Dimensionality reduction for global optimization : Beyond low effective dimension

Details

Supervisors

Massart, Estelle

Faculty

Ecole polytechnique de Louvain

Degree label

Master [120] : ingénieur civil en mathématiques appliquées, à finalité spécialisée

Abstract

enMany fields of engineering require the global minimization of objective functions defined over high-dimensional domains. To ensure tractability, it is crucial to develop optimization methods that exploit structural properties of these functions to accelerate computations. The baseline of this work concerns functions with low effective dimension, which depend only on a linear subspace of the full domain, referred to as the effective subspace. The objective of this thesis is to study generalizations of this concept. The first part of this work focuses on functions with approximate low effective dimension, where deterministic variations remain small in the orthogonal complement of the effective subspace. Theoretical results are established to characterize this extension, and existing algorithms are adapted accordingly. Numerical experiments are performed and demonstrate the competitiveness of the resulting extension. The second part addresses the problem of identifying the effective subspace of functions with low effective dimension in scenarios where only noisy evaluations of the objective function are available. A Riemannian optimization framework is adopted for this purpose. This approach improves estimation accuracy and is more robust to noise, although it incurs higher computational costs and reduced scalability with increasing ambient dimension.