Sharp Threshold Identified for Deep Gaussian Process Degeneracy, Revealing Non-Gaussian Limits
Researchers studying compositional Gaussian processes (GPs) have identified a sharp bandwidth threshold r_c(d) = Θ(√d) that determines whether deep GP priors degenerate or converge to meaningful limit distributions. Previous work showed that GPs degenerate under certain conditions, but this study proves that below the threshold, priors converge to non-degenerate, non-Gaussian distributions with preserved coordinate dependence. This finding advances theoretical understanding of deep Bayesian models and suggests practical regimes where deep GPs can serve as useful probabilistic models.
In studying layered functions in deep Bayesian models, researchers examined compositional Gaussian process priors where each layer is vector-valued. Prior work established that RBF kernel-based GPs degenerate (converge to constant functions) under certain bandwidth conditions, limiting their utility. This paper identifies a precise bandwidth threshold r_c(d) = Θ(√d) above which degeneracy occurs, and crucially demonstrates that below this threshold, the prior converges to non-degenerate, non-Gaussian limit distributions π_Z̄ with non-vanishing coordinate dependence. The authors verified this threshold empirically across multiple dimensions and observed complex multimodal behavior in the limit distributions—a regime that narrows with increasing dimensionality. These results suggest that deep Gaussian process priors can admit non-trivial limits in specific parameter regimes, potentially enabling their use as meaningful probabilistic models in deep learning architectures.
What's missing
The paper does not discuss computational implications or practical applications of these theoretical findings for practitioners using deep GPs. Additionally, the relationship between these results and recent work on neural network Gaussian process limits in other regimes is not detailed in the abstract.
What different sources said
- arXiv cs.AICenter
How Deep Are Deep GPs, Really? A Sharp Threshold and a Non-Gaussian Limit for Compositional GPs
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