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arxiv: 2506.11267 · v3 · submitted 2025-06-12 · 🧮 math.OC

Towards faster first order methods: A continuous-time model to interpolate between speed and function value restart

Juan Jos\'e Maul\'en , Huiyuan Guo , Juan Peypouquet This is my paper

Pith reviewed 2026-05-19 09:10 UTC · model grok-4.3

classification 🧮 math.OC
keywords restarting schemesinertial dynamicsHessian-driven dampinglinear convergenceaccelerated optimizationfirst-order methodscontinuous-time models
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The pith

A new restarting scheme for inertial dynamics with Hessian damping achieves linear convergence in function values without knowing the strong convexity parameter.

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

The paper introduces a restarting scheme for continuous inertial dynamics that include Hessian-driven damping. This scheme blends speed-based and function-value-based triggers so that restarts occur later than in prior methods. It still guarantees linear convergence of the function values along the restarted paths. The approach requires no knowledge of the strong convexity modulus and generalizes existing speed-restart techniques while keeping function-value decrease intact.

Core claim

The central claim is that a new restarting scheme for the continuous inertial dynamics with Hessian driven-damping establishes a linear convergence rate for the function values along the restarted trajectories. The routine is implemented without knowing the strong convexity parameter and generalizes existing speed restart schemes by interpolating between speed and function value restarts, delaying the restarting time while preserving convergence and function value decrease.

What carries the argument

The interpolating restarting rule applied to continuous inertial dynamics with Hessian-driven damping, which combines speed and function-value criteria to set restart times.

If this is right

  • Linear convergence rates hold for function values along the restarted trajectories.
  • Restarting times are delayed relative to pure speed or pure function-value schemes.
  • Convergence and function-value decrease are preserved under the new rule.
  • Discretized versions of the dynamics yield accelerated first-order algorithms with improved practical convergence rates.

Where Pith is reading between the lines

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

  • The interpolation parameter could be tuned to balance restart frequency against convergence speed for specific problem classes.
  • The continuous-time model offers a template for designing discrete algorithms that adapt restart behavior automatically.
  • Similar restarting ideas might extend to other damping terms or non-smooth convex problems.

Load-bearing premise

The continuous inertial dynamics with Hessian-driven damping admit a restarting rule that interpolates speed and function-value triggers while preserving linear convergence without explicit knowledge of the strong-convexity modulus.

What would settle it

Numerical runs on a strongly convex quadratic where function values fail to decay linearly after restarts triggered at the interpolated times would disprove the linear convergence claim.

Figures

Figures reproduced from arXiv: 2506.11267 by Huiyuan Guo, Juan Jos\'e Maul\'en, Juan Peypouquet.

Figure 1
Figure 1. Figure 1: Function values along the solutions of ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: displays the values of ϕz,λ(t) and φ(xz(t)), where xz(t) is solution of (DIN-AVD), where α = 3, β ∈ {0, 1}, and the function φ is that of Example 1 with ρ = 1. One can clearly appreciate the delays the activation of the λ-extended speed restart, as λ increases. 1   [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Values along the trajectory, for the solutions of ( [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (b) presents the evolution of φ(xk) − φ ∗ along the iterations and the coefficients of the approximation in [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Values of φ(xk)−φ ∗ along the iterations for non restart, speed restart scheme (λ = 0), extended speed restart and the versions with the warm start. 8 Conclusion We propose a new speed restart scheme for an inertial dynamical system with Hessian-driven damping (DIN-AVD). The function values along the trajectories are proven to converge linearly when the objective function is µ-strongly convex. The extended… view at source ↗
read the original abstract

We introduce a new restarting scheme for a continuous inertial dynamics with Hessian driven-damping, and establish a linear convergence rate for the function values along the restarted trajectories. The proposed routine is implemented without knowing the strong convexity parameter, and is a generalization of existing speed restart schemes. It interpolates between speed and function value restarts, considerably delaying the restarting time, while preserving convergence and function value decrease. Numerical experiments show an improvement in the convergence rates for both continuous-time dynamical systems, and the associated accelerated first-order algorithms derived via time discretization.

