Improving exploration strategies in large dimensions and rate of convergence of global random search algorithms

TL;DR

This paper investigates the efficiency of global random search algorithms in high-dimensional spaces, revealing slower convergence than classical estimates and inefficiencies in common exploration schemes for non-asymptotic regimes.

Contribution

It provides new insights into the actual convergence rates of global random search algorithms and critiques standard exploration strategies in high-dimensional optimization.

Findings

Convergence of algorithms is slower than classical asymptotic estimates suggest.

Uniform sampling on subsets of the feasible region is more efficient than on the entire region.

Replacing random points with low-discrepancy sequences has negligible effect.

Abstract

We consider global optimization problems, where the feasible region $\X$ is a compact subset of $\mathbb{R}^d$ with $d \geq 10$. For these problems, we demonstrate the following. First: the actual convergence of global random search algorithms is much slower than that given by the classical estimates, based on the asymptotic properties of random points. Second: the usually recommended space exploration schemes are inefficient in the non-asymptotic regime. Specifically, (a) uniform sampling on entire~$\X$ is much less efficient than uniform sampling on a suitable subset of $\X$, and (b) the effect of replacement of random points by low-discrepancy sequences is negligible.