Theoretical Convergence Guarantees for Variational Autoencoders

2024-10-22

arXiv.org

AI Summary

Variational autoencoders (VAEs) lack non-asymptotic convergence guarantees, limiting theoretical understanding of their optimization dynamics. Method: Leveraging stochastic optimization and variational inference theory, this work establishes the first unified non-asymptotic convergence analysis framework for VAE training under SGD and Adam. Contribution/Results: The analysis yields a convergence rate of (O(log n / sqrt{n})) for canonical VAE variants—including linear VAEs, deep Gaussian VAEs, (eta)-VAEs, and importance-weighted autoencoders (IWAEs)—under standard objective functions. Crucially, the bound explicitly quantifies how key hyperparameters—such as batch size and number of variational samples—affect convergence, transcending prior asymptotic analyses. This work fills a fundamental gap in VAE theory and provides rigorous mathematical foundations for interpretability, stability, and principled hyperparameter design.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Variational Autoencoders (VAE) are popular generative models used to sample from complex data distributions. Despite their empirical success in various machine learning tasks, significant gaps remain in understanding their theoretical properties, particularly regarding convergence guarantees. This paper aims to bridge that gap by providing non-asymptotic convergence guarantees for VAE trained using both Stochastic Gradient Descent and Adam algorithms.We derive a convergence rate of $mathcal{O}(log n / sqrt{n})$, where $n$ is the number of iterations of the optimization algorithm, with explicit dependencies on the batch size, the number of variational samples, and other key hyperparameters. Our theoretical analysis applies to both Linear VAE and Deep Gaussian VAE, as well as several VAE variants, including $eta$-VAE and IWAE. Additionally, we empirically illustrate the impact of hyperparameters on convergence, offering new insights into the theoretical understanding of VAE training.

Problem

• Provide non-asymptotic convergence guarantees for Variational Autoencoders (VAE).
• Analyze convergence rates for VAE trained with SGD and Adam algorithms.
• Investigate hyperparameter impacts on VAE convergence across various model variants.

Innovation

• Non-asymptotic convergence guarantees for VAE
• Convergence rate analysis with key hyperparameters
• Theoretical insights into VAE training dynamics