Loss-Parameterized Fisher Width Along Learning Trajectories

What the math gives to ML

The paper supplies a dynamical use of Fisher geometry rather than merely treating the Fisher matrix as a curvature preconditioner: the Gaussian width of a probe set can be tracked along optimization trajectories and compared at matched training loss. Its key transferable asset is the separation between loss level and branch-dependent Fisher geometry, illustrated by the substantially displaced Adam trajectory versus GD/SGD. A practical adaptation is to use matched-loss Fisher width as a trajectory-control signal, regularizing or selecting optimizer updates that remain near a reference branch while preserving the usual loss decrease.