  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
\emph{\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.
