{"id":"16a4bbc2-ad88-47e0-811a-1d1d9e990ca6","arxiv_id":"2505.20484","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"HiPPA's claimed local linear convergence for order p<2 on strongly convex functions is contradicted by a simple quadratic example.","lead":"The paper analyzes convergence rates of high-order proximal-point algorithms on uniformly quasiconvex functions, claiming linear and superlinear rates. A key claimed linear-rate result is false as stated, so the paper's central contribution is not sound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.10(a) rests on an invalid p-norm inequality in Lemma 4.9(a) and fails on the scalar quadratic, so the claimed local linear rate for p<2 is unsupported.","rationale":"The reader's weakest_assumption identifies the same invalid inequality in Lemma 4.9(a), and I agree that this is the core defect. The paper's headline result, Theorem 4.10(a), claims local linear convergence for all p∈(1,2) whenever the modulus has a power lower bound of degree q∈(1,2]. The proof depends on a norm inequality that is plainly false in the small-error regime, and the result itself is contradicted by the scalar quadratic: for p=1.5 the HiPPA error ratio tends to 1, not to a constant below 1. I also found a second, independent algebraic error in the proof of Theorem 4.10(a), where r^{q−1}≥r for r<1 is used as if it were r^{q−1}≤r, so even granting (4.6) the contraction factor would not follow. The abstract's formulation, which requires p∈[q,2), would avoid the quadratic counterexample, but the theorem as stated in the full text omits that condition. The remaining parts of the paper—such as the calculus of uniform quasiconvexity, coercivity, and the global convergence of HiPPA—do not appear to suffer from the same defect, but the central claimed novelty is broken. The reader's REJECT verdict is therefore appropriate; my read does not change it.","tokens_in":25074,"tokens_out":7864,"duration_ms":70215,"concrete_test":"Run a one-dimensional numerical check: take φ(x)=x^2/2, p=1.5, γ=1, x0=0.1, and compute x_{k+1} by solving x_{k+1}+|x_{k+1}-x_k|^{p-2}(x_{k+1}-x_k)=0; record ratios r_k=|x_{k+1}|/|x_k|. If r_k→1 (as the relation r_k=|r_k−1|^{p−1}|x_k|^{p−2} predicts), Theorem 4.10(a)'s '<1' bound fails for a strongly convex, hence uniformly quasiconvex, objective with q=2 and p<2. An equivalent analytic check: substitute a=b, ∥a∥=0.5, p=1.5 into the norm inequality used in Lemma 4.9(a); it gives 0.3536≤0.25, so the lemma's proof step is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Lemma 4.9(a) and its use in Theorem 4.10(a). In the proof of Lemma 4.9(a), for p∈(1,2), the authors effectively replace ∥λa+(1−λ)b∥^p by λ∥a∥^2+(1−λ)∥b∥^2−λ(1−λ)σ_p∥a−b∥^2. This claimed inequality is false: taking a=b with 0<∥a∥<1 gives ∥a∥^p≤∥a∥^2, impossible for p<2. Since ∥x−x_k∥→0 and ∥x_{k+1}−x_k∥→0, the relevant vectors a=x−x_k and b=x_{k+1}−x_k are eventually in exactly the region where the inequality fails. Thus inequality (4.6) is not established. There is a second independent flaw in Theorem 4.10(a): from (4.6) and ρ_q t^q≤φ(t), for r=∥x−x_{k+1}∥<1 and q∈(1,2], one has r^{q−1}≥r, not r^{q−1}≤r, so the displayed deduction of a uniform contraction ratio <1 is invalid. The claim is moreover contradicted by the scalar strongly convex quadratic φ(x)=x^2/2 with p∈(1,2) and fixed γ. The first-order condition yields error ratios satisfying r_k=|r_k−1|^{p−1}e_k^{p−2}, and since p−2<0, r_k→1 as e_k→0. Hence no local linear rate with ratio <1 can hold. The abstract's qualification p∈[q,2), which would exclude this counterexample, is absent from Theorem 4.10(a).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25447,"tokens_out":18660,"duration_ms":162957,"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":[{"comment":"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.","section":"§4.2, Lemma 4.9(a)"},{"comment":"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).","section":"§4.2, Theorem 4.10(a)"},{"comment":"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.","section":"§4.2, Lemma 4.9(a) and Theorem 4.10(a)"}],"minor_comments":[{"comment":"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.","section":"§4.2, Lemma 4.9(a)"},{"comment":"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.","section":"§2 and §4.2"},{"comment":"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.","section":"Table 1"},{"comment":"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.","section":"§4.2, proof of Theorem 4.10(a)"}],"recommendation":"reject","confidential_remarks":"The paper's main claim (Theorem 4.10(a) and the corresponding abstract bullet) is false, as shown by a scalar quadratic counterexample, and the proof rests on an invalid norm inequality. The remaining p=2, p=q>2, and p>q results may be salvageable, but they are not the paper's stated headline. I would encourage the authors to resubmit a substantially revised version that removes the false p<2 claim or restricts it to a genuinely valid regime (e.g., p∈[q,2) with a correct proof), and that either proves or properly cites the key technical lemmas currently deferred to unpublished preprints."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe 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.