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 $n$.
翻译:我们严格地证明,如果目标函数具有构成结构,深高斯进程前期的功能可能优于高斯进程前期的功能。 为此,我们用连续回归模型研究高斯进程后退率的信息理论下限。 我们显示,如果真正的功能是一个普遍添加功能,那么基于任何平均零高斯进程的后端只能以一个在抽样规模上是多亚性不最优的系数(美元),以严格低于最小速率的速率来恢复真相。