  We investigate the unconstrained global optimization of functions with low
effective dimensionality, that are constant along certain (unknown) linear
subspaces. Extending the technique of random subspace embeddings in [Wang et
al., Bayesian optimization in a billion dimensions via random embeddings. JAIR,
55(1): 361--387, 2016], we study a generic Random Embeddings for Global
Optimization (REGO) framework that is compatible with any global minimization
algorithm. Instead of the original, potentially large-scale optimization
problem, within REGO, a Gaussian random, low-dimensional problem with bound
constraints is formulated and solved in a reduced space. We provide novel
probabilistic bounds for the success of REGO in solving the original, low
effective-dimensionality problem, which show its independence of the
(potentially large) ambient dimension and its precise dependence on the
dimensions of the effective and randomly embedding subspaces. These results
significantly improve existing theoretical analyses by providing the exact
distribution of a reduced minimizer and its Euclidean norm and by the general
assumptions required on the problem. We validate our theoretical findings by
extensive numerical testing of REGO with three types of global optimization
solvers, illustrating the improved scalability of REGO compared to the
full-dimensional application of the respective solvers.
