  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 {\em
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 $O(\sqrt{T})$ term (with time horizon $T$)
resulting from zooming extremely close towards the function maximum. In
addition, we complement these upper bounds with algorithm-independent 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.
