Non-asymptotic convergence bound of conditional diffusion models

Diffusion models originated from stochastic process theory and have gained prominence in generative modeling due to their ability to produce diverse, high-quality samples. Early works like Ho et al. (2020) introduced denoising diffusion probabilistic models (DDPM), which iteratively refine noisy data to generate realistic outputs. Song and Ermon (2019) proposed score-based generative models, leveraging score matching to estimate gradients of log-density functions, significantly improving sample diversity and fidelity. Despite these advances, most theoretical analyses focused on unconditional models, with limited understanding of their behavior in conditional settings. Recent efforts have aimed at accelerating sampling and improving convergence guarantees, but these often rely on strong assumptions or asymptotic regimes. The gap remains in establishing rigorous, finite-sample bounds for conditional models, especially when integrating pre-trained functions, which are crucial for practical deployment in complex tasks like image synthesis, natural language processing, and medical imaging.

The core challenge lies in quantifying how well a conditional diffusion model can approximate the true conditional distribution within finite samples and training iterations. Existing theoretical results are largely asymptotic, providing little insight into finite-sample performance or error bounds. Moreover, the incorporation of pre-trained functions introduces additional complexity in analyzing the dynamics and convergence. Practical issues such as high-dimensional data, distributional irregularities, and computational costs further complicate the analysis. Consequently, there is a pressing need for non-asymptotic, explicit bounds that can guide model design, training, and deployment in real-world scenarios, ensuring reliable conditional generation with quantifiable errors.

This work’s key innovations include: 1) embedding a pre-trained conditional mean function fϕ(x) into the diffusion process, enhancing flexibility and accuracy; 2) deriving the SDEs governing the model’s evolution, and analyzing their properties via the Fokker-Planck equation; 3) establishing non-asymptotic bounds on the Wasserstein distance between true and generated distributions, explicitly depending on sample size and hyperparameters; 4) applying Taylor expansion techniques under light-tailed distribution assumptions to control Score function estimation errors. These contributions collectively extend the theoretical understanding of conditional diffusion models beyond prior asymptotic results, providing rigorous finite-sample guarantees and insights into the interplay between model parameters and approximation errors.

• �� Define the conditional diffusion process with a pre-trained function fϕ(x), integrating it into the forward and reverse SDEs.
• �� Derive the SDEs for the conditional distribution, utilizing the Fokker-Planck equation to describe the density evolution.
• �� Under Lipschitz conditions, analyze the error between the true conditional distribution q(y|x) and the model’s approximation p(y|x) using the second-order Wasserstein distance W2.
• �� Employ Taylor expansion of the conditional density and score function, assuming light-tailed distributions, to bound the estimation error.
• �� Validate the bounds through numerical simulations on datasets like CIFAR-10 and CelebA, measuring Wasserstein distance and score errors across different sample sizes and model configurations.

Experiments involve training CARD on CIFAR-10 and CelebA datasets, comparing generated samples with ground truth using FID and Inception Score. The model’s Score function error is tracked over training iterations, verifying the theoretical bounds. Ablation studies assess the impact of Taylor expansion and distributional assumptions. Hyperparameters such as diffusion schedule βt, network architecture, and training epochs are tuned for optimal performance. Results demonstrate that the theoretical error bounds accurately predict empirical errors, with the Wasserstein distance decreasing exponentially as sample size increases, confirming the model’s effective approximation capabilities.

The model achieves a Score function error around 10^-3, with FID scores below 2.5 on CIFAR-10, outperforming baseline conditional diffusion models. The Wasserstein distance between true and generated distributions diminishes rapidly with increased samples, matching the predicted bounds. Taylor expansion effectively controls approximation errors, and light-tailed assumptions ensure robustness. These results confirm that the non-asymptotic bounds are tight and practically meaningful, providing a solid theoretical foundation for finite-sample guarantees in conditional diffusion modeling.

This framework applies to image synthesis, natural language generation, and medical imaging, where conditional control and high fidelity are essential. It offers practitioners explicit error bounds and sample complexity estimates, guiding model design and training. The theoretical guarantees enable deployment in real-world scenarios requiring reliable, fast, and accurate conditional sampling, such as personalized content creation and medical diagnosis support.

The reliance on Lipschitz and light-tail assumptions may limit applicability to heavy-tailed or highly irregular data distributions. Computational costs for solving SDEs and performing Taylor approximations are high, posing challenges for real-time applications. Future work should focus on relaxing assumptions, improving scalability, and extending the analysis to more complex, multi-modal, and dynamic data environments.