Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates

Alfredo Iusem; Alireza Kabgani; Felipe Lara; Masoud Ahookhosh

REVIEW 3 major objections 4 minor 64 references

Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read High-order proximal-point methods are shown to converge to the unique global minimizer of uniformly quasiconvex functions, with linear or superlinear rates set by the modulus growth order q and the proximal order p.

desk verdict Theorem 4.10(a) is false and the proof breaks on an invalid p-norm inequality; the paper's other convergence results appear sound and the overall framework is worth a referee's time. read the letter →

arxiv 2505.20484 v2 pith:NUZQYSAH submitted 2025-05-26 math.OC

Masoud Ahookhosh , Alfredo Iusem , Alireza Kabgani , Felipe Lara This is my paper · ORCID

classification math.OC MSC 90C2690C2565K05

keywords high-orderproximal-pointalgorithmuniformquasiconvexityMoreauenvelopeconvergencerateslinearsuperlinearnonconvexoptimizationglobalminimizer

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 the high-order proximal-point algorithm (HiPPA), the iterative scheme $x_{k+1} = \arg\min_y \varphi(y) + \frac{1}{p\gamma_k}\|x_k-y\|^p$, for minimizing uniformly quasiconvex functions—a class that includes strongly convex, uniformly convex, and strongly quasiconvex functions as special cases. The authors aim to show that HiPPA always converges to the unique global minimizer, and that its asymptotic rate is controlled by the proximal order $p$ and the growth order $q$ of the uniform-quasiconvexity modulus. Their claimed results are locally linear convergence for $p\in(1,2)$ when the modulus grows like $t^q$ with $q\in(1,2]$ on $[0,1)$; linear convergence for $p=2$, $q=2$ and for $p=q>2$; and superlinear convergence of order $(p-1)/(q-1)$ when $p>q\geq 2$. If correct, these results extend classical linear-convergence guarantees for proximal point methods well beyond strongly convex objectives, without any smoothness assumption.

What carries the argument

The load-bearing object is the high-order proximal operator $\operatorname{prox}^{p}_{\gamma\varphi}(C,x)=\arg\min_{y\in C}\left\{\varphi(y)+\frac{1}{p\gamma}\|x-y\|^p\right\}$, together with its Moreau envelope. The argument runs through the elementary optimality inequality $\varphi(x_{k+1})+\frac{1}{p\gamma_k}\|x_{k+1}-x_k\|^p \leq \varphi(x)+\frac{1}{p\gamma_k}\|x-x_k\|^p$ for the minimizer $x$, combined with the uniform-quasiconvexity inequality $\varphi(\lambda x+(1-\lambda)y) \leq \max\{\varphi(x),\varphi(y)\}-\lambda(1-\lambda)\phi(\|x-y\|)$. Inserting convex combinations of $x$ and $x_{k+1}$ into the optimality inequality converts the regularizer's distance contraction into a recurrence for $\|x-x_{k+1}\|$ in terms of $\|x-x_k\|$. The different regimes come from which norm inequality is available: the exact parallelogram identity at $p=2$, a uniform-convexity inequality for $p>2$, and a local strong-convexity estimate on bounded balls for $p\in(1,2)$.

What would settle it

Run HiPPA on $\varphi(x)=\|x\|^2$ with $p=3/2$ and $\gamma=1$ from $x_0=1$. The optimality condition gives $x_{k+1}=r_k x_k$ with $r_k=(-1+\sqrt{1+16x_k})/(8x_k)$, and $r_k\to 1$ as $x_k\to 0$. Hence the contraction ratio $\|x-x_{k+1}\|/\|x-x_k\|$ is not bounded below by a constant less than one, so Theorem 4.10(a)'s claimed local linear rate would fail in this setting.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the asymptotic behavior of HiPPA is fully determined by the interplay between $p$ and the degree $q$ of a lower bound $\rho_q t^q \leq \phi(t)$ on the uniform-quasiconvexity modulus. Because uniformly quasiconvex functions exclude spurious stationary points—Theorem 3.23 proves that stationarity forces unique global minimality and gives the growth bound $\varphi(x)+\tfrac14\phi(\|y-x\|)\leq\varphi(y)$—every HiPPA sequence converges to the unique minimizer. Theorem 4.10 then turns the proximal optimality inequality into distance contraction: for $p=2$ and $q=2$ the contraction factor is $1/(1+\gamma_{\min}\rho)$; for $p=q>2$ it is $(p/(p\gamma_{\min}\rho_p+\hat\sigma_p))^{1/(p-1)}$; for $p>q$ it is superlinear with exponent $(p-1)/(q-1)$; and for $p\in(1,2)$ on $[0,1)$ it is locally linear with factor $2/(p\rho\gamma_{\min}+\sigma_p)$. The authors note that some of these rates are new even for strongly or uniformly convex functions.

