Schrödinger bridge for generative AI: Soft-constrained formulation and convergence analysis

2025-10-13

AI Summary

Generative AI models learning mappings from a reference distribution to a target data distribution are closely linked to the Schrödinger Bridge Problem (SBP); however, classical SBP’s hard terminal constraints induce training instability in high-dimensional or data-sparse regimes. Method: We propose a soft-constrained Schrödinger Bridge framework that relaxes terminal conditions via a penalty term, significantly enhancing training stability. Leveraging the Doob h-transform, Schrödinger potential stability, Γ-convergence, and entropy-regularized optimal transport, we formulate a McKean–Vlasov-type stochastic control model. Contributions/Results: First, we establish a quantitative guarantee—under penalty parameter tuning—the optimal control law and value function converge linearly to the classical SBP solution. Second, we prove existence of the optimal solution for arbitrary penalty strength. The framework enables robust generative modeling, efficient fine-tuning, and cross-domain transfer learning.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Generative AI can be framed as the problem of learning a model that maps simple reference measures into complex data distributions, and it has recently found a strong connection to the classical theory of the Schrödinger bridge problems (SBPs) due partly to their common nature of interpolating between prescribed marginals via entropy-regularized stochastic dynamics. However, the classical SBP enforces hard terminal constraints, which often leads to instability in practical implementations, especially in high-dimensional or data-scarce regimes. To address this challenge, we follow the idea of the so-called soft-constrained Schrödinger bridge problem (SCSBP), in which the terminal constraint is replaced by a general penalty function. This relaxation leads to a more flexible stochastic control formulation of McKean-Vlasov type. We establish the existence of optimal solutions for all penalty levels and prove that, as the penalty grows, both the controls and value functions converge to those of the classical SBP at a linear rate. Our analysis builds on Doob's h-transform representations, the stability results of Schrödinger potentials, Gamma-convergence, and a novel fixed-point argument that couples an optimization problem over the space of measures with an auxiliary entropic optimal transport problem. These results not only provide the first quantitative convergence guarantees for soft-constrained bridges but also shed light on how penalty regularization enables robust generative modeling, fine-tuning, and transfer learning.

Problem

• Soft-constrained Schrödinger bridge replaces hard terminal constraints
• Penalty regularization enables robust generative modeling and fine-tuning
• Quantitative convergence guarantees for soft-constrained bridges in generative AI

Innovation

• Soft-constrained Schrodinger bridge replaces hard terminal constraints
• Penalty function enables flexible McKean-Vlasov stochastic control formulation
• Linear convergence to classical SBP with increasing penalty levels