Finite-Time Convergence Analysis of ODE-based Generative Models for Stochastic Interpolants

2025-08-10

AI Summary

This work addresses the lack of rigorous finite-time numerical convergence guarantees for ordinary differential equation (ODE)-based generative models relying on stochastic interpolation. We systematically analyze error propagation mechanisms of practical discretization schemes—including forward Euler and Heun’s methods. For the first time, we derive an explicit finite-time upper bound on the total variation distance between the discrete and continuous sampling processes, quantifying the interplay among step size, number of iterations, and initial distribution mismatch. Building upon this bound, we propose an adaptive time-stepping schedule that significantly reduces iteration complexity while preserving the theoretical error guarantee. Comprehensive numerical experiments validate our theory: the proposed method maintains generation quality while improving solver efficiency and stability. Our results provide both a rigorous theoretical foundation and a practical optimization framework for interpretable, numerically reliable implementations of diffusion-based generative models.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence guarantees for practical numerical schemes remain largely unexplored. In this work, we address the finite-time convergence analysis of numerical implementations for ordinary differential equations (ODEs) derived from stochastic interpolants. Specifically, we establish novel finite-time error bounds in total variation distance for two widely used numerical integrators: the first-order forward Euler method and the second-order Heun's method. Furthermore, our analysis on the iteration complexity of specific stochastic interpolant constructions provides optimized schedules to enhance computational efficiency. Our theoretical findings are corroborated by numerical experiments, which validate the derived error bounds and complexity analyses.

Problem

• Analyze finite-time convergence of ODE-based generative models
• Establish error bounds for Euler and Heun numerical methods
• Optimize schedules for stochastic interpolant computational efficiency

Innovation

• Finite-time error bounds for ODE integrators
• Optimized schedules improve computational efficiency
• First and second-order numerical integrators analyzed