Convergence Analysis of Probability Flow ODE for Score-based Generative Models

2024-04-15

IEEE Transactions on Information Theory

AI Summary

This work addresses the convergence analysis of deterministic samplers based on probability flow ODEs for score-based generative models. We establish the first theoretical guarantees on total variation (TV) error bounds—previously absent in the literature—by integrating L² score estimation error analysis, probability flow ODE theory, p-th-order Runge–Kutta discretization, and high-dimensional distribution convergence arguments. Our analysis yields explicit, dimension-dependent upper bounds on the TV error in both continuous- and discrete-time settings: (O(d^{3/4}delta^{1/2} + d cdot (dh)^p)), where (d) is the data dimension, (delta) quantifies the score estimation error, (h) is the step size, and (p) is the order of the Runge–Kutta method. The bound reveals a fundamental trade-off between statistical error ((d^{3/4}delta^{1/2})) and numerical discretization error ((d cdot (dh)^p)). Numerical experiments up to dimension 128 validate the theoretical predictions, providing the first dimensionally explicit and numerically interpretable convergence guarantee for deterministic sampling in high-dimensional score-based generative modeling.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to $L^2$-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by $mathcal{O}(d^{3/4}delta^{1/2})$ in the continuous time level, where $d$ denotes the data dimension and $delta$ represents the $L^2$-score matching error. For practical implementations using a $p$-th order Runge-Kutta integrator with step size $h$, we establish error bounds of $mathcal{O}(d^{3/4}delta^{1/2} + dcdot(dh)^p)$ at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.

Problem

• Analyzing convergence of probability flow ODEs in score-based models
• Bounding total variation error in generated data distributions
• Validating theory with numerical studies up to 128 dimensions

Innovation

• Analyzes convergence of probability flow ODEs
• Uses L2-accurate score function estimates
• Employs p-th order Runge-Kutta integrator