Least-time Gradient Flow

arXiv:2610.01426v1 Announce Type: new Abstract: Prescribing the speed of gradient flow on the risk itself, by the dynamics ot w=-u(E(w))nabla E(w)/abs{nabla E(w)}^{2}, makes the risk e(t)=E(w(t)) obey ot e=-u(e) exactly, whatever the landscape~E; the time needed to reach zero risk from e_0 is int_0^{e_0}d e/u(e). Minimizing this time alone is ill posed, and we study the regularized problem inf{int_0^{e_0}(fraclambda2abs{u'}^{2}+1/u),d e: uin H^{1}(0,e_0), uge0, 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^{*}(e)sim(9/(2lambda))^{1/3}e^{2/3} near zero risk: the exponent 2/3 is the one found in ite{betti2026holder} by a power-law ansatz, and it lies in the Holder window (frac12,1) 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.

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Source: arXiv cs.LG | 2026-10-02