Can Representation Learning Decouple from Loss Minimization? Polar Updates Have an Answer

Title:Can Representation Learning Decouple from Loss Minimization? Polar Updates Have an Answer

Abstract:Does representation learning stop when the training loss stops improving? We study this question for matrix Muon, whose polar-normalised updates have a step length set by the gradient's rank rather than its norm. Near the edge of stability, full-batch Muon on teacher-student problems enters approximately period-2 loss oscillations that persist for thousands of steps: the cycle-mean loss stays flat or rises, yet the weights keep moving and the learned features continue to align with the teacher subspace. For linear teacher-student learning toys, we derive explicit cycle and alignment formulas and conditional plateau and decay bounds. For a population mean-field ReLU model, we prove that, under stated dimension, initialisation and small-head conditions, the leading eigenspace of the average gradient outer product (AGOP) recovers the teacher subspace exactly during a loss plateau, before the loss later drops. In all 33 ReLU, GELU and SiLU teacher configurations we study, direction-only alignment metrics show the student AGOP aligned with, or still aligning to, the teacher subspace during the period-2 oscillations; projected head refitting on selected configurations shows that the learned directions are useful for prediction, and further measurements distinguish AGOP alignment from weight-mass concentration. In deep residual ReLU students, freezing the downstream layers while the first layer trains with full-batch exact polar updates recreates a nearly flat cycle-mean loss with improving input-AGOP alignment; freezing and unfreezing switch between this plateau and loss decrease, and the effect is sensitive to momentum and to the choice of orthogonaliser.

Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)

arXiv:2609.36240 [cs.LG]

(or arXiv:2609.36240v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.36240 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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