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Generalized descent converges linearly exactly when the descent order θ equals the reciprocal of the Kurdyka–Łojasiewicz exponent ϑ, and a boosted high-order proximal-point method achieves this for every ϑ.

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load-bearing objection The DEAL framework is a clean, correct unification with rates that check out; the 'linear for any KL exponent' headline is a one-line citation to an unpublished companion preprint. the 4 major comments →

arxiv 2511.10414 v2 pith:YFKR6OE7 submitted 2025-11-13 math.OC

Minimizing Smooth Kurdyka-{L}ojasiewicz Functions via Generalized Descent Methods: Convergence Rate and Complexity

Masoud Ahookhosh , Susan Ghaderi , Alireza Kabgani , Morteza Rahimi This is my paper
classification math.OC MSC 90C2690C3065K05
keywords generalized descentKurdyka-Łojasiewicz inequalitylinear convergenceKL exponentMoreau envelopehigh-order proximal pointiteration complexitynonconvex optimization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper analyzes a broad family of 'generalized descent' methods that reduce a smooth, possibly nonconvex objective by at least ρ times the θ-th power of the gradient norm at each step. Its central result is a sharp threshold: such a method converges linearly if and only if the descent order θ equals the reciprocal of the Kurdyka–Łojasiewicz (KL) exponent ϑ of the objective; when θ exceeds 1/ϑ, convergence is sublinear with explicit polynomial rates. The authors show that several classical algorithms — constant step-size gradient descent, Armijo line search, proximal gradient, and high-order proximal-point — all fit this one inequality, so one theorem governs their rates. In particular, by choosing the regularization power p = 1/(1−ϑ), the boosted high-order proximal-point method converges linearly for arbitrary KL exponents, a result the paper says is new even in the smooth setting. Under a global KL inequality the same framework yields convergence to global minimizers and O(log(1/ε)) iteration complexity.

Core claim

The central claim is Theorem 3.3: for a sequence satisfying the generalized descent inequality f(x_{k+1}) ≤ f(x_k) − ρ‖∇f(x_k)‖^θ together with the relative-error bound ‖x_{k+1}−x_k‖ ≤ c‖∇f(x_k)‖^{θ−1}, if the objective satisfies the KL inequality at a cluster point with exponent ϑ, then θ = 1/ϑ gives linear convergence of the function gap, the gradient norm, and the iterate distance to 0, with rate q = 1 − ρ/τ^θ; while θ > 1/ϑ gives sublinear rates with explicit polynomial exponents. The paper further claims that the boosted high-order proximal-point method, by setting p = 1/(1−ϑ), achieves linear convergence for arbitrary KL exponent ϑ ∈ (0,1) — a result the authors state is new even in th

What carries the argument

The DEAL framework is built on a single sufficient-decrease inequality, f(x_{k+1}) ≤ f(x_k) − ρ‖∇f(x_k)‖^θ, together with the step-size control ‖x_{k+1}−x_k‖ ≤ c‖∇f(x_k)‖^{θ−1}. The KL inequality (f(x)−f(x̄))^ϑ ≤ τ‖∇f(x)‖ converts these into the scalar recurrence (s_k)^{ϑθ} ≤ (τ^θ/ρ)(s_k − s_{k+1}) for the function gap s_k; a known lemma (Fact 2.2) then turns this recurrence into linear or sublinear estimates depending on whether ϑθ equals or exceeds 1. The high-order Moreau envelope and its gradient formula (33) are the bridge that lets nonsmooth objectives and high-order proximal-point methods enter the same framework.

Load-bearing premise

The headline result that boosted high-order proximal-point converges linearly for any KL exponent rests on the paper's assumption — deferred to a companion preprint — that the high-order Moreau envelope is C^1 and inherits exactly the same KL exponent ϑ as the original function.

