Variational Optimality of F\"ollmer Processes in Generative Diffusions

2026-02-11

AI Summary

This work addresses the problem of efficiently generating a target distribution from point masses within finite time while mitigating the impact of estimation errors on the generation path. Building upon a stochastic interpolation framework, the authors propose a novel variational characterization of the Föllmer process by optimizing the diffusion coefficient to minimize the Kullback–Leibler (KL) divergence in path space, thereby achieving an accurate approximation of the target distribution. Theoretical analysis reveals that the optimal diffusion coefficient renders the path-space KL divergence independent of the interpolation schedule, uncovering a statistical equivalence among different scheduling strategies. The method employs a drift estimator expressed as a conditional expectation and refines the diffusion coefficient via posterior tuning, enabling direct estimation of the generative diffusion from independent samples without simulating stochastic processes, and establishing the optimality of the Föllmer process in a variational sense.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Planning, Routing, and Scheduling: Planning with Language Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We construct and analyze generative diffusions that transport a point mass to a prescribed target distribution over a finite time horizon using the stochastic interpolant framework. The drift is expressed as a conditional expectation that can be estimated from independent samples without simulating stochastic processes. We show that the diffusion coefficient can be tuned \emph{a~posteriori} without changing the time-marginal distributions. Among all such tunings, we prove that minimizing the impact of estimation error on the path-space Kullback--Leibler divergence selects, in closed form, a F\"ollmer process -- a diffusion whose path measure minimizes relative entropy with respect to a reference process determined by the interpolation schedules alone. This yields a new variational characterization of F\"ollmer processes, complementing classical formulations via Schr\"odinger bridges and stochastic control. We further establish that, under this optimal diffusion coefficient, the path-space Kullback--Leibler divergence becomes independent of the interpolation schedule, rendering different schedules statistically equivalent in this variational sense.

Problem

• generative diffusions
• Föllmer processes
• Kullback–Leibler divergence
• stochastic interpolant
• path-space optimization

Innovation

• Föllmer process
• generative diffusion
• stochastic interpolant
• path-space KL divergence
• variational characterization