Optimizing Large Language Models with Chained LMOs
Muon has motivated a growing family of optimizers that compose multiple matrix normalizations, but these methods remain fragmented and lack a unified perspective. We introduce chained linear minimization oracles (chained LMOs), which cast these methods as compositions of LMOs. Despite their empirical success, many chains fall outside the standard LMO framework and can diverge on smooth convex objectives. To explain why composition can nevertheless help, we turn to linear associative memory and show that chaining can improve over Muon under anisotropic embeddings. Empirically, we propose TensorChain, a novel optimizer within the framework that stacks compatible weight matrices across diffe...
| Authors | Sungyoon Kim, Kaan Ozkara, Youngsuk Park |
|---|---|
| Primary category | cs.LG |
| Published | 7 October 2026 |
| Last updated at source | 7 October 2026 |
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