A piece-wise constant approximation for non-conjugate Gaussian Process models

TL;DR

This paper introduces a piece-wise constant approximation method for non-conjugate Gaussian Process models, enabling closed-form solutions and data-driven learning of inverse-link functions in Bayesian machine learning.

Contribution

It proposes a novel piece-wise constant approximation for inverse-link functions in non-conjugate GPs, allowing closed-form variational bounds and learnable inverse-link functions from data.

Findings

Closed-form SVGP lower bound with the approximation

Learnable inverse-link functions tailored to data

Enhanced applicability to non-Gaussian likelihoods

Abstract

Gaussian Processes (GPs) are a versatile and popular method in Bayesian Machine Learning. A common modification are Sparse Variational Gaussian Processes (SVGPs) which are well suited to deal with large datasets. While GPs allow to elegantly deal with Gaussian-distributed target variables in closed form, their applicability can be extended to non-Gaussian data as well. These extensions are usually impossible to treat in closed form and hence require approximate solutions. This paper proposes to approximate the inverse-link function, which is necessary when working with non-Gaussian likelihoods, by a piece-wise constant function. It will be shown that this yields a closed form solution for the corresponding SVGP lower bound. In addition, it is demonstrated how the piece-wise constant function itself can be optimized, resulting in an inverse-link function that can be learnt from the data at hand.