# Feature-Learning Networks Are Consistent Across Widths At Realistic Scales

arXiv: https://arxiv.org/abs/2305.18411 · HTML: https://mlsanity.com/paper/2305.18411/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 55 references that are in the map.
- First posted on arXiv 28 May 2023.

## Builds on (most cited first, 30 of 43)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism](https://mlsanity.com/paper/1909.08053/) (2019)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [TRAK: Attributing Model Behavior at Scale](https://mlsanity.com/paper/2303.14186/) (2023)
- [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)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Finite Versus Infinite Neural Networks: an Empirical Study](https://mlsanity.com/paper/2007.15801/) (2020)
- [Finite Depth and Width Corrections to the Neural Tangent Kernel](https://mlsanity.com/paper/1909.05989/) (2019)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Neural Networks as Kernel Learners: The Silent Alignment Effect](https://mlsanity.com/paper/2111.00034/) (2021)
- [Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks](https://mlsanity.com/paper/2205.09653/) (2022)
- [Scaling description of generalization with number of parameters in deep learning](https://mlsanity.com/paper/1901.01608/) (2019)

## Built on by (most cited first, 30 of 41)

- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Don't be lazy: CompleteP enables compute-efficient deep transformers](https://mlsanity.com/paper/2505.01618/) (2025)
- [How Does Critical Batch Size Scale in Pre-training?](https://mlsanity.com/paper/2410.21676/) (2024)
- [Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit](https://mlsanity.com/paper/2309.16620/) (2023)
- [How Feature Learning Can Improve Neural Scaling Laws](https://mlsanity.com/paper/2409.17858/) (2024)
- [Scaling and renormalization in high-dimensional regression](https://mlsanity.com/paper/2405.00592/) (2024)
- [Infinite Limits of Multi-head Transformer Dynamics](https://mlsanity.com/paper/2405.15712/) (2024)
- [Pre-training under infinite compute](https://mlsanity.com/paper/2509.14786/) (2025)
- [Super Consistency of Neural Network Landscapes and Learning Rate Transfer](https://mlsanity.com/paper/2402.17457/) (2024)
- [Scaling Collapse Reveals Universal Dynamics in Compute-Optimally Trained Neural Networks](https://mlsanity.com/paper/2507.02119/) (2025)
- [Hyperparameter Transfer Enables Consistent Gains of Matrix-Preconditioned Optimizers Across Scales](https://mlsanity.com/paper/2512.05620/) (2025)
- [More is Better in Modern Machine Learning: when Infinite Overparameterization is Optimal and Overfitting is Obligatory](https://mlsanity.com/paper/2311.14646/) (2023)
- [On the Parameterization of Second-Order Optimization Effective Towards the Infinite Width](https://mlsanity.com/paper/2312.12226/) (2023)
- [The Optimization Landscape of SGD Across the Feature Learning Strength](https://mlsanity.com/paper/2410.04642/) (2024)
- [Features are fate: a theory of transfer learning in high-dimensional regression](https://mlsanity.com/paper/2410.08194/) (2024)
- [Adaptive kernel predictors from feature-learning infinite limits of neural networks](https://mlsanity.com/paper/2502.07998/) (2025)
- [The Feature Speed Formula: a flexible approach to scale hyper-parameters of deep neural networks](https://mlsanity.com/paper/2311.18718/) (2023)
- [From Kernels to Features: A Multi-Scale Adaptive Theory of Feature Learning](https://mlsanity.com/paper/2502.03210/) (2025)
- [Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention](https://mlsanity.com/paper/2605.29548/) (2026)
- [Scaling with Collapse: Efficient and Predictable Training of LLM Families](https://mlsanity.com/paper/2509.25087/) (2025)
- [Closed-Form Training Dynamics Reveal Learned Features and Linear Structure in Word2Vec-like Models](https://mlsanity.com/paper/2502.09863/) (2025)
- [The Importance of Being Lazy: Scaling Limits of Continual Learning](https://mlsanity.com/paper/2506.16884/) (2025)
- [Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer](https://mlsanity.com/paper/2605.07870/) (2026)
- [Local Loss Optimization in the Infinite Width: Stable Parameterization of Predictive Coding Networks and Target Propagation](https://mlsanity.com/paper/2411.02001/) (2024)
- [Phase Diagram of Dropout for Two-Layer Neural Networks in the Mean-Field Regime](https://mlsanity.com/paper/2510.07554/) (2025)
- [A Generalized Spectral Framework to Expain Neural Scaling and Compression Dynamics](https://mlsanity.com/paper/2511.07892/) (2025)
- [Can Small Training Runs Reliably Guide Data Curation? Rethinking Proxy-Model Practice](https://mlsanity.com/paper/2512.24503/) (2025)
- [The lazy (NTK) and rich (μP) regimes: a gentle tutorial](https://mlsanity.com/paper/2404.19719/) (2024)
- [PhiNets: Brain-inspired Non-contrastive Learning Based on Temporal Prediction Hypothesis](https://mlsanity.com/paper/2405.14650/) (2024)
- [Estimating the Spectral Moments of the Kernel Integral Operator from Finite Sample Matrices](https://mlsanity.com/paper/2410.17998/) (2024)

## Abstract (arXiv)

We study the effect of width on the dynamics of feature-learning neural networks across a variety of architectures and datasets. Early in training, wide neural networks trained on online data have not only identical loss curves but also agree in their point-wise test predictions throughout training. For simple tasks such as CIFAR-5m this holds throughout training for networks of realistic widths. We also show that structural properties of the models, including internal representations, preactivation distributions, edge of stability phenomena, and large learning rate effects are consistent across large widths. This motivates the hypothesis that phenomena seen in realistic models can be captured by infinite-width, feature-learning limits. For harder tasks (such as ImageNet and language modeling), and later training times, finite-width deviations grow systematically. Two distinct effects cause these deviations across widths. First, the network output has initialization-dependent variance scaling inversely with width, which can be removed by ensembling networks. We observe, however, that ensembles of narrower networks perform worse than a single wide network. We call this the bias of narrower width. We conclude with a spectral perspective on the origin of this finite-width bias.