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

TL;DR

This work develops an information-theoretic framework to analyze generalization in VQ-VAEs with discrete latent variables, revealing that generalization and data-generation performance depend primarily on the encoder and latent variables rather than the decoder. By introducing data-dependent priors for latent variables and a permutation symmetric supersample setting, the authors derive decoder-independent generalization bounds and show asymptotic convergence under appropriate regularization. They further bound data-generation quality via a 2-Wasserstein distance bound, linking reconstruction loss and latent-variable complexity to generation performance. Empirically, the results are supported by experiments showing decoder capacity has limited impact on generalization, while encoder/LV design and prior choice significantly influence outcomes. The work highlights practical guidance for regularizing encoders and designing LV priors to improve both reconstruction generalization and synthetic data fidelity in VQ-VAEs.

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.

Figures (8)

• Figure 1: Graphical models illustrating different dependency structures for LVs. The left panel shows the structure considered in the standard supersample setting (Theorem \ref{['naive_it_bound']}). The right panel depicts our proposed structure tailored for unsupervised learning. See Appendix \ref{['app_intuiton']} for further details.
• Figure 2: The behavior of the generalization gap on the MNIST dataset when increasing the number of residual blocks to enlarge the decoder dimension $d_\theta$ ($K=128$, $d_{z}=64$). See Appendix \ref{['app:exp_settings']} for detailed experimental settings.
• Figure 3: The behavior of the generalization gap and the two KL terms from Eq. \ref{['eq_reconstructon_bound1']} on the MNIST dataset ($K=128$, $d_{z}=64$). The three leftmost panels show the asymptotic behavior of the generalization gap, the first KL term, and the second KL term as a function of sample size $n$. The two rightmost panels show scatter plots correlating the generalization gap with the first KL term (fourth panel) and the second KL term (fifth panel). In these plots, the color indicates the number of decoder Residual Blocks (RB=2, 3, 4, or 5) and the marker shape indicates the sample size $n$. (Circle for $n=250$, Square for $n=1000$, Diamond for $n=2000$, and Triangle for $n=4000$).
• Figure 4: Graphical models illustrating the different dependency structures of the random variables considered in the basic IT analysis and in this study. The left figure represents the dependency structure in the basic IT analysis, which simply evaluates the loss function in supervised learning settings, whereas the right figure corresponds to our analysis in the unsupervised learning setting.
• Figure 5: Behavior of the generalization gap and the empirical KL term ($\mathrm{KL}(\mathbf{Q}_{\mathbf{J},U}\|\mathbf{P})/n$) on the CIFAR-10 dataset ($K=128, d_z=64$). The top row shows their asymptotic behavior as a function of sample size $n$. The bottom row shows their behavior as the decoder complexity (number of residual blocks) is increased (for $n=20000$).

Theorems & Definitions (18)

• Theorem 1: hellstrom2022a
• Theorem 2
• Remark 1
• Lemma 1
• Lemma 2
• Lemma 3
• Remark 2
• Theorem 3
• Remark 3
• Theorem 4