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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.16358 |
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| _version_ | 1866916360021344256 |
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| author | Nguyen, Dai Hai Sakurai, Tetsuya |
| author_facet | Nguyen, Dai Hai Sakurai, Tetsuya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16358 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems Nguyen, Dai Hai Sakurai, Tetsuya Machine Learning 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. |
| title | Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2307.16358 |