  We consider the problem of sequentially maximising an unknown function over a
set of actions while ensuring that every sampled point has a function value
below a given safety threshold. We model the function using kernel-based and
Gaussian process methods, while differing from previous works in our assumption
that the function is monotonically increasing with respect to a \emph{safety
variable}. This assumption is motivated by various practical applications such
as adaptive clinical trial design and robotics. Taking inspiration from the
\textsc{\sffamily GP-UCB} and \textsc{\sffamily SafeOpt} algorithms, we propose
an algorithm, monotone safe {\sffamily UCB} (\textsc{\sffamily M-SafeUCB}) for
this task. We show that \textsc{\sffamily M-SafeUCB} enjoys theoretical
guarantees in terms of safety, a suitably-defined regret notion, and
approximately finding the entire safe boundary. In addition, we illustrate that
the monotonicity assumption yields significant benefits in terms of the
guarantees obtained, as well as algorithmic simplicity and efficiency. We
support our theoretical findings by performing empirical evaluations on a
variety of functions, including a simulated clinical trial experiment.
