On the convergence of coordinate ascent variational inference
研究了坐标上升变分推断（CAVI）算法在均值场变分推断中的收敛性，给出了全局或局部指数收敛的条件，并引入广义相关性概念来量化算法收缩率，对大规模贝叶斯模型的应用者有帮助。
As a computational alternative to Markov chain Monte Carlo approaches, variational inference (VI) is becoming more and more popular for approximating intractable posterior distributions in large-scale Bayesian models due to its comparable efficacy and superior efficiency. Several recent works provide theoretical justifications of VI by proving its statistical optimality for parameter estimation under various settings; meanwhile, formal analysis on the algorithmic convergence aspects of VI is still largely lacking. In this paper, we consider the common coordinate ascent variational inference (CAVI) algorithm for implementing the mean-field (MF) VI towards optimizing a Kullback–Leibler divergence objective functional over the space of all factorized distributions. Focusing on the two-block case, we analyze the convergence of CAVI by leveraging the extensive toolbox from functional analysis and optimization. We provide general conditions for certifying global or local exponential convergence of CAVI. Specifically, a new notion of generalized correlation for characterizing the interaction between the constituting blocks in influencing the VI objective functional is introduced, which according to the theory, quantifies the algorithmic contraction rate of two-block CAVI. As illustrations, we apply the theory to a number of examples, and derive explicit problem-dependent upper bounds on the algorithmic contraction rate.