  We study convergence rates of variational posterior distributions for
nonparametric and high-dimensional inference. We formulate general conditions
on prior, likelihood, and variational class that characterize the convergence
rates. Under similar "prior mass and testing" conditions considered in the
literature, the rate is found to be the sum of two terms. The first term stands
for the convergence rate of the true posterior distribution, and the second
term is contributed by the variational approximation error. For a class of
priors that admit the structure of a mixture of product measures, we propose a
novel prior mass condition, under which the variational approximation error of
the mean-field class is dominated by convergence rate of the true posterior. We
demonstrate the applicability of our general results for various models, prior
distributions and variational classes by deriving convergence rates of the
corresponding variational posteriors.
