[Paper Review] Bayesian Optimization with Unknown Constraints

Michael A. Gelbart, Jasper Snoek+1 more|arXiv (Cornell University)|Mar 22, 2014

Machine Learning and Algorithms18 references289 citations

TL;DR

This paper proposes a Bayesian optimization framework for constrained black-box problems where constraints are unknown, noisy, and independently evaluable. It introduces a probabilistic acquisition function that balances exploration, exploitation, and constraint satisfaction, demonstrating superior performance in tuning machine learning algorithms and HMC parameters under strict convergence and efficiency constraints.

ABSTRACT

Recent work on Bayesian optimization has shown its effectiveness in global optimization of difficult black-box objective functions. Many real-world optimization problems of interest also have constraints which are unknown a priori. In this paper, we study Bayesian optimization for constrained problems in the general case that noise may be present in the constraint functions, and the objective and constraints may be evaluated independently. We provide motivating practical examples, and present a general framework to solve such problems. We demonstrate the effectiveness of our approach on optimizing the performance of online latent Dirichlet allocation subject to topic sparsity constraints, tuning a neural network given test-time memory constraints, and optimizing Hamiltonian Monte Carlo to achieve maximal effectiveness in a fixed time, subject to passing standard convergence diagnostics.

Motivation & Objective

• To address constrained optimization problems where both the objective and constraints are expensive-to-evaluate, black-box functions with unknown, potentially noisy constraint functions.
• To develop a general framework that handles probabilistic constraints, allowing users to specify confidence levels (δ) for constraint violations.
• To enable independent evaluation of objective and constraint functions, which is common in real-world applications like product design and machine learning hyperparameter tuning.
• To extend Bayesian optimization beyond unconstrained settings to support practical, real-world optimization under complex feasibility and performance constraints.
• To demonstrate the method's effectiveness on real-world problems such as tuning HMC, neural networks, and online LDA with sparsity and memory constraints.

Proposed method

• Formulates constrained Bayesian optimization using probabilistic constraints, where constraints are modeled as stochastic processes with user-defined confidence levels (δ).
• Proposes a novel acquisition function that extends Expected Improvement to handle constraints by incorporating the probability of constraint satisfaction into the optimization objective.
• Uses Gaussian processes to model both the objective and constraint functions, enabling uncertainty-aware selection of promising evaluation points.
• Integrates independent evaluations of objective and constraint functions, allowing flexible and realistic modeling of real-world optimization pipelines.
• Employs a joint optimization strategy that maximizes expected improvement while ensuring high probability of feasibility, using analytical approximations for tractable computation.
• Applies the framework iteratively: update posterior beliefs after each evaluation, recompute acquisition function, and select next point to evaluate based on the combined objective and constraint utility.

Experimental results

Research questions

• RQ1 How can Bayesian optimization be extended to handle unknown, noisy constraints in black-box optimization problems?
• RQ2 What acquisition function formulation enables effective exploration of the feasible region while balancing objective improvement and constraint satisfaction?
• RQ3 Can the proposed method efficiently optimize complex systems like HMC and neural networks under strict performance and resource constraints?
• RQ4 How does the framework handle independent evaluations of objective and constraint functions, and what advantages does this offer over joint evaluation?
• RQ5 To what extent can the method reduce the number of expensive evaluations required to find high-quality, feasible solutions in real-world applications?

Key findings

• The proposed method successfully optimized Hamiltonian Monte Carlo parameters, achieving 12.5 effective samples per second with 100% constraint satisfaction, outperforming a baseline configuration.
• In HMC tuning, the method selected only τ=2 leapfrog steps and 3.8% burn-in time, significantly reducing computation time while maintaining high effective sample size.
• The optimization achieved a 70% acceptance rate, in contrast to the baseline’s 85%, demonstrating that lower acceptance rates can be acceptable when combined with fewer steps and higher efficiency.
• The framework discovered that numerical integration in HMC diverges for ε > 0.1, which was visualized in constraint surface plots, confirming the method’s ability to identify infeasible regions.
• For online LDA, the method optimized topic sparsity constraints effectively, showing applicability to natural language processing and model regularization.
• The approach demonstrated robustness across diverse problems, including memory-constrained neural network tuning and real-time system optimization, with consistent constraint satisfaction and improved performance.