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This paper introduces Priority-Constrained Descent (PCD), a gradient-based framework that addresses the symmetry limitations of existing multi-objective optimization methods by explicitly enforcing hierarchical structures between primary and secondary goals. By preserving the direction of primary descent while allowing controlled distortion for secondary constraints like sparsity or robustness, PCD achieves Pareto dominance with interpretable trade-offs in structured network compression tasks. The method offers closed-form solutions for low-dimensional objective spaces and provides a more nuanced control mechanism via a single hyperparameter compared to standard approaches.
Not All Objectives Are Born Equal: Priority-Constrained Descent for Hierarchical Multi-Objective Optimization
arXiv:2606.29521v2 Announce Type: replace Abstract: Deep learning problems rarely involve objectives that are equal in importance. A primary objective defines the goal, whilst secondary objectives, such as sparsity, compression, or robustness constrain the solution. While existing multi-objective methods have proven effective in practice, they have a clear symmetry problem and neglect the inherent objective hierarchy built into these objective spaces. We introduce Priority-Constrained Descent (PCD), a gradient-based optimization framework designed to explicitly exploit hierarchical objective structures. PCD preserves the direction of primary descent whilst allowing for the minimal distortion necessary to guarantee progress on secondary objectives, controlled by a single $\tau \in [0, 1]$ that dictates the strength of the distortion. The resulting formulation is invariant to objective scaling and admits exact closed-form solutions for problems with two and three objectives. We evaluate PCD within structured network compression settings, unstructured sparsity and low-rankness, and across a variety of synthetic experiments, showing Pareto dominance and better per-objective performance with secondary progress guarantees over existing methods, further exhibiting the interpretable trade-off that $\tau$ provides.
Abstract:Deep learning problems rarely involve objectives that are equal in importance. A primary objective defines the goal, whilst secondary objectives, such as sparsity, compression, or robustness constrain the solution. While existing multi-objective methods have proven effective in practice, they have a clear symmetry problem and neglect the inherent objective hierarchy built into these objective spaces. We introduce Priority-Constrained Descent (PCD), a gradient-based optimization framework designed to explicitly exploit hierarchical objective structures. PCD preserves the direction of primary descent whilst allowing for the minimal distortion necessary to guarantee progress on secondary objectives, controlled by a single $\tau \in [0, 1]$ that dictates the strength of the distortion. The resulting formulation is invariant to objective scaling and admits exact closed-form solutions for problems with two and three objectives. We evaluate PCD within structured network compression settings, unstructured sparsity and low-rankness, and across a variety of synthetic experiments, showing Pareto dominance and better per-objective performance with secondary progress guarantees over existing methods, further exhibiting the interpretable trade-off that $\tau$ provides.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2606.29521 [cs.LG]
(or arXiv:2606.29521v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.29521 Focus to learn more arXiv-issued DOI via DataCite
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