Avoiding unsafe sets when training with Langevin Dynamics

What the math gives to ML

This paper provides a trajectory-level safety certificate for noisy optimization, rather than only an equilibrium guarantee. The key transferable asset is the explicit decomposition of unsafe probability into an equilibrium mass term and a transient term controlled by the global relaxation rate and the initial chi-square divergence. In a neural-network optimizer, this can become a conservative noise and training-time controller: estimate unsafe-region mass and relaxation rate, then stop or reduce Langevin noise when the certified transient risk falls below a target. The result is most directly testable in Langevin fine-tuning or sampling-based training, first on strongly convex proxy problems where the assumptions can be checked.