Boosting Variational Inference: an Optimization Perspective
This work addresses a gap in theoretical analysis for variational inference, which is important for researchers in machine learning and statistics, though it is incremental as it builds on existing boosting methods.
The authors tackled the lack of theoretical understanding in boosting variational inference by analyzing its convergence properties from an optimization perspective, connecting it to the Frank-Wolfe algorithm and providing conditions for convergence, rates, and simplifications.
Variational inference is a popular technique to approximate a possibly intractable Bayesian posterior with a more tractable one. Recently, boosting variational inference has been proposed as a new paradigm to approximate the posterior by a mixture of densities by greedily adding components to the mixture. However, as is the case with many other variational inference algorithms, its theoretical properties have not been studied. In the present work, we study the convergence properties of this approach from a modern optimization viewpoint by establishing connections to the classic Frank-Wolfe algorithm. Our analyses yields novel theoretical insights regarding the sufficient conditions for convergence, explicit rates, and algorithmic simplifications. Since a lot of focus in previous works for variational inference has been on tractability, our work is especially important as a much needed attempt to bridge the gap between probabilistic models and their corresponding theoretical properties.