On the inability of Gaussian process regression to optimally learn compositional functions

• URL: http://arxiv.org/abs/2205.07764v1
• Date: Mon, 16 May 2022 15:42:25 GMT
• Title: On the inability of Gaussian process regression to optimally learn compositional functions
• Authors: Matteo Giordano and Kolyan Ray and Johannes Schmidt-Hieber
• Abstract summary: Deep Gaussian process priors can outperform Gaussian process priors if the target function has a compositional structure. We show that if the true function is a generalized additive function, then the posterior based on any mean-zero Gaussian process can only recover the truth at a rate that is strictly slower than the minimax rate.
• Score: 3.6525095710982916
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We rigorously prove that deep Gaussian process priors can outperform Gaussian process priors if the target function has a compositional structure. To this end, we study information-theoretic lower bounds for posterior contraction rates for Gaussian process regression in a continuous regression model. We show that if the true function is a generalized additive function, then the posterior based on any mean-zero Gaussian process can only recover the truth at a rate that is strictly slower than the minimax rate by a factor that is polynomially suboptimal in the sample size $n$.

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