What would settle it

Find a concrete semialgebraic KL function with known exponent ϑ whose high-order Moreau envelope φ_γ^p (with p = 1/(1−ϑ)) has a strictly smaller KL exponent, or where C^1 differentiability forces γ so small that the global descent inequality (11) fails; either observation would disprove the claim that boosted HiPPA converges linearly for all ϑ.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • For any DEAL instance — constant step-size, Armijo line search, proximal gradient, or high-order proximal-point — global convergence, convergence rates, and complexity are all governed by the same θ–ϑ relation, so no per-algorithm rate analysis is needed.
  • Linear convergence is achievable for KL exponents beyond the classical ϑ = 1/2 case; the identity θ = 1/ϑ extends linear rates to all ϑ ∈ (0,1).
  • Boosted HiPPA with p = 1/(1−ϑ) is claimed to be the first high-order proximal-point scheme with linear convergence for arbitrary KL exponent, a result the paper says holds even for smooth objectives.
  • Under a global KL inequality, DEAL converges to a global minimizer and reaches ε-accuracy in function value, gradient norm, or iterate distance within O(log(1/ε)) iterations; without the KL property, the gradient-norm complexity is only O(ε^{−θ}).
  • The framework applies to nonsmooth objectives through smoothing (forward-backward and Moreau envelopes), with the smooth surrogate inheriting the KL exponent in the analyzed cases.

Where Pith is reading between the lines

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

  • A testable extension: for any problem where the KL exponent and Hölder smoothness satisfy ϑ = ν/(1+ν) — as the paper proves for least-p inverse problems — both DEAL-C and DEAL-A automatically sit at the linear-convergence threshold θ = 1/ϑ; checking this identity on other Hölder-smooth KL functions would show whether it is a general design principle.
  • If the envelope-inheritance result is confirmed, the same θ–ϑ matching could guide the design of inexact or stochastic variants of DEAL: keeping the effective descent order matched to the smoothed surrogate's exponent should preserve linear convergence.
  • The numerical experiments show heuristic choices of the generalized-descent parameter β (e.g., −0.2) outperforming the theoretically motivated value; a systematic study of the rate behavior away from the exact threshold θ = 1/ϑ could explain this gap and yield better practical parameter rules.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper introduces a generalized descent framework (DEAL), defined by the sufficient-decrease inequality f(x_{k+1}) ≤ f(x_k) − ρ‖∇f(x_k)‖^θ with θ > 1, and analyzes convergence, rates, and complexity under the Kurdyka–Łojasiewicz (KL) property. Section 3 proves, under explicit structural assumptions, that the sequence of gradients is summable, cluster points are critical, and—when the KL exponent ϑ and descent order θ are related—function values, gradient norms, and iterates converge linearly (θ = 1/ϑ) or sublinearly (θ > 1/ϑ); global KL is used to obtain global convergence to minimizers and iteration-complexity bounds. Section 4 instantiates DEAL with constant-step and Armijo line-search versions, and with boosted proximal-gradient and boosted high-order proximal-point methods. The headline claim is that Boosted HiPPA, with regularization order p = 1/(1−ϑ), achieves linear convergence for arbitrary KL exponent ϑ∈(0,1). Numerical experiments on inverse problems and LASSO illustrate the methods.

Significance. If fully established, the paper would provide a clean and reasonably general unification of several descent methods under a single KL-based rate theorem. The core Section 3 arguments are simple, mostly correct, and genuinely useful: the summability of gradient powers, the KL recurrence, and the explicit complexity bounds under global KL are valuable. The paper also gives explicit constants for many of the rates, which is helpful for reproducibility. However, the paper's distinguishing new result—linear convergence of Boosted HiPPA for arbitrary KL exponents—is not proved in this manuscript. It rests on unpublished companion-preprint results about high-order Moreau envelopes that are cited without hypotheses or proof, and the structural hypotheses needed to apply the main theorem to Boosted HiPPA are not verified. The general framework is a solid contribution, but the headline application needs substantial additional justification.

