发表机构
University of Padua; University of Siena; University of Florence（帕多瓦大学; 锡耶纳大学; 佛罗伦萨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出最小时间梯度流，通过正则化使风险按指定动力学精确衰减，证明最优解存在唯一且为线性缩放摆线，并给出接近零风险时的最优速率指数。
AI 中文摘要
通过在风险本身上规定梯度流的速率，由动力学$\dot w=-u(E(w))\nabla E(w)/\abs{\nabla E(w)}^{2}$，使得风险$e(t)=E(w(t))$精确地服从$\dot e=-u(e)$，无论景观$E$如何；从$e_0$达到零风险所需的时间为$\int_0^{e_0}\dd e/u(e)$。仅最小化这个时间是病态的，我们研究正则化问题$\inf\{\int_0^{e_0}(\tfrac\lambda2\abs{u'}^{2}+1/u)\,\dd e:\ u\in H^{1}(0,e_0),\ u\ge0,\ u(0)=0\}$，$\lambda>0$。我们证明最小化器存在、唯一，并且是线性缩放的摆线，并且我们展示了在接近零风险时最优速率表现为$u^{*}(e)\sim(9/(2\lambda))^{1/3}e^{2/3}$：指数$2/3$是在\cite{betti2026holder}中通过幂律假设发现的，它位于Hölder窗口$(\tfrac12,1)$内，在该窗口中到达在有限时间内发生且权重速度消失。证明遵循经典路线：通过直接方法证明存在性，通过严格凸性证明唯一性，最小化器在原点之外为正，以及欧拉-拉格朗日方程的显式积分。
英文摘要
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.
