Fast convex optimization via inertial systems with asymptotically vanishing viscosity and Hessian-driven damping

Juan Peypouquet; Zepeng Wang

REVIEW 2 major objections 4 minor 1 cited by

Fast convex optimization via inertial systems with asymptotically vanishing viscosity and Hessian-driven damping

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read With $r=\alpha\beta$ and $\dot x(0)=-\beta\nabla f(x_0)$, Polyak–Lojasiewicz gives $e^{-2\mu\beta t}$ decay followed by $O(t^{-2\alpha})$ for every $\alpha>0$, and linear rates under strong convexity.

desk verdict Solid theoretical extension of Nesterov-rate analysis: new o(1/t^{2α}) and linear rates for subcritical α under PL, with the main caveat being well-posedness assumptions that are referenced but not stated. read the letter →

arxiv 2506.21730 v2 pith:ENBWYKYK submitted 2025-06-26 math.OC

Zepeng Wang , Juan Peypouquet This is my paper · ORCID

classification math.OC MSC 34D0565K0565K1090C25

keywords asymptoticvanishingdampingHessian-drivenPolyak-Lojasiewiczinequalitystrongconvexityinertialsecond-orderdynamicsacceleratedgradientmethodproximalalgorithmconvexoptimization

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

The paper studies the second-order system $\ddot x+\frac{\alpha}{t}\dot x+\beta\nabla^2 f(x)\dot x+(\gamma+\frac{r}{t})\nabla f(x)=0$ and the accelerated gradient and proximal algorithms obtained from it by discretization. Its central claim is that with $r=\alpha\beta$ and the natural initial velocity $\dot x(0)=-\beta\nabla f(x_0)$, a Polyak–Lojasiewicz objective makes the function gap decay at the exponential rate $e^{-2\mu\beta t}$ up to time $t=\alpha/(\mu\beta)$, and then polynomially as $O(t^{-2\alpha})$, with the stronger $o(t^{-2\alpha})$ when $f$ is convex. For strongly convex objectives the same system gives a linear rate $O(t^{-2\alpha}e^{-\frac{1}{2}\mu\beta t})$, and both properties hold for every $\alpha>0$. This matters because previous inertial analyses of this type generally needed $\alpha\ge 3$ and achieved only $O(t^{-2\alpha/3})$ under strong convexity; the new results extend the accelerated-gradient paradigm to the whole positive range of $\alpha$.

What carries the argument

The load-bearing object is the time-dependent energy function $\varepsilon(t)=\frac{1}{2}\|\dot x+\beta\nabla f(x)\|^2+\frac{\omega}{2}\|\dot x\|^2+\theta(f(x)-f^*)$, with parameters chosen as $r=\alpha\beta/(1+\omega)$ and $\theta=(1+\omega)\gamma-\omega\eta\beta^2/(1+\omega)$. Setting $\omega=0,\eta=\mu$ makes its derivative obey $\dot\varepsilon\le -2\min\{\alpha/t,\mu\beta\}\varepsilon$, which integrates to the exponential-fast-then-polynomial envelope; setting $\omega=1,\eta=\mu$ gives $\dot\varepsilon\le-(2\alpha/t+\mu\beta/2)\varepsilon$, which yields the strong-convexity linear rate. The same idea drives the discrete proofs through the energy sequence $E_k=\frac{1}{2}\|v_k+h\nabla f(x_k)\|^2+\gamma[(1-\omega)\psi_k+\omega(f(y_{k+1})-f^*)]$ and its proximal analogue. These identities are what transfer the rate information from the Hessian-damped flow to the algorithms.

