** Talk 1: A theoretical comparison of deep learning and Bayes with deep Gaussian process priors **

Deep neural networks have received a lot of attention recently and considerable progress has been made to build an underlying mathematical foundation. In a first part of the talk, we summarise some statistical convergence results. Deep Gaussian process priors can be viewed as continuous analogues of Bayesian neural networks, and this raises the question whether there is a closer link with deep learning. In the second part of the talk, we show that the posterior for a suitable deep Gaussian process prior can achieve fast posterior contraction rates and discuss the connection with deep learning.

This is joint work with Gianluca Finocchio.

** Talk 2: On the inability of Gaussian process regression to optimally learn compositional functions **

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.

This is joint work with Kolyan Ray and Johannes Schmidt-Hieber.