pages={505--512},
  year={2003},
  publisher={MIT; 1998}
}

@article{casella1985emp_bayes,
  title={An introduction to empirical {B}ayes data analysis},
  author={Casella, George},
  journal={The American Statistician},
  volume={39},
  number={2},
  pages={83--87},
  year={1985},
  publisher={Taylor \& Francis}
}

@article{sriperumbudur2011universality,
  title={{Universality, Characteristic Kernels and RKHS Embedding of Measures.}},
  author={Sriperumbudur, Bharath K and Fukumizu, Kenji and Lanckriet, Gert RG},
  journal={Journal of Machine Learning Research},
  volume={12},
  number={7},
  year={2011}
}

@book{Hennig2022,
  title={Probabilistic Numerics: Computation as Machine Learning},
  author={Hennig, Philipp and Osborne, Michael A and Kersting, Hans P},
  year={2022},
  publisher={Cambridge University Press}
}


@article{Diaconis1988,
  title={{B}ayesian numerical analysis},
  author={Diaconis, Persi},
  journal={Statistical Decision Theory and Related Topics IV},
  volume={1},
  pages={163--175},
  year={1988}
}

@article{Nishiyama2016,
author = {Nishiyama, Yu and Fukumizu, Kenji},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/Kernel Embeddings/(2016, Nishiyama) Characteristic Kernels and Infinitely Divisible Distributions.pdf:pdf},
journal = {Journal of Machine Learning Research},
keywords = {characteristic kernel,con-,convolution trick,infinitely divisible distribution,jugate kernel,kernel mean},
number = {180},
pages = {1--28},
title = {{Characteristic kernels and infinitely divisible distributions}},
volume = {17},
year = {2016}
}

@article{Ababou1994,
  title={On the condition number of covariance matrices in kriging, estimation, and simulation of random fields},
  author={Ababou, Rachid and Bagtzoglou, Amvrossios C and Wood, Eric F},
  journal={Mathematical Geology},
  volume={26},
  pages={99--133},
  year={1994},
  publisher={Springer}
}


@article{Wendland2005,
abstract = {In this paper, we study the global behavior of a function that is known to be small at a given discrete data set. Such a function might be interpreted as the error function between an unknown function and a given approximant. We will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region. We will apply these results to spline smoothing and show that a specific, a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant. This has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels. {\textcopyright} Springer-Verlag 2005.},
author = {Wendland, Holger and Rieger, Christian},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/Theory of kernel methods/(2005, Wendland) Approximate Interpolation with applications to selecting smoothing parameters.pdf:pdf},
journal = {Numerische Mathematik},
number = {4},
pages = {729--748},
title = {{Approximate interpolation with applications to selecting smoothing parameters}},
volume = {101},
year = {2005}
}

@article{Andrianakis2012,
  title={The effect of the nugget on {G}aussian process emulators of computer models},
  author={Andrianakis, Ioannis and Challenor, Peter G},
  journal={Computational Statistics \& Data Analysis},
  volume={56},
  number={12},
  pages={4215--4228},
  year={2012},
  publisher={Elsevier}
}


@article{Teckentrup2020,
  title={Convergence of {G}aussian process regression with estimated hyper-parameters and applications in {B}ayesian inverse problems},
  author={Teckentrup, Aretha L},
  journal={SIAM/ASA Journal on Uncertainty Quantification},
  volume={8},
  number={4},
  pages={1310--1337},
  year={2020},
  publisher={SIAM}
}

@article{Wenger2021,
  title={{ProbNum: Probabilistic Numerics in Python}},
  author={Wenger, Jonathan and Kr{\"a}mer, Nicholas and Pf{\"o}rtner, Marvin and Schmidt, Jonathan and Bosch, Nathanael and Effenberger, Nina and Zenn, Johannes and Gessner, Alexandra and Karvonen, Toni and Briol, Fran{\c{c}}ois-Xavier and others},
  journal={arXiv preprint arXiv:2112.02100},
  year={2021}
}

@inproceedings{Sun2021,
author={Sun, Zhuo and Barp, Alessandro and Briol, Fran{\c{c}}ois-Xavier},
booktitle = {International Conference on Machine Learning},
pages = {32819--32846},
title = {{Vector-valued control variates}},
year = {2023}
}



@inproceedings{Bharti2023,
  title={Optimally-weighted estimators of the maximum mean discrepancy for likelihood-free inference},
  author={Bharti, Ayush and Naslidnyk, Masha and Key, Oscar and Kaski, Samuel and Briol, Fran{\c{c}}ois-Xavier},
  booktitle = {International Conference on Machine Learning},
  pages = {2289--2312},
  year={2023}
}



@article{Karvonen2017symmetric,
  title={Fully symmetric kernel quadrature},
  author={Karvonen, Toni and Sarkka, Simo},
  journal={SIAM Journal on Scientific Computing},
  volume={40},
  number={2},
  pages={A697--A720},
  year={2018},
  publisher={SIAM}
}

@article{Karvonen2019,
  title={Symmetry exploits for {B}ayesian cubature methods},
  author={Karvonen, Toni and S{\"a}rkk{\"a}, Simo and Oates, Chris},
  journal={arXiv preprint arXiv:1809.10227},
  year={2018}
}

@article{alfonsi2022many,
  title={How many inner simulations to compute conditional expectations with least-square {M}onte {C}arlo?},
  author={Alfonsi, Aur{\'e}lien and Lapeyre, Bernard and Lelong, J{\'e}r{\^o}me},
  journal={arXiv preprint arXiv:2209.04153},
  year={2022}
}

@article{Karvonen2019positivity,
  title={On the positivity and magnitudes of {B}ayesian quadrature weights},
  author={Karvonen, Toni and Kanagawa, Motonobu and S{\"a}rkk{\"a}, Simo},
  journal={Statistics and Computing},
  volume={29},
  pages={1317--1333},
  year={2019},
  publisher={Springer}
}

