Non-asymptotic Analysis of Diffusion Annealed Langevin Monte Carlo for Generative Modelling (2502.09306v1)

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.

• The paper’s main contribution is deriving non-asymptotic convergence bounds for DALMC using both Gaussian and heavy-tailed diffusion paths.
• It employs log-Sobolev inequalities and Lipschitz continuity to rigorously analyze sampling efficiency in generative modelling.
• The work extends theoretical insights into score-based generative models by providing concrete complexity estimates for practical algorithm design.

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

This paper explores the theoretical framework of a novel algorithm, Diffusion Annealed Langevin Monte Carlo (DALMC), within the context of score-based generative models (SGMs). SGMs have gained traction due to their capacity to generate high-quality data across various tasks such as image and audio generation. The authors aim to analyze and provide guarantees for the performance of DALMC, especially when dealing with different diffusion paths that bridge the data distribution with simpler base distributions like Gaussian or heavy-tailed ones.

Theoretical Foundations and Contributions

The primary focus of the work is the non-asymptotic error analysis of DALMC, a Langevin Monte Carlo approximation, when applied under weak assumptions on the data distribution. The paper distinguishes itself by examining both Gaussian and recently proposed heavy-tailed diffusion paths (e.g., Student's t-distributions). Key theoretical contributions include:

1. Gaussian Diffusion Paths: The authors first consider Gaussian base distributions and derive non-asymptotic convergence bounds in KL divergence under assumptions such as finite second-order moments of the data distribution and Lipschitz continuity of the scores. They specifically focus on the log-Sobolev inequalities that guarantee efficient sampling from Gaussian mixture models.
2. Heavy-Tailed Diffusion Paths: Extending their analysis, the paper also caters to recent trends in employing heavy-tailed distributions as the base. In such cases, the data's potential needs to show Lipshitz properties, and the derived complexity bounds are comparable to those achieved for Gaussian distributions.
3. Convergence Guarantees: By leveraging properties of the diffusion paths, the authors show how DALMC achieves convergence with explicit complexity estimates. For Gaussian paths, the paper claims the DALMC requires $$\mathcal{O}(d(M_2 \vee d)^2 L_{\max}^2/\varepsilon^6)$$ steps. Similarly, for heavy-tailed paths, comparable bounds are obtained, attesting to the framework's flexibility.
4. Analysis for Mixtures of Gaussians: A distinct result derived is the validation that certain mixtures of Gaussians satisfy the conditions needed for log-Sobolev inequalities under convexity assumptions—situations often not well-covered in existing literature.

Implications and Future Directions

The implications of these findings are twofold: practically, they outline the performance expectations of DALMC across different scenarios of data distribution assumptions; theoretically, they contribute to understanding the convergence properties of generative models that do not conventionally rely on symmetric Gaussian assumptions. The generalizations to heavy-tailed paths open possibilities to cater to data distributions exhibiting significant kurtosis or outlier presence, which might be more common in real-world scenarios.

Further advancements could explore more sophisticated numerical schemes beyond Euler-Maruyama to reduce step complexity and mitigate discretization bias. Also, there is room to investigate optimized schedules that balance the trade-off between convergence rate and discretization error further, especially as the algorithm scales with higher-dimensional data.

Overall, this work represents a significant academic endeavor in formalizing and enriching the theoretical tools available for analyzing score-based generative models through diffusion processes. It concretely suggests how incorporating Langevin dynamics into the generative process framework can be theoretically substantiated and practically optimized.