  When learning to ride a bike, a child falls down a number of times before
achieving the first success. As falling down usually has only mild
consequences, it can be seen as a tolerable failure in exchange for a faster
learning process, as it provides rich information about an undesired behavior.
In the context of Bayesian optimization under unknown constraints (BOC),
typical strategies for safe learning explore conservatively and avoid failures
by all means. On the other side of the spectrum, non conservative BOC
algorithms that allow failing may fail an unbounded number of times before
reaching the optimum. In this work, we propose a novel decision maker grounded
in control theory that controls the amount of risk we allow in the search as a
function of a given budget of failures. Empirical validation shows that our
algorithm uses the failures budget more efficiently in a variety of
optimization experiments, and generally achieves lower regret, than
state-of-the-art methods. In addition, we propose an original algorithm for
unconstrained Bayesian optimization inspired by the notion of excursion sets in
stochastic processes, upon which the failures-aware algorithm is built.
