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

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 the geometry of the scalarization path. As the weight varies, the corresponding solutions traverse the PF at a generally nonuniform speed. Integrating this speed gives cumulative arc length. After normalization by the total length, it defines a cumulative distribution function (CDF) on weights whose inverse selects points at equal arc-length intervals. Building on this insight, we propose SURF (Sampling Uniformly along the Pareto Front). For structured problems, including bi-objective bandits, we derive analytical 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 for bi-objective LS, SURF's CDF reconstruction error converges linearly to an $O(N^{-2})$ discretization floor. Empirically, experiments on bandits, multi-objective-gymnasium, and multi-objective LLM alignment demonstrate that SURF efficiently achieves more uniform PF coverage than baselines.