Three Costs of Amortizing Gaussian Process Inference with Neural Processes

• URL: http://arxiv.org/abs/2605.21798v1
• Date: Wed, 20 May 2026 22:48:23 GMT
• Title: Three Costs of Amortizing Gaussian Process Inference with Neural Processes
• Abstract summary: Neural processes amortize Gaussian process inference, replacing the exact $O(n3)$ posterior with a learned $O(n)$ map from context sets to predictive distributions.<n>For a class of latent neural processes, we bound the Kullback--Leibler (KL) divergence between the GP and LNPs, decomposing it into three interpretable sources.<n>These results characterize the costs of amortization within the analyzed class and yield architectural recommendations to predict variance from context locations alone in the GP-amortization regime.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Neural processes amortize Gaussian process inference, replacing the exact $O(n^3)$ posterior with a learned $O(n)$ map from context sets to predictive distributions. For a class of latent neural processes, we bound the Kullback--Leibler (KL) divergence between the GP and LNP predictives, decomposing it into three interpretable sources, namely label contamination as the neural process uses label values to estimate a quantity that is label-independent in the exact GP, an information bottleneck because the finite-dimensional representation cannot resolve the full context geometry, and amortization error from a single encoder network shared across all contexts. The bottleneck truncation term decays in the representation dimension $d$ as $O(e^{-cd^{2/d_x}})$ for squared-exponential kernels on $\mathbb{R}^{d_x}$ where $c > 0$ is a kernel-dependent constant and as $O(d^{-2ν/d_x})$ for Matérn-$ν$ kernels, directly linking architecture sizing to kernel smoothness and input dimension. The label contamination term is $O(1)$ in general, with only the observation-noise component decaying as $O(1/n)$, identifying a persistent cost of routing uncertainty estimation through a label-dependent representation. These results characterize the costs of amortization within the analyzed class and yield architectural recommendations to predict variance from context locations alone in the GP-amortization regime, and replace mean aggregation with second-order pooling to close the dominant amortization gap.

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