Percolation Dynamics in Optimization : Variance Cascades and Discrete Scale Invariance

• URL: http://arxiv.org/abs/2609.02373v1
• Date: Wed, 02 Sep 2026 09:44:47 GMT
• Title: Percolation Dynamics in Optimization : Variance Cascades and Discrete Scale Invariance
• Abstract summary: We study the dynamics of Gradient Descent (SGD), which is known to steer deep neural networks toward invariant sets that correspond to simplerworks.<n>We answer this by modeling a gradient flow (SGF) as a percolation process, in which architectural symmetries forceworks to merge in discrete simultaneous blocks rather than one at a time.
• Score: 20.34299008085434
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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