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REVIEW 6 major objections 6 minor 67 references

General Proximal Quasi-Newton Methods based on model functions for nonsmooth nonconvex problems

T0 review · 6 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A proximal quasi-Newton algorithm for nonsmooth nonconvex problems converges to a stationary point without assuming the Hessian approximations stay bounded, deriving their boundedness from the Kurdyka–Łojasiewicz property.

desk verdict The local-model setup is a real contribution, but the theorem that drops boundedness of the variable metric relies on an unstated global version of the model-error bound, so the paper is not yet ready as written. read the letter →

arxiv 2507.18363 v2 pith:JZODAZZL submitted 2025-07-24 math.OC

Xiaoxi Jia This is my paper · ORCID
classification math.OC MSC 49J5265K0590C2690C30
keywords proximalquasi-NewtonmethodslocalmodelfunctionnonsmoothnonconvexoptimizationKurdyka-Łojasiewiczpropertyvariablemetricboundednessbacktrackinglinesearchcompositequadraticinverseproblems
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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 proposes a proximal quasi-Newton algorithm for minimizing proper, lower semicontinuous functions that may be both nonsmooth and nonconvex. In place of a fixed global model of the objective, each iteration uses a local model function centered at the current iterate, and the next iterate is obtained by minimizing that model plus a quadratic proximity term weighted by a variable metric $H_k$ generated by a continuous matrix-valued map. The central claim is that the Hessian approximations $H_k$ need not be assumed bounded: under a mild first-order control on the model error, every accumulation point of the generated sequence is stationary even while $\|H_k\|$ may grow, and if the objective has the Kurdyka–Łojasiewicz property at an accumulation point, the whole sequence converges to a stationary point and the variable metrics become uniformly bounded as a consequence. If this is right, a standard boundedness assumption in proximal quasi-Newton theory is not a prerequisite but a byproduct of convergence, which matters for problems where Hessian approximations legitimately blow up near sharp minima.

What carries the argument

The load-bearing object is the local model function (Definition 3.2): a proper lsc function $f_{\bar x}$ with domain equal to that of $f$, whose approximation error to $f$ is controlled near the model center by a growth function, $|f(x)-f_{\bar x}(x)|\le \omega_{\bar x}(\|x-\bar x\|)$ for $x$ approaching $\bar x$. Together with Assumption 4.2(H2), $\partial g_{\bar x}(\cdot)\subset L B_{\|\cdot-\bar x\|}(0)$, Proposition 3.3 converts this into the subdifferential inclusion $\partial f_{\bar x}(\tilde x)\subset \partial f(\tilde x)+L B_{\|\bar x-\tilde x\|}(0)$, which is what transfers stationarity from the subproblem solution to the objective. The algorithm's acceptance criterion $|f(x_{k,i})-f_{x_k}(x_{k,i})|\le \delta\gamma_{k,i}\|x_{k,i}-x_k\|^2_{H_k}/2$ guarantees sufficient descent, and the continuity of $H$ plus the quadratic proximity term makes the subproblems coercive.

What would settle it

Run Algorithm 1 on $f(x)=|x|^{3/2}$ with the first-order Taylor model $f_{\bar x}(x)=|\bar x|^{3/2}+\tfrac32\operatorname{sgn}(\bar x)|\bar x|^{1/2}(x-\bar x)$ and starting points approaching 0. Along accepted steps $x_{k+1}\to 0$, compute the ratio $|f_{x_k}(x_{k+1})-f(x_{k+1})|/\|x_{k+1}-x_k\|^2$; because the model-error derivative near 0 behaves like $|\bar x|^{-1/2}\|x-\bar x\|$, this ratio should be unbounded even when the acceptance test passes. A nonstationary accumulation point in such a run would refute the claimed reach of the convergence theorem without the fixed-$L$ control, while a stationary one would show that the local model alone is doing the work attributed to Assumption 4.2(H2).

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Extended reading notes

Core claim

The paper's core result is Theorem 4.11: under Assumption 4.1 (lower boundedness, affine lower bounds on model functions, a continuous uniformly positive definite matrix generator $H$, and existence of an accumulation point), Assumption 4.2 with smooth model errors $g_{\bar x}=f_{\bar x}-f$, and the Kurdyka–Łojasiewicz property at an accumulation point, the sequence $\{x_k\}$ generated by Algorithm 1 converges to a stationary point $x^*$ of $f$, and the variable metric sequence $\{H(x_k)\}$ is uniformly bounded. Before invoking the KL property, Theorem 4.7 already establishes that every accumulation point is stationary while $\{H(x_k)\}$ need not be bounded. The paper reads this as evidence that the boundedness of the variable metric should be a problem-tailored consequence of convergence rather than an assumption imposed a priori, and it backs the claim with convergence-rate statements (Theorem 4.12) under standard desingularization functions and with experiments on polytope feasibility and (sparse) quadratic inverse problems.

