Soft-constrained Schrodinger Bridge: a Stochastic Control Approach
<|reference_start|>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 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 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 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.<|reference_end|>
@article{garg2024soft-constrained,
    title={Soft-constrained Schrodinger Bridge: a Stochastic Control Approach},
    author={Jhanvi Garg, Xianyang Zhang, Quan Zhou},
    journal={arXiv preprint arXiv:2403.01717},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.01717},
    primaryClass={stat.ML cs.LG math.OC stat.CO}
}