A Proximal Descent Method for Minimizing Weakly Convex Optimization

Feng-Yi Liao; Yang Zheng

REVIEW 2 major objections 4 minor 2 cited by

A Proximal Descent Method for Minimizing Weakly Convex Optimization

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that a bundle-based proximal descent method finds (η,ε)-inexact stationary points of weakly convex functions at the best-known complexity, and adapts automatically under smoothness and quadratic growth.

desk verdict A clean, honest bundle-method paper with real adaptivity results; the global Lipschitz assumption in the main theorem doesn't cover its own experiments, but the theoretical core is sound. read the letter →

arxiv 2509.02804 v1 pith:E4PKHHF3 submitted 2025-09-02 math.OC cs.SYeess.SY

Feng-Yi Liao , Yang Zheng This is my paper

classification math.OCcs.SYeess.SY MSC 90C2690C2549J52

keywords weaklyconvexoptimizationproximaldescentbundlemethodnonsmoothnonconvexMoreauenvelopeinexactstationarityconvergenceratesadaptiveacceleration

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 tries to show that a simple, deterministic first-order method can minimize any m-weakly convex function—convex, nonsmooth, or composite nonconvex—with explicit worst-case guarantees. It combines the inexact proximal point method with classical convex bundle cutting planes, solving each proximal subproblem approximately rather than exactly. The central claim is that this proximal descent method reaches an (η,ε)-inexact stationary point in O((1/η² + 1/ε) max{1/η², 1/ε}) function and subgradient evaluations, which yields the best-known O(1/δ⁴) rate for Moreau stationarity. A distinctive further claim is that the very same algorithm, without any parameter change, automatically accelerates under smoothness and quadratic growth. This matters because existing weakly convex solvers either need careful stepsize tuning, use fixed inner-loop budgets, or have more complicated algorithmic machinery.

What carries the argument

The central object is the (η,ε)-inexact stationarity measure dist(0, ∂_ε f(x)), where ∂_ε f(x) is an ε-inexact subdifferential defined by allowing an ε slack in the quadratic underestimators of the weakly convex function. Each outer step seeks a point x_{k+1} satisfying a descent condition f(x_{k+1}) ≤ f(x_k) − (m+βρ)/(2α²)||~g_{k+1}||² with ~g_{k+1} ∈ ∂_{ε_{k+1}} f(x_{k+1}), and the inner bundle subroutine builds convex lower approximations of the locally convexified model f(·) + (m/2)||·−x_k||², generating trial points whose proximal steps yield such inexact subgradients. The key identity is that ~g_{k+1} = α(x_k − z_{k+1}) with α = m+ρ, and the inexactness ε_{k+1} equals the approximation

What would settle it

Run Algorithm 4 on the phase-retrieval objective f(x) = (1/n)Σ|⟨a_i,x⟩² − b_i| with Gaussian data and a ground-truth signal, recording G_k = sup_j ||g_j|| in each outer iteration. If G_k grows without bound as x_k approaches a solution, then no global Lipschitz constant L exists and the uniform bound (1+m/ρ)L in Lemma 7 fails for this instance, so the O((1/η² + 1/ε) max{1/η², 1/ε}) total complexity cannot transfer directly to non-Lipschitz weakly convex problems.

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Extended reading notes

Core claim

The paper's central claim is that a proximal descent method—outer inexact proximal point iterations combined with an inner convex bundle routine—finds an (η,ε)-inexact stationary point of an m-weakly convex function in at most O((1/η² + 1/ε) max{1/η², 1/ε}) function and subgradient evaluations, provided the objective is globally Lipschitz. A direct corollary is an O(1/δ⁴) complexity for finding a point whose Moreau-envelope gradient norm is at most δ, matching the best-known rate in the weakly convex literature. The method is adaptive in a strong sense: under M-smoothness the same algorithm achieves O(1/δ²) gradient stationarity, and under quadratic growth with parameter above the weak-conve

Load-bearing premise

The main rate theorem assumes the objective is globally Lipschitz continuous, which is needed to bound the subgradient norms inside the bundle subroutine; many natural weakly convex objectives, including the ones tested in the paper, are not globally Lipschitz.