@article{Kanagawa2019,
  title={Convergence guarantees for adaptive {B}ayesian quadrature methods},
  author={Kanagawa, Motonobu and Hennig, Philipp},
  journal={Advances in Neural Information Processing Systems},
  volume={32},
  year={2019}
}
@article{Karvonen2017classical,
author = {Karvonen, T. and S{\"{a}}rkk{\"{a}}, S.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Karvonen, S{\"{a}}rkk{\"{a}} - 2017 - Classical quadrature rules via Gaussian processes.pdf:pdf},
isbn = {9781509063413},
journal = {IEEE International Workshop on Machine Learning for Signal Processing (MLSP)},
title = {{Classical quadrature rules via Gaussian processes}},
year = {2017}
}


@inproceedings{Sun2023,
author={Sun, Zhuo and Oates, Chris J and Briol, Fran{\c{c}}ois-Xavier},
booktitle = {Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, PMLR 216},
pages = {2047--2057},
title = {{Meta-learning control variates: Variance reduction with limited data}},
year = {2023}
}





@article{He2014,
abstract = {In multiple importance sampling we combine samples from a finite list of proposal distributions. When those proposal distributions are used to create control variates, it is possible (Owen and Zhou, 2000) to bound the ratio of the resulting variance to that of the unknown best proposal distribution in our list. The minimax regret arises by taking a uniform mixture of proposals, but that is conservative when there are many components. In this paper we optimize the mixture component sampling rates to gain further efficiency. We show that the sampling variance of mixture importance sampling with control variates is jointly convex in the mixture probabilities and control variate regression coefficients. We also give a sequential importance sampling algorithm to estimate the optimal mixture from the sample data.},
author = {He, H. Y. and Owen, A. B.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/standard MC/(2014, He) Optimal mixture weights in multiple importance sampling.pdf:pdf},
journal = {arXiv:1411.3954},
title = {{Optimal mixture weights in multiple importance sampling}},
year = {2014}
}
@inproceedings{Bardenet2019,
author = {Belhadji, A. and Bardenet, R. and Chainais, P.},
booktitle = {Neural Information Processing Systems},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2019, Belhadji) Kernel quadrature with DPPs.pdf:pdf},
keywords = {determinantal point processes,kernel hilbert spaces,Monte Carlo integration,reproducing},
pages = {12927--12937},
title = {{Kernel quadrature with DPPs}},
year = {2019}
}

@article{Marques2013,
  title={A spherical {G}aussian framework for {B}ayesian {M}onte {C}arlo rendering of glossy surfaces},
  author={Marques, Ricardo and Bouville, Christian and Ribardi{\`e}re, Micka{\"e}l and Santos, Lu{\'\i}s Paulo and Bouatouch, Kadi},
  journal={IEEE Transactions on Visualization and Computer Graphics},
  volume={19},
  number={10},
  pages={1619--1632},
  year={2013},
  publisher={IEEE}
}

@article{Gunter2014,
  title={Sampling for inference in probabilistic models with fast {B}ayesian quadrature},
  author={Gunter, Tom and Osborne, Michael A and Garnett, Roman and Hennig, Philipp and Roberts, Stephen J},
  journal={Advances in Neural Information Processing Systems},
  volume={27},
  year={2014}
}

@article{Acerbi2018,
  title={Variational {B}ayesian {M}onte {C}arlo},
  author={Acerbi, Luigi},
  journal={Advances in Neural Information Processing Systems},
  volume={31},
  year={2018}
}

@article{Oates2017heart,
  title={Probabilistic models for integration error in the assessment of functional cardiac models},
  author={Oates, Chris J. and Niederer, Steven and Lee, Angela and Briol, Fran{\c{c}}ois-Xavier and Girolami, Mark},
  journal={Advances in Neural Information Processing Systems},
  volume={30},
  year={2017}
}


@article{Chai2019,
abstract = {We present a novel technique for tailoring {B}ayesian 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 {B}ayesian quadrature and Monte Carlo estimators, as we demonstrate on synthetic and real-world examples.},
author = {Chai, H. and Ton, J-F. and Osborne, M. A. and Garnett, R.},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2019, Chai) Automated Model Selection with {B}ayesian Quadrature.pdf:pdf},
isbn = {9781510886988},
journal = {International Conference on Machine Learning},
pages = {931--940},
title = {{Automated model selection with {B}ayesian quadrature}},
year = {2019}
}

@article{Jagadeeswaran2018,
  title={Fast automatic {B}ayesian cubature using lattice sampling},
  author={Jagadeeswaran, Rathinavel and Hickernell, Fred J},
  journal={Statistics and Computing},
  volume={29},
  number={6},
  pages={1215--1229},
  year={2019},
  publisher={Springer}
}

@software{jax2018github,
  author = {James Bradbury and Roy Frostig and Peter Hawkins and Matthew James Johnson and Chris Leary and Dougal Maclaurin and George Necula and Adam Paszke and Jake Vander{P}las and Skye Wanderman-{M}ilne and Qiao Zhang},
  title = {{JAX}: composable transformations of {P}ython+{N}um{P}y programs},
  version = {0.3.13},
  year = {2018},
}

@inproceedings{lemieux2004randomized,
  title={Randomized quasi-{M}onte {C}arlo: A tool for improving the efficiency of simulations in finance},
  author={Lemieux, Christiane},
  booktitle={Proceedings of the 2004 Winter Simulation Conference, 2004.},
  volume={2},
  pages={1565--1573},
  year={2004},
  organization={IEEE}
}

@article{niu2023discrepancy,
  title={Discrepancy-based inference for intractable generative models using quasi-{M}onte {C}arlo},
  author={Niu, Ziang and Meier, Johanna and Briol, Fran{\c{c}}ois-Xavier},
  journal={Electronic Journal of Statistics},
  volume={17},
  number={1},
  pages={1411--1456},
  year={2023},
  publisher={The Institute of Mathematical Statistics and the Bernoulli Society}
}