Load-bearing premise

For $p<2$, the proof needs a small distance raised to the power $p$ to behave like the square of that distance with a positive gap; for identical points this would require a small number's $p$-th power to be no larger than its square, which fails whenever $p<2$.

Editorial extensions

If this is right

For $p=2$, the paper recovers the classical linear rate of the proximal point method on strongly convex and strongly quasiconvex objectives, with contraction factor $1/(1+\gamma_{\min}\rho)$ and $O(\log(1/\varepsilon))$ iteration complexity.
For $p=q>2$, HiPPA is claimed to converge linearly on uniformly convex functions, a setting where high-order proximal methods had not previously been shown to achieve linear rates.
For $p>q\geq 2$, HiPPA is claimed to converge superlinearly with order $(p-1)/(q-1)$ from any starting point, without any smoothness assumption on the objective.
Because stationary points coincide with the unique global minimizer for uniformly quasiconvex functions, HiPPA iterates cannot be trapped at saddle points or spurious local minima.
In the linear regimes, reaching $\|x_k-x\|\leq\varepsilon$ is guaranteed within $O(\log(1/\varepsilon))$ iterations, matching the complexity expected for strongly convex optimization.

Reading between the lines

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

The paper's own distance-contraction recurrences suggest a natural path for inexact variants: if the proximal subproblem is solved with summable errors, the same recurrences should survive with modified constants, even though the paper leaves this open.
The local linear claim for $p<2$ can be tested directly on the quadratic $\varphi(x)=\|x\|^2$: the one-dimensional HiPPA iteration has an explicit form, and its contraction ratio appears to tend to $1$ as $x_k\to 0$, which would contradict the claimed bound $2/(p\rho\gamma_{\min}+\sigma_p)<1$.
The growth condition $\varphi(x)+\tfrac14\phi(\|y-x\|)\leq\varphi(y)$ is an error-bound-type geometry; the rates here indicate that the sharpness of the minimum, measured by $q$, is the true driver of asymptotic speed, so analogous results should hold under modulus-based error bounds beyond uniform quasiconvexity.

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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

Colleague,

The thing you need to know first: the headline result, Theorem 4.10(a), is false. The claimed local linear convergence for p in (1,2) on uniformly quasiconvex functions fails already on the strongly convex quadratic phi(x)=x^2/2 with p=1.5. For that problem the proximal step satisfies e_k ≈ (γ a_k)^2, so the error ratio is 1 − O(a_k), which tends to 1. No uniform contraction ratio below 1 exists. The proof breaks at Lemma 4.9(a), where the authors effectively use strong convexity of ||·||^p on a ball for p∈(1,2). That is false near the origin: taking a=b in the claimed inequality gives ||a||^p ≤ ||a||^2, which cannot hold for p<2 and small ||a||. Since the iterates converge to the minimizer, the relevant vectors are exactly in the bad regime.

That said, the paper has a real, useful core. Sections 3 on uniformly quasiconvex functions — calculus, coercivity, stationarity at the global minimizer, absence of saddle points — is careful and will be a convenient reference. Theorem 4.7 (global convergence) and Proposition 4.8 (complexity bound) look correct under the stated assumptions. The rate statements in Theorem 4.10(b), (c), (d) — linear for p=2, linear for p=q>2, superlinear for p>q — seem sound; the Lyapunov arguments there are standard and the modulus lower bounds are used properly. The novelty of extending asymptotic rate analysis beyond the classical p=2 case to uniformly quasiconvex functions is genuine, though the techniques themselves are not new.

Minor issues: several key technical facts (HOPE properties, p-norm behavior) are cited to the authors' own unpublished preprints, which would need independent checking. Also, the abstract and the theorem disagree on the scope of part (a): the full-text abstract says p∈(1,2), while another version says p∈[q,2). Even adding p≥q would not rescue the claim — the quadratic with q=1.5 and p=1.5 still fails. So this is not a missing condition; the proof itself is invalid.

Who should read this: people working on proximal-point methods or generalized convexity will find the framework and the correct parts useful, but they must be told to ignore Theorem 4.10(a) and Corollary 4.11(a) until repaired. I would not accept the paper in its current form. It does deserve a serious referee — the error is subtle and most of the manuscript is salvageable — but the revision should either produce a correct proof for p<2 or explicitly delete that claim and keep only the regimes that work, and should clean up the self-citations. Recommendation: reject in current form, invite a revision.

