pith. sign in

arxiv: 2506.15968 · v3 · submitted 2025-06-19 · 🧮 math.OC

Convergence analysis of a Tikhonov regularized inertial dynamical system and algorithm for convex optimization problems

Xiangkai Sun , Guoxiang Tian , Huan Zhang This is my paper

Pith reviewed 2026-05-19 09:39 UTC · model grok-4.3

classification 🧮 math.OC
keywords Tikhonov regularizationinertial dynamical systemsconvex optimizationconvergence analysisproximal gradient algorithmHessian-driven dampingminimum-norm solution
0
0 comments X

The pith

A Tikhonov-regularized inertial dynamical system with time scaling and damping achieves fast convergence of function values and strong convergence to the minimum-norm minimizer under suitable parameter choices.

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

The paper analyzes a second-order inertial dynamical system that adds Tikhonov regularization to time scaling, asymptotically vanishing damping, and Hessian-driven damping. Under appropriate parameter settings it proves fast decay of the objective value along trajectories and weak convergence of the trajectory to a minimizer of a convex problem. Proper tuning of the regularization strength and damping coefficients yields both the accelerated rates and strong convergence specifically to the minimum-norm solution. The same analysis carries over to an inertial proximal gradient algorithm obtained by discretizing the continuous dynamics.

Core claim

Under suitable choices of the time-scaling function, damping coefficients, and Tikhonov regularization parameter, the trajectories of the proposed dynamical system satisfy fast convergence of the function value to the minimum while converging weakly to a minimizer; further tuning simultaneously delivers strong convergence of the trajectory to the minimum-norm solution of the convex optimization problem.

What carries the argument

Tikhonov-regularized second-order inertial dynamical system incorporating time scaling, asymptotically vanishing damping, and Hessian-driven damping.

If this is right

  • Fast convergence rates for the objective function value hold along trajectories of the continuous system.
  • Trajectories converge weakly to some minimizer of the convex problem.
  • Simultaneous fast rates and strong convergence to the minimum-norm solution are obtained by suitable parameter tuning.
  • The corresponding inertial proximal gradient algorithm inherits comparable convergence guarantees from the discretization.

Where Pith is reading between the lines

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

  • The continuous-time analysis supplies a principled way to select the discrete step-size and regularization sequence so that the algorithm automatically favors the minimum-norm solution.
  • The combination of vanishing damping and Hessian-driven terms may extend to other inertial schemes for problems with additional structure such as strong convexity or smoothness.
  • Numerical validation on standard test problems indicates that the predicted rates are observable in practice once the continuous parameters are mapped to discrete ones.

Load-bearing premise

The objective function is convex and the parameters can be chosen to satisfy the technical conditions required by the convergence theorems.

What would settle it

A concrete counter-example would be a convex function and a choice of parameters satisfying the stated conditions for which the generated trajectory fails to exhibit the claimed fast function-value decay or fails to converge weakly (or strongly, under the tuned regime) to the minimum-norm minimizer.

Figures

Figures reproduced from arXiv: 2506.15968 by Guoxiang Tian, Huan Zhang, Xiangkai Sun.

Figure 1
Figure 1. Figure 1: The behaviors of the trajectory generated by the dyna [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The behaviors of the trajectory generated by the dyna [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
read the original abstract

This paper deals with a Tikhonov regularized second-order inertial dynamical system that incorporates time scaling, asymptotically vanishing damping and Hessian-driven damping for solving convex optimization problems. Under appropriate setting of the parameters, we first obtain fast convergence results of the function value along the trajectory generated by the dynamical system. Then, we show that the trajectory generated by the dynamical system converges weakly to a minimizer of the convex optimization problem. We also demonstrate that, by properly tuning these parameters, both fast convergence rates of the function value and strong convergence of the trajectory towards the minimum norm solution of the convex optimization problem can be achieved simultaneously. Furthermore, we study convergence properties of an inertial proximal gradient algorithm obtained by the temporal discretization of the dynamical system. Finally, we present numerical experiments to illustrate the obtained 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

1 major / 2 minor

Summary. The manuscript analyzes a Tikhonov-regularized second-order inertial dynamical system incorporating time scaling, asymptotically vanishing damping, and Hessian-driven damping for convex optimization problems. It establishes fast convergence of the objective function value along trajectories, weak convergence of the trajectory to a minimizer, and claims that proper tuning of the parameters simultaneously yields both fast rates and strong convergence to the minimum-norm solution. The paper further derives an inertial proximal gradient algorithm via temporal discretization and presents numerical experiments.

