SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front

2026-05-19

AI Summary

This work addresses the challenge in multi-objective optimization where conventional uniform sampling of scalarization weights often yields uneven coverage of the Pareto front (PF), failing to align with diverse user preferences. The study establishes, for the first time, an analytical relationship between the geometric traversal speed of scalarization paths along the PF and the underlying weight distribution. Building on this insight, it proposes a PF-aware sampling strategy based on the arc-length cumulative distribution function (CDF) and its inverse mapping, which can be implemented either analytically or iteratively. The method guarantees uniform PF coverage and exhibits linear convergence. Empirical evaluations across bi-objective bandit problems, multi-objective Gymnasium environments, and large language model alignment tasks demonstrate significant performance gains over existing approaches, offering both strong theoretical guarantees and practical efficacy.

Abstract

Scalarization is widely used in multi-objective optimization owing to its simplicity and scalability. In many applications, the goal is to generate solutions that represent diverse user preferences, ideally with uniform coverage of the Pareto front (PF). However, uniformly sampling scalarization weights usually induces non-uniform coverage of the PF. We explain this mismatch through a geometric analysis of the scalarization path. As the scalarization weight varies, the corresponding solutions trace the PF with a generally non-uniform traversal speed. This speed induces an arc-length cumulative distribution function (CDF); inverting this CDF map yields a principled rule for selecting weights that produce uniform PF coverage. Building on this insight, we propose SURF (Sampling Uniformly along the PaReto Front). For structured problems, including bi-objective bandits, we derive closed-form expressions for this CDF map and the resulting PF-aware weight sampling rule. For general problems, SURF alternates between CDF reconstruction and weight sampling. Theoretically, we show that under provable conditions, SURF converges linearly to an unavoidable finite-sampling floor. Empirically, experiments on bandits, multi-objective-gymnasium, and multi-objective LLM alignment demonstrate that SURF efficiently achieves more uniform PF coverage than baselines.

Problem

• multi-objective optimization
• Pareto front
• scalarization
• uniform coverage
• weight sampling

Innovation

• Pareto front
• scalarization
• uniform sampling
• multi-objective optimization
• arc-length CDF