  Recently, a series of papers proposed deep learning-based approaches to
sample from target distributions using controlled diffusion processes, being
trained only on the unnormalized target densities without access to samples.
Building on previous work, we identify these approaches as special cases of a
generalized Schr\"odinger bridge problem, seeking a 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.
