ParetoM$^3$: Learning on the Pareto Set under Preference Guidance via Min-Max-Min Optimization

Learning across multiple inherently conflicting objectives requires selecting desirable trade-offs from the Pareto set. In this work, we study preference-guided multi-objective optimization, where a predefined preference function is minimized over the weak Pareto set of multiple nonconvex objectives. Leveraging a merit function that vanishes on the weak Pareto set, we propose a penalty-based min-max-min formulation, termed the M$^3$ problem, which unifies preference optimization and the enforcement of weak Pareto optimality in a single objective. We further develop ParetoM$^3$, a single-loop first-order algorithm for computing stationary solutions of the resulting M$^3$ problem. Under a local Kurdyka-{\L}ojasiewicz assumption, we prove that ParetoM$^3$ finds an $\epsilon$-stationary point of the M$^3$ problem that is also approximately weakly Pareto optimal for the original multi-objective problem when the penalty parameter is sufficiently large, with explicit non-asymptotic convergence rates. We also show that this stationarity notion recovers existing optimality criteria, including approximate preference stationarity and approximate Karush-Kuhn-Tucker conditions, under the same assumptions used in prior works. Experiments on a synthetic benchmark, image classification, regression, and math reasoning demonstrate that ParetoM$^3$ reliably attains preferred weak Pareto optimal solutions while achieving strong practical performance.