@article{hickernell1998generalized,
  title={A generalized discrepancy and quadrature error bound},
  author={Hickernell, Fred},
  journal={Mathematics of Computation},
  volume={67},
  number={221},
  pages={299--322},
  year={1998}
}

@article{gerber2015sequential,
  title={Sequential quasi {M}onte {C}arlo},
  author={Gerber, Mathieu and Chopin, Nicolas},
  journal={Journal of the Royal Statistical Society Series B: Statistical Methodology},
  volume={77},
  number={3},
  pages={509--579},
  year={2015},
  publisher={Oxford University Press}
}

@article{heath2017review,
  title={A review of methods for analysis of the expected value of information},
  author={Heath, Anna and Manolopoulou, Ioanna and Baio, Gianluca},
  journal={Medical Decision Making},
  volume={37},
  number={7},
  pages={747--758},
  year={2017},
  publisher={Sage Publications Sage CA: Los Angeles, CA}
}

@article{brennan2007calculating,
  title={Calculating partial expected value of perfect information via {M}onte {C}arlo sampling algorithms},
  author={Brennan, Alan and Kharroubi, Samer and O'Hagan, Anthony and Chilcott, Jim},
  journal={Medical Decision Making},
  volume={27},
  number={4},
  pages={448--470},
  year={2007},
  publisher={Sage Publications Sage CA: Los Angeles, CA}
}

@article{Giles2019,
  title={Decision-making under uncertainty: using {MLMC} for efficient estimation of {EVPPI}},
  author={Giles, Michael B and Goda, Takashi},
  journal={Statistics and Computing},
  volume={29},
  pages={739--751},
  year={2019},
  publisher={Springer}
}

@inproceedings{Zhu2020,
abstract = {Bayesian quadrature (BQ) is a method for solving numerical integration problems in a Bayesian manner, which allows users to quantify their uncertainty about the solution. The standard approach to BQ is based on a Gaussian process (GP) approximation of the integrand. As a result, BQ is inherently limited to cases where GP approximations can be done in an efficient manner, thus often prohibiting very high-dimensional or non-smooth target functions. This paper proposes to tackle this issue with a new Bayesian numerical integration algorithm based on Bayesian Additive Regression Trees (BART) priors, which we call BART-Int. BART priors are easy to tune and well-suited for discontinuous functions. We demonstrate that they also lend themselves naturally to a sequential design setting and that explicit convergence rates can be obtained in a variety of settings. The advantages and disadvantages of this new methodology are highlighted on a set of benchmark tests including the Genz functions, and on a Bayesian survey design problem.},
author = {Zhu, Harrison and Liu, Xing and Kang, Ruya and Shen, Zhichao and Flaxman, Seth and Briol, Fran\c{c}ois-Xavier},
booktitle = {Neural Information Processing Systems},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2020, Zhu) Bayesian probabilistic numerical integration with tree-based models.pdf:pdf},
pages = {5837--5849},
title = {{Bayesian probabilistic numerical integration with tree-based models}},
year = {2020}
}

@inproceedings{Hayakawa2023,
abstract = {We analyze the Nystr{\"{o}}m approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nystr{\"{o}}m approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nystr{\"{o}}m approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations.},
author = {Hayakawa, Satoshi and Oberhauser, Harald and Lyons, Terry},
booktitle = {International Conference on Machine Learning},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2023, Hayakawa) Sampling-based Nystrom approximation and kernel quadrature.pdf:pdf},
pages = {12678--12699},
title = {{Sampling-based Nystr{\"{o}}m approximation and kernel quadrature}},
volume = {202},
year = {2023}
}



@article{Hamid2023,
author = {Hamid, Saad and Wan, Xinghchen and Jorgensen, Martin and Ru, Binxin and Osborne, Michael},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2023, Hamid) Bayesian quadrature for neural ensemble search.pdf:pdf},
journal = {Transactions in Machine Learning Research},
title = {{Bayesian quadrature for neural ensemble search}},
year = {2023}
}


@inproceedings{Ott2023,
abstract = {Bayesian probabilistic numerical methods for numerical integration offer significant advantages over their non-Bayesian counterparts: they can encode prior information about the integrand, and can quantify uncertainty over estimates of an integral. However, the most popular algorithm in this class, Bayesian quadrature, is based on Gaussian process models and is therefore associated with a high computational cost. To improve scalability, we propose an alternative approach based on Bayesian neural networks which we call Bayesian Stein networks. The key ingredients are a neural network architecture based on Stein operators, and an approximation of the Bayesian posterior based on the Laplace approximation. We show that this leads to orders of magnitude speed-ups on the popular Genz functions benchmark, and on challenging problems arising in the Bayesian analysis of dynamical systems, and the prediction of energy production for a large-scale wind farm.},
author = {Ott, Katharina and Tiemann, Michael and Hennig, Philipp and Briol, Fran\c{c}ois-Xavier},
booktitle = {Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, PMLR 216},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2023, Ott) Bayesian Numerical Integration with Neural Networks.pdf:pdf},
pages = {1606--1617},
title = {{Bayesian numerical integration with neural networks}},
year = {2023}
}


@article{Glynn1989,
author = {Glynn, Peter and Igelhart, Donald},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Importance sampling/(1989, Glynn) Importance sampling for stochastic simulations.pdf:pdf},
journal = {Management Science},
number = {1367-1392},
title = {{Importance sampling for stochastic simulations}},
volume = {35},
year = {1989}
}

@book{Lee1998,
  title={{M}onte {C}arlo computation of conditional expectation quantiles},
  author={Lee, Shing-Hoi},
  year={1998},
  publisher={Stanford University}
}


