Convergence of adaptive mixtures of importance sampling schemes

TL;DR

This paper establishes convergence conditions for adaptive importance sampling algorithms, demonstrating that Rao-Blackwellized versions asymptotically optimize a divergence criterion, unlike simpler methods.

Contribution

It provides theoretical convergence guarantees for adaptive mixtures of importance sampling schemes, especially highlighting the effectiveness of Rao-Blackwellization.

Findings

Rao-Blackwellized adaptive schemes achieve asymptotic optimality.

Simpler adaptive schemes do not benefit from repeated updates.

Convergence conditions are explicitly derived for these algorithms.

Abstract

In the design of efficient simulation algorithms, one is often beset with a poor choice of proposal distributions. Although the performance of a given simulation kernel can clarify a posteriori how adequate this kernel is for the problem at hand, a permanent on-line modification of kernels causes concerns about the validity of the resulting algorithm. While the issue is most often intractable for MCMC algorithms, the equivalent version for importance sampling algorithms can be validated quite precisely. We derive sufficient convergence conditions for adaptive mixtures of population Monte Carlo algorithms and show that Rao--Blackwellized versions asymptotically achieve an optimum in terms of a Kullback divergence criterion, while more rudimentary versions do not benefit from repeated updating.