What would settle it

Fix $\alpha,\beta,\gamma>0$ with $r=\alpha\beta$, choose a twice differentiable convex $f$ satisfying the Polyak–Lojasiewicz inequality with known $\mu$ (for example $f(x)=\frac{1}{2}x^2$ with $\mu=L=1$), set $\dot x(0)=-\beta f'(x_0)$, and integrate (AVD-H) numerically. The theorem is settled if the value gap ever exceeds the claimed envelope $e^{-2\mu\beta t}(f(x_0)-f^*)$ on $t\le \alpha/(\mu\beta)$, if $t^{2\alpha}(f(x(t))-f^*)$ fails to tend to $0$, or if any admissible $f$ yields no global solution.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a two-phase function-value estimate for trajectories of the inertial system (AVD-H). For a twice differentiable $f$ satisfying the Polyak–Lojasiewicz inequality $\|\nabla f(x)\|^2\ge 2\mu(f(x)-f^*)$, with $\alpha,\beta,\gamma>0$, $r=\alpha\beta$, and $\dot x(0)=-\beta\nabla f(x_0)$, the bound is $f(x(t))-f^*\le (f(x_0)-f^*)e^{-2\mu\beta t}$ for $t\le \alpha/(\mu\beta)$, and $f(x(t))-f^*\le (f(x_0)-f^*)(\alpha/(\mu\beta e))^{2\alpha}t^{-2\alpha}$ afterwards; convexity upgrades the tail to $o(t^{-2\alpha})$. For $\mu$-strongly convex $f$, the system satisfies $f(x(t))-f^*\le e^{-\frac{1}{2}\mu\beta(t-T)}/(t/T)^{2\alpha}C_T$ for $t\ge T=4\alpha/(\mu\beta)+4\alpha\beta/(4\gamma-\mu\beta^2)$, provided $\gamma>\frac{1}{4}\mu\beta^2$ and $r=\frac{1}{2}\alpha\beta$. The discrete companion (AGM) reproduces the same linear-then-$O(k^{-2\alpha})$ envelope, with $o(k^{-2\alpha})$ under convexity, and the proximal scheme (APM) achieves linear convergence under strong convexity. The message is that Hessian-driven damping, rather than a large $\alpha$, is what removes the obstacle to fast convergence in this family.

Load-bearing premise

The whole rate picture assumes that the differential equation (AVD-H) always has a unique global solution for every twice differentiable objective whose gradient is $L$-Lipschitz; the paper cites prior work for this rather than proving it, so a finite-time blow-up of any admissible trajectory would leave the rates without meaning.

Editorial extensions

If this is right

For any $\alpha>0$, the continuous dynamics (AVD-H) reaches $O(t^{-2\alpha})$ under Polyak–Lojasiewicz, with the exponential $e^{-2\mu\beta t}$ phase ending at $t=\alpha/(\mu\beta)$.
Hessian-driven damping removes the strong-convexity obstruction: a linear rate proportional to $e^{-\frac{1}{2}\mu\beta t}$ (up to a polynomial factor) holds, whereas the undamped system $\ddot x+\frac{\alpha}{t}\dot x+\nabla f(x)=0$ cannot converge linearly even for strongly convex $f$.
The discretization (AGM) inherits the two-phase structure: the value gap stays within $O((1+\mu h^2(2-\gamma))^{-k})$ until an explicit index $K$, then within $O(k^{-2\alpha})$, and vanishes as $o(k^{-2\alpha})$ for convex $f$.
The proximal algorithm (APM) for composite problems $f+g$ satisfies the same sublinear $O(k^{-2\alpha})$ under Polyak–Lojasiewicz and a linear rate $(1+\rho)^{-k}$ with $\rho=\mu\min\{s/2,\,s(1-sL)/(1+\mu L s^2)\}$ when $f$ is strongly convex.
With the exact initial velocity, the damping coefficient $\gamma$ drops out of the rate constants in the continuous theorem, so $\gamma$ can be tuned freely to control oscillation without changing the order of convergence.

Reading between the lines

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

Editorial extension: the paper fixes the initial velocity to $-\beta\nabla f(x_0)$ exactly; the energy argument indicates that any nonzero initial error $\dot x(0)+\beta\nabla f(x_0)$ should weaken the exponential phase, and a quantitative perturbation bound would make this precise.
Editorial extension: because the switch time $\alpha/(\mu\beta)$ is explicit, the natural algorithmic corollary is a restart schedule near that threshold, although the paper itself does not study restarts.
Editorial extension: since $\gamma$ does not affect the rates under the exact initial condition, one can treat $\gamma$ as an oscillation-suppression dial and test numerically whether larger $\gamma$ smooths trajectories without losing the envelope.

