Sharp Threshold Identified for Deep Gaussian Process Degeneracy, Revealing Non-Gaussian Limits
Researchers have established a sharp bandwidth threshold for deep Gaussian processes (GPs), above which the prior degenerates to constant functions and below which it converges to a non-trivial, non-Gaussian limit distribution. Deep GPs are layered probabilistic models used in Bayesian machine learning, and understanding their limiting behavior as depth increases is fundamental to their theoretical justification. The findings suggest that deep GP priors can admit meaningful, complex limit distributions under the right conditions, with implications for model design and prior selection.
A new preprint posted to arXiv investigates the limiting behavior of compositional Gaussian process (GP) priors as the number of layers grows in deep Bayesian models. The authors identify a sharp bandwidth threshold r_c(d) = Θ(√d) for the RBF kernel: above this threshold, the prior degenerates to the set of constant functions, strengthening previously known bounds; below it, the prior converges to a well-defined limit distribution. Crucially, these limit distributions are proven to be both non-degenerate and non-Gaussian, exhibiting non-vanishing dependence between coordinates — a qualitatively richer behavior than previously characterized regimes. Empirical experiments verify the threshold across a range of input dimensions d and reveal complex multimodal structure in the limit distributions, a regime that becomes increasingly narrow as d grows and would be difficult to detect without knowledge of the threshold. These results provide a more complete theoretical picture of deep GP priors and may inform practical choices of kernel bandwidth in deep Bayesian architectures.
What's missing
The results currently focus on the RBF kernel; whether analogous thresholds and non-Gaussian limits arise for other common kernels remains an open question. The practical downstream impact on posterior inference and predictive performance in real deep GP models is not addressed.
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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