Spectral optimizers such as Muon have recently shown strong empirical performance in large-scale language model training, but the source and extent of their advantage remain poorly understood. We study this question through the linear associative memory problem, a tractable model for factual recall in transformer-based models. In particular, we go beyond orthogonal embeddings and consider Gaussian inputs and outputs, which allows the number of stored associations to greatly exceed the embedding dimension. Our main result sharply characterizes the recovery rates of one step of Muon and SGD on the logistic regression loss under a power law frequency distribution. We show that the storage capacity of Muon significantly exceeds that of SGD, and moreover Muon saturates at a larger critical batch size. We further analyze the multi-step dynamics under a thresholded gradient approximation and show that Muon achieves a substantially faster initial recovery rate than SGD, while both methods eventually converge to the information-theoretic limit at comparable speeds. Experiments on synthetic tasks validate the predicted scaling laws. Our analysis provides a quantitative understanding of the signal amplification of Muon and lays the groundwork for establishing scaling laws across more practical language modeling tasks and optimizers.

Similarly,thetexture ofacloth mayconfuse classifying clothing categories (shirts vs. skirts), but it is crucial for classifying fashion attributes (spottedvs.

Using the method of auxiliary coordinates, we derive a training algorithm that works by alternating steps that train an auxiliary embeddingwith steps that train the mapping.

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We thank all reviewers for their valuable comments. Below, we first address the common concerns and then answer the detailed questions. It leads to smaller bias (see Fig. A), which also agrees with Thm. 2. First, introducing latent variables can improve the sample quality (w.r.t. Indeed, we update Tab. 2 and obtain Tab. As stated in L290, a similar protocol is adopted in MDSM [34].