Information-Theoretic Generalization Bounds for VAEs: A Role of Encoder and Latent Variable

Despite their remarkable success, a rigorous theoretical understanding of how latent variables (LVs) govern the generalization performance of Variational Autoencoders (VAEs) remains largely elusive. Existing theoretical analyses are confined to supervised learning or models with discrete latent spaces, leaving their role in standard VAEs with continuous LVs poorly understood. This paper establishes the first information-theoretic analysis for VAEs by adapting a theoretical framework from supervised learning---the leave-one-out conditional mutual information framework---to the unsupervised, continuous latent space of these models. Our analysis reveals that their generalization error is bounded solely by the information complexity of the encoder and LVs, independent of the decoder. The versatility of our framework is demonstrated through its extension to both hierarchical VAEs, for which we provide layer-wise bounds, and data generation, where we link our information-theoretic principles to a novel bound on the 2-Wasserstein distance between true and generated distributions.

Many AI systems learn by compressing complex data, such as images, into a smaller hidden code and then using that code to reconstruct or generate data. Variational autoencoders (VAEs) are a common example, but it has been unclear why they sometimes work well on new data rather than merely remembering the examples they saw during training. This paper gives a mathematical explanation of that behavior. It shows that the difference between training performance and performance on new data is controlled mainly by two factors: how much example-specific detail is kept in the hidden code, and how strongly the code-making part of the model adapts to the training set. This explanation does not depend directly on how complex the data-reconstruction part of the model is. The paper also extends the explanation to hierarchical VAEs, showing how deeper levels can gradually compress information. Finally, it relates generation quality to two quantities already used during training: how accurately the model reconstructs examples and how much it prevents the hidden code from becoming unnecessarily complicated. These results help clarify what makes such generative models reliable.