Discrete Adjoint Schr\"odinger Bridge Sampler

2026-02-09

AI Summary

Discrete neural samplers are notoriously difficult to train due to the absence of gradients and combinatorial complexity, rendering existing continuous-domain stochastic optimal control methods inapplicable. This work proposes a unified discrete ASBS framework by generalizing the adjoint matching (AM) mechanism and Schrödinger bridge theory to discrete state spaces for the first time. By revealing the state-space-agnostic nature of the AM mechanism and modeling the discrete state space via a cyclic group structure, we derive optimality conditions for the discrete Schrödinger bridge problem. The resulting approach significantly enhances training efficiency and scalability while preserving high sample quality.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Learning discrete neural samplers is challenging due to the lack of gradients and combinatorial complexity. While stochastic optimal control (SOC) and Schr\"odinger bridge (SB) provide principled solutions, efficient SOC solvers like adjoint matching (AM), which excel in continuous domains, remain unexplored for discrete spaces. We bridge this gap by revealing that the core mechanism of AM is $\mathit{state}\text{-}\mathit{space~agnostic}$, and introduce $\mathbf{discrete~ASBS}$, a unified framework that extends AM and adjoint Schr\"odinger bridge sampler (ASBS) to discrete spaces. Theoretically, we analyze the optimality conditions of the discrete SB problem and its connection to SOC, identifying a necessary cyclic group structure on the state space to enable this extension. Empirically, discrete ASBS achieves competitive sample quality with significant advantages in training efficiency and scalability.

Problem

• discrete sampling
• neural samplers
• combinatorial complexity
• gradient-free optimization
• discrete state space

Innovation

• discrete Schr\"odinger bridge
• adjoint matching
• stochastic optimal control
• neural sampling
• cyclic group structure