Least-time Gradient Flow

2026-10-01

AI Summary

This study addresses the challenge of minimizing the time for gradient flow to reach zero risk, which is hindered by the ill-posedness of the primal problem and difficulties in designing optimal rate functions. To overcome this, we propose a regularized variational model that solves for the optimal velocity field, ensuring the risk strictly follows a prescribed dynamical equation. The solution is derived by integrating the direct method of the calculus of variations, strict convexity analysis, explicit integration of the Euler–Lagrange equation, and Hölder continuity theory. We prove that the optimal solution exists uniquely as a scaled cycloid and reveal that the optimal rate exhibits an $e^{2/3}$ power-law decay near zero risk. Furthermore, we verify the finite-time arrival mechanism under specific exponents, thereby completing the convergence theory for non-smooth landscapes.

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 $\inf\{\int_0^{e_0}(\tfrac\lambda2\abs{u'}^{2}+1/u)\,\dd e:\ u\in H^{1}(0,e_0),\ u\ge0,\ u(0)=0\}$, $λ>0$. We prove that the minimizer exists, is unique, and is a linearly scaled cycloid, and we show that the optimal rate behaves like $u^{*}(e)\sim(9/(2λ))^{1/3}e^{2/3}$ 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 $(\tfrac12,1)$ 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.

Problem

• gradient flow
• risk minimization
• optimal rate
• regularization
• calculus of variations

Innovation

• Gradient Flow
• Calculus of Variations
• Cycloid
• Optimal Convergence Rate
• Regularization