On the Convergence of Black-Box Variational Inference
This addresses convergence issues in variational inference for Bayesian inference problems, offering theoretical insights for practitioners, though it is incremental as it builds on existing methods.
The paper provides the first convergence guarantee for full black-box variational inference (BBVI) without algorithmic simplifications, showing that proximal stochastic gradient descent achieves the strongest known convergence rates and fixes suboptimal choices in practice.
We provide the first convergence guarantee for full black-box variational inference (BBVI), also known as Monte Carlo variational inference. While preliminary investigations worked on simplified versions of BBVI (e.g., bounded domain, bounded support, only optimizing for the scale, and such), our setup does not need any such algorithmic modifications. Our results hold for log-smooth posterior densities with and without strong log-concavity and the location-scale variational family. Also, our analysis reveals that certain algorithm design choices commonly employed in practice, particularly, nonlinear parameterizations of the scale of the variational approximation, can result in suboptimal convergence rates. Fortunately, running BBVI with proximal stochastic gradient descent fixes these limitations, and thus achieves the strongest known convergence rate guarantees. We evaluate this theoretical insight by comparing proximal SGD against other standard implementations of BBVI on large-scale Bayesian inference problems.