Editorial extensions

If this is right

For Lipschitz weakly convex functions, there is now a deterministic algorithm that certifies (η,ε)-inexact stationarity with a worst-case bound matching the best known, using quantities computable during the run rather than an implicit Moreau envelope.
The O(1/δ⁴) rate for Moreau stationarity is achieved with a simpler algorithmic structure than prior weakly convex proximal bundle methods, and without requiring a predetermined inner-loop horizon.
Under smoothness, the same method automatically reaches O(1/δ²) gradient stationarity—matching gradient descent—without knowing the smoothness modulus in its step rule.
Under quadratic growth with µq > m, the method automatically reaches (η,ε)-stationarity in O(max{1/η², 1/ε}) evaluations, and O(log(1/δ)) gradient evaluations when the function is also smooth.
Because Algorithm 3 is equivalent to a single-loop proximal bundle method (Algorithm 5), the complexity guarantees transfer directly to that more standard presentation.

Reading between the lines

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

Editorial: The dynamic inner-loop acceptance test is likely to be more efficient in practice than fixed-budget subgradient inner solvers, because it spends few iterations when the proximal gap is large; the numerical comparisons with the proximally guided subgradient method support this intuition but are not a theorem.
Editorial: If the global Lipschitz assumption is replaced by a bounded-subgradient or local-Lipschitz condition that keeps the quantity G_k in Proposition 4.1 uniformly bounded, the main theorem would directly extend to the phase retrieval and blind deconvolution instances tested in the paper, which are not globally Lipschitz.
Editorial: The (η,ε)-inexact stationarity measure gives a practical stopping rule for weakly convex optimization, and exploiting it could make other proximal-point-style algorithms easier to certify in practice.
Editorial: The method still requires knowing the weak-convexity parameter m; an adaptive online estimate of m would eliminate the last user-chosen structural parameter and is a natural next step.

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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

Short version: this is a legit contribution to weakly convex optimization. It adapts the classical proximal bundle method to m-weakly convex functions, proves an O((1/η^2+1/ε)max{1/η^2,1/ε}) evaluation bound for (η,ε)-inexact stationarity, matches the known O(1/δ^4) Moreau rate, and then shows the same algorithm accelerates to O(1/δ^2) under smoothness and linear rates under quadratic growth. The acceleration results (Theorems 3, 5, 6) are the genuinely new part; the (η,ε) rate was already in Liang-Monteiro-Zhang [20], as the paper says. The algorithmic realization as a two-line bundle subroutine with a descent test is simpler than [20], and the analysis is careful.

On substance: I went through Lemmas 4-7 and Propositions 4.1-4.2. They are standard bundle inequalities adapted to weak convexity, and they check out. The telescoping argument for Theorem 1 is fine. The adaptivity claims are conditional on the same machinery, and the proofs are in the appendix.

Soft spots: the main theorem assumes f is globally L-Lipschitz. That assumption enters Lemma 7 and is what makes the inner-loop count finite and independent of the outer iteration. But both numerical sections use phase retrieval and blind deconvolution objectives, which grow superlinearly and are not globally Lipschitz. The paper doesn't discuss this mismatch or supply a bounded-domain/local-Lipschitz argument. That's a real gap between the stated scope and the experiments, though not a flaw in the theory itself. The paper also ships no code or data, and the tables have no error bars; for an algorithm paper this is a weakness but not a fatal one.

One more caveat: Theorem 5's O(max{1/η^2,1/ε}) still retains 1/η^2 and 1/ε, so the "linear in log" sense only shows up in Theorem 6. The reader's summary is accurate there.

Who this is for: anyone working on bundle methods or weakly convex optimization. It deserves a serious referee, and the main results should stand after the Lipschitz/experiment mismatch is addressed and code/data are shared.

Referee Report

2 major / 4 minor

Summary. The paper proposes a proximal descent method (Algorithm 3) for minimizing m-weakly convex, possibly nonsmooth functions. The method is an inexact proximal point algorithm in which each proximal subproblem is solved by a convex bundle subroutine (Algorithm 4). The main result, Theorem 2, states that under a global Lipschitz assumption on the objective, the method finds an (η, ε)-inexact stationary point in O((η^{-2}+ε^{-1}) max{η^{-2}, ε^{-1}}) function and subgradient evaluations. This yields O(δ^{-4}) complexity for an approximate Moreau stationary point (Corollary 4.2). The paper further proves automatic acceleration: O(δ^{-2}) under smoothness (Theorem 3) and linear convergence under a quadratic-growth condition with μ_q > m (Theorems 4-6). Numerical experiments on phase retrieval and blind deconvolution compare the method with the proximally guided stochastic subgradient method.

