Continuous-state Lipschitz-critic formulation

Develop a continuous-state realization of the Wasserstein-robust Bellman dual using Lipschitz critics, thereby extending the belief-dependent robust value computation beyond finite state spaces.

Background

For finite state spaces, the inner Wasserstein-robust optimization is a finite linear program and admits a Kantorovich–Rubinstein dual formulation. The paper suggests using this dual to construct scalable approximations when the belief is represented by particles or variational methods.

The extension to continuous state spaces is not developed in the paper. The proposed direction is a Lipschitz-critic implementation of the dual representation, which would make RATTL applicable to continuous-control and other non-tabular settings.

References

the continuous-state case (a Lipschitz-critic realization of eq:dual) we leave as an explicit open item (\S\ref{sec:outlook}).

— Quantifying Risk Under Evolving Uncertainty: Belief-Dependent Robustness for Safe Sequential Decision Making  (2608.17574 - Ganguly et al., 18 Aug 2026) in Section 3, paragraph “Tractability”