Regularised Zero-Variance Control Variates for High-Dimensional Variance Reduction

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 is 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. We propose to exploit penalised regression to make the method more flexible and feasible, particularly in higher dimensions. Connections between this penalised ZV-CV approach and control functionals are made, providing additional motivation for our approach. Another type of regularisation based on using subsets of derivatives, or a priori regularisation as we refer to it in this paper, is also proposed to reduce computational and storage requirements. Methods for applying ZV-CV and regularised ZV-CV to sequential Monte Carlo (SMC) are described and a new estimator for the normalising constant of the posterior is developed to aid Bayesian model choice. Several examples showing the utility and limitations of regularised ZV-CV for Bayesian inference are given. The methods proposed in this paper are accessible through the R package ZVCV available at https://github.com/LeahPrice/ZVCV.

Post-Processing of MCMC

“…This section focused on the application of gradient-based control variates to approximate an integral of interest based on output from MCMC. Applications to other sampling algorithms, such as population MCMC (Oates et al 2016), stochastic gradient Langevin dynamics (Baker et al 2019), sequential Monte Carlo (South et al 2018) and unbiased MCMC with couplings (South et al 2019), have also been considered and much of our discussion applies unchanged. A current weakness of control variate methodology is that it is under-developed from a theoretical perspective; our focus was on sets of control variates that form linear subspaces of L 2 (P ), for which some limited theoretical understanding has been achieved, but more sophisticated sets of control variates have also been empirically considered.…”

“…This limits the variance reduction that can be achieved. To improve convergence rates, one could consider increasing the size of Φ with increasing M , in the spirit of Portier & Segers (2018), South et al (2018), or using an infinite-dimensional basis with regularisation, as described next.…”

“…However, the regression perspective in Equation ( 19) suggests that, by analogy with high-dimensional regression modelling (Bühlmann & Van De Geer 2011), it may be possible to construct control variates for functions f whose effective dimension is small, despite a high ambient dimension of X . Additional regularisation can be introduced to this effect (South et al 2018, Wan et al 2019, with positive results reported for d ≤ 100. For even larger d, it may be sensible to pursue nonlinear approximation (DeVore 1998), where the basis Φ is restricted to allow dependence only on a subset of the parameters (so-called a priori regularisation in South et al 2018) .…”

“…Additional regularisation can be introduced to this effect (South et al 2018, Wan et al 2019, with positive results reported for d ≤ 100. For even larger d, it may be sensible to pursue nonlinear approximation (DeVore 1998), where the basis Φ is restricted to allow dependence only on a subset of the parameters (so-called a priori regularisation in South et al 2018) .…”

Semi-Exact Control Functionals From Sard's Method

“…The development of non-parametric approaches to the choice of g m has to-date focused on kernel methods (Oates et al, 2017;Barp et al, 2018), piecewise constant approximations (Mijatović et al, 2018) and non-linear approximations based on selecting basis functions from a dictionary (Belomestny et al, 2017;South et al, 2019). Non-parametric approaches can be motivated using the "double descent" phenomena recently exposed in Belkin et al (2019), where a regression model based on a large number of features which are sufficiently regularized can, in some circumstances, achieve better predictive performance compared to regression models with a smaller number of predictors selected based on a bias-variance trade-off.…”