Load-bearing premise

The load-bearing premise is Assumption 4.2(H2): the first-order variation of the model error must be bounded by a fixed constant $L$ times the distance to the model center at every iterate, and without that control the quadratic model-error bound and the transfer of stationarity to the objective both break.

Editorial extensions

If this is right

  • Boundedness of the Hessian approximations becomes a conclusion reachable from convergence: if the objective has the KL property at an accumulation point, $\{H(x_k)\}$ is uniformly bounded without being assumed so.
  • Subsequential stationarity does not require bounded iterates or bounded metrics; it follows from the first-order control on the model error alone.
  • The rate theorem gives concrete behavior depending on the desingularization exponent: finite termination, superlinear, linear, or polynomial convergence, so the method inherits the classical KL landscape of rates.
  • Because $H_k=I$ is an allowed generator, the framework covers proximal gradient and proximal descent methods, and the convergence results apply to additive composite problems and nonlinear composite problems of the form $q(x)+h(A(x))$.
  • On the tested quadratic inverse problems, the model quasi-Newton variant used fewer outer and inner iterations and produced sparser solutions than the model gradient variant, and the gradient method failed on some instances.

Reading between the lines

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

  • The same proof skeleton suggests a design principle for other quasi-Newton schemes: any Hessian update that can be written as a continuous matrix-valued function of the current iterate, plus a model-error control like Assumption 4.2(H2), should admit convergence analysis without a priori metric bounds; this points toward testing limited-memory and learned Hessian generators in the same framework.
  • Since only the KL property at a single accumulation point is required, the theory should apply to functions definable in o-minimal structures, including semialgebraic objectives like polytope feasibility, where KL holds automatically; a natural extension is to constraints or Bregman proximities using the same local-model acceptance test.
  • A testable consequence of the paper's philosophy: on objectives with sharp minima where second derivatives blow up, allowing $H_k$ to grow along the nonsmooth directions should improve conditioning and speed rather than hurt, exactly because the theory tolerates unbounded metrics; this could be benchmarked against fixed-metric proximal gradient on functions like $|x|^p$ with $p\in(1,2)$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 6 minor

Summary. The paper proposes a model-based proximal quasi-Newton method (Algorithm 1) for minimizing proper lower semicontinuous functions. The key algorithmic ingredients are a local model function f_xbar, a variable metric generated by a continuous matrix-valued map H(x), and a backtracking line search based on the model error. Under an assumption on the subdifferential of the model error (Assumption 4.2(H2)), the paper claims subsequential stationarity of all accumulation points (Theorem 4.7), and, adding a Kurdyka-Łojasiewicz property, convergence of the whole sequence to a stationary point (Theorem 4.11) together with convergence rates (Theorem 4.12). The stated goal is to avoid assuming uniform boundedness of the variable metric a priori. Numerical experiments on polytope feasibility and quadratic inverse problems are reported.

Significance. If the gaps listed below are repaired, the framework would be a valuable contribution: it targets a genuine restriction in nonsmooth proximal quasi-Newton analysis (the uniform boundedness of the Hessian approximation) and combines it with a flexible local-model framework. The continuity of the metric generator and the local model definition are natural and the worked examples in Section 5 are useful. The paper also ships a large set of detailed auxiliary proofs and numerical experiments, which is a strength. At present, however, the central results are not supported as written: the key subdifferential transfer result is unproved and stated too broadly, the global formulation of the model-error assumption is not matched by the local examples, and the KL-based proof and the rate theorem rely on omitted arguments.

