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Least-time Gradient Flow
Alessandro Betti, Marco Gori, Stefano Melacci, Jinwei Zhao
TL;DR
Ordinary gradient flow does not reach a critical point in finite time, and minimizing arrival time alone allows schedules with infinite weight speed. The paper regularizes the schedule optimization and proves that its unique minimizer is a cycloid, with a 2/3 rate exponent near zero risk.
Problem
Ordinary gradient flow cannot reach a critical point in finite time, while minimizing arrival time alone admits constant schedules whose arrival time tends to zero as weight speed becomes infinite.
Source: §1 · pp. 1–2Method
The paper minimizes a schedule functional combining arrival time and a penalty on abrupt rate changes, proving existence and uniqueness through the direct method and strict convexity.
Source: §1 · pp. 2–3Results
The regularized functional has exactly one minimizer, a linearly scaled cycloid whose near-zero-risk rate has exponent 2/3.
Source: §1 · pp. 2–3Takeaways & Limitations
The optimal schedule has finite arrival time and vanishing rate at arrival, a combination described as a soft landing for the weights.
Source: §4 · p. 11The paper does not establish discretization performance or minimum-time guarantees in gradient evaluations; numerical cost depends on risk curvature near the minimum.
Source: §4 · p. 11
Abstract
from arXiv · showhide
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 $u^{*}(e)\sim(9/(2\lambda))^{1/3}e^{2/3}$ near zero risk: the exponent $2/3$ is the one found in \cite{betti2026holder} by a power-law ansatz, and it lies in the H\"older 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.
1. Introduction
Ordinary gradient flow does not reach a critical point in finite time, while minimizing arrival time alone permits arbitrarily fast schedules with infinite weight speed. The paper regularizes the schedule and proves a unique cycloidal minimizer with a 2/3 near-zero-risk rate.
1. Introduction
- Ordinary gradient flow can approach a critical point without reaching it in finite time, and a constant schedule's arrival time tends to zero only as its weight speed becomes infinite.The finite-time obstruction is stated for risks with locally Lipschitz gradients.Source: §1 · pp. 1–2
- The regularizer trades arrival time against roughness: its time term is 1/u, while λ prices abrupt rate changes.The schedule is constrained by u(0) = 0 so its rate vanishes at arrival.Source: §1 · p. 2
- The regularized problem has exactly one minimizer, a linearly scaled cycloid satisfying λu′′(e) = −1/u(e)^2, u(0) = 0, and u′(e0) = 0.The minimizer is continuous and strictly increasing on [0, e0], and smooth on (0, e0].Source: §1 · pp. 2–3
- The proof derives and verifies a candidate, then uses the direct method, strict convexity, positivity away from zero, and explicit Euler–Lagrange integration.The direct-method argument addresses the potential term when a minimizing sequence could converge to a function vanishing on a positive-measure set.Source: §1 · p. 3
- Near zero risk, the optimal rate has exponent 2/3, within the Hӧlder window (1/2, 1).The exponent was previously obtained by a power-law ansatz in.Source: §1 · pp. 2–3
2. The extremal: a cycloid
The Euler–Lagrange equation and endpoint condition reduce the optimization to a separable equation whose solution is a parametrized cycloid. The resulting profile is admissible, increases from zero to its maximum, and has a cusp whose behavior can be read from the parameterization.
2. The extremal: a cycloid
- For positive solutions, a minimizer should satisfy λu'' = −1/u^2 and the natural boundary condition at e0; since φ(e0) is free, that condition fixes the constant.Source: §2 · pp. 3–4
- Integrating that equation gives the parametric profile (1.7), a scaled cycloid identified with the brachistochrone curve.After rescaling, its differential equation matches the first integral for a bead sliding under gravity.Source: §2 · p. 4
- The parameterization defines a continuous, strictly increasing profile from u*(0) = 0 to u*(e0) = A, with finite energy.Its derivative is square integrable, and its reciprocal is integrable.Source: §2 · pp. 4–5
- Near zero risk, inverting the parameterization gives θ = (3πe/(4e0))^(1/3)(1 + O(e^(2/3))).This inversion follows from the small-θ expansion of sin 2θ.Source: §2 · p. 5
3. Proof of Theorem 1.1
The proof establishes existence and uniqueness, then shows positivity away from zero so the Euler–Lagrange equation applies and can be integrated. This identifies the minimizer as the parametrized cycloid and relates its time course to its landing condition.
