[Paper Review] Pareto Set Learning for Expensive Multi-Objective Optimization

Xi Lin, Zhiyuan Yang+2 more|arXiv (Cornell University)|Oct 16, 2022

Advanced Multi-Objective Optimization Algorithms23 citations

TL;DR

PSL learns a parametric mapping from trade-off preferences to the entire Pareto set, enabling efficient batch MOBO and flexible decision-making under limited evaluations.

ABSTRACT

Expensive multi-objective optimization problems can be found in many real-world applications, where their objective function evaluations involve expensive computations or physical experiments. It is desirable to obtain an approximate Pareto front with a limited evaluation budget. Multi-objective Bayesian optimization (MOBO) has been widely used for finding a finite set of Pareto optimal solutions. However, it is well-known that the whole Pareto set is on a continuous manifold and can contain infinite solutions. The structural properties of the Pareto set are not well exploited in existing MOBO methods, and the finite-set approximation may not contain the most preferred solution(s) for decision-makers. This paper develops a novel learning-based method to approximate the whole Pareto set for MOBO, which generalizes the decomposition-based multi-objective optimization algorithm (MOEA/D) from finite populations to models. We design a simple and powerful acquisition search method based on the learned Pareto set, which naturally supports batch evaluation. In addition, with our proposed model, decision-makers can readily explore any trade-off area in the approximate Pareto set for flexible decision-making. This work represents the first attempt to model the Pareto set for expensive multi-objective optimization. Experimental results on different synthetic and real-world problems demonstrate the effectiveness of our proposed method.

Motivation & Objective

• Motivate expensive multi-objective optimization problems where evaluations are costly.
• Propose a method to approximate the whole Pareto set rather than a finite subset.
• Enable flexible decision-making by mapping preferences to Pareto solutions.
• Provide a lightweight, batch-capable acquisition procedure.
• Demonstrate effectiveness on synthetic benchmarks and real-world engineering problems.

Proposed method

• Introduce a set model that maps trade-off preferences to Pareto solutions via augmented Tchebycheff scalarization.
• Use a neural network (MLP) as the Pareto set model to output x(lambda) for any lambda in the preference simplex.
• Train the model with Gaussian process surrogates for each objective and gradient-based optimization of the theta parameters.
• Optimize theta by minimizing the expected augmented Tchebycheff scalarization over sampled preferences (Monte Carlo).
• Adopt a lower confidence bound surrogate for f to balance exploration/exploitation in MOBO.
• Develop a batched selection procedure that samples preferences, generates candidate solutions from the learned Pareto set, and selects a batch by maximizing hypervolume improvement.

Experimental results

Research questions

• RQ1 Can the Pareto set be learned as a continuous mapping from preferences to Pareto-optimal solutions in expensive MOBO?
• RQ2 Does PSL enable efficient batched candidate selection and better exploration of the Pareto set compared to finite-population MOBO methods?
• RQ3 How well does the learned Pareto set approximate the true Pareto set/front on synthetic and real-world problems?
• RQ4 What is the computational overhead of PSL relative to existing MOBO approaches?
• RQ5 Does PSL support flexible decision-making by allowing users to navigate trade-offs across the Pareto set?

Key findings

Problem	#objs	MOEA/D-EGO	TSEMO	USeMO-EI	DGEMO	qEHVI	PSL: Model + Selection
F1	2	40.95	4.82	6.12	61.48	36.71	6.59
DTLZ2	3	71.83	7.28	8.76	83.57	75.92	8.61

• PSL can approximate the whole Pareto set and its front, enabling exploration of trade-offs via preference control.
• The PSL-based MOBO achieves competitive or superior hypervolume progression compared to several MOBO baselines across synthetic benchmarks and real-world problems.
• The learned Pareto set is visualizable and usable for decision-makers to select preferred trade-offs.
• PSL demonstrates lower or comparable per-iteration runtime relative to competing MOBO methods, with additional gains from batched selection.
• The batched PSL approach supports efficient batch evaluations while maintaining performance gains.
• Experiments show PSL can learn Pareto fronts that closely match ground-truth fronts on synthetic tasks and approximate fronts on real-world designs.