Least-time Gradient Flow

Prescribing the speed of gradient flow on the risk itself, by the dynamics dot w=-u(E(w))nabla E(w)/absnabla E(w)2, makes the risk e(t)=E(w(t)) obey dot e=-u(e) exactly, whatever the landscape~E; the time needed to reach zero risk from e0 is int0e0dd e/u(e). Minimizing this time alone is ill posed, and we study the regularized problem infint0e0(tfracλ2absu'2+1/u),dd e: u∈ H1(0,e0), u≥0, u(0)=0, λ>0. We prove that the minimizer exists, is unique, and is a linearly scaled cycloid, and we show that the optimal rate behaves like u*(e)~(9/(2λ))1/3e2/3 near zero risk: the exponent 2/3 is the one found in citebetti2026holder by a power-law ansatz, and it lies in the H"older window (tfrac12,1) where the arrival is in finite time with vanishing weight speed. The proof follows the classical route: existence by the direct method, uniqueness by strict convexity, positivity of the minimizer away from the origin, and the explicit integration of the Euler-Lagrange equation.