A tutorial on Gaussian process regression: Modelling, exploring, and exploiting functions

Abstract

This tutorial introduces the reader to Gaussian process regression as an expressive tool to model, actively explore and exploit unknown functions.Gaussian process regression is a powerful, non-parametric Bayesian approach towards regression problems that can be utilized in exploration and exploitation scenarios. This tutorial aims to provide an accessible introduction to these techniques. We will introduce Gaussian processes which generate distributions over functions used for Bayesian non-parametric regression, and demonstrate their use in applications and didactic examples including simple regression problems, a demonstration of kernel-encoded prior assumptions and compositions, a pure exploration scenario within an optimal design framework, and a bandit-like exploration-exploitation scenario where the goal is to recommend movies. Beyond that, we describe a situation modelling risk-averse exploration in which an additional constraint (not to sample below a certain threshold) needs to be accounted for.Lastly, we summarize recent psychological experiments utilizing Gaussian processes. Software and literature pointers are also provided.

Generalization and Search in Risky Environments

“…We assume that generalization within spatially correlated multi‐armed bandits can be described as a function learning mechanism that learns a function mapping the spatial context of each arm to expectations of reward. We use Gaussian process regression (Rasmussen, ; Schulz, Speekenbrink, & Krause, ) as an expressive model of human function learning. Gaussian process regression is a non‐parametric Bayesian approach toward function learning which can perform generalization by making inductive inferences about unobserved outcomes.…”

“…A Gaussian process defines a distribution over functions (see Rasmussen, ; Schulz et al., , for an introduction). Let denote a function over input space (i.e., options or arms) that maps to real‐valued scalar outputs (i.e., rewards).…”

“…Given a learned representation of a function at time t , this knowledge can be used to choose the next input at time t + 1 in a close‐to‐rational way. This is done through a decision strategy that takes the predicted mean μ( x ) and uncertainty σ( x ) for each input and produces a criterion governing which input to choose next in order to balance exploration and exploitation (Brochu, Cora, & De Freitas, ; Schulz et al., ).…”

Generalization as diffusion: human function learning on graphs

“…More recently, a theory of function learning based on GP regression was proposed to unite both accounts (Lucas et al, 2015), because of its inherent duality as both a rule-based and a similarity-based model. GP regression is a non-parametric method for performing Bayesian function learning (Schulz, Speekenbrink, & Krause, 2018), has successfully described human behavior across a range of traditional function learning paradigms (Lucas et al, 2015), and can account for compositional inductive biases (e.g., combining periodic and long range trends; Schulz et al, 2017).…”

Optimization of de‐bittering process of mosambi (Citrus limetta) peel: Artificial neural network, Gaussian process regression and support vector machine modeling approach

“…The Gaussian process regression (GPR) models are a powerful nonparametric Bayesian approach to regression. It uses theoretically infinite number of parameters to understand the inherent function in the training data (Schulz, Speekenbrink, & Krause, 2018). These models are simpler and easier to understand and can help in understanding the underlying function describing the data (Chaurasia et al, 2018).…”