Ivan Kozyrev

Alexey Kravatskiy, Ivan Kozyrev, Nikolai Kozlov et al.

In this article, we explore the use of various matrix norms for optimizing functions of weight matrices, a crucial problem in training large language models. Moving beyond the spectral norm underlying the Muon update, we leverage duals of the Ky Fan $k$-norms to introduce a family of Muon-like algorithms we name Fanions, which are closely related to Dion. By working with duals of convex combinations of the Ky Fan $k$-norms with either the Frobenius norm or the $l_\infty$ norm, we construct the families of F-Fanions and S-Fanions, respectively. Their most prominent members are F-Muon and S-Muon. We complement our theoretical analysis with an extensive empirical study of these algorithms across a wide range of tasks and settings, demonstrating that F-Muon and S-Muon consistently match Muon's performance, while outperforming vanilla Muon on a synthetic linear least squares problem.

Alexander Osinsky, Ivan Kozyrev

The paper considers the problem of finding a submatrix $X_{\mathcal{S}} \in \mathbb{R}^{m \times k}$ in a matrix $X \in \mathbb{R}^{m \times n}$, such that the spectral or Frobenius norm of $X_{\mathcal{S}}^† X$ is limited, which guarantees it provides a good representation of the whole matrix. Such bounds can be reached by applying greedy algorithms, maximizing the submatrix volume. We suggest a modification of a greedy volume maximization, which performs column exchanges asymptotically faster for $n \gg m$ than the known alternatives, while guaranteeing the same bounds on $X_{\mathcal{S}}^† X$. In addition, we prove a new upper bound on the number of required exchanges, which is applicable to the new algorithm as well as to other greedy volume maximization algorithms.