An optimal control perspective on diffusion-based generative modeling

Julius Berner, Lorenz Richter, Karen Ullrich

Keywords: optimal control, diffusion models, deep learning, sampling, partial differential equations, stochastic differential equations

TL;DR: We provide a connection between stochastic optimal control and diffusion-based generative modeling, allowing to transfer methods from one to the respective other field.

Abstract: 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 develop a novel diffusion-based method for sampling from unnormalized densities -- a problem frequently occurring in statistics and computational sciences.

Student Paper: Yes