Soft-constrained Schrodinger Bridge: a Stochastic Control Approach
This work addresses a theoretical extension in stochastic control for generative modeling, with incremental contributions to robust diffusion models.
The authors tackled the problem of generalizing the Schrödinger bridge by allowing a mismatch between the terminal and target distributions, penalized by KL divergence, and derived a theoretical solution showing the terminal distribution is a geometric mixture, with applications in robust generative diffusion models demonstrated on MNIST.
Schrödinger bridge can be viewed as a continuous-time stochastic control problem where the goal is to find an optimally controlled diffusion process whose terminal distribution coincides with a pre-specified target distribution. We propose to generalize this problem by allowing the terminal distribution to differ from the target but penalizing the Kullback-Leibler divergence between the two distributions. We call this new control problem soft-constrained Schrödinger bridge (SSB). The main contribution of this work is a theoretical derivation of the solution to SSB, which shows that the terminal distribution of the optimally controlled process is a geometric mixture of the target and some other distribution. This result is further extended to a time series setting. One application is the development of robust generative diffusion models. We propose a score matching-based algorithm for sampling from geometric mixtures and showcase its use via a numerical example for the MNIST data set.