\n\nThat 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.\n\nMinor 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.\n\nWho 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.","headline":"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.","tokens_in":26014,"tokens_out":7856,"would_cite":false,"duration_ms":70029,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C25","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["high-order proximal-point algorithm","uniform quasiconvexity","high-order Moreau envelope","convergence rates","linear convergence","superlinear convergence","nonconvex optimization","global minimizer"],"falsifier":"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.","tokens_in":24837,"feed_emoji":"📉","tokens_out":12923,"duration_ms":128783,"temperature":0.7,"pith_summary":"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.","feed_headline":"High-order proximal method gets linear rates beyond convexity","feed_subtitle":"For uniformly quasiconvex functions, convergence speed is set by the modulus growth q and the proximal order p.","key_machinery":"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 $p\\in(1,2)$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the convex-analysis toolkit: strong convexity of $\\|x\\|^2$, uniform convexity of $\\|x\\|^q$ for $q>2$, and level-boundedness facts used throughout the proof.","marker":"[6]"},{"why":"Defines the high-order Moreau envelope and proximal operator whose continuity and boundedness properties underpin HiPPA's well-posedness.","marker":"[29]"},{"why":"Establishes strong convexity of $\\|x\\|^q$ on bounded balls for $q\\in(1,2)$, the local estimate behind Lemma 4.9(a).","marker":"[32]"},{"why":"Baseline result that proximal-point methods converge to global minimizers of strongly quasiconvex functions, extended here to uniformly quasiconvex functions.","marker":"[36]"},{"why":"Provides the strong-convexity lemma for $\\|x\\|^q$ on balls used in Example 3.2.","marker":"[48]"},{"why":"Classical proximal point algorithm and its linear convergence under strong convexity, the baseline that Theorem 4.10 generalizes.","marker":"[54]"},{"why":"Gives the variational-analysis results on level boundedness and outer semicontinuity used to prove cluster-point properties of HiPPA.","marker":"[55]"},{"why":"Provides the differential characterization of uniform quasiconvexity used in Theorem 3.16 and the stationarity-to-global-minimizer argument.","marker":"[61]"}],"fun_headline_variants":["HiPPA converges linearly beyond convexity","Proximal method hits superlinear rates for quasiconvex","New convergence rates for high-order proximal point","Asymptotic rates for HiPPA: linear to superlinear","Beyond sublinear: HiPPA rates for quasiconvex objectives"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $p<2$.","fun_headline_variants_meta":{"raw":{"variants":["HiPPA converges linearly beyond convexity","Proximal method hits superlinear rates for quasiconvex","New convergence rates for high-order proximal point","Asymptotic rates for HiPPA: linear to superlinear","Beyond sublinear: HiPPA rates for quasiconvex objectives"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1513,"prompt_tokens":1128,"completion_tokens":385,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":744,"completion_tokens_details":{"reasoning_tokens":305}},"tokens_in":744,"tokens_out":385,"duration_ms":4783,"temperature":1.0,"reasoning_tokens":305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:56:12.109782+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bauschke and P.-L","cited_arxiv_id":null,"evidence_quote":"Supplies the convex-analysis toolkit: strong convexity of $\\|x\\|^2$, uniform convexity of $\\|x\\|^q$ for $q>2$, and level-boundedness facts used throughout the proof."},{"cited_title":"Kabgani and M","cited_arxiv_id":null,"evidence_quote":"Defines the high-order Moreau envelope and proximal operator whose continuity and boundedness properties underpin HiPPA's well-posedness."},{"cited_title":"Kabgani and M","cited_arxiv_id":null,"evidence_quote":"Establishes strong convexity of $\\|x\\|^q$ on bounded balls for $q\\in(1,2)$, the local estimate behind Lemma 4.9(a)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Baseline result that proximal-point methods converge to global minimizers of strongly quasiconvex functions, extended here to uniformly quasiconvex functions."},{"cited_title":"Nesterov","cited_arxiv_id":null,"evidence_quote":"Provides the strong-convexity lemma for $\\|x\\|^q$ on balls used in Example 3.2."},{"cited_title":"Rockafellar","cited_arxiv_id":null,"evidence_quote":"Classical proximal point algorithm and its linear convergence under strong convexity, the baseline that Theorem 4.10 generalizes."},{"cited_title":"Rockafellar and R","cited_arxiv_id":null,"evidence_quote":"Gives the variational-analysis results on level boundedness and outer semicontinuity used to prove cluster-point properties of HiPPA."},{"cited_title":"Vladimirov, Y","cited_arxiv_id":null,"evidence_quote":"Provides the differential characterization of uniform quasiconvexity used in Theorem 3.16 and the stationarity-to-global-minimizer argument."}],"review_version":1}