major comments (6)
  1. [Section 3, Proposition 3.3] Proposition 3.3 is stated without proof and is false as stated. Its hypothesis (3.6) controls ∂g_xbar(x) only for x in a neighborhood of xbar, yet the conclusion (3.4) is asserted for an arbitrary fixed x~ in dom f. For f(x)=x^4, xbar=1, L=1, and the model f_xbar(x)=max{0,1+4(x−1)}, the point x~=100 satisfies (3.5) and (3.6) (the latter on a neighborhood of 1) but violates (3.4): ∂f_xbar(100)={4} is not contained in ∂f(100)+B_{99}(0)={40000}+[−99,99]. The proposition must be restated with x~ lying in the neighborhood where (3.6) is valid, and a proof must be supplied. The subsequent uses in Theorem 4.7 (via (4.19)), Lemma 4.10, and Theorem 4.11 all depend on this correction.
  2. [Section 4, Assumption 4.2(H2)] Assumption 4.2(H2) is a global condition: ∂g_xbar(·)⊂L B_{||·−xbar||}(0) is required for every x and every model center xbar. This is not a consequence of the local model definition (3.3), and the appendix only establishes the local version: Propositions 8.1 and 8.2 verify (3.6) 'for x close to xbar', not globally. The motivating example in Section 2, f(x)=x^4 with f_xbar(x)=max{0,xbar^4+4xbar^3(x−xbar)}, fails the global bound: for fixed xbar≠0, |∇g_xbar(x)|/|x−xbar|→∞ as |x|→∞. The proofs of Proposition 4.6, Proposition 4.9, and Lemma 4.10 apply H2 at points x_{k+1} or xhat_k for which only ||x_{k+1}−x_k||→0 is known. The author should either replace H2 by a localized version with a uniform L on a relevant compact set (with a complete argument), or give examples that genuinely satisfy the global assumption.
  3. [Section 4.3, Lemma 4.10] The proof of Lemma 4.10 is omitted with the justification that it is 'highly similar' to Theorem 4.7. This lemma is load-bearing for Theorem 4.11: the relative error estimate (4.29) is exactly the inequality that feeds the KL inequality in the finite-length induction. Theorem 4.7 alone does not provide the uniform bound γ_k≤γbar_ρ on I_ρ, nor the uniform constant L over the compact set Cρ, so the claimed similarity is not immediate. A complete proof must be written out.
  4. [Section 4.3, Theorem 4.12] The four rate statements (i)–(iv) — finite termination, superlinear convergence, Q-/R-linear convergence, and sublinear rates — are asserted without proof. Since Sections 2 and 7 explicitly claim rate-of-convergence results, these statements are unsupported in the present version. The author should supply at least a complete proof sketch or a precise derivation of the standard KL-based inequalities in the current setting, including the dependence of the constants on the stepsize bounds and the metric bound M.
  5. [Section 4.3, Theorem 4.11] The induction argument in Theorem 4.11 contains a constant mismatch. The quantity α in (4.26) is defined with the term 2γbar_ρ M φ(...)/((1−δ)μγmin), but in the induction step for statement (a) the proof arrives at the bound 2(γbar_ρ M + L) φ(...)/((1−δ)μγmin) and then asserts equality with α. This equality is false unless L=0. The definition of α must be corrected to incorporate L (or the proof adjusted accordingly); otherwise the claimed induction that x_k stays in B_α(x*) is not established. This is a central step in the proof of the main sequential convergence result.
  6. [Section 4.2, Proposition 4.6] The proof of Proposition 4.6 needs a regular subgradient η_k of g_xk at xhat_k with ||η_k||≤L||xhat_k−x_k||, and it then uses the two displayed lim-inf inequalities with this η_k. The existence of such η_k is not justified from the stated assumptions: the optimality of xhat_k for the subproblem yields a limiting subgradient of g_xk, not necessarily a regular subgradient, and the regular subdifferential can be empty at a point while the limiting subdifferential is nonempty. Without η_k, the bound |g_xk(xhat_k)|≤L||xhat_k−x_k||^2+ε||xhat_k−x_k|| is not derived, and hence the conclusion γ_k||x_{k+1}−x_k||→0 is not supported. This gap affects Theorem 4.7, which uses Proposition 4.6 directly.
minor comments (6)
  1. [Abstract and Section 2] The phrase 'the sequence of the variable metric is proved not to be bounded' misstates the results: Theorem 4.7 proves convergence without assuming boundedness, and along the convergent subsequence H(x_k)→H(x*) by continuity; Theorem 4.11 then proves uniform boundedness under KL. The intended meaning is 'not required to be bounded a priori', and the wording should be corrected.
  2. [Theorem 4.7 proof] In the proof of Theorem 4.7, the sentence 'Hence, by Proposition 4.9, we have dist(0,∂f(x_{k+1}))≤...' should refer to Proposition 4.6 (or directly to the preceding inclusion (4.19)), since it is the convergence γ_k||x_{k+1}−x_k||→0 along K that is needed.
  3. [Proposition 4.6 proof] The proof of Proposition 4.6 references '(4.20)' before that equation is defined (it appears only in Section 4.3) and invokes 'Definition 3.1' where the relevant model error bound comes from Definition 3.2. These cross-reference errors should be fixed.
  4. [Various sections] There are numerous typos and grammar issues: 'matric' for 'metric', 'disingularization' for 'desingularization', 'respective' for 'perspective', 'terninate' for 'terminate', 'closed to' for 'close to', and 'decent' for 'descent'. A thorough editorial pass is needed.
  5. [Section 4, Algorithm 1] Algorithm 1 requires an exact global minimizer of each subproblem (4.1). The numerical experiments, however, solve the subproblems with ADMM or PDHG, which are inexact heuristics. The gap between the theoretical exact-solver oracle and the practical implementation should be acknowledged, and ideally the theory should be extended to inexact solves.
  6. [Section 6.1] In the polytope feasibility experiments, only the results of Algorithm 1 are reported because the gradient method returned 'NaN' for all tested p. Since the comparison is claimed as a demonstration of effectiveness, omitting the gradient method's failure trajectories makes the empirical comparison less informative.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence theorems are derived from explicit regularity assumptions on the model error, not from the conclusions they establish.

