SGD Optimization Dynamics Show Percolation and Variance Cascades

Sai Niranjan Ramachandran, Suvrit Sra· September 3, 2026 View original

Key takeaways

  • SGD optimization involves subnetworks merging in discrete, simultaneous blocks.
  • These structural transitions cause variance spikes, similar to physical phase transitions.
  • This phenomenon extends to Adam and AdamW optimizers under specific noise models.
  • Understanding these dynamics can improve deep neural network training stability and efficiency.

Who benefits

AI DevelopmentMachine Learning ResearchData ScienceHigh-Performance Computing

Summary

This research models Stochastic Gradient Descent (SGD) as a percolation process, revealing that deep neural networks merge into simpler subnetworks in discrete, simultaneous blocks. These structural transitions manifest as variance spikes in a macroscopic order parameter, akin to physical phase transitions, and extend to Adam and AdamW optimizers.

This paper investigates the underlying dynamics of Stochastic Gradient Descent (SGD), a widely used optimization algorithm for deep neural networks. The researchers propose a novel perspective, modeling the stochastic gradient flow (SGF) as a percolation process. This model suggests that during optimization, architectural symmetries cause subnetworks within the neural network to merge not gradually, but in distinct, simultaneous blocks. These structural changes are observed as sharp increases, or "variance spikes," in a macroscopic order parameter, drawing parallels to phase transitions seen in physical systems. The study further demonstrates that this trapping mechanism and the associated scaling cascade are not exclusive to SGD but also apply to other popular optimizers like Adam and AdamW, particularly under a heavy-tailed noise model. This provides a deeper understanding of how optimizers navigate the complex loss landscapes of deep learning.

Why it matters

Understanding the dynamics of optimizers like SGD, Adam, and AdamW at a fundamental level can lead to the development of more stable, efficient, and performant training strategies for deep neural networks.

How to implement this in your domain

  1. 1Review current deep learning training protocols to consider the implications of percolation dynamics on convergence and stability.
  2. 2Experiment with noise models and regularization techniques that might influence or leverage these observed variance cascades.
  3. 3Develop diagnostic tools to monitor macroscopic order parameters during training, identifying potential structural transitions.
  4. 4Explore how insights into subnetwork merging could inform architectural design or pruning strategies for neural networks.

Original post by Sai Niranjan Ramachandran, Suvrit Sra

"arXiv:2609.02373v1 Announce Type: new Abstract: We study the dynamics of Stochastic Gradient Descent (SGD), which is known to steer deep neural networks toward invariant sets that correspond to simpler subnetworks. How this steering unfolds over time remains poorly understood. We…"

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Originally posted by Sai Niranjan Ramachandran, Suvrit Sra on X · view source

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