@phdthesis{Tang2013,
author = {Tang, Xiaojin},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/least-squares MC/(2013, Tang) Importance sampling for efficient parametric simulation.pdf:pdf},
school = {Boston University},
title = {{Importance sampling for efficient parametric simulation}},
year = {2013}
}

@article{Krumscheid2018,
  title={Multilevel {M}onte {C}arlo approximation of functions},
  author={Krumscheid, Sebastian and Nobile, Fabio},
  journal={SIAM/ASA Journal on Uncertainty Quantification},
  volume={6},
  number={3},
  pages={1256--1293},
  year={2018},
  publisher={SIAM}
}

@inproceedings{Warren2022,
  title={Generalized {B}ayesian quadrature with spectral kernels},
  author={Warren, Houston and Oliveira, Rafael and Ramos, Fabio},
  booktitle={Uncertainty in Artificial Intelligence},
  pages={2085--2095},
  year={2022},
  
}


@inproceedings{Adachi2022,
abstract = {Calculation of {B}ayesian posteriors and model evidences typically requires numerical integration. {B}ayesian quadrature (BQ), a surrogate-model-based approach to numerical integration, is capable of superb sample efficiency, but its lack of parallelisation has hindered its practical applications. In this work, we propose a parallelised (batch) BQ method, employing techniques from kernel quadrature, that possesses an empirically exponential convergence rate. Additionally, just as with Nested Sampling, our method permits simultaneous inference of both posteriors and model evidence. Samples from our BQ surrogate model are re-selected to give a sparse set of samples, via a kernel recombination algorithm, requiring negligible additional time to increase the batch size. Empirically, we find that our approach significantly outperforms the sampling efficiency of both state-of-the-art BQ techniques and Nested Sampling in various real-world datasets, including lithium-ion battery analytics.},
author = {Adachi, M. and Hayakawa, S. and Oberhauser, H. and Jorgensen, M. and Osborne, M.A.},
booktitle = {Neural Information Processing Systems},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2022, Adachi) Fast {B}ayesian inference with batch {B}ayesian quadrature via kernel recombination.pdf:pdf},
keywords = {Gauss-Hermite quadratures,Gaussian kernels,Univariate integration},
title = {{Fast {B}ayesian inference with batch {B}ayesian quadrature via kernel recombination}},
year = {2022}
}


@article{Pronzato2020,
abstract = {A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) best linear unbiased estimator (BLUE) for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties.},
author = {Pronzato, L. and Zhigljavsky, A.},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2020, Pronzanto) {B}ayesian quadrature energy minimization and space filling desing.pdf:pdf},
journal = {SIAM-ASA Journal on Uncertainty Quantification},
keywords = {BLUE,{B}ayesian quadrature,Discrepancy,Energy minimization,Potential,Space-filling design},
number = {3},
pages = {959--1011},
title = {{{B}ayesian quadrature, energy minimization, and space-filling design}},
volume = {8},
year = {2020}
}

@article{Cockayne2017BPNM,
  title={{B}ayesian probabilistic numerical methods},
  author={Cockayne, Jon and Oates, Chris J and Sullivan, Timothy John and Girolami, Mark},
  journal={SIAM review},
  volume={61},
  number={4},
  pages={756--789},
  year={2019},
  publisher={SIAM}
}

@inproceedings{Kersting2016,
abstract = {There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are 'well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.},
  author={Kersting, Hans and Hennig, Philipp},
booktitle = {Uncertainty in Artificial Intelligence},
pages = {309--318},
title = {{Active uncertainty calibration in Bayesian ODE solvers}},
year = {2016}
}


@article{Demange-Chryst2022,
  title={Efficient estimation of multiple expectations with the same sample by adaptive importance sampling and control variates},
  author={Demange-Chryst, Julien and Bachoc, Fran{\c{c}}ois and Morio, J{\'e}r{\^o}me},
  journal={arXiv preprint arXiv:2212.00568},
  year={2022}
}




@phdthesis{de2017scalable,
  title={Scalable {G}aussian process inference using variational methods},
  author={de Garis Matthews, Alexander Graeme},
  year={2017},
  school={University of Cambridge}
}

@misc{belkin2019reconciling,
      title={Reconciling modern machine learning practice and the bias-variance trade-off}, 
      author={Mikhail Belkin and Daniel Hsu and Siyuan Ma and Soumik Mandal},
      year={2019},
      eprint={1812.11118},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@INPROCEEDINGS{Lecun98gradient-basedlearning,
    author = {Yann Lecun and Léon Bottou and Yoshua Bengio and Patrick Haffner},
    title = {Gradient-based learning applied to document recognition},
    booktitle = {Proceedings of the IEEE},
    year = {1998},
    pages = {2278--2324}
}

@misc{he2015deep,
      title={Deep Residual Learning for Image Recognition}, 
      author={Kaiming He and Xiangyu Zhang and Shaoqing Ren and Jian Sun},
      year={2015},
      eprint={1512.03385},
      archivePrefix={arXiv},
      primaryClass={cs.CV}
}

@book{LampertiJohn1977,
series = {Applied mathematical sciences (Springer-Verlag New York Inc.); v. 23},
publisher = {Springer-Verlag},
booktitle = {Stochastic processes : a survey of the mathematical theory},
isbn = {0387902759},
year = {1977},
title = {Stochastic processes : a survey of the mathematical theory / J. Lamperti.},
language = {eng},
address = {New York},
author = {Lamperti, John},
keywords = {Stochastic processes; Stationary processes; Markov processes},
lccn = {77-324321},
}

@InCollection{sep-bayes-theorem,
	author       =	{Joyce, James},
	title        =	{{Bayes’ Theorem}},
	booktitle    =	{The {Stanford} Encyclopedia of Philosophy},
	editor       =	{Edward N. Zalta},
	howpublished =	{\url{https://plato.stanford.edu/archives/fall2021/entries/bayes-theorem/}},
	year         =	{2021},
	edition      =	{{F}all 2021},
	publisher    =	{Metaphysics Research Lab, Stanford University}
}

