Derivation of Output Correlation Inferences for Multi-Output (aka Multi-Task) Gaussian Process

TL;DR

This paper provides clear derivations of multi-task Gaussian process formulations and their gradients, enhancing understanding for applications like Bayesian optimization involving multiple outputs.

Contribution

It offers accessible derivations of multi-task GP formulations and gradients, clarifying complex prior literature for practical use.

Findings

Provides detailed derivations of MTGP formulations

Clarifies gradient computations for MTGP

Facilitates application of MTGP in Bayesian optimization

Abstract

Gaussian process (GP) is arguably one of the most widely used machine learning algorithms in practice. One of its prominent applications is Bayesian optimization (BO). Although the vanilla GP itself is already a powerful tool for BO, it is often beneficial to be able to consider the dependencies of multiple outputs. To do so, Multi-task GP (MTGP) is formulated, but it is not trivial to fully understand the derivations of its formulations and their gradients from the previous literature. This paper serves friendly derivations of the MTGP formulations and their gradients.