Information-theoretic Generalization Analysis for VQ-VAEs: A Role of Latent Variables

TL;DR

This paper extends information-theoretic analysis to VQ-VAEs, deriving bounds on generalization error and data distribution divergence, highlighting the role of latent variables in unsupervised learning.

Contribution

It introduces a novel data-dependent prior and derives new bounds linking latent variables, generalization, and data generation in VQ-VAEs.

Findings

Generalization error bound depends on latent variables and encoder complexity.

Upper bound on Wasserstein distance explains the impact of LV regularization.

Latent variables significantly influence data generation quality.

Abstract

Latent variables (LVs) play a crucial role in encoder-decoder models by enabling effective data compression, prediction, and generation. Although their theoretical properties, such as generalization, have been extensively studied in supervised learning, similar analyses for unsupervised models such as variational autoencoders (VAEs) remain insufficiently underexplored. In this work, we extend information-theoretic generalization analysis to vector-quantized (VQ) VAEs with discrete latent spaces, introducing a novel data-dependent prior to rigorously analyze the relationship among LVs, generalization, and data generation. We derive a novel generalization error bound of the reconstruction loss of VQ-VAEs, which depends solely on the complexity of LVs and the encoder, independent of the decoder. Additionally, we provide the upper bound of the 2-Wasserstein distance between the distributions of the true data and the generated data, explaining how the regularization of the LVs contributes to the data generation performance.