Instance-dependent Convergence Theory for Diffusion Models

• URL: http://arxiv.org/abs/2410.13738v2
• Date: Thu, 29 May 2025 05:33:03 GMT
• Title: Instance-dependent Convergence Theory for Diffusion Models
• Authors: Yuchen Jiao, Gen Li,
• Abstract summary: We develop a convergence rate that is adaptive to the smoothness of different target distributions, referred to as instance-dependent bound.<n>In addition, $L$ represents a relaxed Lipschitz constant, which, in the case of Gaussian mixture models, scales only logarithmically with the number of components.
• Score: 7.237817437521988
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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