@Book{           bishop:2006:PRML,
  author = 	 "Christopher M. Bishop",
  title = 	 "Pattern Recognition and Machine Learning",
  publisher = 	 "Springer",
  year = 	 "2006",
}

@book{murphy2013machine,
  added-at = {2017-02-27T11:22:42.000+0100},
  address = {Cambridge, Mass. [u.a.]},
  author = {Murphy, Kevin P.},
  biburl = {https://www.bibsonomy.org/bibtex/270148d65a6a66e0ae962bf22c5f66148/hotho},
  description = {Machine Learning: A Probabilistic Perspective (Adaptive Computation and Machine Learning series): Kevin P. Murphy: 9780262018029: Amazon.com: Books},
  interhash = {e99d8a06cc36507b05c38192ab80573e},
  intrahash = {70148d65a6a66e0ae962bf22c5f66148},
  isbn = {9780262018029 0262018020},
  keywords = {hmm lda learning machine statistics},
  publisher = {MIT Press},
  refid = {904442949},
  timestamp = {2017-02-27T11:22:42.000+0100},
  title = {Machine learning : a probabilistic perspective},
  url = {https://www.amazon.com/Machine-Learning-Probabilistic-Perspective-Computation/dp/0262018020/ref=sr_1_2?ie=UTF8&qid=1336857747&sr=8-2},
  year = 2013
}



@article{cortes2009learning,
  title={Learning non-linear combinations of kernels},
  author={Cortes, Corinna and Mohri, Mehryar and Rostamizadeh, Afshin},
  year={2009}
}

@article{lanckriet2004learning,
  title={Learning the kernel matrix with semidefinite programming},
  author={Lanckriet, Gert RG and Cristianini, Nello and Bartlett, Peter and Ghaoui, Laurent El and Jordan, Michael I},
  journal={Journal of Machine learning research},
  volume={5},
  number={Jan},
  pages={27--72},
  year={2004}
}

@article{williams1998computation,
  title={Computation with infinite neural networks},
  author={Williams, Christopher KI},
  journal={Neural Computation},
  volume={10},
  number={5},
  pages={1203--1216},
  year={1998},
  publisher={MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}
}

@article{ong2005learning,
  title={Learning the kernel with hyperkernels},
  author={Ong, Cheng Soon and Smola, Alexander and Williamson, Robert and others},
  year={2005},
  publisher={MIT Press}
}

@article{micchelli2005learning,
  title={Learning the Kernel Function via Regularization.},
  author={Micchelli, Charles A and Pontil, Massimiliano and Bartlett, Peter},
  journal={Journal of machine learning research},
  volume={6},
  number={7},
  year={2005}
}

@article{yang2012nystrom,
  title={Nystr{\"o}m method vs random fourier features: A theoretical and empirical comparison},
  author={Yang, Tianbao and Li, Yu-Feng and Mahdavi, Mehrdad and Jin, Rong and Zhou, Zhi-Hua},
  journal={Advances in Neural Information Processing Systems},
  volume={25},
  pages={476--484},
  year={2012}
}

@inproceedings{williams2001using,
  title={Using the Nystr{\"o}m method to speed up kernel machines},
  author={Williams, Christopher and Seeger, Matthias},
  booktitle={Advances in Neural Information Processing Systems},
  pages={682--688},
  year={2001}
}

@inproceedings{rahimi2007random,
  title={Random Features for Large-Scale Kernel Machines.},
  author={Rahimi, Ali and Recht, Benjamin and others},
  booktitle={NIPS},
  volume={3},
  number={4},
  pages={5},
  year={2007},
  organization={Citeseer}
}

@book{scholkopf2002learning,
  title={Learning with kernels: support vector machines, regularization, optimization, and beyond},
  author={Sch{\"o}lkopf, Bernhard and Smola, Alexander J and Bach, Francis and others},
  year={2002},
  publisher={MIT press}
}

@inproceedings{arora2019exact,
  title={On exact computation with an infinitely wide neural net},
  author={Arora, Sanjeev and Du, Simon S and Hu, Wei and Li, Zhiyuan and Salakhutdinov, Ruslan and Wang, Ruosong},
  booktitle={Advances in Neural Information Processing Systems},
  pages={8141--8150},
  year={2019}
}

@inproceedings{EvolveLinear,
 author = {Lee, Jaehoon and Xiao, Lechao and Schoenholz, Samuel and Bahri, Yasaman and Novak, Roman and Sohl-Dickstein, Jascha and Pennington, Jeffrey},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
 pages = {8572--8583},
 publisher = {Curran Associates, Inc.},
 title = {Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent},
 url = {https://proceedings.neurips.cc/paper/2019/file/0d1a9651497a38d8b1c3871c84528bd4-Paper.pdf},
 volume = {32},
 year = {2019}
}

@misc{kawaguchi2020generalization,
      title={Generalization in Deep Learning}, 
      author={Kenji Kawaguchi and Leslie Pack Kaelbling and Yoshua Bengio},
      year={2020},
      eprint={1710.05468},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@inproceedings{ntk,
  title={Neural tangent kernel: Convergence and generalization in neural networks},
  author={Jacot, Arthur and Gabriel, Franck and Hongler, Cl{\'e}ment},
  booktitle={Advances in Neural Information Processing Systems},
  pages={8571--8580},
  year={2018}
}

@book{GPML,
  title = {{G}aussian Processes for Machine Learning},
  author = {Rasmussen, Carl E. and Williams, Chris},
  pages = {248},
  series = {Adaptive Computation and Machine Learning},
  publisher = {MIT Press},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  address = {Cambridge, MA, USA},
  month = jan,
  year = {2006},
  doi = {},
  month_numeric = {1}
}

