On the choice of the low-dimensional domain for global optimization via random embeddings
This work tackles a specific technical bottleneck in optimization for time-consuming black-box functions with limited evaluations, offering incremental improvements to existing random embedding methods.
The paper addresses the challenge of selecting appropriate low-dimensional domain bounds for global optimization using random embeddings, particularly under box constraints, by analyzing embedding properties and proposing an alternative embedding procedure that simplifies domain definition and improves performance. The proposed enhancements are shown to yield performance and robustness gains in Bayesian optimization on numerical examples.
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.