Strong convergence and fast rates for systems with Tikhonov regularization

Ern\"o Robert Csetnek; Szil\'ard Csaba L\'aszl\'o

REVIEW 2 major objections 5 minor 23 references

Strong convergence and fast rates for systems with Tikhonov regularization

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For a monotone equation Ax=0 in a Hilbert space, the paper proves that a second-order Tikhonov-regularized dynamics converges strongly to the minimal-norm solution at rates close to those of unregularized accelerated flows.

desk verdict The new system is worth attention, but the main proof has a missing β in the discriminant analysis that breaks the negativity step for αβ≤1. read the letter →

arxiv 2411.17329 v1 pith:73OUTTT6 submitted 2024-11-26 math.OC math.DSmath.FA

Ernö Robert Csetnek , Szilárd Csaba László This is my paper · ORCID

classification math.OCmath.DSmath.FA MSC 37N4046N1049M3065K0565K1090B5090C25

keywords monotoneequationTikhonovregularizationstrongconvergenceminimal-normsolutionsecond-orderdynamicalsystemprimal-dualdynamicsratesNewton-typecorrection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a second-order damped dynamical system designed to solve the monotone equation $Ax=0$ in a real Hilbert space, where $A$ is monotone and continuous and its zero set is nonempty. The system adds a Tikhonov term $c t^{-(2q+s)}x(t)$, a vanishing damping $\alpha/t^q$, and a Newton-type correction $\beta t^q \frac{d}{dt}A(x(t))$. The main theorem claims that the trajectory converges strongly to the minimal-norm solution $x^*$, with rates $\|A(x(t))\|=O(t^{-1-q+s})+O(t^{-(4q+s)/2})$ and $\|\dot{x}(t)\|=O(t^{-1+s})+O(t^{-(2q+s)/2})$. These rates are close to those known for the corresponding system without Tikhonov regularization, while the strong convergence selects the minimal-norm solution. The same result is applied to a primal-dual system for linearly constrained convex optimization, yielding strong convergence of primal and dual trajectories together with rates for the feasibility measure and the objective gap.

What carries the argument

The carrying object is the energy functional $E(t)$, a Lyapunov function whose main ingredients are $v(t)=\frac{1}{2}\|b(x(t)-x_t)+t^q(2\dot{x}(t)+\beta t^q A(x(t)))\|^2$, a monotonicity term $u(t)$ built from $\langle A(x(t)),x(t)-x_t\rangle$, and weighted squared norms of $A(x(t))$, $x(t)$, and $x(t)-x_t$. Differentiating $E(t)$ along solutions of (DS) produces a differential inequality of the form $\dot{E}(t)+\frac{K}{t^{q+s}}E(t)\le C t^{q+s-2}+C t^{-q-2s}-\frac{d}{dt}(\cdot)$, and the algebraic core of the proof is choosing auxiliary constants so that the quadratic form in $\|\dot{x}\|^2$, $\langle A,\dot{x}\rangle$, and $\|A\|^2$ is negative for large $t$. The Tikhonov path $x_t$, the unique zero of $A+\frac{c}{t^{2q+s}}\mathrm{Id}$, is what carries the strong convergence toward the minimal-norm solution.

What would settle it

Take $A(x,y)=(-y,x)$ on $\mathbb{R}^2$, a monotone Lipschitz, non-cocoercive operator whose only zero is the origin, fix parameters such as $\alpha=2$, $q=0.5$, $s=0.25$, $\beta=1$, $\gamma=1$, and $c$ below the stated bound, and integrate (DS) from different initial conditions; if $\|(x(t),y(t))\|$ does not decay as $O(t^{q+s-1})+O(t^{-s/2})$ or $\|A(x(t))\|$ does not match the stated rate, Theorem 1 fails. Alternatively, exhibiting a monotone continuous operator for which no strong global solution of (DS) exists would falsify the regularity premise under which the rates are derived.