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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 punchline is that this paper does what it says and then some. For the ODE (AVD-H), it proves exponential decay up to t = α/(μβ) and then O(t^{-2α}) under only a PL inequality, for every α>0, with an o(t^{-2α}) improvement in the convex case. For the discrete algorithms, it gives the first linear-then-polynomial rate for subcritical α in Nesterov's method, and extends the known o(1/k^{2α}) to all α>0 and to the critical step size. The proofs are careful and the Lyapunov construction with the extra ω term is genuinely useful; the observation that r=αβ and v0=-β∇f(x0) make the rates independent of γ is nice and not obvious.

The soft spots are real but minor. The main one is global well-posedness: the paper says existence is "referred to" [17] and never states the hypotheses needed. Since Theorem 2.2 explicitly allows nonconvex PL functions, the reader can't tell whether [17] covers that case. The paper's own Lyapunov estimates, together with the standing L-smoothness assumption from the introduction, actually give uniform bounds on ∇f and the velocity, which would rule out finite-time blow-up; so the gap is a missing sentence, not a serious flaw. Still, the authors should pin down which theorem in [17] applies and state the standing assumptions in the theorems that rely on them. Relatedly, Theorem 2.2 never states L-smoothness but uses it in the o(1/t^{2α}) proof via 1/(2L)||∇f||² ≤ f-f*; it's clearly inherited from the intro, but should be explicit. There are also a few typos (e.g., "xk+1−xk+1" in Proposition 4.3) that don't affect the mathematics, and the paper doesn't directly compare with the general Hessian-driven system in [3], which would help the reader see exactly what's new.

Who should read it: people who care about rate analysis for accelerated methods under PL/strong convexity, especially the subcritical regime. It's a theory paper; it won't change what you compute, but it resolves a genuine gap in the rates and the technique is reusable. The central claims hold up. I would send it to a serious referee, with a request to sharpen the well-posedness discussion and flag the standing assumptions.

Referee Report

2 major / 4 minor

Summary. The paper studies the inertial differential system (AVD-H) combining asymptotic vanishing viscosity, Hessian-driven damping, and a time-dependent gradient coefficient. The main continuous-time results are Lyapunov-energy estimates giving, under a Polyak-Łojasiewicz inequality, a function-value gap that decays like e^{-2μβt} for t ≤ α/(μβ) and like O(t^{-2α}) afterward, with an o(t^{-2α}) improvement under additional convexity; for strongly convex objectives the paper proves a rate of the form O(t^{-2α}e^{-μβt/2}). The paper then discretizes the system into an accelerated gradient algorithm (AGM) and an accelerated proximal-gradient algorithm (APM), proving analogous O(k^{-2α})/o(k^{-2α}) rates under convexity plus PL, and linear rates under strong convexity.

Significance. The claimed rates are substantially faster than the previously known O(t^{-2α/3}) for this class of inertial systems, and they are obtained for all α>0 rather than only for α≥3. The proofs are based on explicit Lyapunov energies with no fitted parameters, which is a genuine strength, and the extension to proximal-gradient algorithms broadens the applicability. If the well-posedness and smoothness-assumption gaps identified below are closed, the continuous-time results would constitute a clear advance over the existing literature and give useful insight into Nesterov acceleration. The discrete-time results are also interesting, although the improvements there are more incremental relative to known O(k^{-2}) and o(k^{-2α}) bounds.

major comments (2)

