Uncertainty quantification using martingales for misspecified Gaussian processes

TL;DR

This paper introduces a frequentist method using martingales to create confidence sequences for Gaussian process functions, enhancing robustness in Bayesian Optimization when priors are misspecified.

Contribution

It proposes a novel martingale-based confidence sequence approach for GPs that does not rely on correct prior assumptions, improving robustness in Bayesian Optimization.

Findings

The confidence sequences are statistically valid even under prior misspecification.

The method empirically outperforms standard GP uncertainty estimates in coverage and utility.

Powered likelihoods further enhance robustness against model misspecification.

Abstract

We address uncertainty quantification for Gaussian processes (GPs) under misspecified priors, with an eye towards Bayesian Optimization (BO). GPs are widely used in BO because they easily enable exploration based on posterior uncertainty bands. However, this convenience comes at the cost of robustness: a typical function encountered in practice is unlikely to have been drawn from the data scientist's prior, in which case uncertainty estimates can be misleading, and the resulting exploration can be suboptimal. We present a frequentist approach to GP/BO uncertainty quantification. We utilize the GP framework as a working model, but do not assume correctness of the prior. We instead construct a confidence sequence (CS) for the unknown function using martingale techniques. There is a necessary cost to achieving robustness: if the prior was correct, posterior GP bands are narrower than our CS. Nevertheless, when the prior is wrong, our CS is statistically valid and empirically outperforms standard GP methods, in terms of both coverage and utility for BO. Additionally, we demonstrate that powered likelihoods provide robustness against model misspecification.