# Adaptive Online Mirror Descent for Tchebycheff Scalarization in Multi-Objective Learning

arXiv: https://arxiv.org/abs/2410.21764 · HTML: https://mlsanity.com/paper/2410.21764/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 57 references that are in the map.
- First posted on arXiv 29 Oct 2024.

## Builds on (most cited first, 26 of 26)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [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)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Exploiting Shared Representations for Personalized Federated Learning](https://mlsanity.com/paper/2102.07078/) (2021)
- [An Efficient Framework for Clustered Federated Learning](https://mlsanity.com/paper/2006.04088/) (2020)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Agnostic Federated Learning](https://mlsanity.com/paper/1902.00146/) (2019)
- [Fair Resource Allocation in Federated Learning](https://mlsanity.com/paper/1905.10497/) (2019)
- [Pareto Multi-Task Learning](https://mlsanity.com/paper/1912.12854/) (2019)
- [MARS: Markov Molecular Sampling for Multi-objective Drug Discovery](https://mlsanity.com/paper/2103.10432/) (2021)
- [Tilted Empirical Risk Minimization](https://mlsanity.com/paper/2007.01162/) (2020)
- [Federated Learning with Fair Averaging](https://mlsanity.com/paper/2104.14937/) (2021)
- [Smooth Tchebycheff Scalarization for Multi-Objective Optimization](https://mlsanity.com/paper/2402.19078/) (2024)
- [Federated Learning Meets Multi-objective Optimization](https://mlsanity.com/paper/2006.11489/) (2020)
- [Pareto Set Learning for Expensive Multi-Objective Optimization](https://mlsanity.com/paper/2210.08495/) (2022)
- [Revisiting Scalarization in Multi-Task Learning: A Theoretical Perspective](https://mlsanity.com/paper/2308.13985/) (2023)
- [Three-Way Trade-Off in Multi-Objective Learning: Optimization, Generalization and Conflict-Avoidance](https://mlsanity.com/paper/2305.20057/) (2023)
- [Robust Multi-Task Learning with Excess Risks](https://mlsanity.com/paper/2402.02009/) (2024)
- [Proportional Fairness in Federated Learning](https://mlsanity.com/paper/2202.01666/) (2022)
- [LibMOON: A Gradient-based MultiObjective OptimizatioN Library in PyTorch](https://mlsanity.com/paper/2409.02969/) (2024)
- [Pareto Efficient Fairness in Supervised Learning: From Extraction to Tracing](https://mlsanity.com/paper/2104.01634/) (2021)
- [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, 4 of 4)

- [Gradient-Based Multi-Objective Deep Learning: Algorithms, Theories, Applications, and Beyond](https://mlsanity.com/paper/2501.10945/) (2025)
- [Multi-objective Large Language Model Alignment with Hierarchical Experts](https://mlsanity.com/paper/2505.20925/) (2025)
- [AReUReDi: Annealed Rectified Updates for Refining Discrete Flows with Multi-Objective Guidance](https://mlsanity.com/paper/2510.00352/) (2025)
- [AutoScale: Linear Scalarization Guided by Multi-Task Optimization Metrics](https://mlsanity.com/paper/2508.13979/) (2025)

## Abstract (arXiv)

Multi-objective learning (MOL) aims to learn under multiple potentially conflicting objectives and strike a proper balance. While recent preference-guided MOL methods often rely on additional optimization objectives or constraints, we consider the classic Tchebycheff scalarization (TCH) that naturally allows for locating solutions with user-specified trade-offs. Due to its minimax formulation, directly optimizing TCH often leads to training oscillation and stagnation. In light of this limitation, we propose an adaptive online mirror descent algorithm for TCH, called (Ada)OMD-TCH. One of our main ingredients is an adaptive online-to-batch conversion that significantly improves solution optimality over traditional conversion in practice while maintaining the same theoretical convergence guarantees. We show that (Ada)OMD-TCH achieves a convergence rate of mathcal O(√log m/T), where m is the number of objectives and T is the number of rounds, providing a tighter dependency on m in the offline setting compared to existing work. Empirically, we demonstrate on both synthetic problems and federated learning tasks that (Ada)OMD-TCH effectively smooths the training process and yields preference-guided, specific, diverse, and fair solutions.