@misc{izmailov2019averaging,
      title={Averaging Weights Leads to Wider Optima and Better Generalization}, 
      author={Pavel Izmailov and Dmitrii Podoprikhin and Timur Garipov and Dmitry Vetrov and Andrew Gordon Wilson},
      year={2019},
      eprint={1803.05407},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{zhang2017understanding,
      title={Understanding deep learning requires rethinking generalization}, 
      author={Chiyuan Zhang and Samy Bengio and Moritz Hardt and Benjamin Recht and Oriol Vinyals},
      year={2017},
      eprint={1611.03530},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{benjamin2019measuring,
      title={Measuring and regularizing networks in function space}, 
      author={Ari S. Benjamin and David Rolnick and Konrad Kording},
      year={2019},
      eprint={1805.08289},
      archivePrefix={arXiv},
      primaryClass={cs.NE}
}

@misc{arora2019harnessing,
      title={Harnessing the Power of Infinitely Wide Deep Nets on Small-data Tasks}, 
      author={Sanjeev Arora and Simon S. Du and Zhiyuan Li and Ruslan Salakhutdinov and Ruosong Wang and Dingli Yu},
      year={2019},
      eprint={1910.01663},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{bietti2019kernel,
      title={A Kernel Perspective for Regularizing Deep Neural Networks}, 
      author={Alberto Bietti and Grégoire Mialon and Dexiong Chen and Julien Mairal},
      year={2019},
      eprint={1810.00363},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}


@misc{rice2020overfitting,
      title={Overfitting in adversarially robust deep learning}, 
      author={Leslie Rice and Eric Wong and J. Zico Kolter},
      year={2020},
      eprint={2002.11569},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{tolstikhin2021mlpmixer,
      title={MLP-Mixer: An all-MLP Architecture for Vision}, 
      author={Ilya Tolstikhin and Neil Houlsby and Alexander Kolesnikov and Lucas Beyer and Xiaohua Zhai and Thomas Unterthiner and Jessica Yung and Andreas Steiner and Daniel Keysers and Jakob Uszkoreit and Mario Lucic and Alexey Dosovitskiy},
      year={2021},
      eprint={2105.01601},
      archivePrefix={arXiv},
      primaryClass={cs.CV}
}

@misc{izmailov2019averaging,
      title={Averaging Weights Leads to Wider Optima and Better Generalization}, 
      author={Pavel Izmailov and Dmitrii Podoprikhin and Timur Garipov and Dmitry Vetrov and Andrew Gordon Wilson},
      year={2019},
      eprint={1803.05407},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{arora2019harnessing,
      title={Harnessing the Power of Infinitely Wide Deep Nets on Small-data Tasks}, 
      author={Sanjeev Arora and Simon S. Du and Zhiyuan Li and Ruslan Salakhutdinov and Ruosong Wang and Dingli Yu},
      year={2019},
      eprint={1910.01663},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{rice2020overfitting,
      title={Overfitting in adversarially robust deep learning}, 
      author={Leslie Rice and Eric Wong and J. Zico Kolter},
      year={2020},
      eprint={2002.11569},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{kirkpatrick2017overcoming,
      title={Overcoming catastrophic forgetting in neural networks}, 
      author={James Kirkpatrick and Razvan Pascanu and Neil Rabinowitz and Joel Veness and Guillaume Desjardins and Andrei A. Rusu and Kieran Milan and John Quan and Tiago Ramalho and Agnieszka Grabska-Barwinska and Demis Hassabis and Claudia Clopath and Dharshan Kumaran and Raia Hadsell},
      year={2017},
      eprint={1612.00796},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{nguyen2018variational,
      title={Variational Continual Learning}, 
      author={Cuong V. Nguyen and Yingzhen Li and Thang D. Bui and Richard E. Turner},
      year={2018},
      eprint={1710.10628},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@misc{zenke2017continual,
      title={Continual Learning Through Synaptic Intelligence}, 
      author={Friedemann Zenke and Ben Poole and Surya Ganguli},
      year={2017},
      eprint={1703.04200},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{pan2021continual,
      title={Continual Deep Learning by Functional Regularisation of Memorable Past}, 
      author={Pingbo Pan and Siddharth Swaroop and Alexander Immer and Runa Eschenhagen and Richard E. Turner and Mohammad Emtiyaz Khan},
      year={2021},
      eprint={2004.14070},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@misc{titsias2020functional,
      title={Functional Regularisation for Continual Learning with {G}aussian Processes}, 
      author={Michalis K. Titsias and Jonathan Schwarz and Alexander G. de G. Matthews and Razvan Pascanu and Yee Whye Teh},
      year={2020},
      eprint={1901.11356},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@inproceedings{Krogh1991ASW,
  title={A Simple Weight Decay Can Improve Generalization},
  author={Anders Krogh and John A. Hertz},
  booktitle={NIPS},
  year={1991}
}

@book{Foundationsofml,
author = {Mohri, Mehryar and Rostamizadeh, Afshin and Talwalkar, Ameet},
title = {Foundations of Machine Learning},
year = {2012},
isbn = {026201825X},
publisher = {The MIT Press},
}

