# Training instability in deep learning follows low-dimensional dynamical principles

arXiv: https://arxiv.org/abs/2601.13160 · HTML: https://mlsanity.com/paper/2601.13160/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 19 Jan 2026.

## Builds on (most cited first, 14 of 14)

- [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)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [Deep Reinforcement Learning that Matters](https://mlsanity.com/paper/1709.06560/) (2017)

## Built on by (most cited first, 1 of 1)

- [Silent Inconsistency in Data-Parallel Full Fine-Tuning: Diagnosing Worker-Level Optimization Misalignment](https://mlsanity.com/paper/2602.14462/) (2026)

## Abstract (arXiv)

Deep learning systems achieve remarkable empirical performance, yet the stability of the training process itself remains poorly understood. Training unfolds as a high-dimensional dynamical system in which small perturbations to optimization, data, parameters, or learning signals can induce abrupt and irreversible collapse, undermining reproducibility and scalability. We propose a unified dynamical perspective that characterizes training stability as an intrinsic property of learning systems, organized along four interacting dimensions: optimization, environmental/data, parametric, and learning-signal stability. We operationalize this perspective through controlled perturbation auditing of training trajectories, probing how learning dynamics respond to structured disturbances without modifying learning algorithms. Across reinforcement learning and large language model training, we identify three recurring regularities: high final performance is frequently decoupled from training stability; controlled stochasticity consistently buffers learning dynamics across paradigms; and deviations in low-dimensional latent meta-states systematically precede observable performance collapse. Together, these findings establish training stability as a measurable and comparable dynamical property of learning systems, providing a descriptive foundation for studying learning dynamics beyond final performance outcomes.