Control functionals for Monte Carlo integration

By Chris J. Oates, Mark A. Girolami, Nicolas Chopin. 2014

275 citations recorded by the source corpus.

A non‐parametric extension of control variates is presented. These leverage gradient information on the sampling density to achieve substantial variance reduction. It is not required that the sampling density be normalized. The novel contribution of this work is based on two important insights: a trade‐off between random sampling and deterministic approximation and a new gradient‐based function space derived from Stein's identity. Unlike classical control variates, our estimators improve rates of convergence, often requiring orders of magnitude fewer simulations to achieve a fixed level of precision. Theoretical and empirical results are presented, the latter focusing on integration problems arising in hierarchical models and models based on non‐linear ordinary differential equations.

Subfields: Numerical Analysis, Statistics and Probability, Numerical Methods & Scientific Computing

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Liu et al. (2016) and Chwialkowski et al. (2016) independently and simultaneously develop a goodness-of-fit test based on Stein’s identity applied to functions in RKHS (Oates et al. 2016).”
Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, background): “For instance, in situations where the evaluation of the integrand requires heavy computations, n should be small and Monte Carlo would perform poorly; such situations typically appear in modern scientific and engineering applications, and thus quadratures with faster convergence rates are desirable ”
A Kernelized Stein Discrepancy for Goodness-of-fit Tests (2016, methodology): “The technique of combining Stein’s identity with RKHS was ﬁrst developed by Oates et al. (2014; 2017; 2016) for variance reduction.”
Batch Stationary Distribution Estimation (2020, methodology): “Unlike VPM, other post-processing meth-ods (Oates et al., 2017) require additional information about the target distribution (Robert and Casella, 2004).”
Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning (2016, methodology): “One exception is a very recent paper (Ranganath et al., 2016) that avoids calculating q ( x ) using an idea related to Stein discrepancy (Gorham & Mackey, 2015; Liu et al., 2016; Oates et al., 2014; Chwialkowski et al., 2016).”
Measuring Sample Quality with Stein's Method (2015, methodology): “(2)The operator TP has also found fruitful application in the design of Monte Carlo control variates [9].”
Monte Carlo Gradient Estimation in Machine Learning (2019, methodology): “ only once, we can still learn a control variate that results in non-trivial variance reduction as long as we generate suciently many samples to train it and use a sample-ecient learning algorithm. Oates et al. (2017) demonstrated the feasibility of this approach using kernel-based non-parametric co”
Advances in Variational Inference (2017, background): “Many other control variates are used in practice [140], [146], [203].”
Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees (2015, methodology): “For the task of computing integrals, Bayesian Quadrature (BQ) [22] and more recent work by [20] provide probabilistic numerics methods that produce a full posterior distribution on the output of numerical schemes.”
Monte Carlo with determinantal point processes (2016, background): “Also, our importance reweighted estimator in Theorem 4 bypasses the need to know the moments of the target measure, which is akin to removing the constraint of knowing the integral of the control approximation to the integrand in [Oates et al., 2017].”
Black-box Importance Sampling (2016, methodology): “…based on minimizing a recently proposed kernelized Stein discrepancy that measures the goodness of a sample ﬁt with an unnor-malized distribution (Oates et al., 2017; Chwialkowski et al., 2016; Liu et al., 2016); this makes our method widely applicable for unnormalized distributions that widely…”
Stein Variational Gradient Descent as Moment Matching (2018, methodology): “We may deﬁne a kernelized Stein discrepancy (KSD) (Liu et al., 2016; Chwialkowski et al., 2016; Oates et al., 2017): The optimal φ in (1) can be solved in closed form: (2) which yields a simple kernel-based representation of KSD: where x and x (cid:48) are i.i.d. draws from q , and κ p ( x , x…”
Action-depedent Control Variates for Policy Optimization via Stein's Identity (2017, methodology): “An incomplete list includes Gorham & Mackey (2015), Oates et al. (2017), Oates et al. (2016), Chwialkowski et al. (2016), Liu et al. (2016), Sedghi et al. (2016), Liu & Wang (2016), Feng et al. (2017), Liu & Lee (2017).”