  Bayesian optimisation is a popular approach for optimising expensive
black-box functions. The next location to be evaluated is selected via
maximising an acquisition function that balances exploitation and exploration.
Gaussian processes, the surrogate models of choice in Bayesian optimisation,
are often used with a constant prior mean function equal to the arithmetic mean
of the observed function values. We show that the rate of convergence can
depend sensitively on the choice of mean function. We empirically investigate 8
mean functions (constant functions equal to the arithmetic mean, minimum,
median and maximum of the observed function evaluations, linear, quadratic
polynomials, random forests and RBF networks), using 10 synthetic test problems
and two real-world problems, and using the Expected Improvement and Upper
Confidence Bound acquisition functions. We find that for design dimensions
$\ge5$ using a constant mean function equal to the worst observed quality value
is consistently the best choice on the synthetic problems considered. We argue
that this worst-observed-quality function promotes exploitation leading to more
rapid convergence. However, for the real-world tasks the more complex mean
functions capable of modelling the fitness landscape may be effective, although
there is no clearly optimum choice.
