Examples of inconsistency in optimization by expected improvement

TL;DR

This paper demonstrates that in 1D Expected Improvement optimization with certain Gaussian process kernels, the optimization trajectory can fail to be dense and may not converge to the true optimum, highlighting limitations of this method.

Contribution

It provides specific examples showing that expected improvement optimization can fail to explore the entire space and may not find the global optimum under certain conditions.

Findings

Optimization trajectories can be non-dense in the design space.

Expected improvement may not converge to the true optimum.

Gaussian kernels can lead to non-convergent optimization for smooth functions.

Abstract

We consider the 1D Expected Improvement optimization based on Gaussian processes having spectral densities converging to zero faster than exponentially. We give examples of problems where the optimization trajectory is not dense in the design space. In particular, we prove that for Gaussian kernels there exist smooth objective functions for which the optimization does not converge on the optimum.