Least-time Gradient Flow

Abstract

Prescribing the speed of gradient flow on the risk itself, by the dynamics $\dot w=-u(E(w))\nabla E(w)/\abs{\nabla E(w)}^{2}$, 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 $e_0$ is $\int_0^{e_0}\dd e/u(e)$. Minimizing this time alone is ill posed, and we study the regularized problem $\inf\{\int_0^{e_0}(\tfrac\lambda2\abs{u'}^{2}+1/u)\,\dd e:\ u\in H^{1}(0,e_0),\ u\ge0,\ u(0)=0\}$, $\lambda >0$. We prove that the minimizer exists, is unique, and is a linearly scaled cycloid, and we show that the optimal rate behaves like $u^{\ast }(e)\sim (9/(2\lambda ){)}^{1/3}{e}^{2/3}$ near zero risk: the exponent $2/3$ is the one found in \cite{betti2026holder} by a power-law ansatz, and it lies in the Hölder window $(\frac{1}{2},1)$ 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.