# Loss Landscape Degeneracy and Stagewise Development in Transformers

arXiv: https://arxiv.org/abs/2402.02364 · HTML: https://mlsanity.com/paper/2402.02364/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 4 Feb 2024.

## Builds on (most cited first, 26 of 26)

- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [TinyStories: How Small Can Language Models Be and Still Speak Coherent English?](https://mlsanity.com/paper/2305.07759/) (2023)
- [Data Selection for Language Models via Importance Resampling](https://mlsanity.com/paper/2302.03169/) (2023)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Analyzing the Structure of Attention in a Transformer Language Model](https://mlsanity.com/paper/1906.04284/) (2019)
- [Qualitatively characterizing neural network optimization problems](https://mlsanity.com/paper/1412.6544/) (2014)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [Pretraining task diversity and the emergence of non-Bayesian in-context learning for regression](https://mlsanity.com/paper/2306.15063/) (2023)
- [Sudden Drops in the Loss: Syntax Acquisition, Phase Transitions, and Simplicity Bias in MLMs](https://mlsanity.com/paper/2309.07311/) (2023)
- [Acquisition of Chess Knowledge in AlphaZero](https://mlsanity.com/paper/2111.09259/) (2021)
- [Shortformer: Better Language Modeling using Shorter Inputs](https://mlsanity.com/paper/2012.15832/) (2020)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [Predicting Grokking Long Before it Happens: A look into the loss landscape of models which grok](https://mlsanity.com/paper/2306.13253/) (2023)
- [What does Attention in Neural Machine Translation Pay Attention to?](https://mlsanity.com/paper/1710.03348/) (2017)
- [Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition](https://mlsanity.com/paper/2310.06301/) (2023)
- [PCA of high dimensional random walks with comparison to neural network training](https://mlsanity.com/paper/1806.08805/) (2018)
- [Stuck in a What? Adventures in Weight Space](https://mlsanity.com/paper/1602.07320/) (2016)

## Built on by (most cited first, 30 of 40)

- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Model Organisms for Emergent Misalignment](https://mlsanity.com/paper/2506.11613/) (2025)
- [Differentiation and Specialization of Attention Heads via the Refined Local Learning Coefficient](https://mlsanity.com/paper/2410.02984/) (2024)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Dynamics of Transient Structure in In-Context Linear Regression Transformers](https://mlsanity.com/paper/2501.17745/) (2025)
- [Beyond Induction Heads: In-Context Meta Learning Induces Multi-Phase Circuit Emergence](https://mlsanity.com/paper/2505.16694/) (2025)
- [Survival of the Fittest Representation: A Case Study with Modular Addition](https://mlsanity.com/paper/2405.17420/) (2024)
- [Influence Dynamics and Stagewise Data Attribution](https://mlsanity.com/paper/2510.12071/) (2025)
- [Patterning: The Dual of Interpretability](https://mlsanity.com/paper/2601.13548/) (2026)
- [Estimating the Local Learning Coefficient at Scale](https://mlsanity.com/paper/2402.03698/) (2024)
- [Identifying Sparsely Active Circuits Through Local Loss Landscape Decomposition](https://mlsanity.com/paper/2504.00194/) (2025)
- [In-Context Algebra](https://mlsanity.com/paper/2512.16902/) (2025)
- [Stagewise Reinforcement Learning and the Geometry of the Regret Landscape](https://mlsanity.com/paper/2601.07524/) (2026)
- [A Basin-Selection Perspective on Grokking via Singular Learning Theory](https://mlsanity.com/paper/2603.01192/) (2026)
- [Towards Worst-Case Guarantees with Scale-Aware Interpretability](https://mlsanity.com/paper/2602.05184/) (2026)
- [Algebraic Dead Directions in LayerNorm Transformers: A Forward-Pass-Only Diagnostic at LLM Scale](https://mlsanity.com/paper/2606.19491/) (2026)
- [Energy-Entropy Regularization: The True Power of Minimal Looped Transformers](https://mlsanity.com/paper/2601.09588/) (2026)
- [Specialization of softmax attention heads: insights from the high-dimensional single-location model](https://mlsanity.com/paper/2603.03993/) (2026)
- [Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning](https://mlsanity.com/paper/2605.07980/) (2026)
- [Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity](https://mlsanity.com/paper/2606.21158/) (2026)
- [Statistical Learning Analysis of Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.11097/) (2026)
- [Upper Bounds for Local Learning Coefficients of Three-Layer Neural Networks](https://mlsanity.com/paper/2603.12785/) (2026)
- [Mechanistic Anomaly Detection via Functional Attribution](https://mlsanity.com/paper/2604.18970/) (2026)
- [Unveiling Multi-regime Patterns in SciML: Distinct Failure Modes and Regime-specific Optimization](https://mlsanity.com/paper/2605.29153/) (2026)
- [Dead Directions: Geometric Singular Learning](https://mlsanity.com/paper/2606.05957/) (2026)
- [The Unified Cognitive Consciousness Theory for Language Models: Anchoring Semantics, Thresholds of Activation, and Emergent Reasoning](https://mlsanity.com/paper/2506.02139/) (2025)
- [Closing the Curvature Gap: Full Transformer Hessians](https://mlsanity.com/paper/2510.16927/) (2025)
- [A Family of LLMs Liberated from Static Vocabularies](https://mlsanity.com/paper/2603.15953/) (2026)
- [The Geometric Cost of Normalization: Affine Bounds on the Bayesian Complexity of Neural Networks](https://mlsanity.com/paper/2603.27432/) (2026)
- [When Are Two Networks the Same? Tensor Similarity for Mechanistic Interpretability](https://mlsanity.com/paper/2605.15183/) (2026)

## Abstract (arXiv)

Deep learning involves navigating a high-dimensional loss landscape over the neural network parameter space. Over the course of training, complex computational structures form and re-form inside the neural network, leading to shifts in input/output behavior. It is a priority for the science of deep learning to uncover principles governing the development of neural network structure and behavior. Drawing on the framework of singular learning theory, we propose that model development is deeply linked to degeneracy in the local geometry of the loss landscape. We investigate this link by monitoring loss landscape degeneracy throughout training, as quantified by the local learning coefficient, for a transformer language model and an in-context linear regression transformer. We show that training can be divided into distinct periods of change in loss landscape degeneracy, and that these changes in degeneracy coincide with significant changes in the internal computational structure and the input/output behavior of the transformers. This finding provides suggestive evidence that degeneracy and development are linked in transformers, underscoring the potential of a degeneracy-based perspective for understanding modern deep learning.