Moreau-Yoshida variational transport: a general framework for solving regularized distributional optimization problems

We address a general optimization problem involving the minimization of a composite objective functional defined over a class of probability distributions. The objective function consists of two components: one assumed to have a variational representation, and the other expressed in terms of the expectation operator of a possibly nonsmooth convex regularizer function. Such a regularized distributional optimization problem widely appears in machine learning and statistics, including proximal Monte-Carlo sampling, Bayesian inference, and generative modeling for regularized estimation and generation. Our proposed method, named Moreau-Yoshida Variational Transport (MYVT), introduces a novel approach to tackle this regularized distributional optimization problem. First, as the name suggests, our method utilizes the Moreau-Yoshida envelope to provide a smooth approximation of the nonsmooth function in the objective. Second, we reformulate the approximate problem as a concave-convex saddle point problem by leveraging the variational representation. Subsequently, we develop an efficient primal–dual algorithm to approximate the saddle point. Furthermore, we provide theoretical analyses and present experimental results to showcase the effectiveness of the proposed method.