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Extended reading notes

Core claim

The central assertion is Theorem 1: for parameters $\alpha>1$, $q,\beta,\gamma,s,c>0$ with $0<q+s<1$ and $c<\frac{8\alpha(\alpha-1)\gamma}{\alpha^2\beta^2+8(\alpha-1)\beta}$, every trajectory of the system (DS) satisfies $\|x(t)-x_t\|=O(t^{q+s-1})+O(t^{-s/2})$, and consequently $x(t)\to x^*$ strongly as $t\to+\infty$, where $x_t$ is the unique zero of $A+\frac{c}{t^{2q+s}}\mathrm{Id}$ and $x^*$ is the projection of $0$ onto the zero set of $A$. The proof constructs a Lyapunov functional $E(t)$ built from the velocity, the operator value, and the distance to the Tikhonov path $x_t$, and it shows that along the trajectory $\dot{E}(t)+\frac{K}{t^{q+s}}E(t)$ is bounded by integrable remainder terms. Integrating this differential inequality yields the stated rates for $\|A(x(t))\|$ and $\|\dot{x}(t)\|$. Because the dynamics evaluates $A$ in a forward way rather than through a resolvent, it is directly implementable whenever the operator can be evaluated along a trajectory.

Load-bearing premise

The proof needs the trajectory $x(t)$ to exist for all $t\ge t_0$ with $t\mapsto A(x(t))$ absolutely continuous, but the paper only guarantees such a strong global solution when $A$ is Lipschitz continuous, while the abstract and theorem state the weaker hypothesis that $A$ is monotone and continuous. If a monotone continuous but non-Lipschitz operator does not admit such a solution, the claimed rates have no trajectory to apply to.

Editorial extensions

If this is right

Choosing $s=\frac{2(1-q)}{3}$ gives $\|\dot{x}(t)\|=O(t^{-(1+2q)/3})$ and $\|A(x(t))\|=O(t^{-(1+5q)/3})$; for $q$ close to $1$ these approach the rates $O(t^{-1})$ and $O(t^{-1-q})$ of the unregularized system in [15].
For the primal-dual system (34), both the primal trajectory $x(t)$ and the dual trajectory $y(t)$ converge strongly to the minimal-norm saddle point, with $\|Bx(t)-b\|$ and $|f(x(t))-f(x^*)|$ decaying at the orders stated in Theorem 2.
Because the dynamics uses forward evaluations of $A$ instead of resolvent-based regularizations, it offers a tractable continuous-time model for monotone inclusions where cocoercivity fails, such as saddle-point and skew-operator problems.
The Tikhonov term acts as a selection device: irrespective of the starting point, the strong limit is always the projection of $0$ onto the zero set of $A$, the minimal-norm solution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: the same Tikhonov-plus-Newton structure could be discretized into an algorithm with variable step sizes; the continuous rates suggest iterate convergence at similar orders, but the correction term $\beta t^q\frac{d}{dt}A(x(t))$ would require numerical differentiation or a discrete analogue.
Testable prediction: for a monotone affine map $A(x)=Mx$ with symmetric positive semidefinite $M$, the theorem predicts that the trajectory approaches the projection of $0$ onto $\ker M$ at exactly the stated rates; this case can be solved in closed form and checked numerically.
Connection: in saddle-point problems, weak convergence alone leaves the limit ambiguous, so the strong convergence to the minimal-norm saddle point gives a principled selection mechanism that could be transferred to discrete primal-dual algorithms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper has a genuinely new combination: forward-evaluated Newton-type monotone dynamics with Tikhonov regularization and correlated vanishing damping. The claimed strong convergence to the minimal-norm solution with rates is not in the cited papers, and the Lyapunov framework is laid out clearly. The application to primal-dual systems is natural. Credit where due: this is a serious attempt at a useful result.

The main problem is in the proof of the key negativity step leading to (30). Equation (26) is missing a factor β in the R1R3 term of the discriminant. When the β is restored, the leading coefficient of R2²−4R1R3 becomes ((2b−2α)β−4)² − 4β(2b+s3b+s5b−4α)(−2+s1+bs4+Ks6b/2). Taking b=α and s1=s3=s4=s6=0 gives 16(1−αβ)+8αβs5. This is strictly positive for every allowed s5>0 when αβ<1, and nonnegative when αβ=1. The theorem's assumptions only require α>1 and β>0, so αβ≤1 is an allowed region. For example, α=2, β=0.1. In that region the quadratic form in ẋ and A(x) cannot be negative definite, so the claimed estimate (30) is not established. This is not a harmless typo; it changes the sign of the discriminant as β varies. The proof could possibly be repaired by restricting to αβ>1 or redesigning the Lyapunov function, but as written Theorem 1 is not proven for a substantial parameter set.

