Lipschitz-Guided Design of Interpolation Schedules in Generative Models

Abstract: We study the design of interpolation schedules in the stochastic interpolants framework for flow and diffusion-based generative models. We show that while all scalar interpolation schedules achieve identical statistical efficiency under Kullback-Leibler divergence in path space after optimal diffusion coefficient tuning, their numerical efficiency can differ substantially. This observation motivates focusing on numerical properties of the resulting drift fields rather than statistical criteria for schedule design. We propose averaged squared Lipschitzness minimization as a principled criterion for numerical optimization, providing an alternative to kinetic energy minimization used in optimal transport approaches. A transfer formula is derived that enables conversion between different schedules at inference time without retraining neural networks. For Gaussian distributions, our optimized schedules achieve exponential improvements in Lipschitz constants over standard linear schedules, while for Gaussian mixtures, they reduce mode collapse in few-step sampling. We also validate our approach on high-dimensional invariant distributions from stochastic Allen-Cahn equations and Navier-Stokes equations, demonstrating robust performance improvements across resolutions.