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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.01717 |
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| _version_ | 1866917646455275520 |
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| author | Garg, Jhanvi Zhang, Xianyang Zhou, Quan |
| author_facet | Garg, Jhanvi Zhang, Xianyang Zhou, Quan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01717 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Soft-constrained Schrodinger Bridge: a Stochastic Control Approach Garg, Jhanvi Zhang, Xianyang Zhou, Quan Machine Learning Optimization and Control Computation 60J60, 60J70, 93E20 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. |
| title | Soft-constrained Schrodinger Bridge: a Stochastic Control Approach |
| topic | Machine Learning Optimization and Control Computation 60J60, 60J70, 93E20 |
| url | https://arxiv.org/abs/2403.01717 |