How should we understand memorization vs. generalization in in-context learning (ICL) tasks, and when does one happen vs. the other? To investigate, we study how rotation-invariant kernel machines learn the ICL linear regression task proposed by Raventos et al. (2023). By decomposing the task in the kernel eigenbasis, we obtain a spectral picture of learning in this ICL setting. This picture reveals that as sample size grows, kernel machines learn successively-higher-degree components of the memorizing solution, which depend upon successively-higher-degree statistics of the training task set. The generalizing solution is approximately a low-degree truncation of the memorizing solution, and so the model first generalizes, then degrades into memorization as training proceeds. The takeaways are threefold. First, if this case is indicative, learning a "generalizing" solution on an ICL task is really just underfitting the memorizing solution! Second, the fact that kernel machines can generalize on this task means that feature learning is inessential to ICL. Third, transformers empirically trained to do ICL are shown to memorize tasks through the higher-order statistics of the training tasks that the kernel-based theory predicts, suggesting it applies even to models capable of feature learning.
Matrix factorization (i.e., problems of the form $\min_{\mathbf{P},\mathbf{Q}} \|\mathbf{M}^\star - \mathbf{P}^\top\mathbf{Q}\|_\mathrm{F}^2$) is a minimal learning problem that exhibits both nonlinear parameter dynamics and representation learning. In this setting, we study how parameter trajectories under the Muon optimizer differ from those of gradient descent. We identify three main dynamical differences: (1) Muon avoids the slow saddle-to-saddle dynamics of GD from small initialization. Muon instead learns all the top singular vectors of $\mathbf{M}^\star$ at the same rate, and they saturate in order of increasing (rather than decreasing) singular value. (2) Muon does not diverge even when the learning rate exceeds the critical threshold set by the local loss sharpness. This frees the learning rate from the condition number of the problem, enabling rapid convergence via exponential learning rate annealing. (3) Once the weights are aligned with each other and the target, Muon flow conserves the matrix quantity $\sqrt{\mathbf{P}^\top \mathbf{P}}-\sqrt{\mathbf{Q}^\top \mathbf{Q}}$, while gradient flow is known to conserve the matrix $\mathbf{P}^\top\mathbf{P} - \mathbf{Q}^\top\mathbf{Q}$. Despite having distinct conserved quantities, both optimizers find the so-called balanced solution from vanishing initialization. When training from small random initialization, the weights spontaneously align early in training. We derive the alignment rates in simple settings and show that they predict the empirical alignment rates in general. Finally, we exploit structural properties of Muon to construct a learning rate schedule that achieves near-perfect alignment in only two optimization steps.