major comments (4)
  1. [§4.3, Corollary 4.11] The headline claim—Boosted HiPPA converges linearly for arbitrary KL exponent ϑ—is not proved in this manuscript. The proof of Corollary 4.11 is a single citation to [21, Theorem 3.9], an unpublished same-author preprint. To apply Theorem 3.3 to f = φ_γ^p, one needs (i) φ_γ^p ∈ C^1 and (ii) φ_γ^p satisfies the KL inequality at the relevant cluster point with the same exponent ϑ and a usable constant τ_γ. Neither is established here; the paper explicitly says in Section 4.3 'To avoid technicalities, we assume φ_γ^p ∈ C1 whenever differentiability is needed', and Fact 4.8 also cites [21, 23]. No hypotheses on prox-regularity, weak convexity, admissible range of γ, or bounded level sets are stated, and the dependence of τ_γ on γ and p is not given. This is load-bearing because p = 1/(1−ϑ) is chosen exactly so that θ = p/(p−1) = 1/ϑ; if the envelope's KL exponent is not ϑ, or if C^1 regulari
  2. [§4.3, Algorithm 4 / Corollary 4.11] Even granting the envelope calculus, Corollary 4.11 does not verify the structural hypotheses needed to invoke Theorems 3.2(d) and 3.3. First, Assumption II (existence of a convergent subsequence) is not implied by the DEAL descent inequality or by the KL property; without coercivity or bounded level sets, an unbounded sequence can have decreasing function values and vanishing gradients. Second, the displacement condition ‖x_{k+1}−x_k‖ ≤ c‖∇φ_γ^p(x_k)‖^{θ−1} is never checked for the Boosted HiPPA line search. With x_{k+1} = (1−κ_k)prox^p_{γφ}(x_k) + κ_k(x_k + d_k), we get ‖x_{k+1}−x_k‖ ≤ γ^{1/(p−1)}‖∇φ_γ^p(x_k)‖^{θ−1} + κ_k‖d_k‖; for arbitrary d_k and the 'largest κ' acceptance rule, the extra term κ_k‖d_k‖ need not be dominated by a multiple of ‖∇φ_γ^p(x_k)‖^{θ−1}. Corollary 4.11 therefore overclaims that all DEAL theorems hold for Boosted HiPPA without additional assumptions on d_k or
  3. [§3, Assumption II and Theorem 3.2] Assumption II is nontrivial and should be highlighted as a limitation. Algorithm 1, as stated, can generate divergent sequences: inequality (11) only controls the decrease of f and the sum of ‖∇f(x_k)‖^θ; it does not imply boundedness of the iterates unless f has bounded sublevel sets or some coercivity is assumed. Thus Theorems 3.2(c)–(d), 3.3, 3.4(c), 3.6, and the corollaries inherit an unverified boundedness/continuity condition. For DEAL-C and DEAL-A the bounded-step structure makes this condition plausible under mild assumptions, but for Boosted HiPPA and Boosted PGA no argument is supplied. The phrase 'global convergence' should be qualified accordingly, or an explicit coercivity/level-set condition should be added to the instances where it is needed.
  4. [§4.3, Algorithm 4 initialization] The acceptance inequality preceding Algorithm 4 requires σ < 1. Indeed, the exact decrease at the prox point is φ_γ^p(prox) = φ_γ^p(x_k) − (1/(pγ))‖x_k−prox‖^p = φ_γ^p(x_k) − (1/p)γ^{1/(p−1)}‖∇φ_γ^p(x_k)‖^{p/(p−1)}, so the claimed inequality with coefficient σγ^{1/(p−1)}/p is valid only if σ < 1. Algorithm 4 states σ ∈ (0, 1/(pγ)); if pγ < 1, this interval includes values of σ ≥ 1, for which the displayed descent inequality cannot hold and the backtracking loop may never terminate. The parameter range should be σ ∈ (0,1), or an equivalent condition ensuring σ < 1, and the statement of Lemma 4.10 should be updated accordingly.
minor comments (5)
  1. [§4.3, Lemma 4.10 proof] The proof refers to 'from (40)', but at that point (40) is not yet defined; (40) is later used for BPGA. The reference should be to the line-search inequality introduced just above Algorithm 4, or a new equation number should be assigned.
  2. [§3, Theorem 3.3 proof] The bounds in (15) and (16) use f(x_0)−f(x̄) even though the linear-rate inequality is proved only for k ≥ k_0. This is harmless if one replaces x_0 by x_{k_0} or absorbs the finite initial segment into the constant, but the text should say so.
  3. [§5.3, Remark 5.2] The KL property for the Moreau envelope is asserted on {x : φ(x) ≤ φ(x_0)}, but the natural level set for the envelope is {x : φ_γ(x) ≤ φ_γ(x_0)}. The cited result [43, Remark 5.1] should be stated with its actual hypotheses, since the inclusion of level sets may require justification.
  4. [§4.3, general] The blanket statement 'To avoid technicalities, we assume φ_γ^p ∈ C1 whenever differentiability is needed' is too vague. The paper should specify whether C^1 regularity is global or on a neighborhood of the level set, and how γ is chosen relative to the problem data (e.g., weak-convexity constant).
  5. [References] Reference [45] appears to contain a typo in the arXiv identifier ('1803.002259'); please verify. Some other references to unpublished arXiv preprints by the same authors ([21], [22], [23]) are used for crucial facts; if these remain unpublished, the main text should include the precise statements used.

