An Optimal Control Perspective on Diffusion-Based Generative Modeling | Julius Berner

Paper: "An Optimal Control Perspective on Diffusion-Based Generative Modeling"

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.

Authors: Julius Berner, Lorenz Richter, and Karen Ullrich

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Chapters

00:00 - Intro
00:53 - The Task of Generative Modeling
03:20 - Overview of the Talk
05:07 - SDE-based Modeling
9:59 - The Fokker-Planck Equation
12:48 - Deriving the ELBO + Verification Theorem
26:06 - Denoting Score Matching
30:53 - Summary
36:31 - Time Reversed Diffusion Sampler (DIS)
39:59 - Path Integral Sampler (PIS)
42:38 - DIS vs. PIS
53:14 - Q+A