Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory
This work provides a quantitative understanding of spectral optimizers for researchers in machine learning, though it is incremental as it builds on existing methods in a tractable model.
The paper tackled the problem of understanding the performance advantage of spectral optimizers like Muon in large-scale language model training by analyzing the linear associative memory problem, showing that Muon achieves significantly higher storage capacity and faster initial recovery rates than SGD, with experiments validating the predicted scaling laws.
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.