@misc{dinh2017sharp,
      title={Sharp Minima Can Generalize For Deep Nets}, 
      author={Laurent Dinh and Razvan Pascanu and Samy Bengio and Yoshua Bengio},
      year={2017},
      eprint={1703.04933},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{triki2018function,
      title={Function Norms and Regularization in Deep Networks}, 
      author={Amal Rannen Triki and Maxim Berman and Matthew B. Blaschko},
      year={2018},
      eprint={1710.06703},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{baratin2021implicit,
      title={Implicit Regularization via Neural Feature Alignment}, 
      author={Aristide Baratin and Thomas George and César Laurent and R Devon Hjelm and Guillaume Lajoie and Pascal Vincent and Simon Lacoste-Julien},
      year={2021},
      eprint={2008.00938},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{blundell2015weight,
      title={Weight Uncertainty in Neural Networks}, 
      author={Charles Blundell and Julien Cornebise and Koray Kavukcuoglu and Daan Wierstra},
      year={2015},
      eprint={1505.05424},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@misc{sun2019functional,
      title={Functional Variational {B}ayesian Neural Networks}, 
      author={Shengyang Sun and Guodong Zhang and Jiaxin Shi and Roger Grosse},
      year={2019},
      eprint={1903.05779},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@inproceedings{
    Rudner2021fsvi,
    title={{R}ethinking {F}unction-{S}pace {V}ariational {I}nference in {B}ayesian {N}eural {N}etworks},
    author={Tim G. J. Rudner and Zonghao Chen and Yee Whye Teh and Yarin Gal},
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@article{Lopes2011,
abstract = {In this review we explore issues of the sensitivity of Bayes estimates to the prior and form of the likelihood. With respect to the prior, we argue that non-{B}ayesian analyses also incorporate prior information, illustrate that the Bayes posterior mean and the frequentist maximum likelihood estimator are often asymptotically equivalent, review a simple computational strategy for analyzing sensitivity to the prior in practice, and finally document the potentially important role of the prior in {B}ayesian model comparison. With respect to issues of likelihood robustness, we review a variety of computational strategies for significantly expanding the maintained sampling model, including the use of finite Gaussian mixture models and models based on Dirichlet process priors. Copyright {\textcopyright} 2011 by Annual Reviews. All rights reserved.},
author = {Lopes, Hedibert F. and Tobias, Justin L.},
journal = {Annual Review of Economics},
keywords = {{B}ayesian methods,Dirichlet process mixture,Factor models,Gibbs sampler,Marginal likelihood,Markov chain Monte Carlo,Scale mixture of normals,Sequential Monte Carlo},
pages = {107--131},
title = {{Confronting prior convictions: On issues of prior sensitivity and likelihood robustness in {B}ayesian analysis}},
volume = {3},
year = {2011}
}

@article{oakley2004probabilistic,
  title={Probabilistic sensitivity analysis of complex models: a {B}ayesian approach},
  author={Oakley, Jeremy E and O'Hagan, Anthony},
  journal={Journal of the Royal Statistical Society: Series B},
  volume={66},
  number={3},
  pages={751--769},
  year={2004},
  publisher={Wiley Online Library}
}

@article{Sobol2001,
  title={Global sensitivity indices for nonlinear mathematical models and their {M}onte {C}arlo estimates},
  author={Sobol, Ilya M},
  journal={Mathematics and Computers in Simulation},
  volume={55},
  number={1-3},
  pages={271--280},
  year={2001},
  publisher={Elsevier}
}

@article{Kallioinen2021,
  title={Detecting and diagnosing prior and likelihood sensitivity with power-scaling},
  author={Kallioinen, Noa and Paananen, Topi and B{\"u}rkner, Paul-Christian and Vehtari, Aki},
  journal={arXiv preprint arXiv:2107.14054},
  year={2021}
}


@article{Lindqvist2022,
abstract = {Conditional {M}onte {C}arlo refers to sampling from the conditional distribution of a random vector (Formula presented.) given the value (Formula presented.) for a function (Formula presented.). Classical conditional Monte Carlo methods were designed for estimating conditional expectations of functions (Formula presented.) by sampling from unconditional distributions obtained by certain weighting schemes. The basic ingredients were the use of importance sampling and change of variables. In the present paper we reformulate the problem by introducing an artificial parametric model in which (Formula presented.) is a pivotal quantity, and next representing the conditional distribution of (Formula presented.) given (Formula presented.) within this new model. The approach is illustrated by several examples, including a short simulation study and an application to goodness-of-fit testing of real data. The connection to a related approach based on sufficient statistics is briefly discussed.},
author = {Lindqvist, B. H. and Erlemann, R. and Taraldsen, G.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Conditional Monte Carlo/(2020, Lindqvist) Conditional Monte Carlo revisited.pdf:pdf},
journal = {Scandinavian Journal of Statistics},
keywords = {Monte Carlo simulation,change of variables,conditional distribution,exponential family,goodness-of-fit testing,pivotal quantity,sufficiency},
number = {3},
pages = {943--968},
title = {{Conditional Monte Carlo revisited}},
volume = {49},
year = {2022}
}

@article{Chaloner1995,
  title={{B}ayesian experimental design: A review},
  author={Chaloner, Kathryn and Verdinelli, Isabella},
  journal={Statistical Science},
  pages={273--304},
  year={1995},
  publisher={JSTOR}
}

@ARTICLE{2020SciPy-NMeth,
  author  = {Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and
            Haberland, Matt and Reddy, Tyler and Cournapeau, David and
            Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and
            Bright, Jonathan and {van der Walt}, St{\'e}fan J. and
            Brett, Matthew and Wilson, Joshua and Millman, K. Jarrod and
            Mayorov, Nikolay and Nelson, Andrew R. J. and Jones, Eric and
            Kern, Robert and Larson, Eric and Carey, C J and
            Polat, {\.I}lhan and Feng, Yu and Moore, Eric W. and
            {VanderPlas}, Jake and Laxalde, Denis and Perktold, Josef and
            Cimrman, Robert and Henriksen, Ian and Quintero, E. A. and
            Harris, Charles R. and Archibald, Anne M. and
            Ribeiro, Ant{\^o}nio H. and Pedregosa, Fabian and
            {van Mulbregt}, Paul and {SciPy 1.0 Contributors}},
  title   = {{{SciPy} 1.0: Fundamental Algorithms for Scientific
            Computing in Python}},
  journal = {Nature Methods},
  year    = {2020},
  volume  = {17},
  pages   = {261--272},
  adsurl  = {https://rdcu.be/b08Wh},
  doi     = {10.1038/s41592-019-0686-2},
}

