Beyond Single Points: Candidate-Set Serving for Amortized Multi-Objective Optimization

Amortized multi-objective optimization (MOO) accelerates test-time inference by learning a preference-conditioned mapping to solutions. Standard methods typically return a single solution per preference. However, this point-estimate approach fails when a preference admits multiple distinct high-quality solutions (leading to poor compromises due to squared-loss averaging) or when the optimal solution jumps across disconnected Pareto branches. To address this, we formalize Candidate-Set Post-training Amortized Scalarized Optimization (PASO), a precommit-then-score serving interface. Instead of a single point, a frozen model generates a sequence of candidates which are exactly scored under the queried preference at test time. Crucially, this framework allows us to theoretically and empirically decouple the effects of supervision, output cardinality, and proposal quality. Our analysis proves that while squared-loss point maps suffer from mean collapse and branch-switching obstructions, candidate sets overcome these limits—improving both the selected solution quality and the coverage of diverse alternatives. Furthermore, our theoretical bounds and experiments demonstrate that simply increasing cardinality is insufficient; candidate sets only succeed when the underlying proposal quality is high. Experiments across continuous, discontinuous, and real-world tasks confirm that candidate sets are essential for multimodal and discontinuous MOO landscapes.