Uncertainty quantification using martingales for misspecified Gaussian processes
This addresses robustness issues in uncertainty quantification for practitioners using GPs in BO, though it is incremental as it builds on existing GP frameworks with a new statistical approach.
The paper tackles the problem of unreliable uncertainty estimates in Gaussian processes (GPs) when priors are misspecified, particularly in Bayesian Optimization (BO), by developing a frequentist confidence sequence using martingales that ensures statistical validity and improves coverage and utility compared to standard GP methods.
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.