Percolation Dynamics in Optimization : Variance Cascades and Discrete Scale Invariance

MCML Authors

Sai Niranjan Ramachandran

→ Group Suvrit Sra
 Resource Aware Machine Learning

Suvrit Sra

Prof. Dr.

Core PI

Resource Aware Machine Learning

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 answer this by modeling the stochastic gradient flow (SGF) as a percolation process, in which architectural symmetries force subnetworks to merge in discrete simultaneous blocks rather than one at a time. These structural transitions register as variance spikes in a macroscopic order parameter, echoing physical phase transitions. We further show this trapping mechanism and its associated scaling cascade extend to Adam and AdamW under an explicit heavy-tailed noise model.

misc RS26a

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Preprint

Sep. 2026

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Authors

S. N. Ramachandran • S. Sra

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Links

arXiv

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Research Area

 A2 | Mathematical Foundations