Soft-constrained Schrodinger Bridge: a Stochastic Control Approach

Schr\"{o}dinger bridge can be viewed as a continuous-time stochastic control problem where the goal is to find an optimally controlled diffusion process with a pre-specified terminal distribution $\mu_T$. We propose to generalize this stochastic control problem by allowing the terminal distribution to differ from $\mu_T$ but penalizing the Kullback-Leibler divergence between the two distributions. We call this new control problem soft-constrained Schr\"{o}dinger 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 $\mu_T$ and some other distribution. This result is further extended to a time series setting. One application of SSB 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.