Convergence Analysis of Flow Matching in Latent Space with Transformers (2404.02538v2)

Abstract: We present theoretical convergence guarantees for ODE-based generative models, specifically flow matching. We use a pre-trained autoencoder network to map high-dimensional original inputs to a low-dimensional latent space, where a transformer network is trained to predict the velocity field of the transformation from a standard normal distribution to the target latent distribution. Our error analysis demonstrates the effectiveness of this approach, showing that the distribution of samples generated via estimated ODE flow converges to the target distribution in the Wasserstein-2 distance under mild and practical assumptions. Furthermore, we show that arbitrary smooth functions can be effectively approximated by transformer networks with Lipschitz continuity, which may be of independent interest.

• The paper demonstrates that transformer networks, under Lipschitz continuity constraints, robustly approximate smooth functions for flow matching.
• It establishes statistical guarantees showing that pre-trained autoencoders converge in reconstruction error based on sample size and data smoothness.
• A comprehensive error analysis confirms that proper discretization and early stopping yield convergence to the target distribution under the Wasserstein-2 metric.

Convergence Analysis of Flow Matching in Latent Space with Transformers

Introduction

The exploration of generative models, notably Generative Adversarial Networks (GANs) and diffusion models, dictates a significant portion of current machine learning research. GANs, despite their success, often suffer from training instability and evaluation difficulties, whereas diffusion models, especially those based on Stochastic Differential Equations (SDE) or Ordinary Differential Equations (ODE), offer a promising direction due to their superior sample quality and stability. The paper by Yuling Jiao, Yanming Lai, Yang Wang, and Bokai Yan titled "Convergence Analysis of Flow Matching in Latent Space with Transformers" presents an insightful analysis into the theoretical underpinnings of flow matching, a pivotal component in ODE-based generative models.

Theoretical Insights into Transformer Approximation

The paper contributes significantly to the approximation capabilities of transformer networks within the context of flow matching. A core finding is that transformer networks, subject to Lipschitz continuity constraints, can effectively approximate any smooth function. Notably, the Lipschitz continuity of the transformer network remains independent of the approximation error, which is pivotal for maintaining the robustness and generalization capabilities of the network. This insight extends the current understanding of transformers, primarily known for their success in natural language processing, to a broader range of applications in generative modeling.

Statistical Guarantees for Pre-training

A critical aspect of the research focuses on the pre-trained autoencoder network, which is utilized for mapping high-dimensional inputs to a lower-dimensional latent space. The analysis demonstrates that with a properly chosen transformer network as the encoder and decoder, the excess risk of reconstruction loss converges at a rate that is only dependent on the pre-training sample size and the smoothness properties of the target and pre-trained data distributions. This result establishes a statistical foundation for the use of autoencoders in reducing dimensionality before generative modeling, ensuring that the quality of reconstruction does not significantly degrade with complex distributions.

Error Analysis in Estimating Target Distribution

The paper provides a rigorous error analysis for estimating the target distribution through flow matching. It reveals that with appropriate discretization and early stopping in sampling, the distribution of the generated samples converges to the target distribution under the Wasserstein-2 distance. This convergence is quantified through the end-to-end error analysis, combining the errors from the pre-trained autoencoder, the approximation and estimation of the transformer network, and the discretization of the flow ODE. Such comprehensive error analysis furnishes a theoretical guarantee for the effectiveness of ODE-based generative models in capturing complex target distributions.

Implications and Future Work

The convergence analysis carried out in this paper has far-reaching implications for both practical and theoretical aspects of generative modeling. Practically, it provides a theoretical justification for the application of transformer networks in generative models, particularly in scenarios where robust and stable generation of high-quality samples is desired. Theoretically, it opens avenues for further research into improving the approximation and estimation capabilities of transformer networks, potentially leading to the development of more efficient and accurate generative models.

In future work, extending the analysis to include conditional generative models and exploring optimization strategies for minimizing the empirically observed error would be instrumental. Additionally, investigating the potential of transformer networks in other types of generative frameworks could further solidify their role in the advancement of generative modeling.

Conclusion

This paper represents a significant step forward in understanding the convergence properties of flow matching with transformers. By bridging the gap between theoretical analysis and practical application, this research contributes to the development of more stable, high-quality generative models, fostering advancements in machine learning and artificial intelligence.