3.1. Existence.
- A bounded-energy minimizing sequence has a uniformly convergent subsequence, and the zero-set lemma preserves the potential-term lower bound needed for existence.Lower semicontinuity of both terms yields a minimizer in X.Source: §3.1 · pp. 5–7
3.2. Uniqueness.
- Strict convexity of the reciprocal potential term makes F strictly convex on X, so its minimizer is unique.If two candidates differ, their midpoint has strictly lower energy.Source: §3.2 · p. 7
3.3. Monotonicity and positivity.
- The minimizer is nondecreasing and positive on (0, e0], which makes the Euler–Lagrange equation available away from zero.An interior zero would force it to vanish on an interval, contradicting finite energy.Source: §3.3 · pp. 7–8
3.4. The Euler–Lagrange equation.
- Positivity permits variations away from zero, yielding λu0'' = −1/u0^2 and smoothness on (0, e0].The proof differentiates the energy under such variations and then bootstraps regularity.Source: §3.4 · p. 8
3.5. Identification with the cycloid, and end of the proof
- The unique minimizer is the parametrized cycloid (1.7), as shown by integrating the Euler–Lagrange equation and matching the endpoint relation.The theorem also establishes continuity, strict monotonicity, and smoothness on (0, e0].Source: §3.5 · p. 9
- Along the optimal risk dynamics, the cycloid parameter θ decreases linearly in time as it runs from π/2 to zero.The risk reaches zero with zero speed; the cycloid takes twice as long as a constant schedule at its peak rate, the price of the landing condition.Source: §3.5 · p. 9
4. Consistency with the H¨older regime
The minimizing rate has a 2/3 cusp exponent, placing its terminal behavior inside the finite-energy Hӧlder window for power-law schedules. The paper relates this choice to soft weight arrival, the bowl geometry near a minimum, and scaling with λ.
4. Consistency with the H¨older regime
- Across power laws, α < 1/2 gives blowing-up arrival speed, α = 1/2 gives bounded positive speed, and α ≥ 1 yields only asymptotic arrival.In the interval 1/2 < α < 1, arrival is finite with vanishing speed and a Hӧlder terminal field.Source: §4 · pp. 9–10
- Near a nondegenerate minimum, the relation e ≍ r^2 halves the terminal-field exponent, making the 1/2 threshold invisible in risk dynamics.The power-law weight speed scales as e^(α−1/2).Source: §4 · pp. 9–10
- At the cusp, u*(e) ∼ (9/(2λ))^1/3 e^2/3, matching the power-law exponent found in.The minimizing profile is not itself a power law on the finite interval.Source: §4 · p. 10
- For α = 2/3, arrival is finite, weight speed vanishes, and the terminal field has Hӧlder exponent 1/3.The rate exponent lies in the interval (1/2, 1).Source: §4 · p. 10
- Changing λ rescales the rate by λ^-1/3 and time by λ^1/3 without changing the cusp exponent.Source: §4 · p. 10
- A power-law schedule has finite functional energy exactly when 1/2 < α < 1.The Dirichlet term excludes α ≤ 1/2, while the time term excludes α ≥ 1.Source: §4 · p. 10
- The pure power-law solution with exponent 2/3 misses the finite-interval boundary condition, so the minimizer follows its cusp then bends to a maximum at e0.The pure power law arrives sooner but incurs a larger penalty than the finite-interval minimizer.Source: §4 · p. 10
- The amplitude-penalized variant is outside the theorem: its existence argument remains, but its closed-form identification and cusp exponent are unresolved.Source: §4 · p. 11