Variational Auto-Encoders (VAEs) have emerged as powerful probabilistic models for generative tasks. However, their convergence properties have not been rigorously proven. The challenge of proving convergence is inherently difficult due to the highly non-convex nature of the training objective and the implementation of a Stochastic Neural Network (SNN) within VAE architectures. This paper addresses these challenges by characterizing the optimization trajectory of SNNs utilized in VAEs through the lens of Neural Tangent Kernel (NTK) techniques. These techniques govern the optimization and generalization behaviors of ultra-wide neural networks. We provide a mathematical proof of VAE convergence under mild assumptions, thus advancing the theoretical understanding of VAE optimization dynamics. Furthermore, we establish a novel connection between the optimization problem faced by over-parameterized SNNs and the Kernel Ridge Regression (KRR) problem. Our findings not only contribute to the theoretical foundation of VAEs but also open new avenues for investigating the optimization of generative models using advanced kernel methods. Our theoretical claims are verified by experimental simulations.
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