The secondary issue the reader raised is also real: Theorem 1 assumes a strong global solution without restating the regularity conditions. The paragraph before the theorem mentions Lipschitz continuity suffices, but the theorem only assumes A continuous and monotone. That should be fixed in any revision.

The paper is worth a serious referee because the question is meaningful and the approach has merit, but the current proof does not support the claims for the stated parameters. A referee should demand a corrected discriminant calculation and an explicit resolution of the αβ≤1 case, plus a clean existence statement.

Who is this for? People working on continuous-time dynamics for monotone inclusions and Tikhonov regularization. They will find the system and rates interesting, but they should not rely on the proof until the gap is fixed.

Referee Report

2 major / 5 minor

Summary. The manuscript studies the second-order dynamical system (DS) with vanishing damping α/t^q, a Newton-type correction βt^q (d/dt)A(x(t)), and Tikhonov regularization (c/t^{2q+s})x(t) for a monotone equation Ax=0 on a Hilbert space. The main result (Theorem 1) claims that, for α>1 and 0<q+s<1, the trajectory satisfies ||x(t)-x_t||=O(t^{q+s-1})+O(t^{-s/2}), whence it converges strongly to the minimal-norm solution, together with rates for ||A(x(t))|| and ||ẋ(t)||. Section 3 applies this to a primal-dual system for linearly constrained convex optimization, obtaining strong convergence and rates for the feasibility measure and objective values. The proof is based on a Lyapunov functional with coefficients analyzed through asymptotic expansions.

Significance. If the central estimate is correct, the paper provides a forward-evaluated Newton-like monotone dynamics with strong convergence to the minimal-norm solution of a monotone inclusion, at rates close to the known accelerated rates for systems without Tikhonov regularization. The Lyapunov construction is explicit and the coefficient analysis is largely detailed; in particular, the proof does not rely on fitted constants or on a conclusion that is equivalent to an input assumption. The application to linearly constrained convex optimization is natural and the derived rates would be a useful addition to the literature. However, the proof contains a sign-error in the discriminant calculation that invalidates the key negativity step for a substantial parameter region, so the main theorem is not established as stated.

major comments (2)

[§2, Eq. (26)] The displayed leading coefficient of R2^2−4R1R3 omits the factor β that is present in R3(t) in (25). With R3(t) = β(−2+s1+bs4+K s6 b/2)t^{3q}+O(t^{3q−s}), the correct leading coefficient is D0 = ((2b−2α)β−4)^2 −4β(2b+s3b+s5b−4α)(−2+s1+bs4+K s6 b/2). Setting b=α and s1=s3=s4=s6=0 gives D0=16(1−αβ)+8αβs5, which is strictly positive for αβ≤1 and every allowed s5>0. More generally, if R1<0 then u=2−s3−s5>0 and, writing w=s1+αs4+Kαs6/2>0, one has D0=16−4αβu(2−w); since u<2 and 2−w<2, this is positive whenever αβ≤1. Thus for β≤1/α, a parameter region allowed by Theorem 1, the negativity criterion leading to (30) cannot hold. The missing factor is therefore not a harmless typo: it changes the sign of the discriminant as β varies. Please correct (26), add a lower bound on β (for example β>1/α) if that is the intended parameter regime, and restate Theorem 1 and Theorem 2 accordingly, or provide an alternative argument covering all stated parameters.
[§2, Theorem 1 and preceding paragraph] The theorem is stated for a monotone continuous operator A, but the system (DS) contains the term βt^q (d/dt)A(x(t)). The paragraph before Theorem 1 guarantees existence of a strong global solution only when A is Lipschitz (citing [15]), and the proof additionally requires t↦A(x(t)) to be absolutely continuous. As written, the hypotheses of Theorem 1 do not ensure that there is any trajectory generated by (DS) satisfying the regularity used in the proof. Please either include the Lipschitz or other sufficient regularity condition in the statement of Theorem 1, or formulate the result conditionally for any strong global solution with the stated absolute-continuity property, and adjust the abstract so that the claimed scope matches the hypotheses.

