Discrete Adjoint Schrödinger Bridge Sampler

Learning discrete neural samplers is challenging due to the lack of gradients and combinatorial complexity. While stochastic optimal control (SOC) and Schrödinger 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 $\mathit{state}\text{-}\mathit{spaceagnostic}$, and introduce $\mathbf{discreteASBS}$, a unified framework that extends AM and adjoint Schrödinger 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.

💡 Research Summary

The paper introduces a novel framework called Discrete Adjoint Schrödinger Bridge Sampler (D‑ASBS) that extends the powerful continuous‑time adjoint matching (AM) technique to discrete state spaces. Sampling from unnormalized discrete distributions is notoriously difficult because gradients are unavailable and the combinatorial explosion makes learning neural samplers costly. The authors first formalize the Schrödinger bridge (SB) problem for continuous‑time Markov chains (CTMCs) on a discrete domain X =