Title: On the choice of the low-dimensional domain for global optimization via random embeddings,
Abstract: The challenge of taking many variables into account in optimization problems
may be overcome under the hypothesis of low effective dimensionality. Then, the
search of solutions can be reduced to the random embedding of a low dimensional
space into the original one, resulting in a more manageable optimization
problem. Specifically, in the case of time consuming black-box functions and
when the budget of evaluations is severely limited, global optimization with
random embeddings appears as a sound alternative to random search. Yet, in the
case of box constraints on the native variables, defining suitable bounds on a
low dimensional domain appears to be complex. Indeed, a small search domain
does not guarantee to find a solution even under restrictive hypotheses about
the function, while a larger one may slow down convergence dramatically. Here
we tackle the issue of low-dimensional domain selection based on a detailed
study of the properties of the random embedding, giving insight on the
aforementioned difficulties. In particular, we describe a minimal
low-dimensional set in correspondence with the embedded search space. We
additionally show that an alternative equivalent embedding procedure yields
simultaneously a simpler definition of the low-dimensional minimal set and
better properties in practice. Finally, the performance and robustness gains of
the proposed enhancements for Bayesian optimization are illustrated on
numerical examples.