Circularity Check

1 steps flagged

Headline arbitrary-KL linear convergence for Boosted HiPPA is delegated to the authors' own unpublished [21]; the DEAL rate derivation itself is self-contained.

specific steps
  1. self citation load bearing [Corollary 4.11 proof; Section 4.3 after Fact 4.8; paragraph after Corollary 4.14]
    "Proof The claim follows from Lemma 4.10 and [21, Theorem 3.9]. ... To avoid technicalities, we assume φ p γ ∈ C1 whenever differentiability is needed. ... A key advantage of Boosted HiPPA lies in its flexibility: by tuning p, it achieves linear convergence for any KL exponent ϑ∈(0,1). Indeed, if φ p γ satisfies the KL inequality with exponent ϑ, choosing θ= 1/ϑ = p/p−1, i.e., p= 1/1−ϑ, guarantees global and linear convergence."

    To apply Theorem 3.3 to f=φ_p^γ, the function must be C^1 and must satisfy KL at the cluster point with exponent ϑ. The C^1 property is merely assumed ('to avoid technicalities'), and the KL-exponent preservation is not proved in this manuscript; the proof of Corollary 4.11 is a one-line citation to [21, Theorem 3.9], a same-author arXiv preprint. Thus the paper's strongest advertised result — linear convergence for arbitrary ϑ — is not derived from assumptions stated in this paper but imported from the authors' own companion work, with no statement of the extra conditions (e.g., prox-regularity, weak convexity, small γ) or of how the KL constant depends on γ. If [21, Thm 3.9] requires conditions not met here, the arbitrary-ϑ conclusion reduces to a conditional statement.

full rationale

The core DEAL analysis in Theorems 3.2-3.6 is not circular: Theorem 3.3 derives the recurrence s_k^{ϑθ} ≤ (τ^θ/ρ)(s_k-s_{k+1}) from the assumed descent inequality (11), the KL inequality, and the displacement condition, then applies the external Fact 2.2. The convergence rate q=1−ρ/τ^θ and the polynomial rates follow algebraically; no target rate is inserted into the constants. DEAL-C and DEAL-A are also reduced to (11) from Hölderian descent and the Armijo step-size lower bound, without assuming the conclusions. The circularity concern is localized to the smoothed high-order instance: the differentiability and KL-preservation of φ_p^γ are taken from the same authors' preprints [21,22,23], and Corollary 4.11's proof is a one-line citation. This is a self-citation/verification burden rather than a by-construction equivalence: if [21, Thm 3.9] is independently valid, the application of DEAL to Boosted HiPPA goes through. Separately (a gap, not circularity), the displacement condition ∥x_{k+1}-x_k∥≤c∥∇φ_p^γ∥^{θ−1} required by Theorem 3.3 is not verified for Algorithm 4's update. Numerical experiments do not repair this theoretical import.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 0 invented entities

The central results are conditional on the KL geometry of the objective and on algorithmic regularity conditions. No physical entities or new mathematical objects with independent falsifiable handles are introduced; 'DEAL' and 'generalized descent direction' are definitions, not entities requiring independent evidence. The main external inputs are the KL exponent/constants and smoothness assumptions, which are assumed rather than derived.

