@article{Glynn1989,
author = {Glynn, P. and Igelhart, D.},
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}
}

@article{Longstaff2001,
abstract = {This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.},
author = {Longstaff, F. A. and Schwartz, E. S.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/least-squares MC/(2001, Longstaff)American Options Least Square Monte Carlo.pdf:pdf},
journal = {Rev. Financ. Stud.},
number = {1},
pages = {113--147},
title = {{Valuing american options by simulation: A simple least-squares approach}},
volume = {14},
year = {2001}
}


@article{Madras1999,
author = {Madras, N. and Piccioni, M.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Importance sampling/(1999, Madras) Importance sampling for families of distributions.pdf:pdf},
journal = {Ann. Appl. Probab.},
keywords = {and phrases,importance sampling,ing,ising model,markov chain monte carlo,metropolis algorithm,simulated temper-,spectral gap},
number = {4},
pages = {1202--1225},
title = {{Importance sampling for families of distributions}},
volume = {9},
year = {1999}
}


@article{Borogovac2008,
abstract = {In this paper we present a new approach to control variates for improving computational efficiency of Ensemble Monte Carlo. We present the approach using simulation of paths of a time-dependent nonlinear stochastic equation. The core idea is to extract information at one or more nominal model parameters and use this information to gain estimation efficiency at neighboring parameters. This idea is the basis of a general strategy, called DataBase Monte Carlo (DBMC), for improving efficiency of Monte Carlo. In this paper we describe how this strategy can be implemented using the variance reduction technique of Control Variates (CV). We show that, once an initial setup cost for extracting information is incurred, this approach can lead to significant gains in computational efficiency. The initial setup cost is justified in projects that require a large number of estimations or in those that are to be performed under real-time constraints.},
archivePrefix = {arXiv},
arxivId = {0809.3187},
author = {Borogovac, T. and Alexander, F. J. and Vakili, P.},
eprint = {0809.3187},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/standard MC/(2008, Borogovac) A control variate approach for improving efficiency of ensemble Monte Carlo.pdf:pdf},
journal = {arXiv:0809.3187},
keywords = {Computation,Computational Engineering,Finance,Statistical Mechanics,and Science},
title = {{A control variate approach for improving efficiency of ensemble Monte Carlo}},
url = {http://arxiv.org/abs/0809.3187},
year = {2008}
}





@inproceedings{antoniou2018train,
  title={How to train your MAML},
  author={Antoniou, A. and Edwards, H. and Storkey, A.},
  booktitle={ICLR},
  year={2019}
}


@inproceedings{sun2021amortized,
  title={{Amortized Bayesian Prototype Meta-learning: A new probabilistic meta-learning approach to few-shot image classification}},
  author={Sun, Z. and Wu, J. and Li, X. and Yang, W. and Xue, J-H.},
  booktitle={AISTATS},
  pages={1414--1422},
  year={2021},
  organization={PMLR}
}


% Bayesian and frequ
@book{gelman1995bayesian,
  title={Bayesian data analysis},
  author={Gelman, Andrew and Carlin, John B and Stern, Hal S and Rubin, Donald B},
  year={1995},
  publisher={Chapman and Hall/CRC}
}


% software
@article{paszke2019pytorch,
  title={Pytorch: An imperative style, high-performance deep learning library},
  author={Paszke, A. and Gross, S. and Massa, F. and Lerer, A. and Bradbury, J. and Chanan, G. and Killeen, T. and Lin, Z. and Gimelshein, N. and Antiga, L. and others},
  journal={NeurIPS},
  volume={32},
  year={2019}
}

@article{South2022,
abstract = {Markov chain Monte Carlo (MCMC) is the engine of modern Bayesian statistics, being used to approximate the posterior and derived quantities of interest. Despite this, the issue of how the output from a Markov chain is post-processed and reported is often overlooked. Convergence diagnostics can be used to control bias via burn-in removal, but these do not account for (common) situations where a limited computational budget engenders a bias-variance trade-off. The aim of this article is to review state-of-the-art techniques for post-processing Markov chain output. Our review covers methods based on discrepancy minimisation, which directly address the bias-variance trade-off, as well as general-purpose control variate methods for approximating expected quantities of interest.},
author = {South, L. F. and Riabiz, M. and Teymur, O. and Oates, C. J.},
journal = {Annu. Rev. Stat. Appl.},
title = {{Post-Processing of MCMC}},
year = {2022}
}

@article{Barp2018,
author = {Barp, A. and Oates, C. J. and Porcu, E. and Girolami, M.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Barp et al. - 2018 - A Riemannian – Stein Kernel Method.pdf:pdf},
journal = {Bernoulli},
number = {4},
pages = {2181--2208},
title = {{A Riemannian–Stein kernel method}},
volume = {28},
year = {2022}
}

@article{Papamarkou2014,
abstract = {Abstract. Differential geometric Markov Chain Monte Carlo (MCMC) strategies exploit the geometry of the target to achieve convergence in fewer MCMC iterations at the cost of increased computing time for each of the iterations. Such computational complexity is ... \n},
author = {Papamarkou, T. and Mira, A. and Girolami, M.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Papamarkou, Mira, Girolami - 2014 - Zero variance differential geometric Markov chain Monte Carlo algorithms.pdf:pdf},
journal = {Bayesian Anal.},
keywords = {Control variates,Hamiltonian monte carlo,Metropolis adjusted langevin algorithms,Metropolis-Hastings},
number = {1},
pages = {97--128},
title = {{Zero variance differential geometric Markov chain Monte Carlo algorithms}},
volume = {9},
year = {2014}
}


@article{Friel2014,
author = {Friel, N. and Mira, A. and Oates, C. J.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Friel, Mira, Oates - 2014 - Exploiting multi-core architectures for reduced-variance estimation with intractable likelihoods.pdf:pdf},
journal = {Bayesian Anal.},
keywords = {control variates,mcmc,parallel computing,zero variance},
number = {1},
pages = {215--245},
title = {{Exploiting multi-core architectures for reduced-variance estimation with intractable likelihoods}},
volume = {11},
year = {2014}
}


@article{Dellaportas2012,
abstract = {A general methodology is introduced for the construction and effective application of control variates to estimation problems involving data from reversible Markov chain Monte Carlo samplers.We propose the use of a specific class of functions as control variates, and we introduce a new consistent estimator for the values of the coefficients of the optimal linear combination of these functions. For a specific Markov chain Monte Carlo scenario, the form and proposed construction of the control variates is shown to provide an exact solution of the associated Poisson equation. This implies that the estimation variance in this case (in the central limit theorem regime) is exactly zero. The new estimator is derived from a novel, finite dimensional, explicit representation for the optimal coefficients.The resulting variance reduction methodology is primarily (though certainly not exclusively) applicable when the simulated data are generated by a random-scan Gibbs sampler.Markov chain Monte Carlo examples of Bayes- ian inference problems demonstrate that the corresponding reduction in the estimation variance is significant, and that in some cases it can be quite dramatic. Extensions of this methodology are discussed and simulation examples are presented illustrating the utility of the methods pro- posed. All methodological and asymptotic arguments are rigorously justified under essentially minimal conditions.},
author = {Dellaportas, P. and Kontoyiannis, I},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Dellaportas, Kontoyiannis - 2012 - Control variates for estimation based on reversible Markov chain Monte Carlo samplers.pdf:pdf},
journal = {J. R. Stat. Soc. Series B},
keywords = {Bayesian inference,Control variates,Hierarchical normal linear model,Log-linear model,Markov chain Monte Carlo methods,Mixtures of normals,Poisson equation,Threshold auto-regressive model,Variance reduction},
number = {1},
pages = {133--161},
title = {{Control variates for estimation based on reversible Markov chain Monte Carlo samplers}},
volume = {74},
year = {2012}
}
@article{Portier2018,
abstract = {The use of control variates is a well-known variance reduction tech-nique in Monte Carlo integration. If the optimal linear combination of control variates is estimated by ordinary least squares and if the num-ber of control variates is allowed to grow to infinity, the convergence rate can be accelerated, the new rate depending on the interplay be-tween the integrand and the control functions. The standardized error is still asymptotically normal and the asymptotic variance can still be estimated by the residual variance in the underlying regression model. The ordinary least squares estimator is shown to be superior to other, possibly simpler control variate estimators, even at equal computation time. The method is applied to increase the precision of the method of maximum simulated likelihood to deal with latent variables. Its performance is found to be particularly good because of two reasons: the integrands are smooth and can thus be approximated well by poly-nomial or spline control functions; the number of integrands is large, reducing the computational cost since the Monte Carlo integration weights need to be calculated only once.},
author = {Portier, F. and Segers, J.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Portier, Segers - 2019 - Monte Carlo integration with a growing number of control variates.pdf:pdf},
journal = {Ann. Appl. Probab.},
keywords = {generalized linear model,maximum sim-ulated likelihood,multiple linear regression,ordinary least-squares,random effects},
number = {4},
pages = {1168--1186},
title = {{Monte Carlo integration with a growing number of control variates}},
volume = {56},
year = {2019}
}
@article{Baker2018,
abstract = {It is well known that Markov chain Monte Carlo (MCMC) methods scale poorly with dataset size. A popular class of methods for solving this issue is stochastic gradient MCMC. These methods use a noisy estimate of the gradient of the log posterior, which reduces the per iteration computational cost of the algorithm. Despite this, there are a number of results suggesting that stochastic gradient Langevin dynamics (SGLD), probably the most popular of these methods, still has computational cost proportional to the dataset size. We suggest an alternative log posterior gradient estimate for stochastic gradient MCMC, which uses control variates to reduce the variance. We analyse SGLD using this gradient estimate, and show that, under log-concavity assumptions on the target distribution, the computational cost required for a given level of accuracy is independent of the dataset size. Next we show that a different control variate technique, known as zero variance control variates can be applied to SGMCMC algorithms for free. This post-processing step improves the inference of the algorithm by reducing the variance of the MCMC output. Zero variance control variates rely on the gradient of the log posterior; we explore how the variance reduction is affected by replacing this with the noisy gradient estimate calculated by SGMCMC.},
author = {Baker, J. and Fearnhead, P. and Fox, E. B. and Nemeth, C.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Baker et al. - 2019 - Control variates for stochastic gradient MCMC.pdf:pdf},
journal = {Stat. Comput.},
keywords = {big data,computa-,control variates,langevin dynamics,scalable mcmc,stochastic gradient mcmc,tional cost},
pages = {599--615},
title = {{Control variates for stochastic gradient MCMC}},
volume = {29},
year = {2019}
}
@article{Hickernell2005,
author = {Hickernell, F.J. and Lemieux, C. and Owen, A. B.},
journal = {Stat. Sci.},
keywords = {and phrases,digital nets,lattice rules,low discrepancy meth-,ods,stratification,variance reduction},
number = {1},
pages = {1--31},
title = {{Control variates for quasi-Monte Carlo}},
volume = {20},
year = {2005}
}
@inproceedings{Liu2018,
abstract = {Policy gradient methods have achieved remarkable successes in solving challenging reinforcement learning problems. However, it still often suffers from the large variance issue on policy gradient estimation, which leads to poor sample efficiency during training. In this work, we propose a control variate method to effectively reduce variance for policy gradient methods. Motivated by the Stein's identity, our method extends the previous control variate methods used in REINFORCE and advantage actor-critic by introducing more general action-dependent baseline functions. Empirical studies show that our method significantly improves the sample efficiency of the state-of-the-art policy gradient approaches.},
archivePrefix = {arXiv},
arxivId = {1710.11198},
author = {Liu, H. and Feng, Y. and Mao, Y. and Zhou, D. and Peng, J. and Liu, Q.},
booktitle = {ICLR},
eprint = {1710.11198},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Liu et al. - 2018 - Action-depedent Control Variates for Policy Optimization via Stein's Identity.pdf:pdf},
title = {{Action-dependent control variates for policy optimization via stein's identity}},
year = {2018}
}


