Optimal Scheduling of Dynamic Transport (2504.14425v2)

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 $\tau: [0,1] \to [0,1]$ 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.

Optimal Scheduling of Dynamic Transport

The paper "Optimal Scheduling of Dynamic Transport" by Panos Tsimpos, Zhi Ren, Jakob Zech, and Youssef Marzouk presents a formal analysis of improving the approximation of transport maps through optimal scheduling in continuous-time dynamical systems. The authors focus on how a specific class of curved trajectories, achieved by re-parameterizing time, can enhance the spatial regularity of the velocity field in flow-based models.

Overview of Methods

The central aim of the paper is to minimize the spatial Lipschitz constant of the velocity field in the corresponding transport map, which is crucial for error control when learning velocity fields from data. The authors consider the interpolation of a given transport map $T$ over unit-time and introduce the concept of a schedule $\tau: [0,1] \to [0,1]$, which modifies the rate at which time progresses in the dynamic model. The choice of schedule has significant implications for the spatial Lipschitz constant, and thus the quality of the approximation that can be achieved through learning.

Theoretical tools deployed in the paper include calculus of variations and $\Gamma$-convergence to frame and solve the optimization problem of finding optimal schedules. By casting the minimization of the uniform Lipschitz bound into a variational problem, the authors solve it by approximating a series of smooth, tractable problems.

Strong Numerical Results and Implications

One of the paper's key findings is that for a broad class of source/target measures and transport maps $T$, the optimal schedule can be computed in closed form, and the optimal Lipschitz constant derived is exponentially smaller than the one induced by a trivial identity schedule, such as one corresponding to straight-line trajectories.

The implications of this work are both practical and theoretical. On the practical side, the optimal schedules can significantly improve the efficacy of learning dynamic transport models, as the reduced spatial Lipschitz constant results in lower approximation errors. On the theoretical front, the existence of a universal structure in optimal schedules, largely independent of the underlying transport map, suggests new pathways for optimizing learning algorithms in flow models.

Future Directions

The authors propose potential extensions, including exploring more general schedules and paths beyond the initial displacement interpolation. The idea of introducing a time re-parameterized schedule may be beneficial not only for neural ODEs but also for other models like flow matching and stochastic interpolants, thus broadening its applicability in AI.

The analytical approach to finding optimal schedules, combined with the sigmoid-like behavior indicated for a universal structure, opens opportunities for designing efficient algorithms that incorporate these schedules into the learning problem or iterative schemes to estimate optimal schedules and refine velocity fields accordingly.

Conclusion

This work is informative for researchers seeking sophisticated methods to optimize dynamic transport processes. While theoretical, the closed-form solution for optimal scheduling provides actionable insights into improving learning algorithms in dynamic systems by controlling the trajectory paths more effectively. Future research might further illuminate how these optimizations can be integrated into existing models, function effectively across disparate applications, and ultimately contribute to the advancement of machine learning methodologies in dynamic generative models.