Sampling from a distribution $p(x) \propto e^{-\mathcal{E}(x)}$ known up to a normalising constant is an important and challenging problem in statistics. Recent years have seen the rise of a new family of amortised sampling algorithms, commonly referred to as diffusion samplers, that enable fast and efficient sampling from an unnormalised density. Such algorithms have been widely studied for continuous-space sampling tasks; however, their application to problems in discrete space remains largely unexplored. Although some progress has been made in this area, discrete diffusion samplers do not take full advantage of ideas commonly used for continuous-space sampling. In this paper, we propose to bridge this gap by introducing off-policy training techniques for discrete diffusion samplers. We show that these techniques improve the performance of discrete samplers on both established and new synthetic benchmarks. Next, we generalise discrete diffusion samplers to the task of bridging between two arbitrary distributions, introducing data-to-energy Schrödinger bridge training for the discrete domain for the first time. Lastly, we showcase the application of the proposed diffusion samplers to data-free posterior sampling in the discrete latent spaces of image generative models.

翻译：从正比于$p(x) \propto e^{-\mathcal{E}(x)}$的分布中采样是统计学中一个重要且具有挑战性的问题。近年来，一类被称为扩散采样器的摊销式采样算法逐渐兴起，能够从未归一化密度函数中实现快速高效采样。此类算法在连续空间采样任务中已得到广泛研究，然而其在离散空间问题中的应用仍鲜有探索。尽管该领域已取得一定进展，但现有离散扩散采样器尚未充分利用连续空间采样中常用的核心思想。本文通过为离散扩散采样器引入离轨策略训练技术来弥合这一差距。实验表明，该技术能提升离散采样器在经典基准与新合成基准上的性能。进一步，我们将离散扩散采样器推广至任意两个分布间的桥接任务，首次在离散域中引入数据-能量薛定谔桥训练方法。最后，我们展示了所提出的扩散采样器在图像生成模型离散隐空间中无数据后验采样的应用潜力。