论文标题
基于扩散的生成建模的最佳控制观点
An optimal control perspective on diffusion-based generative modeling
论文作者
论文摘要
我们基于随机微分方程（SDE）（例如最近开发的扩散概率模型）之间建立了随机最佳控制和生成模型之间的联系。特别是，我们得出了一个汉密尔顿 - 雅各比 - 贝尔曼方程，该方程控制着基础SDE边缘的对数密度的演变。这种观点允许将方法从最佳控制理论转移到生成建模。首先，我们表明证据下限是控制理论众所周知的验证定理的直接结果。此外，我们可以制定基于扩散的生成建模，以最小化路径空间中合适度量之间的kullback-leibler差异。最后，我们开发了一种基于扩散的新方法，用于从非差异密度进行取样 - 统计和计算科学中经常出现这个问题。我们证明，我们的时间转换扩散采样器（DIS）可以在多个数值示例上胜过其他基于扩散的采样方法。
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.
