# Neural networks with late-phase weights

arXiv: https://arxiv.org/abs/2007.12927 · HTML: https://mlsanity.com/paper/2007.12927/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2020.

## Built on by (most cited first, 29 of 29)

- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Model Merging in LLMs, MLLMs, and Beyond: Methods, Theories, Applications and Opportunities](https://mlsanity.com/paper/2408.07666/) (2024)
- [ZipIt! Merging Models from Different Tasks without Training](https://mlsanity.com/paper/2305.03053/) (2023)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [Learning Neural Network Subspaces](https://mlsanity.com/paper/2102.10472/) (2021)
- [Reinforcement Learning, Bit by Bit](https://mlsanity.com/paper/2103.04047/) (2021)
- [Posterior Meta-Replay for Continual Learning](https://mlsanity.com/paper/2103.01133/) (2021)
- [Deep Ensembling with No Overhead for either Training or Testing: The All-Round Blessings of Dynamic Sparsity](https://mlsanity.com/paper/2106.14568/) (2021)
- [Tangent Model Composition for Ensembling and Continual Fine-tuning](https://mlsanity.com/paper/2307.08114/) (2023)
- [Instant Soup: Cheap Pruning Ensembles in A Single Pass Can Draw Lottery Tickets from Large Models](https://mlsanity.com/paper/2306.10460/) (2023)
- [Sparse Model Soups: A Recipe for Improved Pruning via Model Averaging](https://mlsanity.com/paper/2306.16788/) (2023)
- [Mix Data or Merge Models? Optimizing for Diverse Multi-Task Learning](https://mlsanity.com/paper/2410.10801/) (2024)
- [Efficient Self-Ensemble for Semantic Segmentation](https://mlsanity.com/paper/2111.13280/) (2021)
- [Training-Free Model Merging for Multi-target Domain Adaptation](https://mlsanity.com/paper/2407.13771/) (2024)
- [Prune and Tune Ensembles: Low-Cost Ensemble Learning With Sparse Independent Subnetworks](https://mlsanity.com/paper/2202.11782/) (2022)
- [Learning More Generalized Experts by Merging Experts in Mixture-of-Experts](https://mlsanity.com/paper/2405.11530/) (2024)
- [Adapt then Unlearn: Exploring Parameter Space Semantics for Unlearning in Generative Adversarial Networks](https://mlsanity.com/paper/2309.14054/) (2023)
- [Meta-Learning via Classifier(-free) Diffusion Guidance](https://mlsanity.com/paper/2210.08942/) (2022)
- [Understanding the effect of sparsity on neural networks robustness](https://mlsanity.com/paper/2206.10915/) (2022)
- [Low-Loss Subspace Compression for Clean Gains against Multi-Agent Backdoor Attacks](https://mlsanity.com/paper/2203.03692/) (2022)
- [Disentangling the Predictive Variance of Deep Ensembles through the Neural Tangent Kernel](https://mlsanity.com/paper/2210.09818/) (2022)
- [A Survey of Historical Learning: Learning Models with Learning History](https://mlsanity.com/paper/2303.12992/) (2023)
- [DIMAT: Decentralized Iterative Merging-And-Training for Deep Learning Models](https://mlsanity.com/paper/2404.08079/) (2024)
- [State Soup: In-Context Skill Learning, Retrieval and Mixing](https://mlsanity.com/paper/2406.08423/) (2024)
- [Do the Frankenstein, or how to achieve better out-of-distribution performance with manifold mixing model soup](https://mlsanity.com/paper/2309.08610/) (2023)
- [Optimizable Model Averaging for Ensembling Heterogeneous Model Outputs](https://mlsanity.com/paper/2505.21857/) (2025)
- [Improving Robustness of Deep Convolutional Neural Networks via Multiresolution Learning](https://mlsanity.com/paper/2309.13752/) (2023)
- [Neural Network Quantization by Learning Low-Loss Subspaces](https://mlsanity.com/paper/2606.25087/) (2026)
- [A Distributional Optimisation Perspective on Combining Models in Deep Learning](https://mlsanity.com/paper/2609.24328/) (2026)

## Abstract (arXiv)

The largely successful method of training neural networks is to learn their weights using some variant of stochastic gradient descent (SGD). Here, we show that the solutions found by SGD can be further improved by ensembling a subset of the weights in late stages of learning. At the end of learning, we obtain back a single model by taking a spatial average in weight space. To avoid incurring increased computational costs, we investigate a family of low-dimensional late-phase weight models which interact multiplicatively with the remaining parameters. Our results show that augmenting standard models with late-phase weights improves generalization in established benchmarks such as CIFAR-10/100, ImageNet and enwik8. These findings are complemented with a theoretical analysis of a noisy quadratic problem which provides a simplified picture of the late phases of neural network learning.