# Training Dynamics of Transformers to Recognize Word Co-occurrence via Gradient Flow Analysis

arXiv: https://arxiv.org/abs/2410.09605 · HTML: https://mlsanity.com/paper/2410.09605/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 42 references that are in the map.
- First posted on arXiv 12 Oct 2024.

## Builds on (most cited first, 30 of 35)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [PaLM 2 Technical Report](https://mlsanity.com/paper/2305.10403/) (2023)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (2023)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Inductive Biases and Variable Creation in Self-Attention Mechanisms](https://mlsanity.com/paper/2110.10090/) (2021)
- [Representational Strengths and Limitations of Transformers](https://mlsanity.com/paper/2306.02896/) (2023)
- [What and How does In-Context Learning Learn? Bayesian Model Averaging, Parameterization, and Generalization](https://mlsanity.com/paper/2305.19420/) (2023)
- [How Transformers Learn Causal Structure with Gradient Descent](https://mlsanity.com/paper/2402.14735/) (2024)
- [In-Context Convergence of Transformers](https://mlsanity.com/paper/2310.05249/) (2023)
- [Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers](https://mlsanity.com/paper/2107.13163/) (2021)
- [Scan and Snap: Understanding Training Dynamics and Token Composition in 1-layer Transformer](https://mlsanity.com/paper/2305.16380/) (2023)
- [Vision Transformers provably learn spatial structure](https://mlsanity.com/paper/2210.09221/) (2022)
- [How Do Transformers Learn Topic Structure: Towards a Mechanistic Understanding](https://mlsanity.com/paper/2303.04245/) (2023)
- [Why Do Pretrained Language Models Help in Downstream Tasks? An Analysis of Head and Prompt Tuning](https://mlsanity.com/paper/2106.09226/) (2021)
- [A Theoretical Understanding of Shallow Vision Transformers: Learning, Generalization, and Sample Complexity](https://mlsanity.com/paper/2302.06015/) (2023)
- [Training Dynamics of Multi-Head Softmax Attention for In-Context Learning: Emergence, Convergence, and Optimality](https://mlsanity.com/paper/2402.19442/) (2024)
- [Max-Margin Token Selection in Attention Mechanism](https://mlsanity.com/paper/2306.13596/) (2023)
- [JoMA: Demystifying Multilayer Transformers via JOint Dynamics of MLP and Attention](https://mlsanity.com/paper/2310.00535/) (2023)
- [Do Transformers Parse while Predicting the Masked Word?](https://mlsanity.com/paper/2303.08117/) (2023)
- [How Do Nonlinear Transformers Learn and Generalize in In-Context Learning?](https://mlsanity.com/paper/2402.15607/) (2024)

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

- [How Transformers Learn Regular Language Recognition: A Theoretical Study on Training Dynamics and Implicit Bias](https://mlsanity.com/paper/2505.00926/) (2025)
- [Bayes optimal learning of attention-indexed models](https://mlsanity.com/paper/2506.01582/) (2025)
- [How Do Transformers Learn to Associate Tokens: Gradient Leading Terms Bring Mechanistic Interpretability](https://mlsanity.com/paper/2601.19208/) (2026)
- [How Can Mamba Learn In Context with Outliers and Generalize Provably?](https://mlsanity.com/paper/2510.00399/) (2025)
- [Incremental Learning of Sparse Attention Patterns in Transformers](https://mlsanity.com/paper/2602.19143/) (2026)
- [Attention with Trained Embeddings Provably Selects Important Tokens](https://mlsanity.com/paper/2505.17282/) (2025)
- [On the global convergence of gradient flow for wide shallow models beyond homogeneous nonlinearities](https://mlsanity.com/paper/2605.10775/) (2026)
- [Mixture-of-Transformers Learn Faster: A Theoretical Study on Classification Problems](https://mlsanity.com/paper/2510.27004/) (2025)
- [Training Dynamics of Transformers on Unbounded Inputs: Attention Learning Can Lag Output Layer Learning](https://mlsanity.com/paper/2610.20337/) (2026)

## Abstract (arXiv)

Understanding the training dynamics of transformers is important to explain the impressive capabilities behind large language models. In this work, we study the dynamics of training a shallow transformer on a task of recognizing co-occurrence of two designated words. In the literature of studying training dynamics of transformers, several simplifications are commonly adopted such as weight reparameterization, attention linearization, special initialization, and lazy regime. In contrast, we analyze the gradient flow dynamics of simultaneously training three attention matrices and a linear MLP layer from random initialization, and provide a framework of analyzing such dynamics via a coupled dynamical system. We establish near minimum loss and characterize the attention model after training. We discover that gradient flow serves as an inherent mechanism that naturally divide the training process into two phases. In Phase 1, the linear MLP quickly aligns with the two target signals for correct classification, whereas the softmax attention remains almost unchanged. In Phase 2, the attention matrices and the MLP evolve jointly to enlarge the classification margin and reduce the loss to a near minimum value. Technically, we prove a novel property of the gradient flow, termed textitautomatic balancing of gradients, which enables the loss values of different samples to decrease almost at the same rate and further facilitates the proof of near minimum training loss. We also conduct experiments to verify our theoretical results.