# Learning Neural Network Subspaces

arXiv: https://arxiv.org/abs/2102.10472 · HTML: https://mlsanity.com/paper/2102.10472/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 53 references that are in the map.
- First posted on arXiv 20 Feb 2021.

## Builds on (most cited first, 30 of 39)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [When Does Label Smoothing Help?](https://mlsanity.com/paper/1906.02629/) (2019)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Underspecification Presents Challenges for Credibility in Modern Machine Learning](https://mlsanity.com/paper/2011.03395/) (2020)
- [Pruning neural networks without any data by iteratively conserving synaptic flow](https://mlsanity.com/paper/2006.05467/) (2020)
- [Measuring Robustness to Natural Distribution Shifts in Image Classification](https://mlsanity.com/paper/2007.00644/) (2020)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning](https://mlsanity.com/paper/2002.06715/) (2020)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [Snapshot Ensembles: Train 1, get M for free](https://mlsanity.com/paper/1704.00109/) (2017)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)

## Built on by (most cited first, 30 of 82)

- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Robust fine-tuning of zero-shot models](https://mlsanity.com/paper/2109.01903/) (2021)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Diverse Weight Averaging for Out-of-Distribution Generalization](https://mlsanity.com/paper/2205.09739/) (2022)
- [DiLoCo: Distributed Low-Communication Training of Language Models](https://mlsanity.com/paper/2311.08105/) (2023)
- [Deep Model Fusion: A Survey](https://mlsanity.com/paper/2309.15698/) (2023)
- [Loss Surface Simplexes for Mode Connecting Volumes and Fast Ensembling](https://mlsanity.com/paper/2102.13042/) (2021)
- [Diversify and Disambiguate: Learning From Underspecified Data](https://mlsanity.com/paper/2202.03418/) (2022)
- [A Survey on Model MoErging: Recycling and Routing Among Specialized Experts for Collaborative Learning](https://mlsanity.com/paper/2408.07057/) (2024)
- [Mechanistic Mode Connectivity](https://mlsanity.com/paper/2211.08422/) (2022)
- [CoSCL: Cooperation of Small Continual Learners is Stronger than a Big One](https://mlsanity.com/paper/2207.06543/) (2022)
- [Conditional Language Policy: A General Framework for Steerable Multi-Objective Finetuning](https://mlsanity.com/paper/2407.15762/) (2024)
- [Towards Scalable and Versatile Weight Space Learning](https://mlsanity.com/paper/2406.09997/) (2024)
- [LiNeS: Post-training Layer Scaling Prevents Forgetting and Enhances Model Merging](https://mlsanity.com/paper/2410.17146/) (2024)
- [Model Zoos: A Dataset of Diverse Populations of Neural Network Models](https://mlsanity.com/paper/2209.14764/) (2022)
- [Merging by Matching Models in Task Parameter Subspaces](https://mlsanity.com/paper/2312.04339/) (2023)
- [Towards Better Plasticity-Stability Trade-off in Incremental Learning: A Simple Linear Connector](https://mlsanity.com/paper/2110.07905/) (2021)
- [Pareto Manifold Learning: Tackling multiple tasks via ensembles of single-task models](https://mlsanity.com/paper/2210.09759/) (2022)
- [Building a Subspace of Policies for Scalable Continual Learning](https://mlsanity.com/paper/2211.10445/) (2022)
- [Streaming DiLoCo with overlapping communication: Towards a Distributed Free Lunch](https://mlsanity.com/paper/2501.18512/) (2025)
- [Asynchronous Local-SGD Training for Language Modeling](https://mlsanity.com/paper/2401.09135/) (2024)
- [Layer-wise Linear Mode Connectivity](https://mlsanity.com/paper/2307.06966/) (2023)
- [DART: Diversify-Aggregate-Repeat Training Improves Generalization of Neural Networks](https://mlsanity.com/paper/2302.14685/) (2023)
- [lo-fi: distributed fine-tuning without communication](https://mlsanity.com/paper/2210.11948/) (2022)
- [Continual Learning Beyond a Single Model](https://mlsanity.com/paper/2202.09826/) (2022)
- [Learning a subspace of policies for online adaptation in Reinforcement Learning](https://mlsanity.com/paper/2110.05169/) (2021)
- [Improving Online Continual Learning Performance and Stability with Temporal Ensembles](https://mlsanity.com/paper/2306.16817/) (2023)
- [QT-DoG: Quantization-aware Training for Domain Generalization](https://mlsanity.com/paper/2410.06020/) (2024)
- [Connecting Low-Loss Subspace for Personalized Federated Learning](https://mlsanity.com/paper/2109.07628/) (2021)

## Abstract (arXiv)

Recent observations have advanced our understanding of the neural network optimization landscape, revealing the existence of (1) paths of high accuracy containing diverse solutions and (2) wider minima offering improved performance. Previous methods observing diverse paths require multiple training runs. In contrast we aim to leverage both property (1) and (2) with a single method and in a single training run. With a similar computational cost as training one model, we learn lines, curves, and simplexes of high-accuracy neural networks. These neural network subspaces contain diverse solutions that can be ensembled, approaching the ensemble performance of independently trained networks without the training cost. Moreover, using the subspace midpoint boosts accuracy, calibration, and robustness to label noise, outperforming Stochastic Weight Averaging.