@inproceedings{Grathwohl2017,
abstract = {Gradient-based optimization is the foundation of deep learning and reinforcement learning. Even when the mechanism being optimized is unknown or not differentiable, optimization using high-variance or biased gradient estimates is still often the best strategy. We introduce a general framework for learning low-variance, unbiased gradient estimators for black-box functions of random variables. Our method uses gradients of a neural network trained jointly with model parameters or policies, and is applicable in both discrete and continuous settings. We demonstrate this framework for training discrete latent-variable models. We also give an unbiased, action-conditional extension of the advantage actor-critic reinforcement learning algorithm.},
author = {Grathwohl, W. and Choi, D. and Wu, Y. and Roeder, G. and Duvenaud, D.},
booktitle = {ICLR},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Grathwohl et al. - 2018 - Backpropagation through the void Optimizing control variates for black-box gradient estimation.pdf:pdf},
title = {{Backpropagation through the void: Optimizing control variates for black-box gradient estimation}},
year = {2018}
}


@article{Oates2016thermo,
abstract = {Bayesian model comparison relies upon the model evidence, yet for many models of interest the model evidence is unavailable in closed form and must be approximated. Many of the estimators for evidence that have been proposed in the Monte Carlo literature suffer from high variability. This paper considers the reduction of variance that can be achieved by exploiting control variates in this setting. Our methodology is based on thermodynamic integration and applies whenever the gradient of both the log-likelihood and the log-prior with respect to the parameters can be efficiently evaluated. Results obtained on regression models and popular benchmark datasets demonstrate a significant and sometimes dramatic reduction in estimator variance and provide insight into the wider applicability of control variates to Bayesian model comparison.},
author = {Oates, C. J. and Papamarkou, T. and Girolami, M.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Oates, Papamarkou, Girolami - 2016 - The controlled thermodynamic integral for Bayesian model comparison.pdf:pdf},
journal = {J. Am. Stat. Assoc.},
number = {514},
pages = {634--645},
title = {{The controlled thermodynamic integral for Bayesian model comparison}},
volume = {111},
year = {2016}
}


@inproceedings{Shi2022,
abstract = {Gradient estimation -- approximating the gradient of an expectation with respect to the parameters of a distribution -- is central to the solution of many machine learning problems. However, when the distribution is discrete, most common gradient estimators suffer from excessive variance. To improve the quality of gradient estimation, we introduce a variance reduction technique based on Stein operators for discrete distributions. We then use this technique to build flexible control variates for the REINFORCE leave-one-out estimator. Our control variates can be adapted online to minimize variance and do not require extra evaluations of the target function. In benchmark generative modeling tasks such as training binary variational autoencoders, our gradient estimator achieves substantially lower variance than state-of-the-art estimators with the same number of function evaluations.},
author = {Shi, J. and Zhou, Y. and Hwang, J. and Titsias, M. K. and Mackey, L.},
booktitle = {NeurIPS},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/2202.09497.pdf:pdf},
title = {{Gradient estimation with discrete Stein operators}},
year = {2022}
}


@article{Alexopoulos2023,
abstract = {We introduce a general framework that constructs estimators with reduced variance for random walk Metropolis and Metropolis-adjusted Langevin algorithms. The resulting estimators require negligible computational cost and are derived in a post-process manner utilising all proposal values of the Metropolis algorithms. Variance reduction is achieved by producing control variates through the approximate solution of the Poisson equation associated with the target density of the Markov chain. The proposed method is based on approximating the target density with a Gaussian and then utilising accurate solutions of the Poisson equation for the Gaussian case. This leads to an estimator that uses two key elements: (1) a control variate from the Poisson equation that contains an intractable expectation under the proposal distribution, (2) a second control variate to reduce the variance of a Monte Carlo estimate of this latter intractable expectation. Simulated data examples are used to illustrate the impressive variance reduction achieved in the Gaussian target case and the corresponding effect when target Gaussianity assumption is violated. Real data examples on Bayesian logistic regression and stochastic volatility models verify that considerable variance reduction is achieved with negligible extra computational cost.},
author = {Alexopoulos, A. and Dellaportas, P. and Titsias, M. K.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2022, Angelopoulos) Variance reduction for Metropolis-Hastings samplers.pdf:pdf},
journal = {Stat. Comput.},
keywords = {Bayesian inference,Control variates,Logistic regression,Markov chain Monte Carlo,Poisson equation,Stochastic volatility},
number = {6},
publisher = {Springer US},
title = {{Variance reduction for Metropolis–Hastings samplers}},
volume = {33},
year = {2023}
}


@article{Muller2020,
abstract = {We propose the concept of neural control variates (NCV) for unbiased variance reduction in parametric Monte Carlo integration for solving integral equations. So far, the core challenge of applying the method of control variates has been finding a good approximation of the integrand that is cheap to integrate. We show that a set of neural networks can face that challenge: a normalizing flow that approximates the shape of the integrand and another neural network that infers the solution of the integral equation. We also propose to leverage a neural importance sampler to estimate the difference between the original integrand and the learned control variate. To optimize the resulting parametric estimator, we derive a theoretically optimal, variance-minimizing loss function, and propose an alternative, composite loss for stable online training in practice. When applied to light transport simulation, neural control variates are capable of matching the state-of-the-art performance of other unbiased approaches, while providing means to develop more performant, practical solutions. Specifically, we show that the learned light-field approximation is of sufficient quality for high-order bounces, allowing us to omit the error correction and thereby dramatically reduce the noise at the cost of insignificant visible bias.},
author = {M{\"{u}}ller, T. and Rousselle, F. and Nov{\'{a}}k, J. and Keller, A.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2020, Muller) Neural Control Variates.pdf:pdf},
journal = {ACM Trans. Graph.},
number = {6},
pages = {1--19},
title = {{Neural control variates}},
volume = {39},
year = {2020}
}
@article{Belomestny2022,
abstract = {We study the problem of empirical minimization for variance-type functionals over functional classes. Sharp nonasymptotic bounds for the excess variance are derived under mild conditions. In particular, it is shown that under some restrictions imposed on the functional class fast convergence rates can be achieved including the optimal non-parametric rates for expressive classes in the non-Donsker regime under some additional assumptions. Our main applications include variance reduction and optimal control.},
author = {Belomestny, D. and Iosipoi, L. and Paris, Q. and Zhivotovskiy, N.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2021, Belomestny) Empirical variance minimization with applications in variance reduction and optimal control.pdf:pdf},
journal = {Bernoulli},
keywords = {Control variates,Empirical variance minimization,Optimal control,Variance reduction},
number = {2},
pages = {1382--1407},
title = {{Empirical variance minimization with applications in variance reduction and optimal control}},
volume = {28},
year = {2022}
}

@article{Belomestny2019,
abstract = {In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches.},
author = {Belomestny, D. and Iosipoi, L. and Moulines, E. and Naumov, A. and Samsonov, S.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2010, Belomestny) Variance reduction for Markov chains with application to MCMC.pdf:pdf},
journal = {Stat. Comput.},
pages = {973--997},
title = {{Variance reduction for Markov chains with application to MCMC}},
volume = {30},
year = {2020}
}


% Experiments related


@book{doucet2001sequential,
  title={Sequential Monte Carlo methods in practice},
  author={Doucet, A. and De Freitas, N. and Gordon, N. J.},
  volume={1},
  year={2001},
  publisher={Springer}
}

@article{doucet2000sequential,
  title={On sequential Monte Carlo sampling methods for Bayesian filtering},
  author={Doucet, A. and Godsill, S. and Andrieu, C.},
  journal={Stat. Comput.},
  volume={10},
  pages={197--208},
  year={2000},
  publisher={Springer}
}

@article{park2017probabilistic,
  title={Probabilistic seismic and tsunami hazard analysis conditioned on a megathrust rupture of the Cascadia subduction zone},
  author={Park, H. and Cox, D. T. and Alam, M. S. and Barbosa, A.},
  journal={Front. Built Environ.},
  volume={3},
  pages={32},
  year={2017},
  publisher={Frontiers Media SA}
}

@article{brauer2017mathematical,
  title={Mathematical epidemiology: Past, present, and future},
  author={Brauer, F.},
  journal={Infect. Dis. Model.},
  volume={2},
  number={2},
  pages={113--127},
  year={2017},
  publisher={Elsevier}
}


