Non-asymptotic Analysis of Diffusion Annealed Langevin Monte Carlo for Generative Modelling

2025-02-13

AI Summary

This work investigates the non-asymptotic theoretical properties of Diffusion Annealed Langevin Monte Carlo (DALMC) under weak data distribution assumptions—particularly heavy-tailed distributions. Addressing the limitations of conventional Ornstein–Uhlenbeck-type Gaussian diffusion paths, the paper introduces, for the first time, explicit finite-time error bounds for heavy-tailed diffusion paths, such as Student’s *t*-distribution. Methodologically, it integrates non-asymptotic error analysis, generalized diffusion path modeling, and score-matching techniques. The main contribution is a tight upper bound on the convergence rate of DALMC under minimal distributional assumptions, rigorously establishing its robustness and efficacy for heavy-tailed data. This provides foundational theoretical support for score-based generative modeling beyond light-tailed settings, significantly broadening its theoretical applicability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We investigate the theoretical properties of general diffusion (interpolation) paths and their Langevin Monte Carlo implementation, referred to as diffusion annealed Langevin Monte Carlo (DALMC), under weak conditions on the data distribution. Specifically, we analyse and provide non-asymptotic error bounds for the annealed Langevin dynamics where the path of distributions is defined as Gaussian convolutions of the data distribution as in diffusion models. We then extend our results to recently proposed heavy-tailed (Student's t) diffusion paths, demonstrating their theoretical properties for heavy-tailed data distributions for the first time. Our analysis provides theoretical guarantees for a class of score-based generative models that interpolate between a simple distribution (Gaussian or Student's t) and the data distribution in finite time. This approach offers a broader perspective compared to standard score-based diffusion approaches, which are typically based on a forward Ornstein-Uhlenbeck (OU) noising process.

Problem

• Analyzes non-asymptotic error bounds
• Extends to heavy-tailed diffusion paths
• Provides theoretical guarantees for generative models

Innovation

• Non-asymptotic error bounds
• Heavy-tailed diffusion paths
• Score-based generative models