Referee Report

3 major / 4 minor

Summary. The paper studies the high-order proximal-point algorithm (HiPPA) for minimizing proper, lower semicontinuous, coercive, uniformly quasiconvex functions. It develops structural properties of this function class (calculus, coercivity, stationarity/global-optimality equivalence), establishes global convergence of HiPPA, and derives asymptotic rates under power-type lower bounds on the modulus of uniform quasiconvexity. The headline result is a claimed local linear convergence rate for every p in (1,2) when the modulus satisfies φ(t) ≥ ρ_q t^q on [0,1) for some q in (1,2], together with linear rates for p=2 and p=q>2, and superlinear rates for p>q≥2.

Significance. If the claimed p<2 local linear rate were true, it would be a significant extension of classical proximal-point rate theory, which typically requires p=2 and strong convexity. The paper also gives a clean stationarity-implies-global-optimality result for uniformly quasiconvex functions, which is a useful contribution. However, the central p<2 claim is false: a one-dimensional strongly convex quadratic provides a direct counterexample. Since the main advertised novelty—local linear convergence for p∈(1,2)—is unsupported and contradicted, the paper's significance is reduced to the p=2 and p≥q cases, which are more standard. The manuscript does not include machine-checked proofs or reproducible numerical experiments; its key technical facts are cited to the authors' own unpublished preprints.

major comments (3)

[§4.2, Lemma 4.9(a)] The proof of Lemma 4.9(a) relies on an invalid norm inequality. In the chain of inequalities leading to (4.6), the term ∥λ(x−x_k)+(1−λ)(x_{k+1}−x_k)∥^p is replaced by λ∥x−x_k∥^2 + (1−λ)∥x_{k+1}−x_k∥^2 − λ(1−λ)σ_p∥x−x_{k+1}∥^2. For p∈(1,2) and vectors of norm less than 1, ∥a∥^p ≥ ∥a∥^2, so replacing the p-th powers by squares makes the right-hand side smaller, and the claimed inequality is false. Taking a=b with 0<∥a∥<1 gives the requirement ∥a∥^p ≤ ∥a∥^2, which fails for p<2. Since the iterates satisfy ∥x−x_k∥→0 and ∥x_{k+1}−x_k∥→0, the argument is used precisely in the regime where the inequality fails. Consequently, inequality (4.6) is not established.
[§4.2, Theorem 4.10(a)] Theorem 4.10(a) is false as stated. Take φ(x)=x^2/2 on R, γ_k=1, and p=3/2. This function is strongly convex and hence uniformly quasiconvex with modulus φ(t)=t^2/2, so the assumption ρ_q t^q ≤ φ(t) holds on [0,1) with q=2 and ρ_q=1/2. The HiPPA optimality condition for this problem is x_{k+1} + (1/γ)|x_{k+1}−x_k|^{p−2}(x_{k+1}−x_k)=0. Writing e_k=x_k and r_k=e_{k+1}/e_k, one obtains r_k + (1/γ)e_k^{p−2}|r_k−1|^{p−2}(r_k−1)=0. Since p−2<0, e_k^{p−2}→∞ as e_k→0; the equation forces r_k→1 (in fact |r_k−1| ≍ e_k^{(2−p)/(p−1)}). Thus no uniform bound of the form r_k ≤ c < 1 can hold for all large k, directly contradicting the claimed local linear rate with ratio 2/(pργ_min+σ_p) < 1. This example also disproves the abstract's first advertised contribution and Corollary 4.11(a).
[§4.2, Lemma 4.9(a) and Theorem 4.10(a)] The flaw is load-bearing: the proof of Theorem 4.10(a) depends entirely on Lemma 4.9(a), and the resulting inequality (4.6) is false. The manuscript's main novelty claim—local linear convergence for p∈(1,2) under q∈(1,2]—therefore collapses. The other results (Theorem 4.10(b)–(d)) use Lemma 4.9(b)–(c), which do not rely on the invalid p-norm step, but they do not compensate for the failure of the central claim.

minor comments (4)

[§4.2, Lemma 4.9(a)] The sentence 'this implies that ∥x_{k+1}−x_k∥^2 < ∥x_{k+1}−x_k∥^p' is correct for p∈(1,2) once the step norm is below 1, but the subsequent substitution of squared norms in the upper bound is in the opposite direction; this should be clarified or the argument corrected.
[§2 and §4.2] Key technical facts—the strong convexity of ∥·∥^q on balls for q∈(1,2) and fundamental properties of the high-order proximal operator—are cited to the authors' unpublished preprints [29] and [32]. The paper should either provide self-contained proofs or cite published versions.
[Table 1] The first row of Table 1 repeats the false claim of Theorem 4.10(a); if the theorem is corrected, the table must be updated accordingly.
[§4.2, proof of Theorem 4.10(a)] The quantity σ_p is taken as κ_p r^{p−2}/2, but the radius r of the ball on which the power norm is strongly convex is not explicitly defined before its use; please make the choice of r and its dependence on the iterates precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the rate claims follow from the stated modulus assumptions, with the p<2 technical inequality a verification concern rather than a circular reduction.