Significance. If the central claims hold with constructive parameter schedules, the work would strengthen the literature on continuous-time inertial methods by demonstrating a single framework that combines fast function-value decay with strong convergence to the minimum-norm minimizer. The combination of Tikhonov regularization with Hessian-driven damping is technically interesting, and the discretization step plus numerical illustrations provide a bridge to practical algorithms.

major comments (1)
  1. Abstract and the statements of the main convergence theorems (likely §3–4): the claim that 'proper tuning' of time scaling, vanishing damping, Hessian-driven damping, and Tikhonov strength simultaneously achieves fast function-value rates and strong convergence to the minimum-norm solution is not supported by an explicit, verifiable parameter schedule. The underlying differential inequalities impose several coupled decay conditions whose compatibility for arbitrary convex objectives typically requires a priori bounds (e.g., on distance to the solution set or subgradient norms). Without a constructive choice that avoids such knowledge, the simultaneous-convergence result remains an existence statement rather than a practical guarantee.
minor comments (2)
  1. The precise functional forms chosen for the time-scaling and damping coefficients should be stated explicitly at the beginning of the analysis section to facilitate checking the integral conditions.
  2. In the numerical section, the specific parameter functions used in the experiments should be listed so that readers can reproduce the observed behavior.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and for identifying a point that can strengthen the clarity of our results. We address the major comment below and will incorporate the suggested improvements in the revised manuscript.

read point-by-point responses

  1. Referee: Abstract and the statements of the main convergence theorems (likely §3–4): the claim that 'proper tuning' of time scaling, vanishing damping, Hessian-driven damping, and Tikhonov strength simultaneously achieves fast function-value rates and strong convergence to the minimum-norm solution is not supported by an explicit, verifiable parameter schedule. The underlying differential inequalities impose several coupled decay conditions whose compatibility for arbitrary convex objectives typically requires a priori bounds (e.g., on distance to the solution set or subgradient norms). Without a constructive choice that avoids such knowledge, the simultaneous-convergence result remains an existence statement rather than a practical guarantee.

    Authors: We agree that the current statements would benefit from an explicit, constructive parameter schedule that can be verified a priori and does not rely on knowledge of the solution set or subgradient norms. In the revised version we will add a dedicated remark (or subsection) immediately after the statements of the main theorems that supplies one such schedule: for instance, we will take the time-scaling parameter α(t) = t, the vanishing damping coefficient β(t) = 3/t, the Hessian-driven damping coefficient γ(t) = 1/t, and the Tikhonov regularization strength ε(t) = 1/t². We will verify that these choices simultaneously satisfy all the decay conditions required by the differential inequalities while remaining independent of any a priori bounds on the problem data. This change will convert the simultaneous-convergence claim from an existence statement into an explicitly verifiable guarantee. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents a standard convergence analysis for a Tikhonov-regularized inertial dynamical system with time scaling, vanishing damping, and Hessian-driven damping. It derives fast function-value convergence along trajectories, weak convergence to a minimizer, and (under tuned parameters) strong convergence to the minimum-norm solution directly from the system equations, convexity of the objective, and differential inequalities or Lyapunov arguments. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; parameter conditions are stated as technical assumptions satisfied under appropriate choices, without renaming known results or smuggling ansatzes via prior self-work. The analysis remains independent of the target claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The results rest on convexity of the objective and on the ability to select parameters that meet the theorem hypotheses; no new entities are postulated and no data-fitting occurs.

free parameters (1)
  • time-scaling and damping coefficients
    Parameters must be chosen to satisfy the stated conditions for fast rates and strong convergence; they are not derived from the problem data.
axioms (1)
  • domain assumption The objective function is convex.
    Invoked throughout the convergence statements for the optimization problem.

pith-pipeline@v0.9.0 · 5666 in / 1224 out tokens · 34708 ms · 2026-05-19T09:39:00.986022+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Su, W., Boyd, S., Cand` es, E.: A differential equation for m odeling Nesterov’s accelerated gradient method. J. Mach. Learn. Res. 17: 5312-5354 (2016)

    work page 2016

  2. [2]

    Nesterov, Y.: A method of solving a convex programming pro blem with convergence rate O ( 1 k2 ) . Insov. Math. Dokl. 27: 372-376 (1983)

    work page 1983

  3. [3]

