The Muon optimizer was making the first mixture-of-experts layer harder to balance, and it was not the router's fault
AspireAI community · arXiv
Open mixture-of-experts recipes disagree on whether the Muon optimizer should also train the router. Measuring the span of per-expert routing biases in 55M and 108M active-parameter models, this paper finds Muon raised that span at every seed - roughly doubling it on average in the first layer while other layers barely moved. The cause was Muon on the rest of the network rather than the router's optimizer: after warmup the busiest expert was taking four to seven times its fair share. Two cheap fixes worked - subtracting the batch mean from the router inputs over the first 3,000 steps removed the overload with little change in loss, and halving Muon's learning rate removed most of the rise on the smaller model. Moving only the router to AdamW instead made loads shift more during warmup and slightly raised loss, so at this scale the authors favour keeping the router on Muon. The caveat is scale: these are deliberately tiny models, and the paper ships 34 pages of tables, figures and released runs.
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Two sparse-autoencoder analyses have resisted scaling: matching semantically similar features across layers, and compressing large feature circuits into interpretable supernodes. This paper argues both are the same underlying problem, estimating semantic distances between SAE features that sit on different activation manifolds. Its representation treats each feature not as a single decoder vector but as an activation-weighted distribution over the hidden states expressing it, projects those distributions into a shared reference space, and compares them with Wasserstein distance. The representation is proved invariant to activation rescaling and stable under perturbation, and recovers true matches under finite-sample margin conditions. Empirically it outperformed decoder-vector and LLM-based baselines and captured subtle functional distinctions between related features, and it compressed large feature circuits into interpretable supernodes automatically. The limitation is that this is interpretability tooling: the paper demonstrates better feature matching and circuit compression, not that downstream interpretability claims become more accurate.