Abstract
Abstract We investigate the unconstrained global optimization of functions with low effective dimensionality, which are constant along certain (unknown) linear subspaces. Extending the technique of random subspace embeddings in Wang et al. (2016, J. Artificial Intelligence Res., 55, 361–387), 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 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 with the full-dimensional application of the respective solvers.
Highlights
In this paper, we address the unconstrained global optimization problem min f (x), x∈RD (P)where f : RD → R is a real-valued continuous, possibly non-convex, deterministic, function defined on the whole RD
We investigate a general random embeddings framework for unconstrained global optimization of functions with low effective dimensionality, where we allow the effective subspace of the objective function and its dimension to be arbitrary
We describe how to extend the main results to affine random embeddings (which draw random subspaces at any chosen point in RD), which indicate that the probability of success of (RP) is higher if the point of reference is closer to the set of global minimizers
Summary
Alternating learning and optimization steps has been proposed [22], as well as bypassing learning and directly optimizing in randomly chosen lowdimensional subspaces (provided an estimate of the effective dimension is known) [4, 5, 49] For the latter, Wang et al [49] developed the so-called REMBO algorithm, which is a BO framework for problem (P) with box constraints x ∈ X that uses Gaussian random embeddings (namely, A is a Gaussian random matrix) to generate the reduced problem (RP). We investigate a general random embeddings framework for unconstrained global optimization of functions with low effective dimensionality, where we allow the effective subspace of the objective function and its dimension (denoted by de) to be arbitrary (not necessarily aligned with coordinate axes and not limited in dimension by problem constants) This framework allows the use of any global solver to solve the reduced problem. We reserve the letter A to refer to a D × d Gaussian random matrix (see Definition A.1) and write χn to denote a chi-squared random variable with n degrees of freedom (see Definition A.2)
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