Relative Entropy Gradient Sampler for Unnormalized Distribution

Abstract

We propose a relative entropy gradient sampler (REGS) for sampling from unnormalized distributions. REGS is a particle method that seeks a sequence of simple nonlinear transforms iteratively pushing the initial samples from a reference distribution into the samples from an unnormalized target distribution. To determine the nonlinear transforms at each iteration, we consider the Wasserstein gradient flow of relative entropy. This gradient flow determines a path of probability distributions that interpolates the reference distribution and the target distribution. It is characterized by an ordinary differential equation (ODE) system with velocity fields depending on the density ratios of the density of evolving particles and the unnormalized target density. To sample with REGS, we need to estimate the density ratios and simulate the ODE system with particle evolution. We propose a novel nonparametric approach to estimating the logarithmic density ratio using neural networks. Extensive simulation studies on challenging multimodal 1D and 2D mixture distributions and Bayesian logistic regression on real datasets demonstrate that REGS has reasonable performance compared with popular samplers based on Wasserstein gradient flows. Supplementary materials for this article are available online.

Keywords:

• Density ratio
• Gradient flow
• Neural networks
• Nonparametric estimation
• Velocity fields

Acknowledgments

We thank the editor, associate editor and two referees for insightful comments and valuable suggestions, which have led to a substantial improvement of this manuscript.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Feng’s work was partially supported by the National Natural Science Foundation of China (No. 12371270), Shanghai Science and Technology Development Funds (No. 23JC1402100), Shanghai Research Center for Data Science and Decision Technology. The work of J. Huang is supported by the National Natural Science Foundation of China grant (No. 72331005) and research grants from The Hong Kong Polytechnic University. Jiao’s work was supported by the National Nature Science Foundation of China (No.12371441), Fundamental Research Funds for the Central Universities, and the research fund of KLATASDSMOE of China. Liu’s work was supported by the National Natural Science Foundation of China (No. 12271329, 72331005), the Program for Innovative Research Team of SUFE, and the Open Research Fund of Yunnan Key Laboratory of Statistical Modeling and Data Analysis, Yunnan University.