Control, Transport and Sampling: Towards Better Loss Design

Traditionally, the task of sampling from un-normalized densities is largely delegated to MCMC methods. However, modern machine learning developments in optimal transport and generative modeling have greatly expanded the toolbox we have available for performing such tasks, which can further benefit from powerful advancements in deep learning. By reducing the problem to performing empirical risk minimization with neural networks, they hold promise especially for sampling from high-dimensional and multimodal distributions compared to MCMC-based alternatives. In this work, we propose novel training objectives that are amenable to tractable approximations for sampling (without access to data from the target, as typically considered in the MCMC setup), and demonstrate their numerical advantages compared to several related methods that are also built around forward-backward SDEs through time-reversals. Before elaborating on these connections and synthesis in Section 2, we briefly summarize our contributions below. We present a transport/control based sampler using Schrödinger bridge (SB) idea (generalizable to extended state space and nonlinear prior), which compared to MCMC-based approaches has two benefits: (1) alleviate metastability without being confined to make local moves in the state space; (2) based on interpolating path, it comes with the ability to provide low-variance, unbiased normalizing constant estimates using the trajectory information (c.f.