  We establish a connection between stochastic optimal control and generative
models based on stochastic differential equations (SDEs), such as recently
developed diffusion probabilistic models. In particular, we derive a
Hamilton-Jacobi-Bellman equation that governs the evolution of the
log-densities of the underlying SDE marginals. This perspective allows to
transfer methods from optimal control theory to generative modeling. First, we
show that the evidence lower bound is a direct consequence of the well-known
verification theorem from control theory. Further, we can formulate
diffusion-based generative modeling as a minimization of the Kullback-Leibler
divergence between suitable measures in path space. Finally, we develop a novel
diffusion-based method for sampling from unnormalized densities -- a problem
frequently occurring in statistics and computational sciences. We demonstrate
that our time-reversed diffusion sampler (DIS) can outperform other
diffusion-based sampling approaches on multiple numerical examples.

Machine Learning;Optimization and Control;Machine Learning