Statistical Guarantees and Algorithmic Convergence Issues of Variational Boosting

Abstract: We provide statistical guarantees for Bayesian variational boosting by proposing a novel small bandwidth Gaussian mixture variational family. We employ a functional version of Frank-Wolfe optimization as our variational algorithm and study frequentist properties of the iterative boosting updates. Comparisons are drawn to the recent literature on boosting, describing how the choice of the variational family and the discrepancy measure affect both convergence and finite-sample statistical properties of the optimization routine. Specifically, we first demonstrate stochastic boundedness of the boosting iterates with respect to the data generating distribution. We next integrate this within our algorithm to provide an explicit convergence rate, ending with a result on the required number of boosting updates.