[Section 1, after (AVD-H); Theorems 2.2 and 2.5] The paper assumes global existence and uniqueness of C^2 solutions of (AVD-H) for the full parameter range, referring only to 'the discussion in [17]'. This assumption is load-bearing because every rate in Section 2 is stated pointwise for all t>0, and the proof of Theorem 2.2 evaluates the energy at t=0 (ε(0)) and integrates the differential inequality from there. The manuscript does not identify which result in [17] applies, nor does it verify that the hypotheses of that result cover the scope of Theorem 2.2, which explicitly allows nonconvex PL functions and all α>0. Please state the exact existence theorem, verify that its assumptions (e.g., convexity, L-smoothness, or lower bounds on α) are satisfied in the nonconvex PL setting, or add the necessary hypotheses and state the rates for t≥t0 with a local-existence argument. Without this, the claimed rates are conditional on an unverified well-posedness assertion.
[Theorem 2.2, proof of o(1/t^{2α})] The proof of the o(1/t^{2α}) statement invokes the inequality (1/(2L))∥∇f(x)∥^2 ≤ f(x)-f*, which is a consequence of L-smoothness. However, the theorem statement assumes only that f is twice differentiable and satisfies the PL inequality, plus convexity for the o part. Convexity alone does not imply L-smoothness, so as stated the proof has a gap. Either add L-smoothness to the hypotheses of Theorem 2.2 (if it is intended as a standing assumption of Section 1, make that explicit in the theorem), or provide a proof of the o bound that does not use this inequality.

minor comments (4)

[Proposition 4.3] In the displayed formula for E_{k+1}-E_k, the term ∥x_{k+1}-x_{k+1}∥^2 should read ∥x_{k+1}-y_{k+1}∥^2, as is clear from the preceding derivation using (4.2).
[Theorems 3.3 and 4.4] The definitions of K are off by one relative to the range where the linear-contraction term is active: in Theorem 3.3 the condition should be applied to k+1 rather than k, and in Theorem 4.4 similarly. The asymptotic rates are unchanged, but the constants (K-1 or K) should be shifted consistently.
[Remark 2.6] The condition '0<β<2√μ' appears inconsistent with Theorem 2.5, which for γ=1 requires β<2/√μ. Please correct this condition.
[Theorem 2.5] The statement does not specify the initial condition v0 or clarify how the trajectory is assumed to exist up to the time T; please state the existence assumption explicitly so that the bound for t≥T is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rates follow from explicit energy inequalities; the cited existence result [17] is a side condition and not used to force the rate.

full rationale

The derivation chain is self-contained. For the continuous system, Proposition 2.1 computes the derivative of the energy (2.1) by direct algebra from (AVD-H) with r = alpha*beta/(1+omega), and Theorem 2.2 then substitutes omega = 0, eta = mu, and integrates the resulting differential inequality. The rate exponent 2*mu*beta is a consequence of the Polyak-Lojasiewicz input, not an assumption encoded in the energy: the energy only contains the terms needed to make the derivative calculation close. The initial condition v0 = -beta grad f(x0) merely makes epsilon(0) equal to gamma(f(x0)-f*), which is an initialization choice rather than a fitted parameter. The discrete results in Sections 3 and 4 follow the same pattern: the energy sequences are defined explicitly, their differences are computed from the algorithm recursions, and the cited Lemma 4.1 from [9] is an external one-step descent lemma, not a restatement of the target rate. The only self-citation is the existence and uniqueness discussion in [17]; this supports well-posedness of the ODE but is not used to derive the value decay, and no reduction of the main rates to [17] is exhibited. Thus there is no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The parameters α, β, γ define the system and are user-specified, not fitted. The coefficient r is tuned to αβ or αβ/2 to make the energy derivative non-positive; this is a proof choice rather than a data fit. The key domain assumptions are PL, strong convexity, and smoothness. Existence of ODE solutions is assumed via [17].

free parameters (4)

α
User-specified positive coefficient of the viscosity term α/t; all rates hold for any α>0, so it is not fitted.
β
User-specified positive coefficient of the Hessian-driven damping; controls the linear phase length and the exponential decay factor.
γ
User-specified positive coefficient of the ∇f term; for the algorithms it must lie in (0,2).
r = αβ (PL case), αβ/2 (strongly convex case)
Coefficient of the (r/t)∇f term. It is tuned in Proposition 2.1 and Theorems 2.2 and 2.5 to cancel cross terms in the Lyapunov derivative.

