Soft-constrained Schrodinger Bridge: a Stochastic Control Approach
Jhanvi Garg Xianyang Zhang Quan Zhou
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.