# On the Stepwise Nature of Self-Supervised Learning

arXiv: https://arxiv.org/abs/2303.15438 · HTML: https://mlsanity.com/paper/2303.15438/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 75 of its 90 references that are in the map.
- First posted on arXiv 27 Mar 2023.

## Builds on (most cited first, 30 of 75)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Unsupervised Feature Learning via Non-Parametric Instance-level Discrimination](https://mlsanity.com/paper/1805.01978/) (2018)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Learning deep representations by mutual information estimation and maximization](https://mlsanity.com/paper/1808.06670/) (2018)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Self-Supervised Learning from Images with a Joint-Embedding Predictive Architecture](https://mlsanity.com/paper/2301.08243/) (2023)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Learning Representations by Maximizing Mutual Information Across Views](https://mlsanity.com/paper/1906.00910/) (2019)
- [data2vec: A General Framework for Self-supervised Learning in Speech, Vision and Language](https://mlsanity.com/paper/2202.03555/) (2022)
- [Data-Efficient Image Recognition with Contrastive Predictive Coding](https://mlsanity.com/paper/1905.09272/) (2019)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Large Batch Training of Convolutional Networks](https://mlsanity.com/paper/1708.03888/) (2017)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)

## Built on by (most cited first, 30 of 36)

- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Auxiliary task demands mask the capabilities of smaller language models](https://mlsanity.com/paper/2404.02418/) (2024)
- [Transformers learn through gradual rank increase](https://mlsanity.com/paper/2306.07042/) (2023)
- [InfoNCE: Identifying the Gap Between Theory and Practice](https://mlsanity.com/paper/2407.00143/) (2024)
- [Joint Embedding vs Reconstruction: Provable Benefits of Latent Space Prediction for Self Supervised Learning](https://mlsanity.com/paper/2505.12477/) (2025)
- [Tracing the Representation Geometry of Language Models from Pretraining to Post-training](https://mlsanity.com/paper/2509.23024/) (2025)
- [How JEPA Avoids Noisy Features: The Implicit Bias of Deep Linear Self Distillation Networks](https://mlsanity.com/paper/2407.03475/) (2024)
- [Cross-Entropy Is All You Need To Invert the Data Generating Process](https://mlsanity.com/paper/2410.21869/) (2024)
- [When Does LeJEPA Learn a World Model?](https://mlsanity.com/paper/2605.26379/) (2026)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Saddle-to-Saddle Dynamics Explains A Simplicity Bias Across Neural Network Architectures](https://mlsanity.com/paper/2512.20607/) (2025)
- [Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks](https://mlsanity.com/paper/2506.06489/) (2025)
- [The Optimization Landscape of SGD Across the Feature Learning Strength](https://mlsanity.com/paper/2410.04642/) (2024)
- [Parameter Symmetry Potentially Unifies Deep Learning Theory](https://mlsanity.com/paper/2502.05300/) (2025)
- [Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)](https://mlsanity.com/paper/2502.21009/) (2025)
- [Position: An Empirically Grounded Identifiability Theory Will Accelerate Self-Supervised Learning Research](https://mlsanity.com/paper/2504.13101/) (2025)
- [Closed-Form Training Dynamics Reveal Learned Features and Linear Structure in Word2Vec-like Models](https://mlsanity.com/paper/2502.09863/) (2025)
- [Phase Transitions in the Output Distribution of Large Language Models](https://mlsanity.com/paper/2405.17088/) (2024)
- [Harnessing small projectors and multiple views for efficient vision pretraining](https://mlsanity.com/paper/2312.10725/) (2023)
- [Compression of Structured Data with Autoencoders: Provable Benefit of Nonlinearities and Depth](https://mlsanity.com/paper/2402.05013/) (2024)
- [Three Mechanisms of Feature Learning in a Linear Network](https://mlsanity.com/paper/2401.07085/) (2024)
- [A Theoretical Characterization of Optimal Data Augmentations in Self-Supervised Learning](https://mlsanity.com/paper/2411.01767/) (2024)
- [Infinite Width Limits of Self Supervised Neural Networks](https://mlsanity.com/paper/2411.11176/) (2024)
- [AdaDim: Dimensionality Adaptation for SSL Representational Dynamics](https://mlsanity.com/paper/2505.12576/) (2025)
- [Saddle-To-Saddle Dynamics in Deep ReLU Networks: Low-Rank Bias in the First Saddle Escape](https://mlsanity.com/paper/2505.21722/) (2025)
- [Representation Learning Dynamics of Self-Supervised Models](https://mlsanity.com/paper/2309.02011/) (2023)
- [Kernel VICReg for Self-Supervised Learning in Reproducing Kernel Hilbert Space](https://mlsanity.com/paper/2509.07289/) (2025)
- [FroSSL: Frobenius Norm Minimization for Efficient Multiview Self-Supervised Learning](https://mlsanity.com/paper/2310.02903/) (2023)
- [When can we Approximate Wide Contrastive Models with Neural Tangent Kernels and Principal Component Analysis?](https://mlsanity.com/paper/2403.08673/) (2024)
- [Training Dynamics of Nonlinear Contrastive Learning Model in the High Dimensional Limit](https://mlsanity.com/paper/2406.06909/) (2024)

## Abstract (arXiv)

We present a simple picture of the training process of joint embedding self-supervised learning methods. We find that these methods learn their high-dimensional embeddings one dimension at a time in a sequence of discrete, well-separated steps. We arrive at this conclusion via the study of a linearized model of Barlow Twins applicable to the case in which the trained network is infinitely wide. We solve the training dynamics of this model from small initialization, finding that the model learns the top eigenmodes of a certain contrastive kernel in a stepwise fashion, and obtain a closed-form expression for the final learned representations. Remarkably, we then see the same stepwise learning phenomenon when training deep ResNets using the Barlow Twins, SimCLR, and VICReg losses. Our theory suggests that, just as kernel regression can be thought of as a model of supervised learning, kernel PCA may serve as a useful model of self-supervised learning.