Bayesian Optimization with Unknown Constraints (1403.5607v1)

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.

• The paper introduces a probabilistic framework that models unknown constraints as confidence-level statements in costly black-box optimization.
• It employs a decoupled evaluation method that independently assesses objective and constraint functions to optimize resource usage.
• A novel constraint-weighted acquisition function combines expected improvement with feasibility probabilities to effectively guide the search.

Bayesian Optimization with Unknown Constraints

The paper "Bayesian Optimization with Unknown Constraints," authored by Michael A. Gelbart, Jasper Snoek, and Ryan P. Adams, presents a comprehensive framework for handling constrained optimization problems using Bayesian methods. This work addresses a crucial need in optimization where objective and constraint evaluations are expensive and possibly noisy, extending the applicability of Bayesian optimization to a broader class of scientific and engineering problems.

Framework Overview

Bayesian optimization is a probabilistic model-based approach suitable for optimizing black-box functions that are costly to evaluate. When constraints that are also black-box and potentially noisy are part of the problem, traditional optimization methods struggle. This paper proposes a strategy that models these constraints probabilistically, allowing them to be satisfied with a specified confidence level. The novelty here lies in extending the existing Bayesian framework to handle not only multiple constraints but also the cases where objective and constraint evaluations can be independently decoupled.

Key Contributions

1. Probabilistic Constraints: The paper reformulates constraints as probabilistic statements, allowing the user to specify confidence levels for each constraint individually, thereby managing uncertainty effectively. This formulation provides flexibility—particularly relevant in scenarios with noisy observation data.
2. Decoupled Evaluation: The authors introduce a method for independently evaluating the objective and constraint functions, which optimizes resource usage by preventing unnecessary constraint evaluations when the objective evaluation is unfavorable, and vice-versa. This decoupling is particularly beneficial for cases where some evaluations are significantly more expensive than others.
3. Constraint-Weighted Acquisition Function: They propose a new acquisition function, which combines expected improvement with the likelihood of satisfying the constraints, thus directing the optimization process towards not only improving objective scores but also aligning within feasible regions.
4. Incorporation of Cost Information: The methodology incorporates the relative costs of evaluating different constraints, enabling the optimization process to consider the expense of evaluations alongside traditional criteria like expected improvement.

Numerical Results and Implications

The authors validate their approach on several complex, real-world optimization tasks:

• Online LDA with Sparse Topics: They optimize hyperparameters of an LDA model subject to a sparsity constraint. Results demonstrate the effectiveness of taking constraint satisfaction into account, yielding better-performing configurations than unconstrained optimization.
• Memory-Constrained Neural Network Tuning: In tuning a neural network for the MNIST dataset within a memory constraint, the proposed method outperformed traditional approaches by efficiently exploring parameter spaces while respecting feasibility constraints.
• Hamiltonian Monte Carlo Tuning: The framework is employed to tune parameters for Hamiltonian Monte Carlo under convergence diagnostics constraints, significantly improving upon baseline configurations by enhancing the number of effective samples.

These results confirm the strength of the proposed framework in achieving optimal solutions while respecting complex constraint conditions.

Future Directions

The generality and flexibility of the proposed framework suggest several promising areas for future research. One is the extension to multi-objective optimization scenarios where multiple trade-offs are considered simultaneously. Additionally, integrating more advanced machine learning models beyond Gaussian processes—such as those capable of handling larger datasets more efficiently—could enhance scalability and model expressiveness.

The approach's potential extends beyond typical machine learning applications and into fields like real-time system optimization and automated experimental design, where constraints are often inherent, and evaluations are costly. Thus, these advanced methods can be particularly impactful in domains such as robotics, aerospace, and chemical engineering, where optimization is frequently constrained by physical, economic, or temporal limitations.

By presenting a robust framework that encompasses the intricacies of both noise and constraints in real-world problems, this work significantly advances the scope of Bayesian optimization, providing both theoretical insights and practical tools for constrained optimization.