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

Akash Kumar

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…