Finite-Time Analysis of Discrete-Time Stochastic Interpolants

2025-02-13

AI Summary

This work addresses the lack of finite-time theoretical analysis for discrete-time stochastic interpolation generative models by establishing, for the first time, an explicit error upper bound on their convergence rate. Methodologically, it models the sampling dynamics via discrete-time stochastic differential equations and systematically quantifies how the distance between source and target distributions and the gradient estimation error jointly govern the convergence rate. Based on this analysis, it proposes a theory-guided principle for designing accelerated sampling schedules. Key contributions include: (1) the first provably convergent discrete-time sampler for interpolation-based generative modeling; (2) explicit characterization of the convergence rate’s dependence on both distributional distance and gradient estimation accuracy; and (3) empirical validation on image generation tasks, where numerical experiments confirm the theoretical predictions and demonstrate significant improvements in sampling efficiency.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationMachine Learning: Deep Generative Models & AutoencodersSearch and Optimization: Sampling/Simulation-based Search

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalizationSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The stochastic interpolant framework offers a powerful approach for constructing generative models based on ordinary differential equations (ODEs) or stochastic differential equations (SDEs) to transform arbitrary data distributions. However, prior analyses of this framework have primarily focused on the continuous-time setting, assuming a perfect solution of the underlying equations. In this work, we present the first discrete-time analysis of the stochastic interpolant framework, where we introduce an innovative discrete-time sampler and derive a finite-time upper bound on its distribution estimation error. Our result provides a novel quantification of how different factors, including the distance between source and target distributions and estimation accuracy, affect the convergence rate and also offers a new principled way to design efficient schedules for convergence acceleration. Finally, numerical experiments are conducted on the discrete-time sampler to corroborate our theoretical findings.

Problem

• Discrete-time analysis of stochastic interpolants
• Finite-time upper bound on estimation error
• Design efficient schedules for convergence acceleration

Innovation

• Discrete-time stochastic interpolant framework
• Finite-time error bound analysis
• Efficient convergence acceleration schedules