# Scaling Adaptive Depth with Norm-Agnostic Residual Networks

arXiv: https://arxiv.org/abs/2606.16112 · HTML: https://mlsanity.com/paper/2606.16112/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 51 of its 60 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 30 of 51)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale](https://mlsanity.com/paper/2208.07339/) (2022)
- [Going deeper with Image Transformers](https://mlsanity.com/paper/2103.17239/) (2021)
- [CogView: Mastering Text-to-Image Generation via Transformers](https://mlsanity.com/paper/2105.13290/) (2021)
- [Universal Transformers](https://mlsanity.com/paper/1807.03819/) (2018)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [Transformer Feed-Forward Layers Build Predictions by Promoting Concepts in the Vocabulary Space](https://mlsanity.com/paper/2203.14680/) (2022)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [Reducing Transformer Depth on Demand with Structured Dropout](https://mlsanity.com/paper/1909.11556/) (2019)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [ShortGPT: Layers in Large Language Models are More Redundant Than You Expect](https://mlsanity.com/paper/2403.03853/) (2024)
- [Massive Activations in Large Language Models](https://mlsanity.com/paper/2402.17762/) (2024)
- [Gated Attention for Large Language Models: Non-linearity, Sparsity, and Attention-Sink-Free](https://mlsanity.com/paper/2505.06708/) (2025)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Mixture-of-Depths: Dynamically allocating compute in transformer-based language models](https://mlsanity.com/paper/2404.02258/) (2024)
- [Depth-Adaptive Transformer](https://mlsanity.com/paper/1910.10073/) (2019)
- [The Unreasonable Ineffectiveness of the Deeper Layers](https://mlsanity.com/paper/2403.17887/) (2024)
- [When Attention Sink Emerges in Language Models: An Empirical View](https://mlsanity.com/paper/2410.10781/) (2024)
- [DeepNet: Scaling Transformers to 1,000 Layers](https://mlsanity.com/paper/2203.00555/) (2022)
- [Training Very Deep Networks](https://mlsanity.com/paper/1507.06228/) (2015)
- [Quantizable Transformers: Removing Outliers by Helping Attention Heads Do Nothing](https://mlsanity.com/paper/2306.12929/) (2023)

## Abstract (arXiv)

Residual architectures are ubiquitous in deep learning, but they suffer from a subtle structural limitation: the norm of the residual stream can grow rapidly with depth. As a result, updates from later layers become small relative to the accumulated residual state. This reduces their impact on the representation and limits the benefits of scaling models in depth. To address this, we introduce NAG, a norm-agnostic residual architecture that separates magnitude from directional information in the residual stream, preserving meaningful layer contributions throughout depth and preventing later updates from being systematically suppressed by residual-norm growth. Importantly, NAG introduces only a negligible number of additional parameters and relies on simple operations that are easily kernel-fusible, preserving training efficiency in practice. We show that this architecture outperforms baseline Transformers, with gains that increase substantially as depth grows, enabling effective training of much deeper models. The norm-agnostic formulation also leads to an interpretable Mixture-of-Depths (MoD) mechanism that adaptively skips both attention and MLP layers. Beyond serving as a post-training accuracy-compute tradeoff, this mechanism can be used as a pretraining-time scaling strategy: under iso-FLOP training, compute saved by reducing per-token forward-pass cost can be reinvested into training on more tokens while keeping the total parameter count and KV-cache budget fixed. In our experiments, moderate Mixture-of-Depths rates of approximately 20%-25% match full-depth baseline performance under equal training compute while substantially reducing the number of executed layer parameters and forward-pass FLOPs. These results identify sparsity in depth as a new scaling axis for fixed-compute training, enabling very deep yet FLOP-efficient models.