Significance. If the guarantees hold, the paper makes a useful contribution: it gives a deterministic, parameter-light proximal bundle method for weakly convex optimization with explicit non-asymptotic rates that match the state of the art under Lipschitz assumptions, and it shows automatic acceleration under smoothness and quadratic growth. The proofs are detailed and self-contained, with the central inequalities (Lemmas 4-7, Propositions 4.1-4.2) being careful adaptations of classical bundle analysis. No parameter fitting or circular dependence was found in the main derivation. The main caveat is that the central theorem requires a global Lipschitz constant, while the paper's own numerical objectives are not globally Lipschitz; this narrows the scope of the headline guarantees and leaves the experimental validation outside the proved regime.

major comments (2)

[Theorem 2, Lemma 7 (Appendix B.4), Section 6] The main complexity bound in Eq. (37) is gated by the global L-Lipschitz assumption. Lemma 7 bounds G_k by (1+m/ρ)L, and this bound enters the inner-loop termination count (35) and hence the total evaluation count in Theorem 2. The phase-retrieval objective (44) and the blind-deconvolution objective (45) are not globally Lipschitz (they grow quadratically or bilinearly), so no finite L exists and the theorem cannot be invoked in the experiments of Section 6. The paper needs either a bounded-sublevel or local-Lipschitz extension (with an argument that iterates remain in a bounded set, e.g., by descent and coercivity), or a clear restriction of the main claims to globally Lipschitz objectives. The same caveat applies to Theorem 5 and Corollary 5.1, which also assume f is L-Lipschitz.
[Section 6.2, Eq. (44) and Table 2] The numerical validation is not internally consistent. The phase-retrieval setup states b_i = <a_i, \bar x>, but the objective is |<a_i,x>^2 - b_i|; with the stated formula the ground truth \bar x generally does not have zero residual (since <a_i,\bar x> can be negative). The intended choice is almost certainly b_i = (<a_i,\bar x>)^2 or |<a_i,\bar x>|^2. In addition, the text lists phase-retrieval instances (100,300), (200,600), (300,900), while Table 2 lists (100,300), (150,450), (200,600). These inconsistencies should be corrected and the experiments rerun or re-reported before the numerical claims can be assessed.

minor comments (4)

[Proof of Theorem 1, Section 3.2] The proof refers to 'the sublinear rate (37)', but the bound being proved is displayed as (27). The cross-reference should be corrected.
[Table 1] The first row header contains a typo: 'L-Lipchitz' should be 'L-Lipschitz'.
[Section 6.2, Table 2] The stationarity column reports min_k (ρ+m)^2 ||x_{k+1}-x_k||^2, which is consistent with (ρ+m)^2 ||x_{k+1}-x_k||^2 = ||\tilde g_{k+1}||^2 only when α=ρ+m; this is fine, but the table should state the exact squared quantity used, as the main text often plots ||\tilde g_{k+1}||^2.
[Appendix B.4, Lemma 7] The proof uses only the local bound (B.7), which depends on the global Lipschitz bound through g_1 ∈ ∂f(x_k). If the authors add a local-Lipschitz extension, they should make explicit where the local constant replaces L and how the inner-loop subgradient norms are controlled on the relevant sublevel set.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main rates are derived from the algorithm's own descent and bundle-improvement lemmas, and the authors' prior-work citations are not load-bearing.

full rationale

The derivation is self-contained. The main complexity bounds (Theorem 1, Theorem 2, Corollary 4.2) follow from telescoping sums over the algorithm's own descent condition (24) and from the inner-loop improvement lemma (Lemma 6), whose proof is adapted from classical bundle analysis and reproduced in Appendix B; no parameter is fitted and no target rate is assumed as an input. The (η, ε)-inexact stationarity guarantee is a direct consequence of Lemma 4 (an inexact subgradient generated from the bundle model) plus the pigeonhole counting in Theorem 1, not a restatement of the definition of stationarity. The smoothness and quadratic-growth accelerations are proved independently: Proposition 5.1 uses M-smoothness to bound G_k, and Lemma 9 (Appendix C.2.1) derives the linear decay directly from weak convexity and quadratic growth via Lemmas C.1 and C.2. The citations to the authors' prior work [33] for QG ⇒ PL and PPM linear convergence are motivational and are not used in the proof of Lemma 9, so they are not load-bearing self-citations. The global-Lipschitz assumption in Lemma 7 is a genuine scope restriction: it does not cover the non-Lipschitz phase-retrieval and blind-deconvolution objectives tested in Section 6. This is an applicability gap, not a circular step. Remark 3.1 and Section 7 openly note unresolved issues (efficiency of the alternative criterion (29), unknown weak-convexity parameter m); these are limitations, not evidence of circularity. No equation in the paper reduces to its own input by construction.

