NuMuon: Nuclear-Norm-Constrained Muon for Compressible LLM Training

The rapid progress of large language models (LLMs) is increasingly constrained by memory and deployment costs, motivating compression methods for practical deployment. Many state-of-the-art compression pipelines leverage the low-rank structure of trained weight matrices, a phenomenon often associated with the properties of popular optimizers such as Adam. In this context, Muon is a recently proposed optimizer that improves LLM pretraining via full-rank update steps, but its induced weight-space structure has not been characterized yet. In this work, we report a surprising empirical finding: despite imposing full-rank updates, Muon-trained models exhibit pronounced low-rank structure in their weight matrices and are readily compressible under standard pipelines. Motivated by this insight, we propose NuMuon, which augments Muon with a nuclear-norm constraint on the update direction, further constraining the learned weights toward low-rank structure. Across billion-parameter-scale models, we show that NuMuon increases weight compressibility and improves post-compression model quality under state-of-the-art LLM compression pipelines while retaining Muon's favorable convergence behavior.

Trust, but verify

Decentralized AI agent networks, such as Gaia, allows individuals to run customized LLMs on their own computers and then provide services to the public. However, in order to maintain service quality, the network must verify that individual nodes are running their designated LLMs. In this paper, we demonstrate that in a cluster of mostly honest nodes, we can detect nodes that run unauthorized or incorrect LLM through social consensus of its peers. We will discuss the algorithm and experimental data from the Gaia network. We will also discuss the intersubjective validation system, implemented as an EigenLayer AVS to introduce financial incentives and penalties to encourage honest behavior from LLM nodes.