Non-asymptotic convergence bound of conditional diffusion models

• URL: http://arxiv.org/abs/2508.10944v1
• Date: Wed, 13 Aug 2025 13:35:56 GMT
• Title: Non-asymptotic convergence bound of conditional diffusion models
• Authors: Mengze Li,
• Abstract summary: We develop a conditional diffusion model within the domains of classification and regression.<n>It integrates a pre-trained model f_phi(x) into the original diffusion model framework.<n>When f_phi(x) performs satisfactorily, Y|fphi(x) closely approximates Y|X.
• Score: 2.0410061496886454
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Learning and generating various types of data based on conditional diffusion models has been a research hotspot in recent years. Although conditional diffusion models have made considerable progress in improving acceleration algorithms and enhancing generation quality, the lack of non-asymptotic properties has hindered theoretical research. To address this gap, we focus on a conditional diffusion model within the domains of classification and regression (CARD), which aims to learn the original distribution with given input x (denoted as Y|X). It innovatively integrates a pre-trained model f_{\phi}(x) into the original diffusion model framework, allowing it to precisely capture the original conditional distribution given f (expressed as Y|f_{\phi}(x)). Remarkably, when f_{\phi}(x) performs satisfactorily, Y|f_{\phi}(x) closely approximates Y|X. Theoretically, we deduce the stochastic differential equations of CARD and establish its generalized form predicated on the Fokker-Planck equation, thereby erecting a firm theoretical foundation for analysis. Mainly under the Lipschitz assumptions, we utilize the second-order Wasserstein distance to demonstrate the upper error bound between the original and the generated conditional distributions. Additionally, by appending assumptions such as light-tailedness to the original distribution, we derive the convergence upper bound between the true value analogous to the score function and the corresponding network-estimated value.

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