Equivalence of stochastic control and path methods for Schrödinger bridges

On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups

Artificial Intelligence

Summary

The Schrödinger bridge problem is about finding the most likely way a system changes between two states when randomness affects it. The authors show that two different ways of looking at this problem — one using control strategies and one using paths the system can take — are actually the same when the system moves on certain geometric surfaces called Lie groups. This connection helps prove the problem has a unique best solution, clarifies its meaning as the most probable random path given start and end points, and allows simpler computations using known algorithms. They illustrate their findings on a donut-shaped surface and provide code online.

What this means in practice

  • For control system engineers: Solve optimal control problems for systems evolving on complex geometric spaces using entropy-based path methods for exact endpoint constraints.
  • For computational mathematicians: Compute optimal stochastic processes on Lie groups efficiently using Sinkhorn iterations enabled by the equivalence of control and path formulations.

Authors

Hamza Mahmood, Georgiy A. Bondar, Abhishek Halder, Adeel Akhtar

Abstract

We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP
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