We study how the sign operator interacts with gradient noise to shape the learning dynamics and scaling behavior of sign-based optimizers. In power-law kernel regression with diagonal features, we establish a functional scaling law (FSL) for SignSGD under general learning rate schedules.

The FSL describes the learning dynamics through two processes: signal learning and noise forgetting. The key mechanism is noise-induced preconditioning: gradient noise smooths the sign operator, inducing an effective preconditioned SGD.

Noise-curvature alignment enables this preconditioning to transform the effective Hessian from $\mathbf{H}$ to $\mathbf{H}^{1/2}$. The resulting dynamics exhibit faster signal learning but slower noise forgetting than SGD.

This tradeoff yields a regime-dependent comparison with SGD: (i) SignSGD achieves improved data efficiency in the hard-task regime; (ii) both optimizers achieve the same data efficiency in the easy-task regime, but SignSGD maintains this efficiency at larger batch sizes, requiring fewer optimization steps for a fixed data budget. Power-law kernel regression experiments support the proposed mechanism and predicted scaling behavior.

LLM pre-training experiments further suggest that the SignSGD FSL better captures the batch-size dependence of loss dynamics for sign-based optimizers than the original SGD FSL.