Relative Entropy Gradient Sampler for Unnormalized Distribution

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.

我们提出了一种相对熵梯度采样器（Relative Entropy Gradient Sampler，REGS），用于从非归一化分布中采样。REGS作为一种粒子方法，旨在通过迭代的简单非线性变换序列，将参考分布的初始样本逐步迁移至目标非归一化分布的样本空间。为确定每一轮迭代中的非线性变换，我们依托相对熵的沃瑟斯坦梯度流（Wasserstein gradient flow）展开建模：该梯度流可生成一条连接参考分布与目标分布的概率分布插值路径，其由依赖于演化粒子密度与目标非归一化密度之比的速度场的常微分方程（Ordinary Differential Equation，ODE）系统所刻画。借助REGS进行采样时，需先估计密度比，再通过粒子演化模拟该ODE系统。为此，我们提出了一种基于神经网络估计对数密度比的新型非参数方法。针对极具挑战性的多模态一维、二维混合分布开展了大量仿真实验，并在真实数据集上完成贝叶斯逻辑回归任务，实验结果表明，相较于当前主流的基于沃瑟斯坦梯度流的采样器，REGS的性能表现合理且优异。本文的补充材料可在线获取。