    Attouch, H., Chbani, Z., Riahi, H.: Fast proximal methord s via time scaling of damped inertial dynamics. SIAM J. Optim. 29: 2227-2256 (2019)

    work page 2019

  4. [4]

    Attouch, H., Bot ¸, R.I., Csetnek, E.R.: Fast optimization via inertial dynamics with closed-loop damping. J. Eur. Math. Soc. 25: 1985-2056 (2023)

    work page 1985

  5. [5]

    Attouch, H., Bot ¸, R.I., Nguyen, D.K.: Fast Convex optimi zation via time scale and averaging of the steepest descent. Math. Oper. Res (2024) https://doi.org/ 10.1287/moor.2023.0186

    work page doi:10.1287/moor.2023.0186 2024

  6. [6]

    Attouch, H., Chbani, Z., Riahi, H.: Combining fast inerti al dynamics for convex optimization with Tikhonov regularization. J. Math. Anal. Appl. 457: 1065-10 94 (2018)

    work page 2018

  7. [7]

    Xu, B., W en, B.: On the convergence of a class of inertial dy namical systems with Tikhonov regular- ization. Optim. Lett. 15: 2025-2052 (2021)

    work page 2025

  8. [8]

    Attouch, H., Balhag, A., Chbani, Z., Riahi, H.: Damped ine rtial dynamics with vanishing Tikhonov regularization: Strong asymptotic convergence towards th e minimum norm solution. J. Differ. Equ. 311: 29-58 (2022)

    work page 2022

  9. [9]

    L´ aszl´ o, S.C.: On the strong convergence of the trajecto ries of a Tikhonov regularized second-order dynamical system with asymptotically vanishing damping. J . Differ. Equ. 362: 355-381 (2023)

    work page 2023

  10. [10]

    Bagy, A.C., Chbani, Z., Riahi, H.: Fast convergence rate of values with strong convergence of trajectories via inertial dynamics with Tikhonov regularization terms a nd asymptotically vanishing damping. J. Nonlinear Var. Anal. 8: 691-715 (2024)

    work page 2024

  11. [11]

    Attouch, H., L´ aszl´ o, S.C.: Convex optimization via in ertial algorithms with vanishing Tikhonov regu- larization: Fast convergence to the minimum norm solution. Math. Meth. Oper. Res. 99: 307-347 (2024)

    work page 2024

  12. [12]

    Ren, H., Ge, B., Zhuge, X.: Fast convergence of inertial g radient dynamics with multiscale aspects. J. Optim. Theory Appl. 196: 461-489 (2023)

    work page 2023

  13. [13]

    Csetnek, E.R., Karapetyants, M.A.: Second-order dynam ics featuring Tikhonov regularization and time scaling. J. Optim. Theory Appl. 202: 1385-1240 (2024)

    work page 2024

  14. [14]

    Bot ¸, R.I., Csetnek, E.R., L´ aszl´ o, S.C.: Tikhonov regularization of a second order dynamical system with Hessian damping. Math. Program. 189: 151-186 (2021)

    work page 2021

  15. [15]

    Bagy, A.C., Chbani, Z., Riahi, H.: Strong convergence of trajectories via inertial dynamics combining Hessian driven damping and Tikhonov regularization for gen eral convex minimizations. Numer. Funct. Anal. Optim. 44: 1481-1509 (2023)

    work page 2023

  16. [16]

    Attouch, H., Balhag, A., Chbani, Z., Riahi, H.: Accelera ted gradient methods combining Tikhonov regularization with geometric damping driven by the Hessia n. Appl. Math. Optim. 88: 29 (2023)

    work page 2023

  17. [17]

    Alecsa, C.D., L´ aszl´ o, S.C.: Tikhonov regularization of a perturbed heavy ball system with vanishing damping. SIAM J. Optim. 31: 2921-2954 (2021)

    work page 2021

  18. [18]

    Karapetyants, M.A.: A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique. Comput. Optim. Appl. 8 7: 531-569 (2024)

    work page 2024

  19. [19]

    Zhong, G., Hu, X., Tang, M., Zhong, L.: Fast convex optimi zation via differential equation with Hessian- driven damping and Tikhonov regularization. J. Optim. Theo ry Appl. 203: 42-82 (2024)

    work page 2024

  20. [20]

    L´ aszl´ o, S.C.: Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization. Comput. Opti m. Appl. 90: 113-149 (2025)

    work page 2025