# Convergence and Divergence of Language Models under Different Random Seeds

arXiv: https://arxiv.org/abs/2509.26643 · HTML: https://mlsanity.com/paper/2509.26643/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 28 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 16 of 16)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [BLiMP: The Benchmark of Linguistic Minimal Pairs for English](https://mlsanity.com/paper/1912.00582/) (2019)
- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [Sudden Drops in the Loss: Syntax Acquisition, Phase Transitions, and Simplicity Bias in MLMs](https://mlsanity.com/paper/2309.07311/) (2023)
- [LLM Circuit Analyses Are Consistent Across Training and Scale](https://mlsanity.com/paper/2407.10827/) (2024)
- [Language acquisition: do children and language models follow similar learning stages?](https://mlsanity.com/paper/2306.03586/) (2023)
- [How to Compute the Probability of a Word](https://mlsanity.com/paper/2406.14561/) (2024)
- [Characterizing Learning Curves During Language Model Pre-Training: Learning, Forgetting, and Stability](https://mlsanity.com/paper/2308.15419/) (2023)
- [PolyPythias: Stability and Outliers across Fifty Language Model Pre-Training Runs](https://mlsanity.com/paper/2503.09543/) (2025)
- [Tracing the Representation Geometry of Language Models from Pretraining to Post-training](https://mlsanity.com/paper/2509.23024/) (2025)
- [Neural Networks Learn Statistics of Increasing Complexity](https://mlsanity.com/paper/2402.04362/) (2024)
- [Exact Byte-Level Probabilities from Tokenized Language Models for FIM-Tasks and Model Ensembles](https://mlsanity.com/paper/2410.09303/) (2024)

## Built on by (most cited first, 3 of 3)

- [Start Making Sense(s): A Developmental Probe of Attention Specialization Using Lexical Ambiguity](https://mlsanity.com/paper/2511.21974/) (2025)
- [Do Deep Networks Forget Initialization? A Forgetting-Time View of Practical Inductive Bias](https://mlsanity.com/paper/2605.29152/) (2026)
- [When Do Models Admit They Are Wrong? Failure Disclosure Is Unstable Under Reinforcement Learning](https://mlsanity.com/paper/2609.33220/) (2026)

## Abstract (arXiv)

In this paper, we investigate the convergence of language models (LMs) trained under different random seeds, measuring convergence as the expected per-token Kullback--Leibler (KL) divergence across seeds. By comparing LM convergence as a function of model size and training checkpoint, we identify a four-phase convergence pattern: (i) an initial uniform phase, (ii) a sharp-convergence phase, (iii) a sharp-divergence phase, and (iv) a slow-reconvergence phase. Further, we observe that larger models reconverge faster in later training stages, while smaller models never actually reconverge; these results suggest that a certain model size may be necessary to learn stable distributions. Restricting our analysis to specific token frequencies or part-of-speech (PoS) tags further reveals that convergence is uneven across linguistic categories: frequent tokens and function words converge faster and more reliably than their counterparts (infrequent tokens and content words). Overall, our findings highlight factors that influence the stability of the learned distributions in model training.