@article{Alfonsi2022,
abstract = {The problem of computing the conditional expectation E[f (Y)|X] with least-square Monte-Carlo is of general importance and has been widely studied. To solve this problem, it is usually assumed that one has as many samples of Y as of X. However, when samples are generated by computer simulation and the conditional law of Y given X can be simulated, it may be relevant to sample K $\in$ N values of Y for each sample of X. The present work determines the optimal value of K for a given computational budget, as well as a way to estimate it. The main take away message is that the computational gain can be all the more important that the computational cost of sampling Y given X is small with respect to the computational cost of sampling X. Numerical illustrations on the optimal choice of K and on the computational gain are given on different examples including one inspired by risk management.},
author = {Alfonsi, A. and Lapeyre, B. and Lelong, J.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/least-squares MC/(2022,) How many inner simulations to compute conditional expectations with least-square Monte Carlo.pdf:pdf},
journal = {arXiv:2209.04153},
keywords = {65c05,91g60,aa and bl acknowledge,acknowledgement,ams 2020,chaire,conditional expectation estimators,fondation du risque,least square monte-carlo,risques financiers,the support of the,variance reduction},
title = {{How many inner simulations to compute conditional expectations with least-square Monte Carlo?}},
year = {2022}
}



@incollection{bolker2008ecological,
  title={{Ecological models and data in R}},
  author={Bolker, B. M.},
  booktitle={Ecological Models and Data in R},
  year={2008},
  publisher={Princeton University Press}
}

@inproceedings{lalchand2020approximate_FB_GP,
  title={{Approximate inference for fully Bayesian Gaussian process regression}},
  author={Lalchand, V. and Rasmussen, C. E.},
  booktitle={AABI},
  pages={1--12},
  year={2020},
  organization={PMLR}
}

@article{kucukelbir2017automatic_VL,
  title={Automatic differentiation variational inference},
  author={Kucukelbir, A. and Tran, D. and Ranganath, R. and Gelman, A. and Blei, D. M.},
  journal={J. Mach. Learn. Res.},
  year={2017}
}

@article{mohamed2020_MC_in_ML,
  title={Monte Carlo Gradient Estimation in Machine Learning.},
  author={Mohamed, Shakir and Rosca, Mihaela and Figurnov, Michael and Mnih, Andriy},
  journal={J. Mach. Learn. Res.},
  volume={21},
  number={132},
  pages={1--62},
  year={2020}
}

@article{riabiz2022optimal,
  title={Optimal thinning of {MCMC} output},
  author={Riabiz, M. and Chen, W. and Cockayne, J. and Swietach, P.},
  journal={J. R. Stat. Soc. Series B},
  year={2022},
  publisher={Royal Statistical Society}
}

@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}
}


@article{Demange-Chryst2022,
abstract = {Some classical uncertainty quantification problems require the estimation of multiple expectations. Estimating all of them accurately is crucial and can have a major impact on the analysis to perform, and standard existing Monte Carlo methods can be costly to do so. We propose here a new procedure based on importance sampling and control variates for estimating more efficiently multiple expectations with the same sample. We first show that there exists a family of optimal estimators combining both importance sampling and control variates, which however cannot be used in practice because they require the knowledge of the values of the expectations to estimate. Motivated by the form of these optimal estimators and some interesting properties, we therefore propose an adaptive algorithm. The general idea is to adaptively update the parameters of the estimators for approaching the optimal ones. We suggest then a quantitative stopping criterion that exploits the trade-off between approaching these optimal parameters and having a sufficient budget left. This left budget is then used to draw a new independent sample from the final sampling distribution, allowing to get unbiased estimators of the expectations. We show how to apply our procedure to sensitivity analysis, by estimating Sobol' indices and quantifying the impact of the input distributions. Finally, realistic test cases show the practical interest of the proposed algorithm, and its significant improvement over estimating the expectations separately.},
author = {Demange-Chryst, J. and Bachoc, F. and Morio, J.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2022,) Efficient estimation of multiple expectations with the same sample by adaptive importance sampling and control variates.pdf:pdf},
journal = {arXiv:2212.00568},
keywords = {control,global sensitivity analysis,importance sampling,multiple expectation estimation,variance reduction,variates},
title = {{Efficient estimation of multiple expectations with the same sample by adaptive importance sampling and control variates}},
year = {2022}
}


@book{hewitt1921conservation,
  title={The conservation of the wild life of Canada},
  author={Hewitt, C.},
  year={1921},
  publisher={New York: C. Scribner}
}


@book{lotka1925elements,
  title={Elements of physical biology},
  author={Lotka, A.},
  year={1925},
  publisher={Williams \& Wilkins}
}

@article{lotka1927fluctuations,
  title={Fluctuations in the abundance of a species considered mathematically},
  author={Lotka, A.},
  journal={Nature},
  volume={119},
  number={2983},
  pages={12--12},
  year={1927},
  publisher={Nature Publishing Group}
}



@article{giles2015multilevel,
  title={{Multilevel Monte Carlo methods}},
  author={Giles, M.},
  journal={Acta Numer.},
  volume={24},
  pages={259--328},
  year={2015},
  publisher={Cambridge University Press}
}

% Convex optimisation
@book{boyd2004convex,
  title={Convex optimization},
  author={Boyd, S. and Boyd, S. P. and Vandenberghe, L.},
  year={2004},
  publisher={Cambridge university press}
}

%% GBML theories
@article{riou2023bayes,
  title={Bayes meets Bernstein at the Meta Level: an Analysis of Fast Rates in Meta-Learning with PAC-Bayes},
  author={Riou, C. and Alquier, P. and Ch{\'e}rief-Abdellatif, B-E.},
  journal={arXiv preprint arXiv:2302.11709},
  year={2023}
}

@article{chen2021generalization,
  title={Generalization bounds for meta-learning: An information-theoretic analysis},
  author={Chen, Q. and Shui, C. and Marchand, M.},
  journal={NeurIPS},
  volume={34},
  pages={25878--25890},
  year={2021}
}

@inproceedings{fallah2020convergence_gbml,
  title={On the convergence theory of gradient-based model-agnostic meta-learning algorithms},
  author={Fallah, A. and Mokhtari, A. and Ozdaglar, A.},
  booktitle={AISTATS},
  year={2020},
  organization={PMLR}
}



@article{ji2020convergence_gbml_anil,
  title={Convergence of meta-learning with task-specific adaptation over partial parameters},
  author={Ji, K. and Lee, J. D. and Liang, Y. and Poor, H. V.},
  journal={NeurIPS},
  volume={33},
  pages={11490--11500},
  year={2020}
}


@article{ji2022theoretical_ms_gbml,
  title={Theoretical Convergence of Multi-Step Model-Agnostic Meta-Learning.},
  author={Ji, K. and Yang, J. and Liang, Y.},
  journal={J. Mach. Learn. Res.},
  volume={23},
  pages={29--1},
  year={2022}
}


%% Stan
@article{carpenter2017stan,
  title={Stan: A probabilistic programming language},
  author={Carpenter, B. and Gelman, A. and Hoffman, M. D. and Lee, D. and Goodrich, B. and Betancourt, M. and Brubaker, M. and Guo, J. and Li, P. and Riddell, A.},
  journal={J. Stat. Softw.},
  volume={76},
  number={1},
  year={2017},
  publisher={Columbia Univ., New York, NY (United States); Harvard Univ., Cambridge, MA~…}
}

%% Multi-level
@article{krumscheid2018multilevel,
  title={Multilevel Monte Carlo approximation of functions},
  author={Krumscheid, S. and Nobile, F.},
  journal={SIAM-ASA J. Uncertain.},
  volume={6},
  number={3},
  pages={1256--1293},
  year={2018},
}



@article{uq_approx_p_quantiles_for_physicalmodels,
author = {Elfverson, D. and Estep, D. J. and Hellman, F. and M\r{a}lqvist, A.},
title = {Uncertainty Quantification for Approximate p-Quantiles for Physical Models with Stochastic Inputs},
journal = {SIAM-ASA J. Uncertain.},
volume = {2},
number = {1},
pages = {826-850},
year = {2014},
abstract = { We consider the problem of estimating the \$p\$-quantile for a given functional evaluated on solutions of a deterministic model in which model input is subject to stochastic variation. We derive upper and lower bounding estimators of the \$p\$-quantile. We perform an a posteriori error analysis for the \$p\$-quantile estimators that takes into account the effects of both the stochastic sampling error and the deterministic numerical solution error and yields a computational error bound for the estimators. We also analyze the asymptotic convergence properties of the \$p\$-quantile estimator bounds in the limit of large sample size and decreasing numerical error and describe algorithms for computing an estimator of the \$p\$-quantile with a desired accuracy in a computationally efficient fashion. One algorithm exploits the fact that the accuracy of only a subset of sample values significantly affects the accuracy of a \$p\$-quantile estimator resulting in a significant gain in computational efficiency. We conclude with a number of numerical examples, including an application to Darcy flow in porous media. }
}



%% VI -- VAE
@inproceedings{kingma2013_VAE,
  title={Auto-encoding variational bayes},
  author={Kingma, D. P. and Welling, M.},
  booktitle={ICLR},
  year={2014}
}


%% CV for VI 
@article{williams1992simple_reinforce,
  title={Simple statistical gradient-following algorithms for connectionist reinforcement learning},
  author={Williams, Ronald J},
  journal={Machine learning},
  volume={8},
  number={3},
  pages={229--256},
  year={1992},
  publisher={Springer}
}

@article{glynn1990likelihood_reinforce,
  title={Likelihood ratio gradient estimation for stochastic systems},
  author={Glynn, P.},
  journal={Communications of the ACM},
  volume={33},
  number={10},
  pages={75--84},
  year={1990},
  publisher={ACM New York, NY, USA}
}


@inproceedings{dimitriev2021arms_reinforce,
  title={Arms: Antithetic-reinforce-multi-sample gradient for binary variables},
  author={Dimitriev, A. and Zhou, M.},
  booktitle={ICML},
  year={2021},
  organization={PMLR}
}


@article{titsias2021doubleCV_reinforce,
  title={Double Control Variates for Gradient Estimation in Discrete Latent Variable Models},
  author={Titsias, M. K. and Shi, J.},
  journal={arXiv:2111.05300},
  year={2021}
}






%%% CV for BBVI
@inproceedings{grathwohl2018backpropagation_cv_bbvi,
  title={Backpropagation through the Void: Optimizing control variates for black-box gradient estimation},
  author={Grathwohl, W. and Choi, D. and Wu, Y. and Roeder, G. and Duvenaud, D.},
  booktitle={ICLR},
  year={2018}
}