full rationale

The paper's central derivation is self-contained. Theorem 4.7 derives stationarity of any accumulation point from Assumption 4.1, Assumption 4.2(H2), and the subgradient transfer in Proposition 3.3; Theorem 4.11 adds only the Kurdyka–Łojasiewicz property (Assumption 4.8) to get whole-sequence convergence. Assumption 4.2(H2) is a stated one-sided regularity hypothesis on the model-error functions g_xbar (their subgradients lie in L times the ball of radius ||x−xbar||); it is not an echo of the conclusion 0 ∈ ∂f(x*), and Proposition 3.3 is the independent bridge between model and objective subgradients. No parameter is fitted from the data and then reported as a prediction, and no result is invoked from the author's prior work except [30] as an alignment remark that plays no role in the proofs. The boundedness of the variable metric is genuinely derived: continuity of H gives local boundedness on compact sublevel sets, while the KL argument bounds stepsizes, and absence of a uniform bound outside the KL setting follows from the lack of any uniform bound in Assumption 4.1(c). The local-to-global discrepancy between (3.6) and (H2) is a possible correctness gap or an implicit strengthening of the assumption, but an overly strong assumption is not circular: the conclusions are not contained in the hypotheses by definition. Therefore the derivation chain is independent of its own outputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The convergence theorems rest on standard variational analysis and the KL property, plus the paper-specific assumptions listed above. No free parameters are fitted in the theory; the constants in Section 6 are experimental hyperparameters. No new entities with independent empirical evidence are introduced.

assumptions (6)
  • domain assumption f is proper, lower semicontinuous and bounded below (Assumption 4.1(a))
    Required for sublevel sets and a decreasing objective value sequence; used throughout Section 4.
  • ad hoc to paper The model function f_xbar is bounded below by an affine function for every xbar in dom f (Assumption 4.1(b))
    Used to guarantee subproblems are coercive; not justified by Definition 3.2 and stronger than needed, since a constant lower bound would suffice with the quadratic term.
  • domain assumption H: R^n to positive definite matrices is continuous and λ_min(H) ≥ µ (Assumption 4.1(c))
    Defines the variable metric generator; continuity is used to bound ||H_k|| on convergent subsequences and in the KL argument.
  • ad hoc to paper Assumption 4.2(H2): ∂g_xbar(·) subset of L B_{||·−xbar||}(0) with fixed L for all xbar
    The main modeling assumption; provides the first-order control of model error that drives Propositions 4.6 and 4.9. It is not implied by Definition 3.2.
  • domain assumption Kurdyka-Łojasiewicz property at an accumulation point (Assumption 4.8)
    Standard regularity condition satisfied by semialgebraic functions; used for whole-sequence convergence and rates.
  • ad hoc to paper Exact global minimizer of each subproblem (4.1) is available at every inner iteration
    Algorithm 1 requires solving a generally nonconvex subproblem exactly; no inexact oracle is provided, so implementability is not established.

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Pith. "Pith review of General Proximal Quasi-Newton Methods based on model functions for nonsmooth nonconvex problems." pith.science (2026). https://pith.science/paper/JZODAZZL

@misc{pith2026250718363,
  author       = {Pith},
  title        = {Pith review of: General Proximal Quasi-Newton Methods based on model functions for nonsmooth nonconvex problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JZODAZZL}},
  note         = {Machine review of arXiv:2507.18363}
}
read the original abstract

In this manuscript, we propose a general proximal quasi-Newton method tailored for nonconvex and nonsmooth optimization problems, where we do not require the sequence of the variable metric (or Hessian approximation) to be uniformly bounded as a prerequisite, instead, the variable metric is updated by a continuous matrix generator. From the respective of the algorithm, the objective function is approximated by the so-called local model function and subproblems aim to exploit the proximal point(s) of such model function, which help to achieve the sufficiently decreasing functional sequence along with the backtracking line search principle. Under mild assumptions in terms of the first-order information of the model function, every accumulation point of the generated sequence is stationary and the sequence of the variable metric is proved not to be bounded. Additionally, if the function has the Kurdyka-{\L}ojasiewicz property at the corresponding accumulation point, we find that the whole sequence is convergent to the stationary point, and the sequence of the variable metric is proved to be uniformly bounded. Through the above results, we think that the boundedness of the sequence of the variable metric should depend on the regularity of objectives, rather than being assumed as a prior for nonsmooth optimization problems. Numerical experiments on polytope feasibility problems and (sparse) quadratic inverse problems demonstrate the effectiveness of our proposed model-based proximal quasi-Newton method, in comparison with the associated model-based proximal gradient method.

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