# Latent State Models of Training Dynamics

arXiv: https://arxiv.org/abs/2308.09543 · HTML: https://mlsanity.com/paper/2308.09543/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 67 references that are in the map.
- First posted on arXiv 18 Aug 2023.

## Builds on (most cited first, 30 of 41)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Beyond the Imitation Game: Quantifying and extrapolating the capabilities of language models](https://mlsanity.com/paper/2206.04615/) (2022)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [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)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [The Break-Even Point on Optimization Trajectories of Deep Neural Networks](https://mlsanity.com/paper/2002.09572/) (2020)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [The Early Phase of Neural Network Training](https://mlsanity.com/paper/2002.10365/) (2020)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [The MultiBERTs: BERT Reproductions for Robustness Analysis](https://mlsanity.com/paper/2106.16163/) (2021)
- [BERTs of a feather do not generalize together: Large variability in generalization across models with similar test set performance](https://mlsanity.com/paper/1911.02969/) (2019)
- [Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction](https://mlsanity.com/paper/2206.07085/) (2022)
- [A Tale of Two Circuits: Grokking as Competition of Sparse and Dense Subnetworks](https://mlsanity.com/paper/2303.11873/) (2023)
- [Critical Learning Periods in Deep Neural Networks](https://mlsanity.com/paper/1711.08856/) (2017)
- [Extracting Automata from Recurrent Neural Networks Using Queries and Counterexamples](https://mlsanity.com/paper/1711.09576/) (2017)
- [Theoretical Analysis of Auto Rate-Tuning by Batch Normalization](https://mlsanity.com/paper/1812.03981/) (2018)
- [The Surprising Simplicity of the Early-Time Learning Dynamics of Neural Networks](https://mlsanity.com/paper/2006.14599/) (2020)

## Built on by (most cited first, 16 of 16)

- [Sudden Drops in the Loss: Syntax Acquisition, Phase Transitions, and Simplicity Bias in MLMs](https://mlsanity.com/paper/2309.07311/) (2023)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [LLM Circuit Analyses Are Consistent Across Training and Scale](https://mlsanity.com/paper/2407.10827/) (2024)
- [PolyPythias: Stability and Outliers across Fifty Language Model Pre-Training Runs](https://mlsanity.com/paper/2503.09543/) (2025)
- [(How) Do Language Models Track State?](https://mlsanity.com/paper/2503.02854/) (2025)
- [Sometimes I am a Tree: Data Drives Unstable Hierarchical Generalization](https://mlsanity.com/paper/2412.04619/) (2024)
- [Survival of the Fittest Representation: A Case Study with Modular Addition](https://mlsanity.com/paper/2405.17420/) (2024)
- [Random Scaling of Emergent Capabilities](https://mlsanity.com/paper/2502.17356/) (2025)
- [Curriculum Learning for LLM Pretraining: An Analysis of Learning Dynamics](https://mlsanity.com/paper/2601.21698/) (2026)
- [Toward a Theory of Generalizability in LLM Mechanistic Interpretability Research](https://mlsanity.com/paper/2509.22831/) (2025)
- [Start Making Sense(s): A Developmental Probe of Attention Specialization Using Lexical Ambiguity](https://mlsanity.com/paper/2511.21974/) (2025)
- [Neural Neural Scaling Laws](https://mlsanity.com/paper/2601.19831/) (2026)
- [Interactive Training: Feedback-Driven Neural Network Optimization](https://mlsanity.com/paper/2510.02297/) (2025)
- [Implicitly Normalized Online PCA: A Regularized Algorithm with Exact High-Dimensional Dynamics](https://mlsanity.com/paper/2512.01231/) (2025)
- [Time Course MechInterp: Analyzing the Evolution of Components and Knowledge in Large Language Models](https://mlsanity.com/paper/2506.03434/) (2025)
- [Latent States in Neural Networks: Recovering the Temporal Structure of Drifting Data from Model Weights](https://mlsanity.com/paper/2607.27482/) (2026)

## Abstract (arXiv)

The impact of randomness on model training is poorly understood. How do differences in data order and initialization actually manifest in the model, such that some training runs outperform others or converge faster? Furthermore, how can we interpret the resulting training dynamics and the phase transitions that characterize different trajectories? To understand the effect of randomness on the dynamics and outcomes of neural network training, we train models multiple times with different random seeds and compute a variety of metrics throughout training, such as the L2 norm, mean, and variance of the neural network's weights. We then fit a hidden Markov model (HMM) over the resulting sequences of metrics. The HMM represents training as a stochastic process of transitions between latent states, providing an intuitive overview of significant changes during training. Using our method, we produce a low-dimensional, discrete representation of training dynamics on grokking tasks, image classification, and masked language modeling. We use the HMM representation to study phase transitions and identify latent "detour" states that slow down convergence.