BMuon: Budgeting Muon's Spectral Geometry for Structured Compressibility

Muon has emerged as a strong optimizer for neural network training and admits an interpretation as steepest descent under a spectral-norm geometry. Yet we find that Muon-trained Transformers can degrade sharply when their MLP hidden channels are pruned after training. To reconcile Muon's optimization performance with structured compressibility, we introduce Budget Muon (BMuon), which modifies Muon's spectral geometry so that hidden channels across blocks compete for a shared allocation budget. The resulting oracle is exactly normalized steepest descent under a family of norms that interpolate between Muon's geometry and structured sparse geometries. Our principal sparsity control is an interpretable global channel-allocation budget, rather than a setting-dependent sparsity-penalty coefficient. For linear factorization models, we show that Muon admits minimum-cost exact representations with redundant hidden width, whereas under the BMuon oracle geometry and a sufficiently scarce budget, every minimum-cost representation is width-minimal. For practical training, we derive a minorization-maximization surrogate for the exact allocation problem. On ImageNet-1K with ViT-S/16, BMuon improves over Muon under gradual structured MLP pruning, and its learned allocations provide a more effective pruning criterion than weight magnitude. On a 1B-parameter LLaMA-style model trained on FineWeb, BMuon improves over Muon under post-training MLP hidden-channel pruning.