Distributionally Robust Schrödinger Bridge

Problem

This work addresses the limitations of learned dynamics in stochastic transport, particularly the failure to maintain performance when the initial distribution shifts at test time. The authors highlight that existing methods do not adequately handle this variability, leading to suboptimal performance in real-world applications. This paper is a preprint and has not undergone peer review.

Method

The authors propose a distributionally robust framework for the Schrödinger Bridge (SB) problem. The core objective is to control the energy and Kullback-Leibler (KL) divergence penalty between the terminal distribution and a target distribution. A single controller is employed to minimize a worst-case objective, which adapts as the initial distribution varies within a defined ambiguity set. The formulation is derived using an exact variational approach, leading to an alternating algorithm that iteratively updates the adversarial initial distribution, estimates the terminal log-density ratio, and trains the controller. Additionally, the authors develop Wasserstein and Sinkhorn variants of the algorithm, leveraging stochastic control optimality conditions for gradient approximation.

Results

The proposed method demonstrates improved robustness to input perturbations compared to the standard Schrödinger Bridge approach. Specifically, it achieves a lower mean sliced Wasserstein distance than fixed-level noise augmentation across both tested unseen noise levels, indicating superior performance in handling distributional shifts. However, the available text does not report quantitative results.

Limitations

The authors note a tradeoff in nominal performance when enhancing robustness, suggesting that while the method improves resilience to distribution shifts, it may compromise performance under nominal conditions. This limitation is critical for applications where maintaining baseline performance is essential.

Why it matters

This research has significant implications for downstream work in robust control and stochastic processes, particularly in scenarios where initial conditions are uncertain or variable. By addressing the shortcomings of existing methods in handling distribution shifts, this approach could lead to more reliable and adaptable systems in various applications, including robotics, finance, and autonomous systems.