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arxiv: 2604.12751 · v1 · submitted 2026-04-14 · 🧮 math.OC · cs.SY· eess.SY

Finite-Time Optimization via Scaled Gradient-Momentum Flows

Yu Zhou , Mengmou Li , Masaaki Nagahara This is my paper

Pith reviewed 2026-05-10 15:35 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords finite-time convergencegradient-momentum flowscontinuous-time optimizationscaled dynamicsgradient dominanceHeavy-Ball methodfinite-time stability
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The pith

A state-dependent scaling mechanism turns classical gradient-momentum flows into globally finite-time convergent optimizers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework that adds state-dependent scaling to gradient-momentum dynamics in continuous time. This scaling enables flows like the Heavy-Ball method and proportional-integral flows to converge to the global minimum in finite time. The key is an explicit bridge between the gradient-dominance property of the objective and the finite-time stability of the scaled system. If this holds, it means existing continuous-time optimization methods can be modified simply to achieve faster, finite-time guarantees without losing their structure.

Core claim

The paper establishes that introducing a state-dependent scaling into gradient-momentum flows, such as Heavy-Ball-type and PI-type dynamics, achieves global finite-time convergence for optimization problems where the objective satisfies a gradient-dominance property. Explicit conditions are provided that connect the dominance parameter to the finite-time stability of the resulting dynamics.

What carries the argument

The state-dependent scaling mechanism applied to the gradient and momentum terms in the flow equations.

If this is right

  • Classical momentum methods can be adapted to finite-time convergence via this scaling.
  • Gradient-dominance functions now have provable finite-time rates in continuous time.
  • The framework provides a general way to modify dynamics for faster stability.
  • Numerical experiments confirm the theoretical finite-time behavior on test problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Discrete-time versions of these scaled flows might yield new accelerated optimization algorithms with finite-step convergence.
  • The approach could extend to other continuous-time methods like Nesterov acceleration if similar scaling is applied.
  • If gradient dominance holds approximately, the flows might still show rapid practical convergence.
  • Connections to Lyapunov-based finite-time stability theory suggest broader applications in control and optimization.

Load-bearing premise

The objective function satisfies a gradient-dominance property that allows the scaling to link it to finite-time stability of the dynamics.

What would settle it

A concrete function satisfying gradient dominance for which the scaled gradient-momentum dynamics fail to reach the minimizer in finite time.

Figures

Figures reproduced from arXiv: 2604.12751 by Masaaki Nagahara, Mengmou Li, Yu Zhou.

Figure 1
Figure 1. Figure 1: Time evolution of ∥θ(t)−θ ⋆∥ for the Rosenbrock function on a logarithmic scale. Left: comparison of different α values with β = γ = 0.5. Right: comparison of different (β, γ) pairs with α = −0.5. 0 2 4 6 8 10 Time 10−9 10−7 10−5 10−3 10−1 101 f(θ(t)) p = 1.5 p = 2 p = 3 0 2 4 6 8 10 Time 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 kθ(t)k p = 1.5 p = 2 p = 3 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates the behavior under different gradient￾dominance orders for the objective function (32). In all cases, the trajectories converge to the optimal solution in finite time. Moreover, the order of gradient dominance directly influences the settling time: smaller values of p lead to shorter settling times. 0 2 4 6 8 10 12 14 Time 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 kθ − θ ?k α = −0.25 α = −0.5… view at source ↗
read the original abstract

In this paper, we develop a scaled gradient-momentum framework for continuous-time optimization that achieves global finite-time convergence. A state-dependent scaling mechanism is introduced to enable classical dynamics, such as Heavy-Ball-type and proportional-integral (PI)-type flows, to attain finite-time convergence. We establish explicit conditions that bridge the gradient-dominance property of the objective function and finite-time stability of the proposed scaled dynamics. Numerical experiments validate the theoretical results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper develops a scaled gradient-momentum framework for continuous-time optimization that achieves global finite-time convergence. A state-dependent scaling mechanism is introduced to enable classical dynamics such as Heavy-Ball-type and proportional-integral (PI)-type flows to attain finite-time convergence. Explicit conditions are established that bridge the gradient-dominance property of the objective function and finite-time stability of the proposed scaled dynamics, with validation through numerical experiments.

Significance. If the bridging conditions and Lyapunov analysis hold, the work provides a systematic method to convert gradient-dominance assumptions into global finite-time stability for momentum-based continuous-time flows. This is significant for the field of continuous-time optimization, as it extends classical Heavy-Ball and PI dynamics with an explicit scaling construction that preserves equilibria while enforcing the finite-time inequality V̇ ≤ −c V^α (α < 1). The explicit conditions and numerical corroboration of settling times represent a concrete advance over qualitative finite-time results.

minor comments (3)
  1. [§3.2] §3.2: The construction of the state-dependent scaling function σ(x) is stated to be positive and continuous away from equilibrium, but the precise functional form and its dependence on the gradient-dominance constants could be made more explicit to facilitate verification of the Lyapunov derivative bound.
  2. [§4] §4: The numerical experiments report settling times that match the predicted finite-time bounds, but the choice of step-size discretization and the precise definition of 'settling time' (e.g., tolerance threshold) should be stated explicitly to allow reproducibility.
  3. [Introduction] The introduction would benefit from a short comparison table contrasting the proposed scaling approach with existing finite-time continuous-time methods (e.g., those based on homogeneous systems or terminal attractors) to clarify the novelty of the bridging conditions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of our contributions, and the recommendation for minor revision. The significance highlighted in the report aligns with our goals in developing the scaled gradient-momentum framework.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript constructs an explicit state-dependent scaling for Heavy-Ball and PI flows such that, under the external gradient-dominance hypothesis on the objective, the closed-loop Lyapunov derivative satisfies the finite-time inequality V̇ ≤ −c V^α. Equilibria remain exactly the critical points of the objective, the scaling is chosen positive and continuous away from equilibrium, and the bridging conditions are derived directly from these assumptions without any fitted parameter being relabeled as a prediction or any load-bearing step reducing to a self-citation. The numerical examples corroborate the predicted settling times but do not define the theoretical claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a state-dependent scaling that converts asymptotic flows into finite-time ones under gradient dominance; no free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption The objective function satisfies a gradient-dominance property.
    Invoked to bridge to finite-time stability of the scaled dynamics.

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