assumptions (5)

domain assumption Global existence and uniqueness of solutions to (AVD-H) for twice differentiable f with L-Lipschitz gradient
Invoked in Section 1 after (AVD-H) with reference to [17]; the paper does not prove it.
domain assumption PL inequality: ||∇f(x)||² ≥ 2μ(f(x)-f*)
Used in Sections 2.1 and 3.2 to control the gradient norm by the function value gap.
domain assumption Strong convexity: f(y) ≥ f(x) + ⟨∇f(x),y-x⟩ + μ/2||y-x||²
Used in Sections 2.2, 3.3, and 4.3 to derive linear convergence.
standard math L-smoothness: f(y) ≤ f(x) + ⟨∇f(x),y-x⟩ + L/2||y-x||²
Standing assumption from the introduction; used throughout the Lyapunov estimates.
standard math Lemma 4.1 from [9]
Externally cited estimate for proximal gradient methods; used in Section 4 without proof.

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Pith. "Pith review of Fast convex optimization via inertial systems with asymptotically vanishing viscosity and Hessian-driven damping." pith.science (2026). https://pith.science/paper/ENBWYKYK

@misc{pith2026250621730,
  author       = {Pith},
  title        = {Pith review of: Fast convex optimization via inertial systems with asymptotically vanishing viscosity and Hessian-driven damping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENBWYKYK}},
  note         = {Machine review of arXiv:2506.21730}
}

read the original abstract

We study the convergence rate of a family of inertial algorithms, which can be obtained by discretization of an inertial system combining asymptotic vanishing viscous and Hessian-driven damping. We establish a fast sublinear convergence rate in case the objective function is convex and satisfies Polyak-\L ojasiewicz inequality. We also establish a linear convergence rate for strongly convex functions. The results can provide more insights into the convergence property of Nesterov's accelerated gradient method.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Inertial Primal Dual Dynamics with Hessian-driven Damping for Saddle Point Problems
math.OC 2026-07 conditional novelty 6.0 of 10

New inertial primal-dual ODEs with Hessian damping achieve O(1/t²) convex rates and O(1/t^{α−1}) strongly-convex rates without knowing the strong convexity moduli.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [17]

J. J. Maul´en and J. Peypouquet, A speed restart scheme for a dynamics with Hessian-driven damping, Journal of Optimization Theory and Applications, 199 (2023), pp. 831–855

work page 2023

[2] [3]

Attouch, Z

H. Attouch, Z. Chbani, J. F adili, and H. Riahi , First-order optimization algorithms via inertial systems with Hessian driven damping , Mathematical Programming, 193 (2022), pp. 113–155

work page 2022

[3] [1]

Apidopoulos, J.-F

V. Apidopoulos, J.-F. Aujol, and C. Dossal, Convergence rate of inertial forward–backward algorithm beyond Nesterov’s rule , Mathematical Programming, 180 (2020), p. 137–156

work page 2020

[4] [2]

Apidopoulos, J.-F

V. Apidopoulos, J.-F. Aujol, C. Dossal, and A. Rondepierre , Convergence rates of an inertial gradient descent algorithm under growth and flatness conditions , Mathematical Programming, 187 (2021), pp. 151–193

work page 2021

[5] [4]

Attouch, Z

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

work page 2018

[6] [5]

Attouch, Z

H. Attouch, Z. Chbani, and H. Riahi , Rate of convergence of the Nesterov accelerated gra- dient method in the subcritical case α ≤ 3, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019)

work page 2019

[7] [6]

Attouch and J

H. Attouch and J. Peypouquet , The rate of convergence of Nesterov’s accelerated forward- backward method is actually faster than 1/k2, SIAM Journal on Optimization, 26 (2016), pp. 1824–1834

work page 2016

[8] [7]

Attouch, J

H. Attouch, J. Peypouquet, and P. Redont, Fast convex optimization via inertial dynamics with Hessian driven damping, Journal of Differential Equations, 261 (2016), pp. 5734–5783

work page 2016