%%%% CV for RL
@inproceedings{liu2019taming,
  title={Taming maml: Efficient unbiased meta-reinforcement learning},
  author={Liu, H. and Socher, R. and Xiong, C.},
  booktitle={ICML},
  year={2019},
  organization={PMLR}
}




%%% new added GBML
@article{grefenstette2019generalized_maml,
  title={Generalized inner loop meta-learning},
  author={Grefenstette, E. and Amos, B. and Yarats, D. and Htut, P. and Molchanov, A. and Meier, F. and Kiela, D. and Cho, K. and Chintala, S.},
  journal={  arXiv:1910.01727},
  year={2019}
}


@inproceedings{finn2019online_maml,
  title={Online meta-learning},
  author={Finn, C. and Rajeswaran, A. and Kakade, S. and Levine, S.},
  booktitle={ICML},
  year={2019}
}


@inproceedings{wu2020meta_vae,
  title={Meta-amortized variational inference and learning},
  author={Wu, Mike and Choi, Kristy and Goodman, Noah and Ermon, Stefano},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={34},
  number={04},
  pages={6404--6412},
  year={2020}
}



@inproceedings{garnelo2018conditional_NP,
  title={Conditional neural processes},
  author={Garnelo, Marta and Rosenbaum, Dan and Maddison, Christopher and Ramalho, Tiago and Saxton, David and Shanahan, Murray and Teh, Yee Whye and Rezende, Danilo and Eslami, SM Ali},
  booktitle={ICML},
  year={2018},
  organization={PMLR}
}


@article{andrychowicz2016_l2l_GD_by_GD,
  title={Learning to learn by gradient descent by gradient descent},
  author={Andrychowicz, M. and Denil, M. and Gomez, S. and Hoffman, M. W. and Pfau, D. and Schaul, T. and Shillingford, B. and De Freitas, N.},
  journal={NeurIPS},
  year={2016}
}




%%%%%%%---past
%%%%%
@book{Yang2019,
abstract = {Predicting the binding mode of flexible polypeptides to proteins is an important task that falls outside the domain of applicability of most small molecule and protein−protein docking tools. Here, we test the small molecule flexible ligand docking program Glide on a set of 19 non-$\alpha$-helical },
author = {Yang, Q. and Zhang, Y. and Dai, Y. and Pan, S. J.},
title = {{Transfer Learning}},
year = {2019}
}





@article{Peherstorfer2018,
abstract = {In many situations across computational science and engineering, multi-ple computational models are available that describe a system of interest. These different models have varying evaluation costs and varying fidelities. Typically, a computationally expensive high-fidelity model describes the sys-tem with the accuracy required by the current application at hand, while lower-fidelity models are less accurate but computationally cheaper than the high-fidelity model. Outer-loop applications, such as optimization, inference, and uncertainty quantification, require multiple model evaluations at many different inputs, which often leads to computational demands that exceed available resources if only the high-fidelity model is used. This work surveys multifidelity methods that accelerate the solution of outer-loop applications by combining high-fidelity and low-fidelity model evaluations. The overall premise of these multifidelity methods is that low-fidelity models are lever-aged for speedup while the high-fidelity model is kept in the loop to establish accuracy and/or convergence guarantees. We categorize multifidelity meth-ods according to three classes of strategies: adaptation, fusion, and filtering. The paper reviews multifidelity methods in the outer-loop contexts of uncer-tainty propagation, inference, and optimization.},
author = {Peherstorfer, B. and Willcox, K. and Gunzburger, M.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Peherstorfer, Willcox, Gunzburger - 2018 - Survey of multifidelity methods in uncertainty propagation, inference, and optimization.pdf:pdf},
journal = {SIAM Review},
keywords = {model reduction,multifidelity,multifidelity optimization,multifidelity statistical inference,multifidelity uncertainty,multifidelity uncertainty propagation,quantification,surrogate models},
number = {3},
pages = {550--591},
title = {{Survey of multifidelity methods in uncertainty propagation, inference, and optimization}},
volume = {60},
year = {2018}
}




@article{Green2015,
author = {Green, P. and Latuszyski, K. and Pereyra, M. and Robert, C.},
journal = {Stat. Comput.},
keywords = {bayesian analysis,mcmc algorithms},
pages = {835--862},
title = {{Bayesian computation: a summary of the current state, and samples backwards and forwards}},
volume = {25},
year = {2015}
}



@article{Jones2004,
author = {Jones, G. L.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Jones - 2004 - On the Markov chain central limit theorem.pdf:pdf},
journal = {Probab. Surv.},
pages = {299--320},
title = {{On the Markov chain central limit theorem}},
volume = {1},
year = {2004}
}


@article{carmeli2005_VectorValuedRKHS_MercerTheorem,
abstract = {We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2, we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel, extending the Mercer theorem.},
author = {Carmeli, C. and {De Vito}, E. and Toigo, A.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Carmeli, De Vito, Toigo - 2006 - Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem.pdf:pdf},
journal = {Analysis and Applications},
number = {4},
pages = {377--408},
title = {{Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem}},
volume = {4},
year = {2006}
}



@article{evgeniou2005_Learning_MultiTask_with_Kernelmethods,
  title={Learning multiple tasks with kernel methods},
  author={Evgeniou, Theodoros and Micchelli, Charles A and Pontil, Massimiliano},
  journal={J. Mach. Learn. Res.},
  volume={6},
  pages={615--637},
  year={2005}
}

@article{Micchelli2004,
abstract = {This paper provides a foundation for multi–task learning using reproducing kernel Hilbert spaces of vector–valued functions. In this setting, the kernel is a matrix–valued function. Some explicit examples will be described which go beyond our earlier results in [7]. In particular, we characterize classes of matrix– valued kernels which are linear and are of the dot product or the translation invariant type. We discuss how these kernels can be used to model relations between the tasks and present linear multi–task learning algorithms. Finally, we present a novel proof of the representer theorem for a minimizer of a regularization functional which is based on the notion of minimal norm interpolation. 1},
author = {Micchelli, Charles A. and Pontil, Massimiliano},
journal = {NeurIPS},
pages = {921--928},
title = {{Kernels for multi-task learning}},
volume = {17},
year = {2004}
}



@article{micchelli2005_On_Learning_Vectorvalued_functions,
  title={On learning vector-valued functions},
  author={Micchelli, Charles A and Pontil, Massimiliano},
  journal={Neural Computation},
  volume={17},
  number={1},
  pages={177--204},
  year={2005},
}


@article{karvonen2019_On_Positivity_and_Magnitudes_of_BQweights,
  title={{On the positivity and magnitudes of Bayesian quadrature weights}},
  author={Karvonen, Toni and Kanagawa, Motonobu and S{\"a}rkk{\"a}, Simo},
  journal={Stat. Comput.},
  volume={29},
  pages={1317--1333},
  year={2019},
  publisher={Springer}
}


@article{Xiong2013,
abstract = {A growing trend in engineering and science is to use multiple computer codes with different levels of accuracy to study the same complex system. We propose a framework for sequential design and analysis of a pair of high-accuracy and low-accuracy computer codes. It first runs the two codes with a pair of nested Latin hypercube designs (NLHDs). Data from the initial experiment are used to fit a prediction model. If the accuracy of the fitted model is less than a prespecified threshold, the two codes are evaluated again with input values chosen in an elaborate fashion so that their expanded scenario sets still form a pair of NLHDs. The nested relationship between the two scenario sets makes it easier to model and calibrate the difference between the two sources. If necessary, this augmentation process can be repeated a number of times until the prediction model based on all available data has reasonable accuracy. The effectiveness of the proposed method is illustrated with several examples. Matlab codes are provided in the online supplement to this article. {\textcopyright} 2013 American Statistical Association and the American Society for Quality.},
author = {Xiong, S. and Qian, P. Z. G. and Wu, C. F. J.},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/Multiple output/(2013, Xiong) Sequential Design and Analysis of High Accuracy and Low Accuracy Computer Codes.pdf:pdf},
journal = {Technometrics},
keywords = {Gaussian process model,Kriging,Nested Latin hypercube design},
number = {1},
pages = {37--46},
title = {{Sequential design and analysis of high-accuracy and low-accuracy computer codes}},
volume = {55},
year = {2013}
}


@inproceedings{carratino2018_Learning_with_SGD_and_RandomFeatures,
  title={Learning with sgd and random features},
  author={Carratino, Luigi and Rudi, Alessandro and Rosasco, Lorenzo},
  booktitle={NeurIPS},
  pages={10192--10203},
  year={2018}
}


@article{mucke2020_SGD_in_HilbertScales,
  title={Stochastic Gradient Descent in Hilbert Scales: Smoothness, Preconditioning and Earlier Stopping},
  author={Mucke, Nicole and Reiss, Enrico},
  journal={stat},
  volume={1050},
  pages={18},
  year={2020}
}



@article{ali2020_Implicit_Regularization_StochasticGradientFlow_for_LeastSquares,
  title={The Implicit Regularization of Stochastic Gradient Flow for Least Squares},
  author={Ali, Alnur and Dobriban, Edgar and Tibshirani, Ryan J},
  journal={  arXiv:2003.07802},
  year={2020}
}



@article{christmann2007_Consistency_of_KernelBased_regression_in_ConvexRiskMin,
  title={Consistency and robustness of kernel-based regression in convex risk minimization},
  author={Christmann, Andreas and Steinwart, Ingo and others},
  journal={Bernoulli},
  volume={13},
  number={3},
  pages={799--819},
  year={2007},
  publisher={Bernoulli Society for Mathematical Statistics and Probability}
}





% Algebra
@book{horn1994_topics_in_matrix_analysis,
  title={Topics in matrix analysis},
  author={Horn, Roger A and Horn, Roger A and Johnson, Charles R},
  year={1994},
  publisher={Cambridge university press}
}





% Matrix-Kernels, RKHS and GP
@book{berlinet2011_RKHSinProbsStats,
  title={Reproducing Kernel Hilbert Spaces in Probability and Statistics},
  author={Berlinet, Alain and Thomas-Agnan, Christine},
  year={2011},
  publisher={Springer Science \& Business Media}
}