minor comments (5)

[§2, below Eq. (16)] The phrase 'for every s1, s2, s3.s4 > 0' contains a typo; it should read 's1, s2, s3, s4'.
[Abstract and Remark 3] The phrase 'very closed to' should be 'very close to'; there is also an editorial spacing error in 'traj ectories' in the abstract.
[§2, Eq. (32)] The negative term −(bβT^{-s}/2)||x_T||^2 is dropped without comment when passing to the final estimate; since the term is nonpositive the inequality is valid, but stating this would improve clarity.
[§2, Eq. (31)] The auxiliary inequality 't^u e^{W t^v} ≤ C d/dt( t^{u−v+1} e^{W t^v} )' is asserted without proof; a one-line verification or a reference would help the reader check the subsequent integration step.
[§1 and §2] The sentence 'the starting time in (DS) is t0 > 0' appears twice with the same wording; one occurrence should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Lyapunov derivation is self-contained; cited prior results are auxiliary lemmas.

full rationale

The central proof of Theorem 1 is a direct Lyapunov estimate. The energy functional E(t) is explicitly built from the trajectory of (DS) and the monotonicity of A, and the chain of inequalities (5)-(29) consists of algebraic manipulations and Young-type estimates with auxiliary constants s1-s6 and K. The negativity of the quadratic form (R1,R2,R3) and the sign conditions on R4-R6 are derived from explicit leading-order formulas, not assumed or fitted from the conclusion. Inequality (30), after multiplication by the exponential weight and integration, yields E(T) ≤ C0 T^{2q+2s-2}+C0T^{-s}; the rates for ||x-x_t||, ||A(x)||, and ||ẋ|| then follow from the definition of E(t). No step invokes Theorem 1 or the desired strong-convergence statement as an input. The Tikhonov path x_t is used only through standard facts recalled from [14] (strong convergence to x*, ||x_t|| ≤ ||x*||, and ||d/dt x_t|| ≤ (p/t)||x_t||); these are stated auxiliary lemmas whose hypotheses do not include the target rates and they are not equivalent to the theorem. The self-citation to [15] concerns existence/uniqueness under a Lipschitz assumption, a regularity premise rather than the convergence claim. There are no fitted parameters renamed as predictions and no self-citation chain that forces the choice of the system. The skeptic's β-factor concern about the discriminant in Eq. (26) is a possible algebraic correctness gap in proving R2^2-4R1R3<0; even if valid, it is a gap in the justification of sign negativity, not a circular dependency, so it does not affect the circularity score.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no empirical parameters and no new entities. The free-parameter entry groups proof-only constants. The axioms are the usual monotone operator framework plus prior Tikhonov regularization results and standard inequalities.

free parameters (1)

Auxiliary Lyapunov constants b, K, s1 through s6 = b=alpha; K<min(2theta/(4+alpha*beta),2c/(3alpha)); s_i small positive
Introduced in the proof to make the coefficient matrix in the differential inequality negative definite and the residual coefficients nonpositive. They are proof artifacts, not empirical fits, and do not enter the theorem hypotheses.

assumptions (5)

domain assumption Existence and uniqueness of a strong global solution of (DS) when A is Lipschitz continuous and monotone.
Invoked in Section 2 before Theorem 1; the proof differentiates A along the trajectory, which requires t to A(x(t)) to be absolutely continuous.
domain assumption Tikhonov path theory: for monotone A, the zero x_epsilon of A+epsilon Id converges strongly to the minimal-norm solution and satisfies the derivative bound in equations (3) and (4).
Quoted from [14]; used throughout the proof with epsilon(t)=c/t^{2q+s} to control the derivative of the Tikhonov path x_t.
domain assumption The operator A is monotone and continuous on a real Hilbert space, and its zero set is nonempty.
This is the framework of the paper, stated in the introduction and used for monotonicity inequalities and the Tikhonov construction.
domain assumption In the application, f is C2 convex and B is a continuous linear operator with b in the strong relative interior of B(H), so that the KKT conditions characterize solutions.
Section 3 relies on standard convex analysis from [11] to encode the constrained problem as a monotone equation.
standard math Standard inequalities (Young, Cauchy-Schwarz) and exponential integration arguments used in the Lyapunov analysis.
Used repeatedly in Section 2 to combine energy estimates and to integrate the differential inequality.

