Statistical Framework Unifies Adam, Shampoo and Muon Optimizers
The practical value lies in the unification: one statistical principle explains Adam, Shampoo and Muon, so practitioners can tune or combine these optimizers with a clearer theoretical map instead of treating them as folklore. Engineers training GPT-scale models get concrete alternatives, since MND beats Distributed Shampoo and KL-Shampoo on stability and runtime, and MND-Muon wins 21 of 24 paired runs against Muon. Researchers in optimizer design can extend the covariance-approximation framework to new preconditioners.