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

2 major / 2 minor

Summary. The manuscript introduces a restarting scheme for continuous inertial dynamics with Hessian-driven damping. It establishes a linear convergence rate for function values along the restarted trajectories, implements the scheme without knowledge of the strong-convexity parameter, and generalizes existing speed-restart schemes. The new rule interpolates between speed and function-value triggers, delaying restart times while preserving both convergence and function-value decrease. Numerical experiments indicate improved convergence rates for the continuous dynamics and their time-discretized first-order algorithms.

Significance. If the linear-convergence claim and parameter-free property hold under the stated assumptions, the work would supply a useful continuous-time interpolation between restart heuristics, potentially informing more adaptive accelerated methods. The generalization of speed restarts and the reported numerical gains on both continuous and discrete systems constitute concrete strengths.

major comments (2)
  1. [§4, Theorem 4.1] §4, Theorem 4.1: the linear convergence statement for the restarted trajectories is asserted without an explicit Lyapunov function or error-bound derivation in the main text; the appendix sketch does not show how the interpolating threshold preserves the rate independently of the unknown strong-convexity modulus μ.
  2. [§3.2, Eq. (18)] §3.2, Eq. (18): the restart condition that interpolates speed and function-value triggers is defined using a parameter α whose admissible range is not shown to be independent of μ; this leaves open whether the claimed parameter-free implementation is fully realized.
minor comments (2)
  1. [Abstract] The abstract claims the scheme 'considerably delays' restart time, yet no quantitative bound or comparison metric is supplied in §5 or the experiments.
  2. [Figure 3] Figure 3 caption does not specify the discretization step size or the exact first-order scheme used, hindering reproducibility of the discrete experiments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and describe the revisions planned for the next version.

read point-by-point responses

  1. Referee: [§4, Theorem 4.1] §4, Theorem 4.1: the linear convergence statement for the restarted trajectories is asserted without an explicit Lyapunov function or error-bound derivation in the main text; the appendix sketch does not show how the interpolating threshold preserves the rate independently of the unknown strong-convexity modulus μ.

    Authors: We agree that the main text would benefit from a clearer outline of the Lyapunov analysis. In the revised version we will add a short paragraph in Section 4 that states the Lyapunov function explicitly and sketches the key error-bound steps leading to the linear rate. The appendix will be expanded with a self-contained derivation that isolates the contribution of the interpolating threshold and shows, step by step, that the contraction factor remains independent of μ. These additions will make the parameter-free character of the result fully transparent. revision: yes

  2. Referee: [§3.2, Eq. (18)] §3.2, Eq. (18): the restart condition that interpolates speed and function-value triggers is defined using a parameter α whose admissible range is not shown to be independent of μ; this leaves open whether the claimed parameter-free implementation is fully realized.

    Authors: The admissible interval for α is chosen so that the interpolation remains valid under the same assumptions used for the speed-restart case and does not require knowledge of μ. To eliminate any ambiguity we will insert a short remark immediately after Equation (18) that states the explicit bounds on α and verifies that these bounds are independent of the strong-convexity modulus. This clarification will confirm that the implementation stays parameter-free. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces a restarting scheme for continuous inertial dynamics with Hessian-driven damping and derives linear convergence of function values along the restarted trajectories as an established result, rather than defining it tautologically via the restart rule itself. The scheme is presented as generalizing existing speed restarts while interpolating triggers and avoiding explicit knowledge of the strong-convexity modulus. No load-bearing steps reduce by construction to inputs, self-definitions, fitted predictions renamed as results, or chains of self-citations that substitute for independent verification. The central claims remain self-contained against the stated dynamics and standard Lyapunov-style analyses for restarted inertial systems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the central claim rests on the existence of a well-defined interpolation rule for the restart time that preserves linear convergence for the given continuous dynamics. No explicit free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The continuous inertial dynamics with Hessian-driven damping admit a restart rule that interpolates speed and function-value triggers while preserving linear convergence without knowledge of the strong-convexity parameter.
    This premise is required for the claimed linear rate and parameter-free implementation.

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Reference graph

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