Soft-constrained Schrödinger Bridge: a Stochastic Control Approach

Jhanvi Garg, Xianyang Zhang, Quan Zhou
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:4429-4437, 2024.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-garg24a, title = { Soft-constrained {S}chrödinger Bridge: a Stochastic Control Approach }, author = {Garg, Jhanvi and Zhang, Xianyang and Zhou, Quan}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {4429--4437}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/garg24a/garg24a.pdf}, url = {https://proceedings.mlr.press/v238/garg24a.html}, abstract = { 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. } }
Endnote
%0 Conference Paper %T Soft-constrained Schrödinger Bridge: a Stochastic Control Approach %A Jhanvi Garg %A Xianyang Zhang %A Quan Zhou %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-garg24a %I PMLR %P 4429--4437 %U https://proceedings.mlr.press/v238/garg24a.html %V 238 %X 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.
APA
Garg, J., Zhang, X. & Zhou, Q.. (2024). Soft-constrained Schrödinger Bridge: a Stochastic Control Approach . Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:4429-4437 Available from https://proceedings.mlr.press/v238/garg24a.html.

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