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

Matteo Giordano · Kolyan Ray · Johannes Schmidt-Hieber

Hall J (level 1) #722

Keywords: [ posterior contraction ] [ large-sample asymptotics ] [ minimax estimation ] [ Gaussian Processes ] [ Bayesian Nonparametrics ]

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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