Lenient Regret and Good-Action Identification in Gaussian Process Bandits

Selwyn Gomes, Jonathan Scarlett, Xu Cai

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Abstract

In this paper, we study the problem of Gaussian process (GP) bandits under relaxed optimization criteria stating that any function value above a certain threshold is "good enough". On the theoretical side, we study various lenient regret notions in which all near-optimal actions incur zero penalty, and provide upper bounds on the lenient regret for GP-UCB and an elimination algorithm, circumventing the usual O( √ T ) term (with time horizon T ) resulting from zooming extremely close towards the function maximum. In addition, we complement these upper bounds with algorithmindependent lower bounds. On the practical side, we consider the problem of finding a single "good action" according to a known pre-specified threshold, and introduce several good-action identification algorithms that exploit knowledge of the threshold. We experimentally find that such algorithms can often find a good action faster than standard optimization-based approaches.

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