# Transformers learn through gradual rank increase

arXiv: https://arxiv.org/abs/2306.07042 · HTML: https://mlsanity.com/paper/2306.07042/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 56 references that are in the map.
- First posted on arXiv 12 Jun 2023.

## Builds on (most cited first, 30 of 40)

- [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)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Neural Networks can Learn Representations with Gradient Descent](https://mlsanity.com/paper/2206.15144/) (2022)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks](https://mlsanity.com/paper/2202.08658/) (2022)
- [Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks](https://mlsanity.com/paper/1904.13262/) (2019)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [Limitations of Lazy Training of Two-layers Neural Networks](https://mlsanity.com/paper/1906.08899/) (2019)
- [Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity](https://mlsanity.com/paper/2106.15933/) (2021)
- [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Implicit Bias in Deep Linear Classification: Initialization Scale vs Training Accuracy](https://mlsanity.com/paper/2007.06738/) (2020)
- [The Implicit Bias of Depth: How Incremental Learning Drives Generalization](https://mlsanity.com/paper/1909.12051/) (2019)
- [Neural Networks Efficiently Learn Low-Dimensional Representations with SGD](https://mlsanity.com/paper/2209.14863/) (2022)

## Built on by (most cited first, 30 of 52)

- [How Transformers Learn Causal Structure with Gradient Descent](https://mlsanity.com/paper/2402.14735/) (2024)
- [Transformers as Support Vector Machines](https://mlsanity.com/paper/2308.16898/) (2023)
- [JoMA: Demystifying Multilayer Transformers via JOint Dynamics of MLP and Attention](https://mlsanity.com/paper/2310.00535/) (2023)
- [LoRA Training in the NTK Regime has No Spurious Local Minima](https://mlsanity.com/paper/2402.11867/) (2024)
- [Training Dynamics of In-Context Learning in Linear Attention](https://mlsanity.com/paper/2501.16265/) (2025)
- [When is Task Vector Provably Effective for Model Editing? A Generalization Analysis of Nonlinear Transformers](https://mlsanity.com/paper/2504.10957/) (2025)
- [Beyond Anti-Forgetting: Multimodal Continual Instruction Tuning with Positive Forward Transfer](https://mlsanity.com/paper/2401.09181/) (2024)
- [How JEPA Avoids Noisy Features: The Implicit Bias of Deep Linear Self Distillation Networks](https://mlsanity.com/paper/2407.03475/) (2024)
- [Approaching Deep Learning through the Spectral Dynamics of Weights](https://mlsanity.com/paper/2408.11804/) (2024)
- [A phase transition between positional and semantic learning in a solvable model of dot-product attention](https://mlsanity.com/paper/2402.03902/) (2024)
- [Towards a Theoretical Understanding of the 'Reversal Curse' via Training Dynamics](https://mlsanity.com/paper/2405.04669/) (2024)
- [Understanding the Expressive Power and Mechanisms of Transformer for Sequence Modeling](https://mlsanity.com/paper/2402.00522/) (2024)
- [A distributional simplicity bias in the learning dynamics of transformers](https://mlsanity.com/paper/2410.19637/) (2024)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [Uncovering hidden geometry in Transformers via disentangling position and context](https://mlsanity.com/paper/2310.04861/) (2023)
- [Geometric Signatures of Compositionality Across a Language Model's Lifetime](https://mlsanity.com/paper/2410.01444/) (2024)
- [Learning In-context n-grams with Transformers: Sub-n-grams Are Near-stationary Points](https://mlsanity.com/paper/2508.12837/) (2025)
- [Towards a theory of model distillation](https://mlsanity.com/paper/2403.09053/) (2024)
- [Saddle-to-Saddle Dynamics Explains A Simplicity Bias Across Neural Network Architectures](https://mlsanity.com/paper/2512.20607/) (2025)
- [Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks](https://mlsanity.com/paper/2506.06489/) (2025)
- [Characterizing Large Language Model Geometry Helps Solve Toxicity Detection and Generation](https://mlsanity.com/paper/2312.01648/) (2023)
- [Subspace Networks: Scaling Decentralized Training with Communication-Efficient Model Parallelism](https://mlsanity.com/paper/2506.01260/) (2025)
- [How Transformers Learn Regular Language Recognition: A Theoretical Study on Training Dynamics and Implicit Bias](https://mlsanity.com/paper/2505.00926/) (2025)
- [Generalization or Hallucination? Understanding Out-of-Context Reasoning in Transformers](https://mlsanity.com/paper/2506.10887/) (2025)
- [To Use or not to Use Muon: How Simplicity Bias in Optimizers Matters](https://mlsanity.com/paper/2603.00742/) (2026)
- [Dissecting the Interplay of Attention Paths in a Statistical Mechanics Theory of Transformers](https://mlsanity.com/paper/2405.15926/) (2024)
- [Learning on Transformers is Provable Low-Rank and Sparse: A One-layer Analysis](https://mlsanity.com/paper/2406.17167/) (2024)
- [How Transformers Get Rich: Approximation and Dynamics Analysis](https://mlsanity.com/paper/2410.11474/) (2024)
- [Optimizing Rank for High-Fidelity Implicit Neural Representations](https://mlsanity.com/paper/2512.14366/) (2025)
- [DiTASK: Multi-Task Fine-Tuning with Diffeomorphic Transformations](https://mlsanity.com/paper/2502.06029/) (2025)

## Abstract (arXiv)

We identify incremental learning dynamics in transformers, where the difference between trained and initial weights progressively increases in rank. We rigorously prove this occurs under the simplifying assumptions of diagonal weight matrices and small initialization. Our experiments support the theory and also show that phenomenon can occur in practice without the simplifying assumptions.