# Auxiliary Learning as an Asymmetric Bargaining Game

arXiv: https://arxiv.org/abs/2301.13501 · HTML: https://mlsanity.com/paper/2301.13501/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 67 references that are in the map.
- First posted on arXiv 31 Jan 2023.

## Builds on (most cited first, 30 of 44)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Cityscapes Dataset for Semantic Urban Scene Understanding](https://mlsanity.com/paper/1604.01685/) (2016)
- [SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation](https://mlsanity.com/paper/1511.00561/) (2015)
- [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)
- [An Overview of Multi-Task Learning in Deep Neural Networks](https://mlsanity.com/paper/1706.05098/) (2017)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [End-to-End Multi-Task Learning with Attention](https://mlsanity.com/paper/1803.10704/) (2018)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Cross-stitch Networks for Multi-task Learning](https://mlsanity.com/paper/1604.03539/) (2016)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Which Tasks Should Be Learned Together in Multi-task Learning?](https://mlsanity.com/paper/1905.07553/) (2019)
- [S4L: Self-Supervised Semi-Supervised Learning](https://mlsanity.com/paper/1905.03670/) (2019)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Multi-Task Learning as a Bargaining Game](https://mlsanity.com/paper/2202.01017/) (2022)
- [Just Pick a Sign: Optimizing Deep Multitask Models with Gradient Sign Dropout](https://mlsanity.com/paper/2010.06808/) (2020)
- [Gradient Vaccine: Investigating and Improving Multi-task Optimization in Massively Multilingual Models](https://mlsanity.com/paper/2010.05874/) (2020)
- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
- [Attentive Single-Tasking of Multiple Tasks](https://mlsanity.com/paper/1904.08918/) (2019)
- [A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications](https://mlsanity.com/paper/1705.06270/) (2017)
- [Instance-aware Semantic Segmentation via Multi-task Network Cascades](https://mlsanity.com/paper/1512.04412/) (2015)
- [Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions](https://mlsanity.com/paper/1903.03088/) (2019)
- [Adapting Auxiliary Losses Using Gradient Similarity](https://mlsanity.com/paper/1812.02224/) (2018)
- [Self-Supervised Learning for Domain Adaptation on Point-Clouds](https://mlsanity.com/paper/2003.12641/) (2020)
- [Self-Supervised Generalisation with Meta Auxiliary Learning](https://mlsanity.com/paper/1901.08933/) (2019)
- [RotoGrad: Gradient Homogenization in Multitask Learning](https://mlsanity.com/paper/2103.02631/) (2021)

## Built on by (most cited first, 12 of 12)

- [LLM-Rubric: A Multidimensional, Calibrated Approach to Automated Evaluation of Natural Language Texts](https://mlsanity.com/paper/2501.00274/) (2024)
- [Bayesian Uncertainty for Gradient Aggregation in Multi-Task Learning](https://mlsanity.com/paper/2402.04005/) (2024)
- [Enhancing Domain Adaptation through Prompt Gradient Alignment](https://mlsanity.com/paper/2406.09353/) (2024)
- [Using dynamic loss weighting to boost improvements in forecast stability](https://mlsanity.com/paper/2409.18267/) (2024)
- [Expert Merging in Sparse Mixture of Experts with Nash Bargaining](https://mlsanity.com/paper/2510.16138/) (2025)
- [Model Merging via Multi-Teacher Knowledge Distillation](https://mlsanity.com/paper/2512.21288/) (2025)
- [Enhancing Molecular Property Prediction with Auxiliary Learning and Task-Specific Adaptation](https://mlsanity.com/paper/2401.16299/) (2024)
- [Rethinking Teacher-Student Curriculum Learning through the Cooperative Mechanics of Experience](https://mlsanity.com/paper/2404.03084/) (2024)
- [Learning Representation for Multitask learning through Self Supervised Auxiliary learning](https://mlsanity.com/paper/2409.16651/) (2024)
- [Not All Objectives Are Born Equal: Priority-Constrained Descent for Hierarchical Multi-Objective Optimization](https://mlsanity.com/paper/2606.29521/) (2026)
- [Two-Stage Multi-task Self-Supervised Learning for Medical Image Segmentation](https://mlsanity.com/paper/2402.07119/) (2024)
- [Generating Auxiliary Tasks with Reinforcement Learning](https://mlsanity.com/paper/2510.22940/) (2025)

## Abstract (arXiv)

Auxiliary learning is an effective method for enhancing the generalization capabilities of trained models, particularly when dealing with small datasets. However, this approach may present several difficulties: (i) optimizing multiple objectives can be more challenging, and (ii) how to balance the auxiliary tasks to best assist the main task is unclear. In this work, we propose a novel approach, named AuxiNash, for balancing tasks in auxiliary learning by formalizing the problem as generalized bargaining game with asymmetric task bargaining power. Furthermore, we describe an efficient procedure for learning the bargaining power of tasks based on their contribution to the performance of the main task and derive theoretical guarantees for its convergence. Finally, we evaluate AuxiNash on multiple multi-task benchmarks and find that it consistently outperforms competing methods.