Posterior contraction rates for constrained deep Gaussian processes in density estimation and classication

TL;DR

This paper establishes how quickly constrained deep Gaussian processes can accurately approximate densities and classifications, providing theoretical guarantees for their convergence rates under various constraints.

Contribution

It introduces a general framework for posterior contraction rates of constrained deep Gaussian processes, applicable to multiple stochastic process models and smoothness classes.

Findings

Derived contraction rates for integrated Brownian motions, Riemann-Liouville, and Matérn processes.

Reproduced known minimax rates for standard smoothness classes.

Provided a new concentration function accounting for constraints.

Abstract

We provide posterior contraction rates for constrained deep Gaussian processes in non-parametric density estimation and classication. The constraints are in the form of bounds on the values and on the derivatives of the Gaussian processes in the layers of the composition structure. The contraction rates are rst given in a general framework, in terms of a new concentration function that we introduce and that takes the constraints into account. Then, the general framework is applied to integrated Brownian motions, Riemann-Liouville processes, and Mat{\'e}rn processes and to standard smoothness classes of functions. In each of these examples, we can recover known minimax rates.