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Multi-objective Optimization by Learning Space Partitions

Yiyang Zhao, Linnan Wang, Kevin Yang, Tianjun Zhang, Tian Guo, Yuandong Tian

TL;DR

LaMOO tackles multi-objective optimization by learning data-driven partitions of the search space guided by dominance information and refined by Monte Carlo Tree Search to balance exploration and exploitation. The method acts as a meta-optimizer that wraps inner MOO/SOO solvers (e.g., qEHVI or CMA-ES) within a learned region hierarchy to focus sampling near the Pareto frontier. Theoretical analysis provides conditions under which learning partitions improves sample efficiency, and empirical results across synthetic benchmarks, NAS on NasBench201, vehicle safety design, and molecule discovery show substantial HV gains and reduced sample counts. The approach yields significant practical impact for expensive black-box MOO problems by reducing evaluations while maintaining Pareto coverage.

Abstract

In contrast to single-objective optimization (SOO), multi-objective optimization (MOO) requires an optimizer to find the Pareto frontier, a subset of feasible solutions that are not dominated by other feasible solutions. In this paper, we propose LaMOO, a novel multi-objective optimizer that learns a model from observed samples to partition the search space and then focus on promising regions that are likely to contain a subset of the Pareto frontier. The partitioning is based on the dominance number, which measures "how close" a data point is to the Pareto frontier among existing samples. To account for possible partition errors due to limited samples and model mismatch, we leverage Monte Carlo Tree Search (MCTS) to exploit promising regions while exploring suboptimal regions that may turn out to contain good solutions later. Theoretically, we prove the efficacy of learning space partitioning via LaMOO under certain assumptions. Empirically, on the HyperVolume (HV) benchmark, a popular MOO metric, LaMOO substantially outperforms strong baselines on multiple real-world MOO tasks, by up to 225% in sample efficiency for neural architecture search on Nasbench201, and up to 10% for molecular design.

Multi-objective Optimization by Learning Space Partitions

TL;DR

LaMOO tackles multi-objective optimization by learning data-driven partitions of the search space guided by dominance information and refined by Monte Carlo Tree Search to balance exploration and exploitation. The method acts as a meta-optimizer that wraps inner MOO/SOO solvers (e.g., qEHVI or CMA-ES) within a learned region hierarchy to focus sampling near the Pareto frontier. Theoretical analysis provides conditions under which learning partitions improves sample efficiency, and empirical results across synthetic benchmarks, NAS on NasBench201, vehicle safety design, and molecule discovery show substantial HV gains and reduced sample counts. The approach yields significant practical impact for expensive black-box MOO problems by reducing evaluations while maintaining Pareto coverage.

Abstract

In contrast to single-objective optimization (SOO), multi-objective optimization (MOO) requires an optimizer to find the Pareto frontier, a subset of feasible solutions that are not dominated by other feasible solutions. In this paper, we propose LaMOO, a novel multi-objective optimizer that learns a model from observed samples to partition the search space and then focus on promising regions that are likely to contain a subset of the Pareto frontier. The partitioning is based on the dominance number, which measures "how close" a data point is to the Pareto frontier among existing samples. To account for possible partition errors due to limited samples and model mismatch, we leverage Monte Carlo Tree Search (MCTS) to exploit promising regions while exploring suboptimal regions that may turn out to contain good solutions later. Theoretically, we prove the efficacy of learning space partitioning via LaMOO under certain assumptions. Empirically, on the HyperVolume (HV) benchmark, a popular MOO metric, LaMOO substantially outperforms strong baselines on multiple real-world MOO tasks, by up to 225% in sample efficiency for neural architecture search on Nasbench201, and up to 10% for molecular design.

Paper Structure

This paper contains 31 sections, 6 theorems, 36 equations, 16 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Learning Space Partitions: A Theoretical Understanding
  4. Problem Setting
  5. Rewards under optimal action sequence
  6. Extension to Multi-Objective Optimization
  7. LaMOO: Latent Action Multi-objective Optimization
  8. Experiments
  9. Small-scale Problems
  10. Synthetic Functions.
  11. Structural Optimization in Automobile Safety Design (vehicle safety)
  12. Nasbench201
  13. Multi-Objective Molecule Discovery
  14. Ablation studies
  15. Conclusion
  16. ...and 16 more sections

Key Result

Lemma 1

The algorithm to uniformly draw $k$ samples in $S$, pick the best and return is a $(1,1)$-oracle.

Figures (16)

  • Figure 1: Left: A basic setting in Multi-objective Optimization (MOO), optimizing $M=2$ objectives in Eqn. \ref{['prob-formulation']}. (a) depicts the objective space $(f_1, f_2)$ and (b) shows the search space $\mathbf{x} \in \Omega$. In (a), $P$ denotes the Pareto frontier, $R$ is the reference point, the hypervolume $HV$ is the space of the shaded area, and $o(\mathbf{x})$ are the dominance numbers. In (b), once a few samples are collected within $\Omega$, LaMOO learns to partition the search space $\Omega$ into sub-regions (i.e. $\Omega_{good}$ and $\Omega_{bad}$) according to the dominance number in objective space, and then focuses future sampling on the good regions that are close to the Pareto Frontier. This procedure can be repeated to further partition $\Omega_{good}$ and $\Omega_{bad}$. Right: A table shows the properties of MOO methods used in experiments.
  • Figure 2: (a) The leaf nodes D and E that correspond to the non-splittable space $\Omega_{D}$ and $\Omega_{E}$. (b). The node selection procedure based on the UCB value. (c). The new samples generation from the selected space $\Omega_{E}$ for bayesian optimization.
  • Figure 3: Left: Branin-Currin with 2 dimensions and 2 objectives. Middle: VehicleSafety with 5 dimensions and 3 objectives. Right: Nasbench201 with 6 dimensions and 2 objectives. We ran each algorithm 7 times (shaded area is $\pm$ std of the mean). Top: Bayesian Optimization w/o LaMOO. Bottom: evolutionary algorithms w/o LaMOO. Note the two algorithm families show very different sample efficiency in MOO tasks.
  • Figure 4: DTLZ2 with many objectives, We ran each algorithm 7 times (shaded area is $\pm$ std of the mean). From left to right: BO with 2 objectives; EA with 2 objectives; BO with 10 objectives; EA with 10 objectives.
  • Figure 5: Molecule Discovery: Left: Molecule discovery with two objectives (GSK3$\beta$+JNK3). Middle: Molecule discovery with three objectives (QED+SA+SARS). Right: Molecule Discovery with four objectives (GSK3$\beta$+JNK3+QED+SA). We ran each algorithm 15 times (shaded area is $\pm$ std of the mean).
  • ...and 11 more figures

Theorems & Definitions (13)

  • Definition 1: $(\alpha, \eta)$-Oracle
  • Lemma 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 3 more