Convergence of the full nested DRSB algorithm

Establish convergence of the full nested Distributionally Robust Schrödinger Bridge algorithm, including convergence of its alternating adversarial initial-distribution, terminal-cost, and controller updates to a stationary point or saddle point.

Background

The Distributionally Robust Schrödinger Bridge objective is optimized through alternating block updates: an adversarial initial distribution is updated, a terminal-cost network estimates the terminal log-density ratio, and a controller is trained using adjoint matching. The paper explicitly notes that these finite optimization steps do not exchange the minimax operators and do not provide a convergence theorem for the resulting nested procedure.

Numerical diagnostics show empirical stabilization of the objective and update quantities on a two-dimensional Gaussian task, but the reported experiments do not establish stationarity of all blocks or convergence to a saddle point. A general convergence result for the complete algorithm therefore remains unresolved.

References

These updates neither exchange the minimax operators nor establish convergence of the full nested problem.

— Distributionally Robust Schrödinger Bridge  (2610.02043 - Sul et al., 1 Oct 2026) in Section 3.2, subsection “Iterative Optimization Scheme for DRSB”; Section 4, subsection “Additional Experimental Analysis,” paragraph “Empirical convergence”; Section 5, “Conclusion and Limitations”