# On the Implicit Bias Towards Minimal Depth of Deep Neural Networks

arXiv: https://arxiv.org/abs/2202.09028 · HTML: https://mlsanity.com/paper/2202.09028/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 49 references that are in the map.
- First posted on arXiv 18 Feb 2022.

## Builds on (most cited first, 30 of 37)

- [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)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Scaling Vision Transformers](https://mlsanity.com/paper/2106.04560/) (2021)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Opening the Black Box of Deep Neural Networks via Information](https://mlsanity.com/paper/1703.00810/) (2017)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [A Geometric Analysis of Neural Collapse with Unconstrained Features](https://mlsanity.com/paper/2105.02375/) (2021)
- [Exploring Deep Neural Networks via Layer-Peeled Model: Minority Collapse in Imbalanced Training](https://mlsanity.com/paper/2101.12699/) (2021)
- [Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path](https://mlsanity.com/paper/2106.02073/) (2021)
- [Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks](https://mlsanity.com/paper/1805.12076/) (2018)
- [Neural collapse with unconstrained features](https://mlsanity.com/paper/2011.11619/) (2020)
- [Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation](https://mlsanity.com/paper/2105.14368/) (2021)
- [Extended Unconstrained Features Model for Exploring Deep Neural Collapse](https://mlsanity.com/paper/2202.08087/) (2022)
- [On the Role of Neural Collapse in Transfer Learning](https://mlsanity.com/paper/2112.15121/) (2021)
- [Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra](https://mlsanity.com/paper/2008.11865/) (2020)
- [Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1709.01953/) (2017)
- [A Continuous-Time View of Early Stopping for Least Squares](https://mlsanity.com/paper/1810.10082/) (2018)
- [Limitations of Neural Collapse for Understanding Generalization in Deep Learning](https://mlsanity.com/paper/2202.08384/) (2022)
- [Revealing the Structure of Deep Neural Networks via Convex Duality](https://mlsanity.com/paper/2002.09773/) (2020)
- [Implicit Regularization Towards Rank Minimization in ReLU Networks](https://mlsanity.com/paper/2201.12760/) (2022)
- [Convolutional Neural Networks Analyzed via Convolutional Sparse Coding](https://mlsanity.com/paper/1607.08194/) (2016)

## Built on by (most cited first, 12 of 12)

- [Deep Neural Collapse Is Provably Optimal for the Deep Unconstrained Features Model](https://mlsanity.com/paper/2305.13165/) (2023)
- [Linguistic Collapse: Neural Collapse in (Large) Language Models](https://mlsanity.com/paper/2405.17767/) (2024)
- [A Law of Data Separation in Deep Learning](https://mlsanity.com/paper/2210.17020/) (2022)
- [Tracing Representation Progression: Analyzing and Enhancing Layer-Wise Similarity](https://mlsanity.com/paper/2406.14479/) (2024)
- [Perturbation Analysis of Neural Collapse](https://mlsanity.com/paper/2210.16658/) (2022)
- [Reverse Engineering Self-Supervised Learning](https://mlsanity.com/paper/2305.15614/) (2023)
- [Neural Collapse versus Low-rank Bias: Is Deep Neural Collapse Really Optimal?](https://mlsanity.com/paper/2405.14468/) (2024)
- [Progressive Feedforward Collapse of ResNet Training](https://mlsanity.com/paper/2405.00985/) (2024)
- [Neural Collapse is Globally Optimal in Deep Regularized ResNets and Transformers](https://mlsanity.com/paper/2505.15239/) (2025)
- [Towards Understanding Neural Collapse: The Effects of Batch Normalization and Weight Decay](https://mlsanity.com/paper/2309.04644/) (2023)
- [Exploring Learned Representations of Neural Networks with Principal Component Analysis](https://mlsanity.com/paper/2309.15328/) (2023)
- [Feature learning is decoupled from generalization in high capacity neural networks](https://mlsanity.com/paper/2507.19680/) (2025)

## Abstract (arXiv)

Recent results in the literature suggest that the penultimate (second-to-last) layer representations of neural networks that are trained for classification exhibit a clustering property called neural collapse (NC). We study the implicit bias of stochastic gradient descent (SGD) in favor of low-depth solutions when training deep neural networks. We characterize a notion of effective depth that measures the first layer for which sample embeddings are separable using the nearest-class center classifier. Furthermore, we hypothesize and empirically show that SGD implicitly selects neural networks of small effective depths. Secondly, while neural collapse emerges even when generalization should be impossible - we argue that the degree of separability in the intermediate layers is related to generalization. We derive a generalization bound based on comparing the effective depth of the network with the minimal depth required to fit the same dataset with partially corrupted labels. Remarkably, this bound provides non-trivial estimations of the test performance. Finally, we empirically show that the effective depth of a trained neural network monotonically increases when increasing the number of random labels in data.