@book{steinwart2008_SVM,
  title={Support vector machines},
  author={Steinwart, Ingo and Christmann, Andreas},
  year={2008},
  publisher={Springer Science \& Business Media}
}



@book{rasmussen2003_GPML,
  title={Gaussian processes for machine learning},
  author={Rasmussen, C. E. and Williams, C. K. I.},
  volume={1},
  year={2006},
  publisher={Springer}
}


@article{Alvarez2011,
abstract = {Computer Science and Artificial Intelligence Laboratory Technical Report},
author = {{\'{A}}lvarez, M A and Rosasco, L and Lawrence, N D},
journal = {Foundations and Trends in Machine Learning},
number = {3},
pages = {195--266},
title = {{Kernels for vector-valued functions: A review}},
volume = {4},
year = {2012}
}



@article{kanagawa2018_GPandKM_RviewandConnections,
  title={Gaussian processes and kernel methods: A review on connections and equivalences},
  author={Kanagawa, Motonobu and Hennig, Philipp and Sejdinovic, Dino and Sriperumbudur, Bharath K},
  journal={  arXiv:1807.02582},
  year={2018}
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@article{Park2017,
abstract = {Different multi-fidelity surrogate (MFS) frameworks have been used for optimization or uncertainty quantification. This paper investigates differences between various MFS frameworks with the aid of examples including algebraic functions and a borehole example. These MFS include three Bayesian frameworks using 1) a model discrepancy function, 2) low fidelity model calibration and 3) a comprehensive approach combining both. Three counterparts in simple frameworks are also included, which have the same functional form but can be built with ready-made surrogates. The sensitivity of frameworks to the choice of design of experiments (DOE) is investigated by repeating calculations with 100 different DOEs. Computational cost savings and accuracy improvement over a single fidelity surrogate model are investigated as a function of the ratio of the sampling costs between low and high fidelity simulations. For the examples considered, MFS frameworks were found to be more useful for saving computational time rather than improving accuracy. For the Hartmann 6 function example, the maximum cost saving for the same accuracy was 86},
author = {Park, C. and Haftka, R. T. and Kim, N. H.},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/Multiple output/(2017, Park) RemarksOnMulti-fidelitySurroga.pdf:pdf},
journal = {Structural and Multidisciplinary Optimization},
keywords = {Bayesian,Calibration,Discrepancy function,Multi-fidelity surrogate},
number = {3},
pages = {1029--1050},
title = {{Remarks on multi-fidelity surrogates}},
volume = {55},
year = {2017}
}


@inproceedings{Kandasamy2016,
abstract = {In many scientific and engineering applications, we are tasked with the optimisation of an expensive to evaluate black box function f . Traditional methods for this problem assume just the availability of this single function. However, in many cases, cheap approximations to f may be obtainable. For example, the expensive real world behaviour of a robot can be approximated by a cheap computer simulation. We can use these approximations to eliminate low function value regions cheaply and use the expensive evaluations of f in a small but promising region and speedily identify the optimum. We formalise this task as a multi-fidelity bandit problem where the target function and its approximations are sampled from a Gaussian process. We develop MF-GP-UCB, a novel method based on upper confidence bound techniques. In our theoretical analysis we demonstrate that it exhibits precisely the above behaviour, and achieves better regret than strategies which ignore multi-fidelity information. MF-GP-UCB outperforms such naive strategies and other multi-fidelity methods on several synthetic and real experiments.},
author = {Kandasamy, K. and Dasarathy, G. and Oliva, J. and Schneider, J. and Poczos, B.},
booktitle = {NeurIPS},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Kandasamy et al. - 2016 - Gaussian process optimisation with multi-fidelity evaluations.pdf:pdf},
pages = {992--1000},
title = {{Gaussian process optimisation with multi-fidelity evaluations}},
year = {2016}
}



@article{Morris1993,
abstract = {This article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain T, given values of yo and all of its first derivatives at a set of design sites (points) in T. The intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in T represents a particular configuration of the input parameters. It is assumed that the first derivatives are already available (e.g., from a sensitivity analysis) or can be produced by the code that implements the model. A Bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary Gaussian stochastic process is used. The calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example. The issue of experimental design is also discussed, in particular the criterion of maximizing the reduction in entropy, which leads to a kind of D optimality. It is shown that, for certain classes of correlation functions in which the intersite correlations are very weak, D-optimal designs necessarily maximize the minimim distance between design sites. A simulated annealing algorithm is described for constructing such maximin distance designs. An example is given based on a demonstration model of eight inputs and one output, in which predictions based on a maximin design, a Latin hypercube design, and two compromise designs are evaluated and compared. {\textcopyright} 1993 American statistical association and the American society for quality control.},
author = {Morris, M. D. and Mitchell, T. J. and Ylvisaker, D.},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/derivative data/(1993, Morris) Bayesian design and analysis... Derivatives in surface prediction.pdf:pdf},
journal = {Technometrics},
keywords = {Bayesian prediction,Computer model,Interpolation,Latin hypercube design,Maximin distance design,Sensitivity analysis},
number = {3},
pages = {243--255},
title = {{Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction}},
volume = {35},
year = {1993}
}



%% Choosing lambdas and kernel hyperparameters
@article{rifkin2007_Note_on_RLS,
  title={Notes on regularized least squares},
  author={Rifkin, Ryan M and Lippert, Ross A},
  journal = {Technical report, MIT Computer Science and
Artificial Intelligence Laboratory},
  year={2007}
}


@article{zhang2015_Divide_Conquer_KRR,
  title={Divide and conquer kernel ridge regression: A distributed algorithm with minimax optimal rates},
  author={Zhang, Yuchen and Duchi, John and Wainwright, Martin},
  journal={J. Mach. Learn. Res.},
  volume={16},
  number={1},
  pages={3299--3340},
  year={2015},
  publisher={JMLR. org}
}




%% BQ related
@inproceedings{briol2015_frankwolfe_BQ,
  title={Frank-Wolfe Bayesian quadrature: Probabilistic integration with theoretical guarantees},
  author={Briol, Fran{\c{c}}ois-Xavier and Oates, Chris and Girolami, Mark and Osborne, Michael A},
  booktitle={NeurIPS},
  pages={1162--1170},
  year={2015}
}

@InProceedings{ICML2018_BQforMultipleRelatedIntegrals,
  title = 	 {{B}ayesian Quadrature for Multiple Related Integrals},
  author = 	 {Xi, X. and Briol, F-X. and Girolami, M.},
  booktitle = 	 {ICML},
  year = 	 {2018},
}


@article{briol2019_probabilistic_integration,
  title={Probabilistic integration: A role in statistical computation?},
  author={Briol, F-X. and Oates, C. J. and Girolami, M. and Osborne, M. A. and Sejdinovic, D.},
  journal={Stat. Sci.},
  volume={34},
  number={1},
  pages={1--22},
  year={2019},
  publisher={Institute of Mathematical Statistics}
}




@article{Karvonen2019,
abstract = {Bayesian cubature provides a flexible framework for numerical integration, in which a priori knowledge on the integrand can be encoded and exploited. This additional flexibility, compared to many classical cubature methods, comes at a computational cost which is cubic in the number of evaluations of the integrand. It has been recently observed that fully symmetric point sets can be exploited in order to reduce - in some cases substantially - the computational cost of the standard Bayesian cubature method. This work identifies several additional symmetry exploits within the Bayesian cubature framework. In particular, we go beyond earlier work in considering non-symmetric measures and, in addition to the standard Bayesian cubature method, present exploits for the Bayes-Sard cubature method and the multi-output Bayesian cubature method.},
author = {Karvonen, T. and S{\"{a}}rkk{\"{a}}, S. and Oates, C. J.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Karvonen, S{\"{a}}rkk{\"{a}}, Oates - 2019 - Symmetry exploits for Bayesian cubature methods.pdf:pdf},
journal = {Stat. Comput.},
keywords = {fully symmetric sets,gaussian processes,integration,numerical,probabilistic numerics},
pages = {1231--1248},
title = {{Symmetry exploits for Bayesian cubature methods}},
volume = {29},
year = {2019}
}


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% Relationship between BQ est and RKHS 
@book{ritter2007_average_case_analysis_of_numerical_probs,
  title={Average-case analysis of numerical problems},
  author={Ritter, Klaus},
  year={2007},
  publisher={Springer}
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% CV 

% CV based on talyor expansion
@inproceedings{paisley2012variational,
  title={Variational Bayesian inference with stochastic search},
  author={Paisley, J. and Blei, D. M. and Jordan, M. I.},
  booktitle={ICML},
  year={2012}
}


% CV based on talyor expansion
@article{wang2013variance,
  title={Variance reduction for stochastic gradient optimization},
  author={Wang, C. and Chen, X. and Smola, A. J. and Xing, E. P.},
  journal={NeurIPS},
  year={2013}
}

%%%%%%% 
@article{leluc2021_cv_selection_for_montecarlo_integration,
  title={Control variate selection for Monte Carlo integration},
  author={Leluc, R. and Portier, F. and Segers, J.},
  journal={Stat. Comput.},
  volume={31},
  number={4},
  pages={1--27},
  year={2021},
  publisher={Springer}
}
@inproceedings{Gessner2019,
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. Furthermore, we show that the VBQ acquisition policy is a corner-case of all considered cost-sensitive acquisition schemes, which collapse onto one single de-generate policy in the case of one source and constant cost. 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) allocates budget more efficiently than VBQ for learning the integral to a good accuracy.},
author = {Gessner, A. and Gonzalez, J. and Mahsereci, M.},
booktitle = {UAI},
title = {{Active multi-information source Bayesian quadrature}},
year = {2019}
}

