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In this paper, we study the problem of Gaussian process (GP) bandits underrelaxed optimization criteria stating that any function value above a certainthreshold is "good enough". On the theoretical side, we study various {\emlenient regret} notions in which all near-optimal actions incur zero penalty,and provide upper bounds on the lenient regret for GP-UCB and an eliminationalgorithm, circumventing the usual $O(\sqrt{T})$ term (with time horizon $T$)resulting from zooming extremely close towards the function maximum. Inaddition, we complement these upper bounds with algorithm-independent lowerbounds. On the practical side, we consider the problem of finding a single"good action" according to a known pre-specified threshold, and introduceseveral good-action identification algorithms that exploit knowledge of thethreshold. We experimentally find that such algorithms can often find a goodaction faster than standard optimization-based approaches.

Lenient Regret and Good-Action Identification in Gaussian Process Bandits