# On the Convergence of Encoder-only Shallow Transformers

arXiv: https://arxiv.org/abs/2311.01575 · HTML: https://mlsanity.com/paper/2311.01575/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 61 references that are in the map.
- First posted on arXiv 2 Nov 2023.

## Builds on (most cited first, 30 of 49)

- [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)
- [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)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Swin Transformer V2: Scaling Up Capacity and Resolution](https://mlsanity.com/paper/2111.09883/) (2021)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [How Do Vision Transformers Work?](https://mlsanity.com/paper/2202.06709/) (2022)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [When Vision Transformers Outperform ResNets without Pre-training or Strong Data Augmentations](https://mlsanity.com/paper/2106.01548/) (2021)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [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)
- [Towards moderate overparameterization: global convergence guarantees for training shallow neural networks](https://mlsanity.com/paper/1902.04674/) (2019)
- [Inductive Biases and Variable Creation in Self-Attention Mechanisms](https://mlsanity.com/paper/2110.10090/) (2021)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [An Improved Analysis of Training Over-parameterized Deep Neural Networks](https://mlsanity.com/paper/1906.04688/) (2019)

## Built on by (most cited first, 13 of 13)

- [Unraveling the Gradient Descent Dynamics of Transformers](https://mlsanity.com/paper/2411.07538/) (2024)
- [Training Dynamics of Transformers to Recognize Word Co-occurrence via Gradient Flow Analysis](https://mlsanity.com/paper/2410.09605/) (2024)
- [Faster WIND: Accelerating Iterative Best-of-N Distillation for LLM Alignment](https://mlsanity.com/paper/2410.20727/) (2024)
- [Asymptotics of SGD in Sequence-Single Index Models and Single-Layer Attention Networks](https://mlsanity.com/paper/2506.02651/) (2025)
- [On the Convergence of Gradient Descent on Learning Transformers with Residual Connections](https://mlsanity.com/paper/2506.05249/) (2025)
- [Towards Understanding the Word Sensitivity of Attention Layers: A Study via Random Features](https://mlsanity.com/paper/2402.02969/) (2024)
- [Learning to Adapt: In-Context Learning Beyond Stationarity](https://mlsanity.com/paper/2604.10946/) (2026)
- [Generalizable autoregressive modeling of time series through functional narratives](https://mlsanity.com/paper/2410.08421/) (2024)
- [Depth-induced NTK: Bridging Over-parameterized Neural Networks and Deep Neural Kernels](https://mlsanity.com/paper/2511.05585/) (2025)
- [Finite-Time Analysis of Gradient Descent for Shallow Transformers](https://mlsanity.com/paper/2601.16514/) (2026)
- [Training Infinitely Deep and Wide Transformers](https://mlsanity.com/paper/2605.17660/) (2026)
- [A Second-Order Perspective on Pruning at Initialization and Knowledge Transfer](https://mlsanity.com/paper/2509.24066/) (2025)
- [Efficient RGB-D Scene Understanding via Multi-task Adaptive Learning and Cross-dimensional Feature Guidance](https://mlsanity.com/paper/2603.07570/) (2026)

## Abstract (arXiv)

In this paper, we aim to build the global convergence theory of encoder-only shallow Transformers under a realistic setting from the perspective of architectures, initialization, and scaling under a finite width regime. The difficulty lies in how to tackle the softmax in self-attention mechanism, the core ingredient of Transformer. In particular, we diagnose the scaling scheme, carefully tackle the input/output of softmax, and prove that quadratic overparameterization is sufficient for global convergence of our shallow Transformers under commonly-used He/LeCun initialization in practice. Besides, neural tangent kernel (NTK) based analysis is also given, which facilitates a comprehensive comparison. Our theory demonstrates the separation on the importance of different scaling schemes and initialization. We believe our results can pave the way for a better understanding of modern Transformers, particularly on training dynamics.