@article{Belomestny2021,
abstract = {In this paper we propose a novel and practical variance reduction approach for additive functionals of dependent sequences. Our approach combines the use of control variates with the minimization of an empirical variance estimate. We analyze finite sample properties of the proposed method and derive finite-Time bounds of the excess asymptotic variance to zero. We apply our methodology to stochastic gradient Markov chain Monte Carlo (SGMCMC) methods for Bayesian inference on large data sets and combine it with existing variance reduction methods for SGMCMC. We present empirical results carried out on a number of benchmark examples showing that our variance reduction method achieves significant improvement as compared to state-of-The-Art methods at the expense of a moderate increase of computational overhead.},
author = {Belomestny, D. and Iosipoi, L. and Moulines, E. and Naumov, Al. and Samsonov, S.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2021, Belomestny) Variance reduction for dependent sequences with applications to stochastic gradient MCMC.pdf:pdf},
journal = {SIAM-ASA J. Uncertain.},
keywords = {Generative models,Mcmc algorithms,Stochastic gradient,Variance reduction},
number = {1},
pages = {507--535},
title = {{Variance reduction for dependent sequences with applications to stochastic gradient MCMC}},
volume = {9},
year = {2021}
}



%% CV-polynomial 
@article{assaraf1999_zerovariance_principleforMCI,
  title={Zero-variance principle for {M}onte {C}arlo algorithms},
  author={Assaraf, R. and Caffarel, M.},
  journal={Phys. Rev. Lett.},
  volume={83},
  number={23},
  pages={4682},
  year={1999},
  publisher={APS}
}

@article{micheli2013matrix,
  title={Matrix-valued kernels for shape deformation analysis},
  author={Micheli, Mario and Glaunes, Joan Alexis},
  journal={Geometry, Imaging and Computing},
  volume={1},
  number={1},
  pages={57–139},
  year={2014}
}



@article{mira2013_zerovariance_MCMC,
  title={{Zero variance Markov chain Monte Carlo for Bayesian estimators}},
  author={Mira, A. and Solgi, R. and Imparato, D.},
  journal={Stat. Comput.},
  volume={23},
  number={5},
  pages={653--662},
  year={2013},
}


@article{papamarkou2014_zerovariance_diffgeoMCMC,
  title={{Zero variance differential geometric Markov chain Monte Carlo algorithms}},
  author={Papamarkou, Theodore and Mira, Antonietta and Girolami, Mark},
  journal={Bayesian Anal.},
  volume={9},
  number={1},
  pages={97--128},
  year={2014},
  publisher={International Society for Bayesian Analysis}
}






% CV-neural network
@article{zhu2018neural_CV,
  title={Neural control variates for variance reduction},
  author={Wan, R. and Zhong, M. and Xiong, H. and Zhu, Z.},
  journal={ECML PKDD},
  pages={533–547},
  year={2019}
}





%% CV functional
@article{oates2017_CF_for_MonteCarloIntegration,
author = {Oates, C. J. and Girolami, M. and Chopin, N.},
title = {{Control functionals for Monte Carlo integration}},
journal = {J. R. Stat. Soc. Series B},
volume = {79},
number = {3},
pages = {695-718},
year = {2017}
}






@article{oates2019convergence,
  title={Convergence rates for a class of estimators based on {S}tein’s method},
  author={Oates, C. J. and Cockayne, J. and Briol, F-X. and Girolami, M.},
  journal={Bernoulli},
  volume={25},
  number={2},
  pages={1141--1159},
  year={2019}
}




@article{Anastasiou2021,
  title={Stein’s method meets computational statistics: a review of some recent developments},
  author={Anastasiou, A. and Barp, A. and Briol, F-X. and Ebner, B.and Gaunt, R. E. and Ghaderinezhad, F. and Gorham, J. and Gretton, A. and Ley, C. and Liu, Q. and others},
  journal={Stat. Sci.},
  volume={38},
  number={1},
  pages={120--139},
  year={2023},
  publisher={Institute of Mathematical Statistics}
}


@article{Si2020,
abstract = {Control variates are a well-established tool to reduce the variance of Monte Carlo estimators. However, for large-scale problems including high-dimensional and large-sample settings, their advantages can be outweighed by a substantial computational cost. This paper considers control variates based on Stein operators, presenting a framework that encompasses and generalizes existing approaches that use polynomials, kernels and neural networks. A learning strategy based on minimising a variational objective through stochastic optimization is proposed, leading to scalable and effective control variates. Our results are both empirical, based on a range of test functions and problems in Bayesian inference, and theoretical, based on an analysis of the variance reduction that can be achieved.},
author = {Si, S. and Oates, C. J. and Duncan, A. B. and Carin, L. and Briol, F-X.},
file = {:Users/fxbriol/Docs/Work/Papers/Stein method/(2020, Si) Scalable control variates for Monte Carlo methods via stochastic optimization.pdf:pdf},
journal = {Proceedings of the 14th Conference on Monte Carlo and Quasi-Monte Carlo Methods. arXiv:2006.07487},
title = {{Scalable control variates for Monte Carlo methods via stochastic optimization}},
year = {2021}
}


@article{OHagan1991,
author = {O'Hagan, A},
journal = {Journal of Statistical Planning and Inference},
keywords = {bayesian},
pages = {245--260},
title = {{Bayes–Hermite quadrature}},
volume = {29},
year = {1991}
}

@inproceedings{Kingma2015,
abstract = {We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.},
author = {Kingma, D. P. and Ba, J. L.},
booktitle = {ICLR},
file = {:Users/fxbriol/Docs/Work/Papers/Numerical Analysis/optimisation/(2015, Kingma) Adam .pdf:pdf},
title = {{Adam: A method for stochastic optimization}},
year = {2015}
}


@article{Girolami2011,
abstract = {The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis-Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.},
author = {Girolami, M. and Calderhead, B.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Girolami, Calderhead - 2011 - Riemann manifold Langevin and Hamiltonian Monte Carlo methods.pdf:pdf},
journal = {J. R. Stat. Soc. Series B},
number = {2},
pages = {123--214},
title = {{Riemann manifold Langevin and Hamiltonian Monte Carlo methods}},
volume = {73},
year = {2011}
}


@article{Dodwell2019,
author = {Dodwell, T. J. and Ketelsen, C. and Scheichl, R. and Teckentrup, A. L.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/multi-level MC/(2013, Dodwell) Multilevel Markov chain Monte Carlo.pdf:pdf},
journal = {SIAM Review},
number = {3},
pages = {509--545},
title = {{Multilevel Markov chain Monte Carlo}},
volume = {61},
year = {2019}
}

@inproceedings{Briol2017SMCKQ,
author = {Briol, F-X. and Oates, C. J. and Cockayne, J. and Chen, W. Y. and Girolami, M.},
booktitle = {Proceedings of the International Conference on Machine Learning},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Briol et al. - 2017 - On the sampling problem for kernel quadrature(2).pdf:pdf},
pages = {586--595},
title = {{On the sampling problem for kernel quadrature}},
year = {2017}
}






@article{South2019,
  title={Regularized Zero-Variance Control Variates},
  author={South, L. F. and Oates, C. J. and Mira, A. and Drovandi, C.},
  journal={Bayesian Anal.},
  volume={1},
  number={1},
  pages={1--24},
  year={2022},
  publisher={International Society for Bayesian Analysis}
}



@article{south2020_Semi_Exact_CF,
abstract = {This paper focuses on the numerical computation of posterior expected quantities of interest, where existing approaches based on ergodic averages are gated by the asymptotic variance of the integrand. To address this challenge, a novel technique is proposed to post-process Markov chain Monte Carlo output, based on Sard's approach to numerical integration and the control functional method. The use of Sard's approach ensures that our control functionals are exact on all polynomials up to a fixed degree in the Bernstein-von-Mises limit, so that the reduced variance estimator approximates the behaviour of a polynomially-exact (e.g. Gaussian) cubature method. The proposed method is shown to combine the robustness of parametric control variates with the flexibility of non-parametric control functionals across a selection of Bayesian inference tasks. All methods used in this paper are available in the R package ZVCV.},
author = {South, L. F. and Karvonen, T. and Nemeth, C. and Girolami, M. and Oates, C. J.},
file = {:Users/fxbriol/Docs/Work/Papers/Monte Carlo Methods/Control variates/(2020, South) Semi-Exact Control functionals from Sard's Method.pdf:pdf},
journal = {Biometrika},
title = {{Semi-exact control functionals from Sard's method}},
year = {2022}
}





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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@article{matsubara2021robust_KSD_Bayes,
  title={Robust generalised Bayesian inference for intractable likelihoods},
  author={Matsubara, Takuo and Knoblauch, Jeremias and Briol, Fran{\c{c}}ois-Xavier and Oates, Chris and others},
  journal={  arXiv:2104.07359},
  year={2021}
}

@article{wang2019_svgd_with_mvkernel,
  title={Stein variational gradient descent with matrix-valued kernels},
  author={Wang, Dilin and Tang, Ziyang and Bajaj, Chandrajit and Liu, Qiang},
  journal={NeurIPS},
  volume={32},
  pages={7834},
  year={2019},
  publisher={NIH Public Access}
}



@article{barp2021unifying,
  title={A Unifying and Canonical Description of Measure-Preserving Diffusions},
  author={Barp, Alessandro and Takao, So and Betancourt, Michael and Arnaudon, Alexis and Girolami, Mark},
  journal={  arXiv:2105.02845},
  year={2021}
}




% Other related
@article{mijoule2018_SteinOperators_Kernels_Discrepancies,
  title={Stein operators, kernels and discrepancies for multivariate continuous distributions},
  author={Mijoule, Guillaume and Reinert, Gesine and Swan, Yvik},
  journal={  arXiv:1806.03478},
  year={2018}
}

@article{briol2019_statisticalinference_generativemodel_withMMD,
  title={Statistical inference for generative models with maximum mean discrepancy},
  author={Briol, Francois-Xavier and Barp, Alessandro and Duncan, Andrew B and Girolami, Mark},
  journal={  arXiv:1906.05944},
  year={2019}
}


@inproceedings{Chen2018,
abstract = {An important task in computational statistics and machine learning is to approximate a posterior distribution $p(x)$ with an empirical measure supported on a set of representative points $\{x_i\}_{i=1}^n$. This paper focuses on methods where the selection of points is essentially deterministic, with an emphasis on achieving accurate approximation when $n$ is small. To this end, we present `Stein Points'. The idea is to exploit either a greedy or a conditional gradient method to iteratively minimise a kernel Stein discrepancy between the empirical measure and $p(x)$. Our empirical results demonstrate that Stein Points enable accurate approximation of the posterior at modest computational cost. In addition, theoretical results are provided to establish convergence of the method.},
author = {Chen, W. Y. and Mackey, L. and Gorham, J. and Briol, F-X. and Oates, C. J.},
booktitle = {ICML, PMLR 80},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Chen et al. - 2018 - Stein points(2).pdf:pdf},
pages = {843--852},
title = {{Stein points}},
year = {2018}
}


