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

arXiv: https://arxiv.org/abs/2605.19306 · HTML: https://mlsanity.com/paper/2605.19306/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 34 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 8 of 8)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [A Reductions Approach to Fair Classification](https://mlsanity.com/paper/1803.02453/) (2018)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Learning the Pareto Front with Hypernetworks](https://mlsanity.com/paper/2010.04104/) (2020)
- [A Monotonic Optimization Approach for Solving Strictly Quasiconvex Multiobjective Programming Problems](https://mlsanity.com/paper/2003.07549/) (2020)

## Abstract (arXiv)

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%.