A dimensionality reduction technique for unconstrained global optimization of functions with low effective dimensionality

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 stu...

Full description

Saved in:
Bibliographic Details
Published in:Information and Inference: A Journal of the IMA Vol. 11; no. 1; pp. 167 - 201
Main Authors: Cartis, Coralia, Otemissov, Adilet
Format: Journal Article
Language:English
Published: Oxford University Press 26.03.2022
Subjects:
random matrix theory
functions with low effective dimensionality
dimensionality reduction techniques
global optimization
ISSN:2049-8772
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2049-8772
DOI:10.1093/imaiai/iaab011