Papers › Convergence of flow-based generative models via proximal gradient descent in Wasserstein space
Convergence of flow-based generative models via proximal gradient descent in Wasserstein space
Xiuyuan Cheng, Jianfeng Lu, Yixin Tan, Yao Xie
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(ε²) when using N ≲log(1/ε) many JKO steps (N Residual Blocks in the flow) where ε 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-W₂ mixed error guarantees. The non-asymptotic convergence rate of the JKO-type W₂-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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