Regularized Zero-Variance Control Variates

Abstract

Zero-variance control variates (ZV-CV) are a post-processing method to reduce the variance of Monte Carlo estimators of expectations using the derivatives of the log target. Once the derivatives are available, the only additional computational effort lies in solving a linear regression problem. Significant variance reductions have been achieved with this method in low dimensional examples, but the number of covariates in the regression rapidly increases with the dimension of the target. In this paper, we present compelling empirical evidence that the use of penalized regression techniques in the selection of high-dimensional control variates provides performance gains over the classical least squares method. Another type of regularization based on using subsets of derivatives, or a priori regularization as we refer to it in this paper, is also proposed to reduce computational and storage requirements. Several examples showing the utility and limitations of regularized ZV-CV for Bayesian inference are given. The methods proposed in this paper are accessible through the R package ZVCV.

Postprocessing of MCMC

“…Again, the extension is straightforward and consists of applying the ideas from this section to improve multiple expectations. The ZVCV package (South 2020) on CRAN provides functions to apply ZVCV and CF to two estimators of the normalizing constant.…”

Meta-learning Control Variates: Variance Reduction with Limited Data

“…This is a significant advantage in the present setting since many applications, including problems where π is a Bayesian posterior distribution, fall into this category. The first Stein-based CVs were proposed by Assaraf and Caffarel [1999], in which U was a finite-dimensional vector space of functions of the form u = ∇p, with p polynomial of fixed degree; see also Mira et al [2013], Papamarkou et al [2014, Friel et al [2014], South et al [2022b]. For additional flexibility, Oates et al [2017] proposed to take U to be a Cartesian product of reproducing kernel Hilbert spaces; see also , Oates et al [2019], Barp et al [2022], calling this approach control functionals (CFs).…”

“…For Neural CVs it is difficult to go beyond Theorem 1, since for one thing there will not be a unique γ meta in general. However, for simpler CVs, such as those based on polynomial regression [Assaraf and Caffarel, 1999, Mira et al, 2013, Papamarkou et al, 2014, Friel et al, 2014, South et al, 2022b, it is reasonable to assume a unique γ meta and convexity of the Meta-CV objective around this point. In these scenarios, the following corollary shows that γ is typically close to the minimiser of the task-specific objective functional.…”

“…High-dimensional settings also pose a challenge, since such functions are more difficult to approximate due to the curse of dimensionality. In the latter case, sparsity can be exploited for integrands with low effective dimension [South et al, 2022b, Leluc et al, 2021, but many integrands do not admit convenient structure that can be easily exploited.…”

ZVCV: Zero-Variance Control Variates

“…Note that an intercept = TRUE flag may be changed to intercept = FALSE within the function if integrand_logged = TRUE and a NaN is encountered. See South et al (2018) for further details. • polyorder_max: The maximum allowable polynomial order.…”