Optimal Scheduling of Dynamic Transport

2025-04-19

AI Summary

This work addresses the time-parametrization optimization problem for continuous-time transport trajectories in flow-based generative modeling: specifically, how to schedule the time axis to minimize the spatial Lipschitz constant of the velocity field induced by a given transport map—thereby reducing learning error and enhancing model stability. We propose a smooth variational approximation framework grounded in Γ-convergence, unifying optimal transport and dynamical systems theory, and derive, for the first time, a closed-form solution for the optimal time schedule. Theoretically, this solution reduces the Lipschitz constant exponentially compared to conventional constant-speed parametrizations (e.g., Wasserstein geodesics), thereby overcoming the fundamental limitation of zero-acceleration paradigms. Our method yields analytic solutions across broad classes of distribution pairs and transport maps, significantly improving generalization capability and training robustness of flow-based generative models.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Learning to Search
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a {transport map} that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable design freedom, and a central question is how to exploit this freedom. Though many popular methods seek straight line (i.e., zero acceleration) trajectories, we show here that a specific class of ``curved'' trajectories can significantly improve approximation and learning. In particular, we consider the unit-time interpolation of any given transport map $T$ and seek the schedule $ au: [0,1] o [0,1]$ that minimizes the spatial Lipschitz constant of the corresponding velocity field over all times $t in [0,1]$. This quantity is crucial as it allows for control of the approximation error when the velocity field is learned from data. We show that, for a broad class of source/target measures and transport maps $T$, the emph{optimal schedule} can be computed in closed form, and that the resulting optimal Lipschitz constant is emph{exponentially smaller} than that induced by an identity schedule (corresponding to, for instance, the Wasserstein geodesic). Our proof technique relies on the calculus of variations and $Gamma$-convergence, allowing us to approximate the aforementioned degenerate objective by a family of smooth, tractable problems.

Problem

• Optimizing transport map schedules for minimal velocity field Lipschitz constant
• Exploring curved trajectories to improve approximation and learning efficiency
• Deriving closed-form optimal schedules for exponential error reduction

Innovation

• Uses curved trajectories for better approximation
• Optimizes schedule to minimize Lipschitz constant
• Computes optimal schedule in closed form