  Variational Bayes (VB) is a popular tool for Bayesian inference in
statistical modeling. Recently, some VB algorithms are proposed to handle
intractable likelihoods with applications such as approximate Bayesian
computation. In this paper, we propose several unbiased estimators based on
multilevel Monte Carlo (MLMC) for the gradient of Kullback-Leibler divergence
between the posterior distribution and the variational distribution when the
likelihood is intractable, but can be estimated unbiasedly. The new VB
algorithm differs from the VB algorithms in the literature which usually render
biased gradient estimators. Moreover, we incorporate randomized quasi-Monte
Carlo (RQMC) sampling within the MLMC-based gradient estimators, which was
known to provide a favorable rate of convergence in numerical integration.
Theoretical guarantees for RQMC are provided in this new setting. Numerical
experiments show that using RQMC in MLMC greatly speeds up the VB algorithm,
and finds a better parameter value than some existing competitors do.
