[Paper Review] Improved Max-value Entropy Search for Multi-objective Bayesian Optimization with Constraints

Daniel Fernández-Sánchez, Eduardo C. Garrido‐Merchán+1 more|arXiv (Cornell University)|Nov 2, 2020

Advanced Multi-Objective Optimization Algorithms4 citations

TL;DR

MESMOC+ is a novel acquisition function for constrained multi-objective Bayesian optimization that minimizes the entropy of the Pareto frontier in function space, enabling efficient, information-theoretic search with linear computational cost. It outperforms prior methods in entropy estimation accuracy and optimization performance, especially in decoupled evaluation settings where it dynamically selects both input locations and black-boxes to evaluate.

ABSTRACT

We present MESMOC+, an improved version of Max-value Entropy search for Multi-Objective Bayesian optimization with Constraints (MESMOC). MESMOC+ can be used to solve constrained multi-objective problems when the objectives and the constraints are expensive to evaluate. MESMOC+ works by minimizing the entropy of the solution of the optimization problem in function space, i.e., the Pareto frontier, to guide the search for the optimum. The cost of MESMOC+ is linear in the number of objectives and constraints. Furthermore, it is often significantly smaller than the cost of alternative methods based on minimizing the entropy of the Pareto set. The reason for this is that it is easier to approximate the required computations in MESMOC+. Moreover, MESMOC+'s acquisition function is expressed as the sum of one acquisition per each black-box (objective or constraint). Thus, it can be used in a decoupled evaluation setting in which one chooses not only the next input location to evaluate, but also which black-box to evaluate there. We compare MESMOC+ with related methods in synthetic and real optimization problems. These experiments show that the entropy estimation provided by MESMOC+ is more accurate than that of previous methods. This leads to better optimization results. MESMOC+ is also competitive with other information-based methods for constrained multi-objective Bayesian optimization, but it is significantly faster.

Motivation & Objective

• Address the challenge of optimizing multiple expensive-to-evaluate objectives under constraints using Bayesian optimization.
• Reduce the computational cost of entropy-based acquisition functions in multi-objective and constrained settings.
• Enable decoupled evaluation by allowing independent selection of input locations and black-boxes to evaluate at each iteration.
• Improve the accuracy of entropy estimation for the Pareto frontier compared to existing methods.
• Achieve better optimization performance with fewer evaluations while maintaining low per-iteration computational cost.

Proposed method

• Proposes an improved max-value entropy search (MES) framework tailored for multi-objective optimization with constraints (MESMOC+).
• Models objectives and constraints as Gaussian processes to estimate uncertainty in unexplored regions of the input space.
• Defines the acquisition function as the expected reduction in entropy of the Pareto frontier in function space.
• Expresses the acquisition function as a sum of individual contributions from each black-box (objective or constraint), enabling decoupled evaluation.
• Uses Monte Carlo sampling to approximate the intractable entropy reduction, with efficient computation linear in the number of objectives and constraints.
• Supports both coupled and decoupled evaluation modes: in the latter, it selects not only where to evaluate but also which black-box to evaluate at each point.

Experimental results

Research questions

• RQ1 Can entropy-based acquisition functions be efficiently extended to multi-objective optimization with constraints while maintaining low computational cost?
• RQ2 How does the accuracy of Pareto frontier entropy estimation in MESMOC+ compare to existing methods in constrained multi-objective settings?
• RQ3 Does the decoupled evaluation capability of MESMOC+ lead to improved optimization performance with the same number of evaluations?
• RQ4 How does MESMOC+ compare in efficiency and convergence speed to state-of-the-art information-theoretic methods for constrained multi-objective Bayesian optimization?
• RQ5 Can MESMOC+ achieve better hyper-volume of the Pareto front than baseline methods like PESMOC and RANDOM in real-world problems?

Key findings

• MESMOC+ achieves significantly more accurate entropy estimation than previous methods, leading to better optimization results.
• The decoupled variant of MESMOC+ outperforms the coupled version, achieving better hyper-volume of the Pareto front with the same number of evaluations.
• In the neural network hyperparameter tuning task, PESMOC${}_{\text{dec}}$ achieved the highest hyper-volume, followed closely by MESMOC+${}_{\text{dec}}$ after 100 evaluations.
• After 100 evaluations, the performance gap between MESMOC+${}_{\text{dec}}$ and PESMOC${}_{\text{dec}}$ was minimal, indicating strong convergence.
• MESMOC+${}_{\text{dec}}$ focused the majority of evaluations on the prediction error black-box, reflecting its higher optimization difficulty.
• MESMOC+ is significantly faster per iteration than competing information-based methods due to its linear cost in the number of objectives and constraints.