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arXiv · 2307.16358

Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems

Dai Hai Nguyen·Tetsuya Sakurai
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Abstract

We consider a general optimization problem of minimizing a composite objective functional defined over a class of probability distributions. The objective is composed of two functionals: one is assumed to possess the variational representation and the other is 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, such as proximal Monte-Carlo sampling, Bayesian inference and generative modeling, for regularized estimation and generation. We propose a novel method, dubbed as Moreau-Yoshida Variational Transport (MYVT), for solving the regularized distributional optimization problem. First, as the name suggests, our method employs the Moreau-Yoshida envelope for 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, and then develope an efficient primal-dual algorithm to approximate the saddle point. Furthermore, we provide theoretical analyses and report experimental results to demonstrate the effectiveness of the proposed method.

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Dai Hai Nguyen, Tetsuya Sakurai. 2024-08-10. Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems. https://doi.org/10.1007/s10994-024-06586-z

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