Least-time Gradient Flow
Abstract
Prescribing the speed of gradient flow on the risk itself, by the dynamics ẇ=−u(E(w))∇ E(w)/abs∇ E(w)2, makes the risk e(t)=E(w(t)) obey ė=−u(e) exactly, whatever the landscape~E; the time needed to reach zero risk from e0 is ∫0e0dd e/u(e). Minimizing this time alone is ill posed, and we study the regularized problem inf{∫0e0(λ/2absu'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 \cite{betti2026holder} by a power-law ansatz, and it lies in the Hölder window (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.
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BibTeX
@misc{betti2026leasttime,
title = {{Least-time Gradient Flow}},
author = {Alessandro Betti and Marco Gori and Stefano Melacci and Jinwei Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01426},
eprint = {2610.01426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01426},
}
RIS
TY - UNPB
TI - Least-time Gradient Flow
AU - Alessandro Betti
AU - Marco Gori
AU - Stefano Melacci
AU - Jinwei Zhao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01426
UR - https://arxiv.org/abs/2610.01426
AB - Prescribing the speed of gradient flow on the risk itself, by the dynamics ẇ=−u(E(w))∇ E(w)/abs∇ E(w)^2, makes the risk e(t)=E(w(t)) obey ė=−u(e) exactly, whatever the landscape~E; the time needed to reach zero risk from e_0 is ∫_0^e_0dd e/u(e). Minimizing this time alone is ill posed, and we study the regularized problem inf{∫_0^e_0(λ/2absu'^2+1/u) dd e: u∈ H^1(0,e_0), 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))^1/3e^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 (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.
ER -