Characterizing Deep Gaussian Processes via Nonlinear Recurrence Systems

TL;DR

This paper analyzes the expressive limitations of deep Gaussian processes by modeling their layer interactions as nonlinear dynamic systems, extending previous work to various kernels and providing theoretical bounds and experimental validation.

Contribution

It introduces a novel dynamic systems perspective to understand DGPs, extending analysis beyond squared exponential kernels and deriving convergence bounds.

Findings

Recurrence relations between layers are analytically derived.

Tighter bounds and convergence rates for different kernels are established.

Experimental results support the theoretical analysis.

Abstract

Recent advances in Deep Gaussian Processes (DGPs) show the potential to have more expressive representation than that of traditional Gaussian Processes (GPs). However, there exists a pathology of deep Gaussian processes that their learning capacities reduce significantly when the number of layers increases. In this paper, we present a new analysis in DGPs by studying its corresponding nonlinear dynamic systems to explain the issue. Existing work reports the pathology for the squared exponential kernel function. We extend our investigation to four types of common stationary kernel functions. The recurrence relations between layers are analytically derived, providing a tighter bound and the rate of convergence of the dynamic systems. We demonstrate our finding with a number of experimental results.