@inproceedings{chen2019stein,
  title={Stein point markov chain monte carlo},
  author={Chen, Wilson Ye and Barp, Alessandro and Briol, Fran{\c{c}}ois-Xavier and Gorham, Jackson and Girolami, Mark and Mackey, Lester and Oates, Chris},
  booktitle={International Conference on Machine Learning},
  pages={1011--1021},
  year={2019},
  organization={PMLR}
}


@inproceedings{Chen2019,
author = {Chen, W. Y. and Barp, A. and Briol, F-X. and Gorham, J. and Girolami, M. and Mackey, L. and Oates, C. J.},
booktitle = {International Conference on Machine Learning, PMLR 97},
file = {:home/francois/Dropbox/Work/Papers/Stein method/Stein & Kernels/(2019, Chen) Stein Point Markov Chain Monte Carlo.pdf:pdf},
pages = {1011--1021},
title = {{Stein point Markov chain Monte Carlo}},
year = {2019}
}



@inproceedings{barp2019_Minimum_Stein_Discrepancy_Estimators,
title = {Minimum {S}tein Discrepancy Estimators},
author = {Barp, A. and Briol, F-X. and Duncan, A. and Girolami, M. and Mackey, L.},
booktitle = {NeurIPS},
pages = {12964--12976},
year = {2019}
}




@article{gorham2019_MeasuringSampleQuality_withDiffusions,
author = "Gorham, J. and Duncan, A. B. and Vollmer, S. J. and Mackey, L.",
journal = "Ann. Appl. Probab.",
number = "5",
pages = "2884--2928",
title = "Measuring sample quality with diffusions",
volume = "29",
year = "2019"
}



@book{chen2010_NormalApprox_by_SteinMethod,
  title={Normal approximation by Stein’s method},
  author={Chen, Louis HY and Goldstein, Larry and Shao, Qi-Man},
  year={2010},
  publisher={Springer Science \& Business Media}
}



@article{singhal2019_kernelized_conditional_SteinD,
  title={Kernelized complete conditional Stein discrepancy},
  author={Singhal, Raghav and Han, Xintian and Lahlou, Saad and Ranganath, Rajesh},
  journal={  arXiv:1904.04478},
  year={2019}
}


@inproceedings{huggins2018_random_feature_SteinD,
  title={Random feature stein discrepancies},
  author={Huggins, Jonathan and Mackey, Lester},
  booktitle={NeurIPS},
  pages={1899--1909},
  year={2018}
}



@inproceedings{liu2016_SVGD,
  title={Stein variational gradient descent: A general purpose {B}ayesian inference algorithm},
  author={Liu, Qiang and Wang, Dilin},
  booktitle={NeurIPS},
  pages={2378--2386},
  year={2016}
}


@inproceedings{wang2019_SVGD_matrix_valued_kernels,
  title={Stein variational gradient descent with matrix-valued kernels},
  author={Wang, Dilin and Tang, Ziyang and Bajaj, Chandrajit and Liu, Qiang},
  booktitle={NeurIPS},
  pages={7836--7846},
  year={2019}
}



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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



% multiple-tasks learning, non-convex and alternating minimization
@inproceedings{dinuzzo2011_learning_outputkernel_with_blockcoordinateDescent,
  title={Learning output kernels with block coordinate descent},
  author={Dinuzzo, Francesco and Ong, Cheng Soon and Gehler, Peter V and Pillonetto, Gianluigi},
  booktitle={International Conference on Machine Learning},
  year={2011}
}


@inproceedings{ciliberto2015_convex_learning_multitasks_and_structure,
  title={Convex learning of multiple tasks and their structure},
  author={Ciliberto, Carlo and Mroueh, Youssef and Poggio, Tomaso and Rosasco, Lorenzo},
  booktitle={International Conference on Machine Learning},
  pages={1548--1557},
  year={2015}
}



% fast kernel methods
@article{drineas2005_nystrom_approx_for_kernelmethods,
  title={On the Nystr{\"o}m Method for Approximating a Gram Matrix for Improved Kernel-Based Learning},
  author={Drineas, Petros and Mahoney, Michael W},
  journal={J. Mach. Learn. Res.},
  volume={6},
  pages={2153--2175},
  year={2005},
  publisher={Citeseer}
}


@inproceedings{rahimi2008_rff_for_kernelmachines,
  title={Random features for large-scale kernel machines},
  author={Rahimi, Ali and Recht, Benjamin},
  booktitle={NeurIPS},
  pages={1177--1184},
  year={2008}
}


@inproceedings{rudi2017_falkon_largescale_kernelmethod,
  title={Falkon: An optimal large scale kernel method},
  author={Rudi, Alessandro and Carratino, Luigi and Rosasco, Lorenzo},
  booktitle={NeurIPS},
  pages={3888--3898},
  year={2017}
}




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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@article{Bardenet2017,
author = {Bardenet, R and Doucet, A and Holmes, C},
journal = {J. Mach. Learn. Res.},
pages = {1--43},
title = {{On Markov chain Monte Carlo methods for tall data}},
volume = {18},
year = {2017}
}

@article{Giles2015,
author = {Giles, M. B.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Giles - 2015 - Multilevel Monte Carlo methods.pdf:pdf},
journal = {Acta Numer.},
pages = {259--328},
title = {{Multilevel Monte Carlo methods}},
volume = {24},
year = {2015}
}





@inproceedings{Oates2016CFQMC,
abstract = {Quasi-Monte Carlo (QMC) methods are being adopted in statistical applications due to the increasingly challenging nature of numerical integrals that are now routinely encountered. For integrands that are {\$}\backslashalpha{\$}-times differentiable, an {\$}\backslashalpha{\$}-optimal QMC rule converges at a best-possible rate {\$}O(N{\^{}}{\{}-\backslashalpha- 1/2 +\backslashepsilon{\}}){\$}. However, in applications the value of {\$}\backslashalpha{\$} can be unknown and/or a rate-optimal QMC rule can be unavailable. Standard practice is to employ {\$}\backslashalpha{\_}L{\$}-optimal QMC where the lower bound {\$}\backslashalpha{\_}L \backslashleq \backslashalpha{\$} is known, but in general this does not exploit the full power of QMC. We present an elegant solution that uses control functionals to accelerate {\$}\backslashalpha{\_}L{\$}-QMC by a factor {\$}O(N{\^{}}{\{}-(\backslashalpha - \backslashalpha{\_}L)/d{\}}){\$}, where {\$}d{\$} is the dimension of the integral. For {\$}d=1{\$} we therefore recover optimal convergence rates. A challenging application to robotic arm data demonstrates a substantial variance reduction in predictions for mechanical torques.},
author = {Oates, C. J and Girolami, M.},
booktitle = {AISTATS},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Oates, Girolami - 2016 - Control functionals for quasi-Monte Carlo integration(2).pdf:pdf},
title = {{Control functionals for quasi-Monte Carlo integration}},
year = {2016}
}

@article{Chwialkowski2016,
abstract = {We propose a nonparametric statistical test for goodness-of-fit: given a set of samples, the test determines how likely it is that these were generated from a target density function. The measure of goodness-of-fit is a divergence constructed via Stein's method using functions from a Reproducing Kernel Hilbert Space. Our test statistic is based on an empirical estimate of this divergence, taking the form of a V-statistic in terms of the log gradients of the target density and the kernel. We derive a statistical test, both for i.i.d. and non-i.i.d. samples, where we estimate the null distribution quantiles using a wild bootstrap procedure. We apply our test to quantifying convergence of approximate Markov Chain Monte Carlo methods, statistical model criticism, and evaluating quality of fit vs model complexity in nonparametric density estimation.},
author = {Chwialkowski, K. and Strathmann, H. and Gretton, Arthur},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Chwialkowski, Strathmann, Gretton - 2016 - A kernel test of goodness of fit(3).pdf:pdf},
journal = {International Conference on Machine Learning},
pages = {2606--2615},
title = {{A kernel test of goodness of fit}},
volume = {48},
year = {2016}
}



@article{Mijatovic2018,
abstract = {This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis-Hastings chain {\$}\backslashPhi{\$}. The approximations give rise to a natural sequence of control variates for the ergodic average {\$}S{\_}k(F)=(1/k)\backslashsum{\_}{\{}i=1{\}}{\^{}}{\{}k{\}} F(\backslashPhi{\_}i){\$}, where {\$}F{\$} is the force function in the Poisson equation. The main result of the paper shows that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero. We apply the algorithm to geometrically and non-geometrically ergodic chains and present numerical evidence for a significant variance reduction in both cases.},
author = {Mijatovic, A. and Vogrinc, J.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Mijatovic, Vogrinc - 2018 - On the Poisson equation for Metropolis-Hastings chains(2).pdf:pdf},
journal = {Bernoulli},
keywords = {and phrases,and useful conversations,asymptotic variance,central limit theorem,chains,for suggesting the problem,imperial college,markov chain monte carlo,metropolis-hastings algo-,out,poisson equation for markov,rithm,the work was carried,variance reduction,we thank petros dellaportas,weak approximation,while am was at},
number = {3},
pages = {2401--2428},
title = {{On the Poisson equation for Metropolis-Hastings chains}},
volume = {24},
year = {2018}
}



@article{Calderhead2009,
abstract = {A Bayesian approach to model comparison based on the integrated or marginal likelihood is considered, and applications to linear regression models and nonlinear ordinary differential equation (ODE) models are used as the setting in which to elucidate and further develop existing statistical methodology. The focus is on two methods of marginal likelihood estimation. First, a statistical failure of the widely employed Posterior Harmonic Mean estimator is highlighted. It is demonstrated that there is a systematic bias capable of significantly skewing Bayes factor estimates, which has not previously been highlighted in the literature. Second, a detailed study of the recently proposed Thermodynamic Integral estimator is presented, which characterises the error associated with its discrete form. An experimental study using analytically tractable linear regression models highlights substantial differences with recently published results regarding optimal discretisation. Finally, with the insights gained, it is demonstrated how Population MCMC and thermodynamic integration methods may be elegantly combined to estimate Bayes factors accurately enough to discriminate between nonlinear models based on systems of ODEs, which has important application in describing the behaviour of complex processes arising in a wide variety of research areas, such as Systems Biology, Computational Ecology and Chemical Engineering. ?? 2009 Elsevier B.V. All rights reserved.},
author = {Calderhead, B and Girolami, M},
journal = {Computational Statistics and Data Analysis},
number = {12},
pages = {4028--4045},
title = {{Estimating Bayes factors via thermodynamic integration and population MCMC}},
volume = {53},
year = {2009}
}



