Soft-constrained Schrodinger Bridge: a Stochastic Control Approach

• URL: http://arxiv.org/abs/2403.01717v2
• Date: Mon, 22 Apr 2024 17:50:48 GMT
• Title: Soft-constrained Schrodinger Bridge: a Stochastic Control Approach
• Authors: Jhanvi Garg, Xianyang Zhang, Quan Zhou,
• Abstract summary: Schr"odinger bridge can be viewed as a continuous-time control problem where the goal is to find an optimally controlled diffusion process. 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. One application is the development of robust generative diffusion models.
• Score: 4.922305511803267
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers [49.97755400231656]
  We present the first performance guarantee with explicit dimensional general score-mismatched diffusion samplers. We show that score mismatches result in an distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions. This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise.
  arXiv  Detail & Related papers  (2024-10-17T16:42:12Z)
• Harmonic Path Integral Diffusion [0.4527270266697462]
  We present a novel approach for sampling from a continuous multivariate probability distribution. Our method constructs a time-dependent bridge from a delta function centered at the origin of the state space.
  arXiv  Detail & Related papers  (2024-09-23T16:20:21Z)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• Generalized Schrödinger Bridge Matching [54.171931505066]
  Generalized Schr"odinger Bridge (GSB) problem setup is prevalent in many scientific areas both within and without machine learning. We propose Generalized Schr"odinger Bridge Matching (GSBM), a new matching algorithm inspired by recent advances. We show that such a generalization can be cast as solving conditional optimal control, for which variational approximations can be used.
  arXiv  Detail & Related papers  (2023-10-03T17:42:11Z)
• Adaptive Annealed Importance Sampling with Constant Rate Progress [68.8204255655161]
  Annealed Importance Sampling (AIS) synthesizes weighted samples from an intractable distribution. We propose the Constant Rate AIS algorithm and its efficient implementation for $alpha$-divergences.
  arXiv  Detail & Related papers  (2023-06-27T08:15:28Z)
• The Schr\"odinger Bridge between Gaussian Measures has a Closed Form [101.79851806388699]
  We focus on the dynamic formulation of OT, also known as the Schr"odinger bridge (SB) problem. In this paper, we provide closed-form expressions for SBs between Gaussian measures.
  arXiv  Detail & Related papers  (2022-02-11T15:59:01Z)
• Path Integral Sampler: a stochastic control approach for sampling [4.94950858749529]
  We present Path Integral Sampler(PIS), a novel algorithm to draw samples from unnormalized probability density functions. The PIS draws samples from the initial distribution and then propagates the samples through the Schr"odinger bridge to reach the terminal distribution. We provide theoretical justification of the sampling quality of PIS in terms of Wasserstein distance when sub-optimal control is used.
  arXiv  Detail & Related papers  (2021-11-30T05:50:12Z)
• Deep Generative Learning via Schr\"{o}dinger Bridge [14.138796631423954]
  We learn a generative model via entropy with a Schr"odinger Bridge. We show that the generative model via Schr"odinger Bridge is comparable with state-of-the-art GANs.
  arXiv  Detail & Related papers  (2021-06-19T03:35:42Z)
• Distributionally Robust Bayesian Quadrature Optimization [60.383252534861136]
  We study BQO under distributional uncertainty in which the underlying probability distribution is unknown except for a limited set of its i.i.d. samples. A standard BQO approach maximizes the Monte Carlo estimate of the true expected objective given the fixed sample set. We propose a novel posterior sampling based algorithm, namely distributionally robust BQO (DRBQO) for this purpose.
  arXiv  Detail & Related papers  (2020-01-19T12:00:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.