# Not All Objectives Are Born Equal: Priority-Constrained Descent for Hierarchical Multi-Objective Optimization

arXiv: https://arxiv.org/abs/2606.29521 · HTML: https://mlsanity.com/paper/2606.29521/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 40 references that are in the map.
- First posted on arXiv 28 Jun 2026.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (2017)
- [Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics](https://mlsanity.com/paper/1705.07115/) (2017)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Multi-Task Learning as a Bargaining Game](https://mlsanity.com/paper/2202.01017/) (2022)
- [Learning the Pareto Front with Hypernetworks](https://mlsanity.com/paper/2010.04104/) (2020)
- [Structured Pruning for Deep Convolutional Neural Networks: A survey](https://mlsanity.com/paper/2303.00566/) (2023)
- [FAMO: Fast Adaptive Multitask Optimization](https://mlsanity.com/paper/2306.03792/) (2023)
- [Independent Component Alignment for Multi-Task Learning](https://mlsanity.com/paper/2305.19000/) (2023)
- [Do Current Multi-Task Optimization Methods in Deep Learning Even Help?](https://mlsanity.com/paper/2209.11379/) (2022)
- [Auxiliary Learning as an Asymmetric Bargaining Game](https://mlsanity.com/paper/2301.13501/) (2023)
- [Cooper: A Library for Constrained Optimization in Deep Learning](https://mlsanity.com/paper/2504.01212/) (2025)
- [PMGDA: A Preference-based Multiple Gradient Descent Algorithm](https://mlsanity.com/paper/2402.09492/) (2024)
- [Neural Network Quantization for Microcontrollers: A Comprehensive Survey of Methods, Platforms, and Applications](https://mlsanity.com/paper/2508.15008/) (2025)
- [Exact Pareto Optimal Search for Multi-Task Learning and Multi-Criteria Decision-Making](https://mlsanity.com/paper/2108.00597/) (2021)

## Built on by (most cited first, 1 of 1)

- [FedPCD: Priority-Constrained Federated Aggregation](https://mlsanity.com/paper/2610.13508/) (2026)

## Abstract (arXiv)

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 τ ∈ [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 τ provides.