On the inability of Gaussian process regression to optimally learn compositional functions. (arXiv:2205.07764v2 [stat.ML] UPDATED)

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 …

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