# The Butterfly Effect: Neural Network Training Trajectories Are Highly Sensitive to Initial Conditions

arXiv: https://arxiv.org/abs/2506.13234 · HTML: https://mlsanity.com/paper/2506.13234/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 61 of its 76 references that are in the map.
- First posted on arXiv 16 Jun 2025.

## Builds on (most cited first, 30 of 61)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (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)
- [Federated Learning with Matched Averaging](https://mlsanity.com/paper/2002.06440/) (2020)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [What is being transferred in transfer learning?](https://mlsanity.com/paper/2008.11687/) (2020)
- [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)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Do Wide and Deep Networks Learn the Same Things? Uncovering How Neural Network Representations Vary with Width and Depth](https://mlsanity.com/paper/2010.15327/) (2020)
- [A disciplined approach to neural network hyper-parameters: Part 1 -- learning rate, batch size, momentum, and weight decay](https://mlsanity.com/paper/1803.09820/) (2018)
- [Model Fusion via Optimal Transport](https://mlsanity.com/paper/1910.05653/) (2019)
- [Qualitatively characterizing neural network optimization problems](https://mlsanity.com/paper/1412.6544/) (2014)
- [Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel](https://mlsanity.com/paper/2010.15110/) (2020)
- [Linear Mode Connectivity in Multitask and Continual Learning](https://mlsanity.com/paper/2010.04495/) (2020)

## Built on by (most cited first, 9 of 9)

- [The Appeal and Reality of Recycling LoRAs with Adaptive Merging](https://mlsanity.com/paper/2602.12323/) (2026)
- [Scaling Linear Mode Connectivity and Merging to Billion Parameter Pretrained Transformers](https://mlsanity.com/paper/2606.23607/) (2026)
- [Entropic Confinement and Mode Connectivity in Overparameterized Neural Networks](https://mlsanity.com/paper/2512.06297/) (2025)
- [How Data Augmentation Shapes Neural Representations](https://mlsanity.com/paper/2605.15306/) (2026)
- [Mapping the Emergence of Regularization-Driven Dynamics in Grokking](https://mlsanity.com/paper/2608.25813/) (2026)
- [Winning the Lottery by Preserving Network Training Dynamics with Concrete Ticket Search](https://mlsanity.com/paper/2512.07142/) (2025)
- [Quasi-Monte Carlo Initialization for Meta-Reinforcement Learning](https://mlsanity.com/paper/2607.21637/) (2026)
- [Learned, Then Lost: A Measured Single-Example Counterfactual in Pre-training](https://mlsanity.com/paper/2608.19168/) (2026)
- [When Do Models Admit They Are Wrong? Failure Disclosure Is Unstable Under Reinforcement Learning](https://mlsanity.com/paper/2609.33220/) (2026)

## Abstract (arXiv)

Neural network training is inherently sensitive to initialization and the randomness induced by stochastic gradient descent. However, it is unclear to what extent such effects lead to meaningfully different networks, either in terms of the models' weights or the underlying functions that were learned. In this work, we show that during the initial "chaotic" phase of training, even extremely small perturbations reliably causes otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time. We quantify this divergence through (i) L2 distance between parameters, (ii) the loss barrier when interpolating between networks, (iii) L2 and barrier between parameters after permutation alignment, and (iv) representational similarity between intermediate activations; revealing how perturbations across different hyperparameter or fine-tuning settings drive training trajectories toward distinct loss minima. Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.