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Convergence Rates of Constrained Expected Improvement

Haowei Wang, Jingyi Wang, Zhongxiang Dai, Nai-Yuan Chiang, Szu Hui Ng, Cosmin G. Petra

TL;DR

The paper provides the first theoretical convergence-rate guarantees for the constrained expected improvement (CEI) acquisition in constrained Bayesian optimization. It analyzes simple regret upper bounds under both frequentist RKHS assumptions and Bayesian GP priors, deriving explicit rates for squared exponential and Matérn kernels, and shows these bounds can be sharpened using maximum information gain. The results demonstrate sublinear convergence of CEI to the best feasible solution, with high-probability bounds under the Bayesian setting. Numerical experiments on synthetic RKHS/GP problems and standard CBO benchmarks validate the theoretical findings and illustrate the practical behavior of CEI. These contributions offer theoretical reassurance for CEI's use in constrained black-box optimization and point toward future work on noisy settings and multi-constraint extensions.

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 $\mathcal{O} \left(t^{-\frac{1}{2}}\log^{\frac{d+1}{2}}(t) \right) \ \text{and }\ \mathcal{O}\left(t^{\frac{-ν}{2ν+d}} \log^{\fracν{2ν+d}}(t)\right)$ for the commonly used squared exponential and Matérn kernels ($ν>\frac{1}{2}$), 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.

Convergence Rates of Constrained Expected Improvement

TL;DR

The paper provides the first theoretical convergence-rate guarantees for the constrained expected improvement (CEI) acquisition in constrained Bayesian optimization. It analyzes simple regret upper bounds under both frequentist RKHS assumptions and Bayesian GP priors, deriving explicit rates for squared exponential and Matérn kernels, and shows these bounds can be sharpened using maximum information gain. The results demonstrate sublinear convergence of CEI to the best feasible solution, with high-probability bounds under the Bayesian setting. Numerical experiments on synthetic RKHS/GP problems and standard CBO benchmarks validate the theoretical findings and illustrate the practical behavior of CEI. These contributions offer theoretical reassurance for CEI's use in constrained black-box optimization and point toward future work on noisy settings and multi-constraint extensions.

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 and constraint function are assumed to each lie in a reproducing kernel Hilbert space (RKHS), CEI achieves the convergence rates of for the commonly used squared exponential and Matérn kernels (), respectively. Second, we show that when 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.
Paper Structure (23 sections, 23 theorems, 102 equations, 5 figures, 1 algorithm)

This paper contains 23 sections, 23 theorems, 102 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Gaussian process models for $f$ and $c$
  4. Constrained Expected Improvement
  5. Convergence rates of CEI
  6. Simple regret upper bound under frequentist assumptions
  7. Technical Challenges under Frequentist Assumptions.
  8. Sketch of Proof for Theorem \ref{['theorem:CEI-convg-nonoise']}.
  9. Improved simple regret upper bound under frequentist assumptions
  10. Sketch of Proof for Theorem \ref{['prop:EI-convg-rate-nonoise-gamma']}.
  11. Simple regret upper bound under Bayesian objective assumption
  12. Technical Challenges under Bayesian Assumptions.
  13. Sketch of Proof for Theorem \ref{['theorem:CEI-convg-noise']}.
  14. Numerical experiments
  15. Synthetic problems
  16. ...and 8 more sections

Key Result

Theorem 3.3

Under Assumption assp:rkhs, the CEI algorithm leads to the simple regret upper bound of for some $t_k\in [\frac{t}{2}-1,t]$, and $c_{\tau B}=\frac{\tau(B_f)}{\tau(-B_f)}$.

Figures (5)

  • Figure 1: The log-log plots for simple regret vs optimization iterations of CEI for the synthetic problems.
  • Figure 2: Simple regret of CEI for five test problems.
  • Figure 3: Contour plots for the objective function (left) and constraint function (right) for Problem 1. The infeasible region is marked on the plots. The global optimum is marked with a star sign.
  • Figure 4: Contour plots for the objective function (left) and the two constraint functions (middle and right) for Problem 2. The infeasible region is marked with black line on the objective contour. The global optimum is marked with a star sign.
  • Figure 5: Contour plots for the objective function (left) and constraint function (right) for Problem 5. The infeasible region is marked on the plots. The global optimum is marked with a star sign.

Theorems & Definitions (48)

  • Definition 3.1
  • Theorem 3.3
  • Remark 3.4: Constraint in the simple regret upper bound
  • Lemma 3.5: bull2011convergence
  • Corollary 3.6
  • Theorem 3.7
  • Remark 3.8: Improved rate of convergence
  • Theorem 3.10
  • Theorem 3.11
  • Remark 3.12: Comparison to the frequentist setting
  • ...and 38 more