On power-law covariance least squares problems, SignSVD (Muon) and SignSGD (Adam proxy) show three phases of relative performance depending on data exponent α and target exponent β.
High-dimensional limit theorems for SGD: Momentum and Adaptive Step-sizes
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Dynamic preconditioning preserves the Polyak-Ruppert CLT for averaged SGD if the preconditioner stabilizes at rate β > (α + 1)/2.
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