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.

翻译：我们建立了基于随机差异方程式的最佳控制和基因化模型之间的联系,例如最近开发的传播概率模型。特别是,我们从汉密尔顿-Jacobi-Bellman方程式中得出一个调节基础SDE边缘的日志密度演变的汉密尔顿-Jacobi-Bellman方程式。这一视角允许将方法从最佳控制理论转移到基因化模型。首先,我们表明,证据的下限是控制理论中众所周知的核查理论的直接后果。此外,我们开发了一种新的基于传播的方法,用于从不规范的密度进行取样,这是统计和计算科学中经常出现的问题。