Convergence Rates of Constrained Expected Improvement

TL;DR

This paper establishes the theoretical convergence rates of the constrained expected improvement (CEI) algorithm in Bayesian optimization, providing bounds under RKHS and Gaussian process assumptions, and validates these with experiments.

Contribution

The paper provides the first theoretical convergence rate analysis of CEI in constrained Bayesian optimization under RKHS and GP assumptions.

Findings

CEI achieves convergence rates of O(t^{-1/2} log^{(d+1)/2}(t)) for squared exponential kernels.

CEI achieves convergence rates of O(t^{- u/(2 u+d)} log^{ u/(2 u+d)}(t)) for Matérn kernels.

Numerical experiments confirm the theoretical convergence bounds.

Abstract

Constrained Bayesian optimization (CBO) methods have seen significant success in black-box optimization with constraints. One of the most commonly used CBO methods is the constrained expected improvement (CEI) algorithm. CEI is a natural extension of expected improvement (EI) when constraints are incorporated. However, the theoretical convergence rate of CEI has not been established. In this work, we study the convergence rate of CEI by analyzing its simple regret upper bound. First, we show that when the objective function $f$ and constraint function $c$ are assumed to each lie in a reproducing kernel Hilbert space (RKHS), CEI achieves the convergence rates of $\mathcal{O} \left(t^{-\frac{1}{2}}\log^{\frac{d+1}{2}}(t) \right) \ \text{and }\ \mathcal{O}\left(t^{\frac{-\nu}{2\nu+d}} \log^{\frac{\nu}{2\nu+d}}(t)\right)$ for the commonly used squared exponential and Mat\'{e}rn kernels ($\nu>\frac{1}{2}$), respectively. Second, we show that when $f$ is assumed to be sampled from Gaussian processes (GPs), CEI achieves similar convergence rates with a high probability. Numerical experiments are performed to validate the theoretical analysis.