@article{OHagan1991BayesHermiteQ,
  title={Bayes–Hermite quadrature},
  author={Anthony O'Hagan},
  journal={Journal of Statistical Planning and Inference},
  year={1991},
  volume={29},
  pages={245-260}
}

@inproceedings{xi2018bayesian,
  title={{B}ayesian quadrature for multiple related integrals},
  author={Xi, Xiaoyue and Briol, Fran{\c{c}}ois-Xavier and Girolami, Mark},
  booktitle={International Conference on Machine Learning},
  pages={5373--5382},
  year={2018},
}
@article{williams2000using,
  title={Using the Nystr{\"o}m method to speed up kernel machines},
  author={Williams, Christopher and Seeger, Matthias},
  journal={Advances in neural information processing systems},
  volume={13},
  year={2000}
}
@InProceedings{osborne12,
  title = 	 {{B}ayesian Quadrature for Ratios},
  author = 	 {Osborne, Michael and Garnett, Roman and Roberts, Stephen and Hart, Christopher and Aigrain, Suzanne and Gibson, Neale},
  booktitle = 	 {Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics},
  pages = 	 {832--840},
  year = 	 {2012},
  editor = 	 {Lawrence, Neil D. and Girolami, Mark},
  volume = 	 {22},
  series = 	 {Proceedings of Machine Learning Research},
  address = 	 {La Palma, Canary Islands},
  month = 	 {21--23 Apr},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v22/osborne12/osborne12.pdf},
  url = 	 {https://proceedings.mlr.press/v22/osborne12.html}
}

@article{muandet2017kernel,
  title={Kernel mean embedding of distributions: A review and beyond},
  author={Muandet, Krikamol and Fukumizu, Kenji and Sriperumbudur, Bharath and Sch{\"o}lkopf, Bernhard and others},
  journal={Foundations and Trends{\textregistered} in Machine Learning},
  volume={10},
  number={1-2},
  pages={1--141},
  year={2017},
  publisher={Now Publishers, Inc.}
}

@inproceedings{li2022optimal,
  title={Optimal Rates for Regularized Conditional Mean Embedding Learning},
  author={Li, Zhu and Meunier, Dimitri and Mollenhauer, Mattes and Gretton, Arthur},
  booktitle={Advances in Neural Information Processing Systems},
  year={2022}
}


@inproceedings{titsias2009variational,
  title={Variational learning of inducing variables in sparse {G}aussian processes},
  author={Titsias, Michalis},
  booktitle={Artificial Intelligence and Statistics},
  pages={567--574},
  year={2009},
}

@inproceedings{liu2016kernelized,
  title={A kernelized {S}tein discrepancy for goodness-of-fit tests},
  author={Liu, Qiang and Lee, Jason and Jordan, Michael},
  booktitle={International conference on machine learning},
  pages={276--284},
  year={2016},
}

@inproceedings{chwialkowski2016kernel,
  title={A kernel test of goodness of fit},
  author={Chwialkowski, Kacper and Strathmann, Heiko and Gretton, Arthur},
  booktitle={International conference on machine learning},
  pages={2606--2615},
  year={2016},
}

@article{gogolashvili2023importance,
  title={When is Importance Weighting Correction Needed for Covariate Shift Adaptation?},
  author={Gogolashvili, Davit and Zecchin, Matteo and Kanagawa, Motonobu and Kountouris, Marios and Filippone, Maurizio},
  journal={arXiv preprint arXiv:2303.04020},
  year={2023}
}

@article{steinwart2012mercer,
  title={Mercer’s theorem on general domains: On the interaction between measures, kernels, and RKHSs},
  author={Steinwart, Ingo and Scovel, Clint},
  journal={Constructive Approximation},
  volume={35},
  pages={363--417},
  year={2012},
  publisher={Springer}
}

@book{fremlin2000measure,
  title={Measure theory},
  author={Fremlin, David Heaver},
  volume={4},
  year={2000},
}

@article{aronszajn1950theory,
  title={Theory of reproducing kernels},
  author={Aronszajn, Nachman},
  journal={Transactions of the American mathematical society},
  volume={68},
  number={3},
  pages={337--404},
  year={1950}
}

@book{edmunds1996function,
  title={Function spaces, entropy numbers, differential operators},
  author={Edmunds, David Eric and Triebel, Hans},
  volume={120},
  year={1996},
  publisher={Cambridge University Press}
}
@article{devore1993besov,
  title={Besov spaces on domains in $\mathbb{R}^d$},
  author={DeVore, Ronald A and Sharpley, Robert C},
  journal={Transactions of the American Mathematical Society},
  volume={335},
  number={2},
  pages={843--864},
  year={1993}
}
@book{adams2003sobolev,
  title={Sobolev spaces},
  author={Adams, Robert A and Fournier, John JF},
  year={2003},
  publisher={Elsevier}
}

@book{novak1988deterministic,
  title={Deterministic and stochastic error bounds in numerical analysis},
  author={Novak, Erich},
  volume={1349},
  year={1988},
  publisher={Springer}
}

@article{shimodaira2000improving,
  title={Improving predictive inference under covariate shift by weighting the log-likelihood function},
  author={Shimodaira, Hidetoshi},
  journal={Journal of statistical planning and inference},
  volume={90},
  number={2},
  pages={227--244},
  year={2000},
  publisher={Elsevier}
}

@article{caponnetto2007optimal,
  title={Optimal rates for the regularized least-squares algorithm},
  author={Caponnetto, Andrea and De Vito, Ernesto},
  journal={Foundations of Computational Mathematics},
  volume={7},
  pages={331--368},
  year={2007},
  publisher={Springer}
}

@book{evans2018measure,
  title={Measure theory and fine properties of functions},
  author={Evans, Lawrence Craig},
  year={2018},
  publisher={Routledge}
}