Optimistic Optimization of a Deterministic Function without the Knowledge of its Smoothness

We consider a global optimization problem of a deterministic function f in a semimetric space, given a finite budget of n evaluations. The function f is assumed to be locally smooth (around one of its global maxima) with respect to a semi-metric . We describe two algorithms based on optimistic exploration that use a hierarchical partitioning of the space at all scales. A first contribution is an algorithm, DOO, that requires the knowledge of . We report a finite-sample performance bound in terms of a measure of the quantity of near-optimal states. We then define a second algorithm, SOO, which does not require the knowledge of the semimetric under which f is smooth, and whose performance is almost as good as DOO optimally-fitted.