Instance-dependent Convergence Theory for Diffusion Models

Yuchen Jiao; Gen Li

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arXiv · 2410.13738

Instance-dependent Convergence Theory for Diffusion Models

Yuchen Jiao·Gen Li

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Abstract

Score-based diffusion models have demonstrated outstanding empirical performance in machine learning and artificial intelligence, particularly in generating high-quality new samples from complex probability distributions. Improving the theoretical understanding of diffusion models, with a particular focus on the convergence analysis, has attracted significant attention. In this work, we develop a convergence rate that is adaptive to the smoothness of different target distributions, referred to as instance-dependent bound. Specifically, we establish an iteration complexity of $\min\{d,d^{2/3}L^{1/3},d^{1/3}L\}\varepsilon^{-2/3}$ (up to logarithmic factors), where $d$ denotes the data dimension, and $\varepsilon$ quantifies the output accuracy in terms of total variation (TV) distance. In addition, $L$ represents a relaxed Lipschitz constant, which, in the case of Gaussian mixture models, scales only logarithmically with the number of components, the dimension and iteration number, demonstrating broad applicability.

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BibTeX RIS

Yuchen Jiao, Gen Li. 2024-10-17. Instance-dependent Convergence Theory for Diffusion Models. https://arxiv.org/abs/2410.13738

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