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Cite this review

Pith. "Pith review of Strong convergence and fast rates for systems with Tikhonov regularization." pith.science (2026). https://pith.science/paper/73OUTTT6

@misc{pith2026241117329,
  author       = {Pith},
  title        = {Pith review of: Strong convergence and fast rates for systems with Tikhonov regularization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73OUTTT6}},
  note         = {Machine review of arXiv:2411.17329}
}

read the original abstract

We introduce and investigate the asymptotic behaviour of the trajectories of a second order dynamical system with Tikhonov regularization for solving a monotone equation with single valued, monotone and continuous operator acting on a real Hilbert space. We consider a vanishing damping which is correlated with the Tikhonov parameter and which is in line with recent developments in the literature on this topic. A correction term which involves the time derivative of the operator along the trajectory is also involved in the system and makes the link with Newton and Levenberg-Marquardt type methods. We obtain strong convergence of the trajectory to the minimal norm solution and fast convergence rates for the velocity and a quantity involving the operator along the trajectory. The rates are very closed to the known fast convergence results for systems without Tikhonov regularization, the novelty with respect to this is that we also obtain strong convergence of the trajectory to the minimal norm solution. As an application we introduce a primal-dual dynamical system for solving linearly constrained convex optimization problems, where the strong convergence of the trajectories is highlighted together with fast convergence rates for the feasibility measure and function values.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [15]

Bot ¸, E.R

R.I. Bot ¸, E.R. Csetnek, D.K. Nguyen, Fast Optimistic Gradient Descent Ascent (OGDA) Method in Continuous and Discrete Time , Found Comput Math 2023, https://doi.org/10.1007/s10208-023 -09636-5

work page doi:10.1007/s10208-023 2023

[2] [1]

Apidopoulos, J.-F

V. Apidopoulos, J.-F. Aujol, Ch. Dossal, The diﬀerential inclusion modeling the FISTA algorithm and optimality of convergence rate in the case b ≤ 3, SIAM J. Optim. 28(1) (2018), 551—574

work page 2018

[3] [2]

Attouch, Z

H. Attouch, Z. Chbani, J. Peypouquet, P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity , Mathematical Programming 168 (1-2) (2018), 123–175

work page 2018

[4] [3]

Attouch, Z

H. Attouch, Z. Chbani, H. Riahi, Combining fast inertial dynamics for convex optimization w ith Tikhonov regularization, J. Math. Anal. Appl 457 (2018), 1065–1094

work page 2018

[5] [4]

Attouch, S

H. Attouch, S. L´ aszl´ o,Convex optimization via inertial algorithms with vanishin g Tikhonov regularization: fast convergence to the minimum norm solution , Math Meth Oper Res 99, 307–347 (2024)

work page 2024

[6] [5]

Alecsa, S.C

C.D. Alecsa, S.C. L´ aszl´ o, Tikhonov Regularization of a Perturbed Heavy Ball System wi th Vanishing Damping, SIAM Journal on Optimization, 31(4), 2921-2954, 2021

work page 2021

[7] [6]

Alecsa, S.C

C.D. Alecsa, S.C. L´ aszl´ o, T. Pint ¸a,An Extension of the Second Order Dynamical System that Model s Nesterov’s Convex Gradient Method , Applied Mathematics and Optimization, 84, 1687-1716, 2021

work page 2021

[8] [7]

Attouch, S.C

H. Attouch, S.C. L´ aszl´ o,Continuous Newton-like Inertial Dynamics for Monotone Inc lusions, Set-Valued Var. Anal 29, 555–581 (2021)

work page 2021