Convergence of Proximal-Gradient Stochastic Variational Inference under Non-Decreasing Step-Size Sequence

Stochastic approximation methods have recently gained popularity for variational inference, but many existing approaches treat them as "black-box" tools. Thus, they often do not take advantage of the geometry of the posterior and usually require a decreasing sequence of step-sizes (which converges slowly in practice). We introduce a new stochastic-approximation method that uses a proximal-gradient framework. Our method exploits the geometry and structure of the variational lower bound, and contains many existing methods, such as stochastic variational inference, as a special case. We establish the convergence of our method under a "non-decreasing" step-size schedule, which has both theoretical and practical advantages. We consider setting the step-size based on the continuity of the objective and the geometry of the posterior, and show that our method gives a faster rate of convergence for variational-Gaussian inference than existing stochastic methods.