Improved sampling via learned diffusions
Lorenz Richter Julius Berner Guan-Horng Liu
Recently, a series of papers proposed deep learning-based approaches to
sample from unnormalized target densities using controlled diffusion processes.
In this work, we identify these approaches as special cases of the
Schr\"odinger bridge problem, seeking the most likely stochastic evolution
between a given prior distribution and the specified target. We further
generalize this framework by introducing a variational formulation based on
divergences between path space measures of time-reversed diffusion processes.
This abstract perspective leads to practical losses that can be optimized by
gradient-based algorithms and includes previous objectives as special cases. At
the same time, it allows us to consider divergences other than the reverse
Kullback-Leibler divergence that is known to suffer from mode collapse. In
particular, we propose the so-called log-variance loss, which exhibits
favorable numerical properties and leads to significantly improved performance
across all considered approaches.