# Subcritical Signal Propagation at Initialization in Normalization-Free Transformers

arXiv: https://arxiv.org/abs/2604.11890 · HTML: https://mlsanity.com/paper/2604.11890/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 25 references that are in the map.
- First posted on arXiv 13 Apr 2026.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Deep Networks with Stochastic Depth](https://mlsanity.com/paper/1603.09382/) (2016)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Understanding the Difficulty of Training Transformers](https://mlsanity.com/paper/2004.08249/) (2020)
- [Sensitivity and Generalization in Neural Networks: an Empirical Study](https://mlsanity.com/paper/1802.08760/) (2018)
- [DeepNet: Scaling Transformers to 1,000 Layers](https://mlsanity.com/paper/2203.00555/) (2022)
- [Stabilizing Transformer Training by Preventing Attention Entropy Collapse](https://mlsanity.com/paper/2303.06296/) (2023)
- [Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse](https://mlsanity.com/paper/2206.03126/) (2022)
- [Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice](https://mlsanity.com/paper/1711.04735/) (2017)
- [Robust Learning with Jacobian Regularization](https://mlsanity.com/paper/1908.02729/) (2019)
- [Mean Field Residual Networks: On the Edge of Chaos](https://mlsanity.com/paper/1712.08969/) (2017)
- [NormFormer: Improved Transformer Pretraining with Extra Normalization](https://mlsanity.com/paper/2110.09456/) (2021)
- [Geometric Dynamics of Signal Propagation Predict Trainability of Transformers](https://mlsanity.com/paper/2403.02579/) (2024)
- [Transformers Get Stable: An End-to-End Signal Propagation Theory for Language Models](https://mlsanity.com/paper/2403.09635/) (2024)
- [Two failure modes of deep transformers and how to avoid them: a unified theory of signal propagation at initialisation](https://mlsanity.com/paper/2505.24333/) (2025)
- [Stronger Normalization-Free Transformers](https://mlsanity.com/paper/2512.10938/) (2025)

## Built on by (most cited first, 1 of 1)

- [When Does Removing LayerNorm Help? Activation Bounding as a Regime-Dependent Implicit Regularizer](https://mlsanity.com/paper/2604.23434/) (2026)

## Abstract (arXiv)

We study signal propagation at initialization in transformers through the averaged partial Jacobian norm (APJN), a measure of gradient amplification across layers. We extend APJN analysis to transformers with bidirectional attention and permutation-symmetric input token configurations by deriving recurrence relations for activation statistics and APJNs across layers. Our theory predicts how attention modifies the asymptotic behavior of the APJN at large depth and matches APJNs measured in deep vision transformers. The criticality picture known from residual networks carries over to transformers: the pre-LayerNorm architecture exhibits power-law APJN growth, whereas transformers with LayerNorm replaced by elementwise tanh-like nonlinearities have stretched-exponential APJN growth, indicating that the latter are subcritical. Applied to Dynamic Tanh (DyT) and Dynamic erf (Derf) transformers, the theory explains why these architectures can be more sensitive to initialization and optimization choices and require careful tuning for stable training.