Derivative-free algorithms seek the minimum value of a given objective
function without using any derivative information. The performance of these
methods often worsen as the dimension increases, a phenomenon predicted by
their worst-case complexity guarantees. Nevertheless, recent algorithmic
proposals have shown that incorporating randomization into otherwise
deterministic frameworks could alleviate this effect for direct-search methods.
The best guarantees and practical performance are obtained when employing a
random vector and its negative, which amounts to drawing directions in a random
one-dimensional subspace. Unlike for other derivative-free schemes, however,
the properties of these subspaces have not been exploited.
In this paper, we study a generic direct-search algorithm in which the
polling directions are defined using random subspaces. Complexity guarantees
for such an approach are derived thanks to probabilistic properties related to
both the subspaces and the directions used within these subspaces. By
leveraging results on random subspace embeddings and sketching matrices, we
show that better complexity bounds are obtained for randomized instances of our
framework. A numerical investigation confirms the benefit of randomization,
particularly when done in subspaces, when solving problems of moderately large
dimension