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

The central claims rest on standard weakly convex analysis (Moreau envelope, subdifferential calculus) and on explicit structural assumptions stated in the theorems. No numbers are fitted to data; beta and rho are algorithm parameters, not free parameters calibrated to results. The method assumes knowledge of m and exact oracles for f and its subgradients.

assumptions (5)

domain assumption f is m-weakly convex
The problem class itself; used throughout, e.g., Definition 2.1 and Lemma 1.
domain assumption f is L-Lipschitz (Theorems 2 and 5)
Used to bound G_k in Lemma 7 (Appendix B.4).
domain assumption f is M-smooth (Theorems 3 and 6)
Used to bound G_k^2/Delta_k in Proposition 5.1 (Appendix C.1).
domain assumption Quadratic growth with mu_q > m (Theorems 4-6)
Ensures the PL inequality via [33, Theorem 3.1] and linear convergence of the proximal point method; see Lemma 9 and Appendix C.2.
domain assumption Exact oracle for function values and subgradients
Algorithm 4 requires exact f(z) and a subgradient g at each trial point; see Algorithm 4 and footnote 2.

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Cite this review

Pith. "Pith review of A Proximal Descent Method for Minimizing Weakly Convex Optimization." pith.science (2026). https://pith.science/paper/E4PKHHF3

@misc{pith2026250902804,
  author       = {Pith},
  title        = {Pith review of: A Proximal Descent Method for Minimizing Weakly Convex Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4PKHHF3}},
  note         = {Machine review of arXiv:2509.02804}
}

abstract

We study the problem of minimizing a $m$-weakly convex and possibly nonsmooth function. Weak convexity provides a broad framework that subsumes convex, smooth, and many composite nonconvex functions. In this work, we propose a $\textit{proximal descent method}$, a simple and efficient first-order algorithm that combines the inexact proximal point method with classical convex bundle techniques. Our analysis establishes explicit non-asymptotic convergence rates in terms of $(\eta,\epsilon)$-inexact stationarity. In particular, the method finds an $(\eta,\epsilon)$-inexact stationary point using at most $\mathcal{O}\!\left( \Big(\tfrac{1}{\eta^2} + \tfrac{1}{\epsilon}\Big) \max\!\left\{\tfrac{1}{\eta^2}, \tfrac{1}{\epsilon}\right\} \right)$ function value and subgradient evaluations. Consequently, the algorithm also achieves the best-known complexity of $\mathcal{O}(1/\delta^4)$ for finding an approximate Moreau stationary point with $\|\nabla f_{2m}(x)\|\leq \delta$. A distinctive feature of our method is its \emph{automatic adaptivity}: with no parameter tuning or algorithmic modification, it accelerates to $\mathcal{O}(1/\delta^2)$ complexity under smoothness and further achieves linear convergence under quadratic growth. Overall, this work bridges convex bundle methods and weakly convex optimization, while providing accelerated guarantees under structural assumptions.

Figures

Figures reproduced from arXiv: 2509.02804 by the authors.

**Figure 1.** Moreau envelops for a 2-weakly convex function x 7→ f(x) = |x 2 − 1| with ρ = 4 and 8. Their stationary points are the same at −1, 0, and 1, and their function values also agree at these stationary points. 2.2 Approximate stationary points When minimizing a weakly convex function, we cannot guarantee the discovery of its global minimum due to the potential nonconvexity. The practical goal is to find a first-order s… view at source ↗

**Figure 2.** Performance of Algorithm 3 on the phase retrieval problem with varying algorithm parameters: β ∈ {0.25, 0.5, 0.75} and ρ ∈ {0.1, 1, 10}. The first row plots the evolution of the quantity kg˜k+1k 2 in (19a). The second row plots the evolution of the quantity ǫk+1 in (19b). left column of [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

**Figure 3.** Performance of Algorithm 3 on the blind deconvolution problem with varying algorithm parameters: β ∈ {0.25, 0.5, 0.75} and ρ ∈ {0.1, 1, 10}. The first row plots the evolution of the quantity kg˜k+1k 2 in (19a). The second row plots the evolution of the quantity ǫk+1 in (19b). Algorithm 3 consistently outperforms PGSG, across different dimensional problems. For example, when the dimension of the problem equals d = 1… view at source ↗

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Cited by 2 Pith papers

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Reviewed August 5, 2026 · model on record in the stance chip above.

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