free parameters (4)
  • descent order θ = user-chosen; for DEAL-C/A θ=1+1/ν; for HiPPA θ=p/(p−1); for PGA θ=2
    Controls the decrease in (11). Linear convergence requires θ=1/ϑ, so θ must be tuned to the (assumed) KL exponent; it is not derived from first principles.
  • generalized descent scaling β = β=(1−ν)/ν in theory; heuristics β=0.5, β=0, β=−0.2 in Section 5
    Definition 4.1 introduces β>−1. The theory fixes β from the Hölder exponent ν, but the numerical section selects β=−0.2 after seeing performance, a post-hoc choice.
  • proximal order p = p=1/(1−ϑ)
    Boosted HiPPA's order is chosen to satisfy θ=p/(p−1)=1/ϑ; for ϑ close to 1, p is very large and the prox subproblem becomes expensive.
  • decrease constant ρ = e.g., ρ=c1α^ν/(1+ν), ρ=σᾱc1, ρ=γ^{1/(p−1)}/p, etc.
    Positive constant in (11); its size affects the rate q=1−ρ/τ^θ, but it is a step-size/design parameter, not fitted to data.
axioms (7)
  • domain assumption Assumption I: f∈C^1 and the minimizer set X* is nonempty.
    Standing assumption for problem (1); used throughout Section 3.
  • domain assumption Assumption II: the generated sequence has a convergent subsequence.
    Used in Theorems 3.2(c,d), 3.3, and 3.4(b,c); not guaranteed for noncoercive f.
  • domain assumption KL inequality holds at a cluster point with exponent ϑ and constant τ.
    Definition 2.4(a); drives all rates in Section 3.
  • domain assumption Global KL inequality holds with constants ϑ, τ.
    Definition 2.4(b); enables global optimality and the complexity bounds in Theorems 3.4–3.6.
  • ad hoc to paper Iterate proximity condition ‖x_{k+1}−x_k‖ ≤ c‖∇f(x_k)‖^{θ−1}.
    Imposed in Theorem 3.2(d), 3.3, 3.5, and 3.6; automatic for the specific instances by Remark 4.2, but not for an arbitrary DEAL implementation.
  • ad hoc to paper For Boosted HiPPA, φ_γ^p∈C^1 and the envelope inherits the KL exponent ϑ from φ.
    Needed for Corollary 4.11; imported from same-author preprint [21, Theorem 3.9].
  • domain assumption Assumption III: f has a ν-Hölder continuous gradient.
    Required for the step-size and convergence analysis of DEAL-C and DEAL-A in Sections 4.1–4.2.

pith-pipeline@v1.3.0-alltime-deepseek · 20839 in / 17766 out tokens · 170164 ms · 2026-08-03T22:26:10.619132+00:00 · methodology

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Cite this review

Pith. "Pith review of Minimizing Smooth Kurdyka-{\L}ojasiewicz Functions via Generalized Descent Methods: Convergence Rate and Complexity." pith.science (2026). https://pith.science/paper/YFKR6OE7

@misc{pith2026251110414,
  author       = {Pith},
  title        = {Pith review of: Minimizing Smooth Kurdyka-\Lojasiewicz Functions via Generalized Descent Methods: Convergence Rate and Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFKR6OE7}},
  note         = {Machine review of arXiv:2511.10414}
}
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read the original abstract

This paper introduces a generalized descent algorithm (DEAL) for minimizing smooth nonconvex functions. If the objective function is nonsmooth, a smoothing technique (e.g., forward-backward and high-order Moreau envelopes) is applied to generate a smooth counterpart. The proposed framework unifies several methods, such as gradient-based methods with constant step-sizes and Armijo line search, and several proximal splitting methods. The method is built around a generalized descent inequality that adapts the amount of decrease to the geometry of the objective function. Under the Kurdyka-{\L}ojasiewicz (KL) property, we establish global convergence of the generated sequence to critical points and provide a unified convergence rate analysis. In particular, we show that the convergence behavior depends jointly on the KL exponent and the descent order, and we identify a precise condition under which generalized descent methods achieve linear convergence. By choosing the order of high-order proximal regularization according to the KL exponent, our boosted high-order proximal-point method achieves linear convergence for arbitrary KL exponents. If the objective function satisfies a global KL inequality, we further strengthen the results by proving convergence to global minimizers and deriving explicit iteration-complexity bounds. Numerical experiments validate our theoretical foundation.

Figures

Figures reproduced from arXiv: 2511.10414 by Alireza Kabgani, Masoud Ahookhosh, Morteza Rahimi, Susan Ghaderi.

Figure 1
Figure 1. Figure 1: Convergence of loss function (log scale) and gradient norm over iterations. First row: DEAL-C and its [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of loss function (log scale) and gradient norm over iterations for BPGA, BPGA-1, BPGA [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

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