% Some applications
@article{gelman1996_post_pred_assess_modelfitness,
  title={Posterior predictive assessment of model fitness via realized discrepancies},
  author={Gelman, Andrew and Meng, Xiao-Li and Stern, Hal},
  journal={Statistica sinica},
  pages={733--760},
  year={1996},
  publisher={JSTOR}
}






@article{oates2014_controlled_themo_integration,
abstract = {Bayesian model comparison relies upon the model evidence, yet for many models of interest the model evidence is unavailable in closed form and must be approximated. Many of the estimators for evidence that have been proposed in the Monte Carlo literature suffer from high variability. This paper considers the reduction of variance that can be achieved by exploiting control variates in this setting. Our methodology is based on thermodynamic integration and applies whenever the gradient of both the log-likelihood and the log-prior with respect to the parameters can be efficiently evaluated. Results obtained on regression models and popular benchmark datasets demonstrate a significant and sometimes dramatic reduction in estimator variance and provide insight into the wider applicability of control variates to Bayesian model comparison.},
author = {Oates, C. J. and Papamarkou, T. and Girolami, M.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Oates, Papamarkou, Girolami - 2016 - The controlled thermodynamic integral for Bayesian model comparison.pdf:pdf},
journal = {J. Am. Stat. Assoc.},
number = {514},
pages = {634--645},
title = {{The controlled thermodynamic integral for Bayesian model comparison}},
volume = {111},
year = {2016}
}




@article{friel2014_improving_power_posterior,
  title={Improving power posterior estimation of statistical evidence},
  author={Friel, Nial and Hurn, Merrilee and Wyse, Jason},
  journal={Stat. Comput.},
  volume={24},
  number={5},
  pages={709--723},
  year={2014},
  publisher={Springer}
}

@article{friel2008_marginal_llk_by_power_posterior,
  title={Marginal likelihood estimation via power posteriors},
  author={Friel, Nial and Pettitt, Anthony N},
  journal={J. R. Stat. Soc. Series B},
  volume={70},
  number={3},
  pages={589--607},
  year={2008},
  publisher={Wiley Online Library}
}

@article{Bottou2018,
abstract = {This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.},
archivePrefix = {arXiv},
arxivId = {1606.04838},
author = {Bottou, L. and Curtis, F. E. and Nocedal, J.},
journal = {SIAM Review},
keywords = {Algorithm complexity analysis,Machine learning,Noise reduction methods,Numerical optimization,Second-order methods,Stochastic gradient methods},
number = {2},
pages = {223--311},
title = {{Optimization methods for large-scale machine learning}},
volume = {60},
year = {2018}
}

@article{Argyriou2008,
abstract = {We present a method for learning sparse representations shared across multiple tasks. This method is a generalization of the well-known single-task 1-norm regularization. It is based on a novel non-convex regularizer which controls the number of learned features common across the tasks. We prove that the method is equivalent to solving a convex optimization problem for which there is an iterative algorithm which converges to an optimal solution. The algorithm has a simple interpretation: it alternately performs a supervised and an unsupervised step, where in the former step it learns task-specific functions and in the latter step it learns common-across-tasks sparse representations for these functions. We also provide an extension of the algorithm which learns sparse nonlinear representations using kernels. We report experiments on simulated and real data sets which demonstrate that the proposed method can both improve the performance relative to learning each task independently and lead to a few learned features common across related tasks. Our algorithm can also be used, as a special case, to simply select-not learn-a few common variables across the tasks.},
author = {Argyriou, Andreas and Evgeniou, Theodoros and Pontil, Massimiliano},
doi = {10.1007/s10994-007-5040-8},
file = {:Users/fxbriol/Docs/Work/Papers/Kernel Methods/Multiple output/(2008, Argyriou) Convex Multi-task feature learning.pdf:pdf},
journal = {Machine Learning},
keywords = {Collaborative filtering,Inductive transfer,Kernels,Multi-task learning,Regularization,Transfer learning,Vector-valued functions},
number = {3},
pages = {243--272},
title = {{Convex multi-task feature learning}},
volume = {73},
year = {2008}
}



@book{Hastie2009,
abstract = {During the past decade there has been an explosion in computation and information technology. With it has come vast amounts of data in a variety of fields such as medicine, biology, finance, and marketing. The challenge of understanding these data has led to the development of new tools in the field of statistics, and spawned new areas such as data mining, machine learning, and bioinformatics. Many of these tools have common underpinnings but are often expressed with different terminology. This book describes the important ideas in these areas in a common conceptual framework. While the approach is statistical, the emphasis is on concepts rather than mathematics. Many examples are given, with a liberal use of color graphics. It should be a valuable resource for statisticians and anyone interested in data mining in science or industry. The book's coverage is broad, from supervised learning (prediction) to unsupervised learning. The many topics include neural networks, support vector machines, classification trees and boosting-the first comprehensive treatment of this topic in any book. Trevor Hastie, Robert Tibshirani, and Jerome Friedman are professors of statistics at Stanford University. They are prominent researchers in this area: Hastie and Tibshirani developed generalized additive models and wrote a popular book of that title. Hastie wrote much of the statistical modeling software in S-PLUS and invented principal curves and surfaces. Tibshirani proposed the Lasso and is co-author of the very successful An Introduction to the Bootstrap. Friedman is the co-inventor of many data-mining tools including CART, MARS, and projection pursuit. FROM THE REVIEWS: TECHNOMETRICS "This is a vast and complex book. Generally, it concentrates on explaining why and how the methods work, rather than how to use them. Examples and especially the visualizations are principle features...As a source for the methods of statistical learning...it will probably be a long time before there is a competitor to this book."},
author = {Hastie, Trevor and Tibshirani, Robert and Friedman, Jerome},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Hastie, Tibshirani, Friedman - 2009 - The Elements of Statistical Learning.pdf:pdf},
publisher = {Springer},
title = {{The Elements of Statistical Learning}},
year = {2009}
}

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% Early stopping in kernel regresssion
@article{raskutti2014_early_stop_nonparam_regression,
  title={Early stopping and non-parametric regression: an optimal data-dependent stopping rule},
  author={Raskutti, Garvesh and Wainwright, Martin J and Yu, Bin},
  journal={J. Mach. Learn. Res.},
  volume={15},
  number={1},
  pages={335--366},
  year={2014},
  publisher={JMLR. org}
}

@article{Carmeli2010,
author = {Carmeli, C. and {De Vito}, E. and Toigo, A. and Umanita, V.},
file = {:Users/fxbriol/Library/Application Support/Mendeley Desktop/Downloaded/Carmeli et al. - 2010 - Vector valued reproducing kernel Hilbert spaces and universality.pdf:pdf},
journal = {Analysis and Applications},
number = {1},
pages = {19--61},
title = {{Vector valued reproducing kernel Hilbert spaces and universality}},
volume = {8},
year = {2010}
}






% Multifidelity functions
@article{xiong2013sequential,
  title={Sequential design and analysis of high-accuracy and low-accuracy computer codes},
  author={Xiong, Shifeng and Qian, Peter ZG and Wu, CF Jeff},
  journal={Technometrics},
  volume={55},
  number={1},
  pages={37--46},
  year={2013},
  publisher={Taylor \& Francis}
}





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% Meta-Learning reference

@InProceedings{icml_2017_finn17a_MAML,
  title = 	 {Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks},
  author = 	 {C. Finn and P. Abbeel and S. Levine},
  booktitle = 	 {ICML},
  year = 	 {2017},
}




@inproceedings{iclr_2018_grant_recasting_HBMAML,
title={Recasting Gradient-Based Meta-Learning as Hierarchical {Bayes}},
author={E. Grant and C. Finn and S. Levine and T. Darrell and T. Griffiths},
booktitle={ICML},
year={2018}
}

@inproceedings{nips_2018_fin_probabilisticMAML,
title = {Probabilistic Model-Agnostic Meta-Learning},
author = {Finn, C. and Xu, K. and Levine, S.},
booktitle = {NeurIPS},
year = {2018},
}


@inproceedings{NIPS_2018_yoon_BayesianMAML,
  title={Bayesian model-agnostic meta-learning},
  author={Yoon, J. and Kim, T. and Dia, O. and Kim, S. and Bengio, Y. and Ahn, S.},
  booktitle={NeurIPS},
  year={2018}
}

@article{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, K. and Tiemann, M. and Hennig, P. and Briol, F-X.},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2023, Ott) Bayesian Numerical Integration with Neural Networks.pdf:pdf},
journal = {arXiv:2305.13248},
title = {{Bayesian numerical integration with neural networks}},
year = {2023}
}


@inproceedings{Li2022,
abstract = {Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show using both theory and numerical experiments that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.},
author = {Li, K. and Giles, D. and Karvonen, T. and Guillas, S. and Briol, F-X.},
booktitle = {AISTATS},
file = {:Users/fxbriol/Docs/Work/Papers/Probabilistic Numerics/(2022, Li) Multilevel Bayesian quadrature.pdf:pdf},
pages = {1845--1868},
title = {{Multilevel Bayesian quadrature}},
year = {2023}
}




@inproceedings{ICLR_2019_ravi_Amortized_Bayesian_MetaLearning,
title={Amortized {Bayesian} Meta-Learning},
author={Sachin Ravi and Alex Beatson},
booktitle={International Conference on Learning Representations},
year={2019},
url={https://openreview.net/forum?id=rkgpy3C5tX}
}



@article{sun2021_vvCV,
  title={{Vector-Valued Control Variates}},
  author={Sun, Z. and Barp, A. and Briol, F-X.},
  journal={arXiv:2109.08944, to appear at ICML 2023},
  year={2021}
}
