Ordinal Optimisation for Continuous Problems under Gaussian Noise

Ordinal optimisation is a probabilistic approach for tackling difficult noisy search problems, and in its original form, was formulated for search on finite sets. We formally extend the method of ordinal optimisation to problems where search is done over uncountable sets, such as subsets of $\mathbb{R}^{d}$. One advantage of doing so is to address the issue of misrepresented success probabilities in smaller sample sizes. In our formulation, we provide distribution-free properties of the ordinal optimisation success probability, and also develop new methods for computing the aforementioned quantity. We also derive an efficient formula for approximating the success probability under normality assumptions, which is proven to lower bound the actual success probability under reasonable conditions. It is also shown that this approximation formula is applicable in particular relaxed settings where the distributions need not necessarily be Gaussian. Lastly, we demonstrate the utility of the newly developed results by using it to solve two applied problems: 1) a variant of the secretary problem; and 2) controller tuning by simulation-based optimisation.