A Two-Phase Adaptive Balanced Penalty Method for Controllable Pareto Front Learning under Split Feasibility Conditions

TL;DR

This paper introduces a novel two-phase adaptive penalty method for controllable Pareto front learning under split feasibility, with theoretical guarantees and improved solution quality demonstrated on benchmarks.

Contribution

It proposes the ABP algorithm with convergence guarantees and a two-phase training strategy for hypernetworks in constrained Pareto optimization.

Findings

ABP achieves full-sequence convergence under standard assumptions.

ABP-HyperNet improves feasibility from 36-49% to 87-100%.

Up to 2.3x higher EFHV compared to unconstrained baselines.

Abstract

We address the open problem of training hypernetworks for Controllable Pareto Front Learning (CPFL) under split feasibility conditions with rigorous theoretical guarantees. We reformulate the constrained Pareto problem as a Bi-Level Scalarized Split Problem (BSSP) and propose the Adaptive Balanced Penalty (ABP) algorithm, whose three gradient components -- optimality, set feasibility, and image feasibility -- are blended through an adaptive indicator driven by a computable lower bound. Using a novel convex surrogate technique, we prove full-sequence convergence under standard convexity and Robbins-Monro step-size assumptions. The ABP penalty structure is then translated into a two-phase, feasibility-first training strategy for Hyper-MLP and HyperTrans architectures (ABP-HyperNet). To evaluate constrained CPFL, we introduce the Expected Feasible Hypervolume (EFHV), which jointly captures solution quality and constraint satisfaction. Experiments on five multi-objective benchmarks validate the ABP solver against ground truth, while three multi-task learning datasets demonstrate that ABP-HyperNet achieves up to 2.3x higher EFHV than unconstrained baselines by raising feasibility from 36-49% to 87-100%.