Uncertainty quantification using martingales for misspecified Gaussian processes

Willie Neiswanger, Aaditya Ramdas
Proceedings of the 32nd International Conference on Algorithmic Learning Theory, PMLR 132:963-982, 2021.

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 \emph{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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v132-neiswanger21a, title = {Uncertainty quantification using martingales for misspecified Gaussian processes}, author = {Neiswanger, Willie and Ramdas, Aaditya}, booktitle = {Proceedings of the 32nd International Conference on Algorithmic Learning Theory}, pages = {963--982}, year = {2021}, editor = {Feldman, Vitaly and Ligett, Katrina and Sabato, Sivan}, volume = {132}, series = {Proceedings of Machine Learning Research}, month = {16--19 Mar}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v132/neiswanger21a/neiswanger21a.pdf}, url = {https://proceedings.mlr.press/v132/neiswanger21a.html}, 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 \emph{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.} }
Endnote
%0 Conference Paper %T Uncertainty quantification using martingales for misspecified Gaussian processes %A Willie Neiswanger %A Aaditya Ramdas %B Proceedings of the 32nd International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Vitaly Feldman %E Katrina Ligett %E Sivan Sabato %F pmlr-v132-neiswanger21a %I PMLR %P 963--982 %U https://proceedings.mlr.press/v132/neiswanger21a.html %V 132 %X 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 \emph{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.
APA
Neiswanger, W. & Ramdas, A.. (2021). Uncertainty quantification using martingales for misspecified Gaussian processes. Proceedings of the 32nd International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 132:963-982 Available from https://proceedings.mlr.press/v132/neiswanger21a.html.

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