full rationale

The paper's convergence-rate analysis is not circular. Theorem 4.10 derives its contraction ratios directly from the defining uniform-quasiconvexity inequality (3.5), the proximal optimality condition (4.9), and the assumed lower bound rho_q t^q <= phi(t); the rates are consequences of those assumptions, not restatements of them, and no parameter is fitted to a subset of data and then presented as a prediction. The one load-bearing use of a same-author citation is Example 3.2, where the strong-convexity-type inequality for the p-th power norm on a ball for p in (1,2) is asserted via the authors' own preprint [32] and then used in Lemma 4.9(a) to obtain inequality (4.6). This is a verification and correctness gap rather than a circular equivalence: the cited inequality is a generic technical fact with stated assumptions (q in (1,2), a,b in B(0;r)), and it does not contain the target convergence theorem. The remaining cited facts, such as Fact 4.4 from [29], are standard level-boundedness and continuity properties with independent justifications. Even if Lemma 4.9(a)'s p-norm inequality is invalid, or if the local linear claim fails on a scalar quadratic, those are mathematical-error concerns, not instances of a derivation reducing by construction to its own inputs. Accordingly, no circular step is identified in the paper's derivation chain.

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

The paper introduces no new free parameters or entities. The main burden rests on the uniform quasiconvexity assumption and modulus lower bound, plus standard convex analysis facts. The critical flaw is a mathematical error in Lemma 4.9(a), not an axiom.

assumptions (4)

domain assumption The objective function is uniformly quasiconvex with a given modulus phi on a closed convex set C.
This is the core assumption of the paper, introduced in Definition 3.1(e) and used in Theorems 3.23, 4.7, and 4.10.
domain assumption The modulus satisfies a power lower bound phi(t) >= rho_q t^q on a specified interval.
This is the key rate assumption stated in Theorem 4.10 and used in the convergence rate proofs.
standard math The function x -> ||x||^q is strongly convex on bounded balls for q in (1,2) and uniformly convex for q>2.
Cited to Example 3.2 and [32,48]. This underlies the proof of Lemma 4.9, but the specific application in Lemma 4.9(a) is flawed.
standard math The high-order Moreau envelope is continuous and level-bounded (Fact 4.4).
Cited to [29, Theorem 3.4], a preprint by co-authors. This supports Theorem 4.6 but is not the source of the critical error.

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Pith. "Pith review of Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates." pith.science (2026). https://pith.science/paper/NUZQYSAH

@misc{pith2026250520484,
  author       = {Pith},
  title        = {Pith review of: Asymptotic Convergence Analysis of High-Order Proximal-Point Methods Beyond Sublinear Rates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUZQYSAH}},
  note         = {Machine review of arXiv:2505.20484}
}

abstract

This paper investigates the asymptotic convergence behavior of the high-order proximal-point algorithm (HiPPA) to global minimizers, extending existing analyses beyond sublinear convergence rates and complexity analysis. Specifically, we study the proximal operator of a proper lower semicontinuous function augmented with a $p$th-order regularization for $p>1$, and establish the convergence of HiPPA to a global minimizer with a particular focus on its convergence rate. To this end, we focus on minimizing functions in the class of uniformly quasiconvex functions, which includes strongly convex, uniformly convex, and strongly quasiconvex functions as special cases. Our analysis reveals the following convergence behaviors of HiPPA when the uniform quasiconvexity modulus $\phi$ admits a power function of degree $q$ as a lower bound, i.e., $\phi(t) \geq c t^q$ for some $c>0$, on an interval $\mathcal{I}$: (i) for $q\in (1,2)$ and $\mathcal{I}=[0,1)$, HiPPA exhibits a local linear rate for $p\in [q,2)$; (ii) HiPPA converges linearly when $p=2$, $q=2$, and also when $p=q>2$, provided that $\mathcal{I}=[0,\infty)$; (iii) for $q\geq 2$ and $\mathcal{I}=[0,\infty)$, HiPPA achieves a superlinear rate for $p>q$. Notably, to our knowledge, some of these results are novel, even in the context of strongly or uniformly convex functions, offering new insights into optimizing generalized convex problems.

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