Convergence of flow-based generative models via proximal gradient descent in Wasserstein space

• URL: http://arxiv.org/abs/2310.17582v3
• Date: Wed, 3 Jul 2024 20:05:43 GMT
• Title: Convergence of flow-based generative models via proximal gradient descent in Wasserstein space
• Authors: Xiuyuan Cheng, Jianfeng Lu, Yixin Tan, Yao Xie,
• Abstract summary: Flow-based generative models enjoy certain advantages in computing the data generation and the likelihood. We provide a theoretical guarantee of generating data distribution by a progressive flow model.
• Score: 20.771897445580723
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Flow-based generative models enjoy certain advantages in computing the data generation and the likelihood, and have recently shown competitive empirical performance. Compared to the accumulating theoretical studies on related score-based diffusion models, analysis of flow-based models, which are deterministic in both forward (data-to-noise) and reverse (noise-to-data) directions, remain sparse. In this paper, we provide a theoretical guarantee of generating data distribution by a progressive flow model, the so-called JKO flow model, which implements the Jordan-Kinderleherer-Otto (JKO) scheme in a normalizing flow network. Leveraging the exponential convergence of the proximal gradient descent (GD) in Wasserstein space, we prove the Kullback-Leibler (KL) guarantee of data generation by a JKO flow model to be $O(\varepsilon^2)$ when using $N \lesssim \log (1/\varepsilon)$ many JKO steps ($N$ Residual Blocks in the flow) where $\varepsilon $ is the error in the per-step first-order condition. The assumption on data density is merely a finite second moment, and the theory extends to data distributions without density and when there are inversion errors in the reverse process where we obtain KL-$W_2$ mixed error guarantees. The non-asymptotic convergence rate of the JKO-type $W_2$-proximal GD is proved for a general class of convex objective functionals that includes the KL divergence as a special case, which can be of independent interest. The analysis framework can extend to other first-order Wasserstein optimization schemes applied to flow-based generative models.

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