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ItsDEAL: Inexact two-level smoothing descent algorithms for weakly convex optimization

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Pith's one-line read The paper claims that weakly convex optimization can be solved by an inexact two-level smoothing scheme, whose iterates converge to proximal fixed points, and linearly under a Kurdyka–Łojasiewicz condition.

desk verdict Solid two-level smoothing framework with likely-correct convergence theory, but the numerics never verify the inexactness assumptions the theorems rely on. read the letter →

arxiv 2501.02155 v1 pith:CTLJQRGW submitted 2025-01-04 math.OC

Alireza Kabgani , Masoud Ahookhosh This is my paper · ORCID
classification math.OC MSC 90C2690C2590C0665K0549J5249J53
keywords weaklyconvexoptimizationhigh-orderMoreauenvelopeinexactproximaloperatorKurdyka–ŁojasiewiczpropertyHöldercontinuousgradientdescentalgorithmrobustsparserecoverytwo-levelsmoothing
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 is about minimizing nonsmooth, weakly convex cost functions—functions that become convex after adding a quadratic term—without ever computing a true proximal point. The authors propose ItsDEAL, a two-level scheme in which the lower level solves a high-order proximal subproblem approximately and the upper level runs an inexact first-order descent method on the resulting high-order Moreau envelope (HOME). They prove that the inexact gradient oracle is good enough: if the prox error is controlled relative to the current inexact gradient and the function-value error sums to a finite tolerance, every cluster point of the iterates is a proximal fixed point. Under the Kurdyka–Łojasiewicz property with a quasi-additivity condition, the whole sequence converges, and when the KL exponent is $1/(1+\vartheta)$, the convergence is linear. Numerical experiments on robust sparse recovery suggest the scheme is competitive with subgradient methods, with the non-quadratic order $p=1.25$ performing best.

What carries the argument

The central object is the high-order Moreau envelope (HOME) and its high-order proximal operator (HOPE), defined for $p>1$ by $\mathrm{prox}^p_{\gamma\varphi}(x)=\arg\min_y\{\varphi(y)+\frac{1}{p\gamma}\|x-y\|^p\}$ and $\varphi^p_\gamma(x)=\min_y\{\varphi(y)+\frac{1}{p\gamma}\|x-y\|^p\}$. The paper restricts to $p\in(1,2]$ because for $p>2$ the prox set need not be single-valued and HOME need not be differentiable. The workhorse results are Theorems 21 and 22: on a ball of radius $r$, with $\gamma$ below a threshold, HOPE is single-valued and Hölder continuous with exponent $1/2$, and the HOME gradient is Hölder continuous with exponent $(p-1)/2$. That Hölder-smoothness makes a Hölderian descent lemma available, which the algorithms exploit through backtracking choices of the Lipschitz estimate $L_{k+1}$ (Algorithm 4) or an inexact Armijo line search (Algorithm 5); the relative-error condition (4.2) is what turns prox noise into a controllable gradient error.

What would settle it

On a small random instance of the robust sparse recovery problem, compute the approximate proximal point produced by Algorithm 6 after 200 iterations and compare it with a very accurate proximal solve. If at any iteration the observed pair violates $\delta_k \le \mu\|x_k-\mathrm{prox}^{p,\varepsilon_k}_{\gamma\varphi}(x_k)\|$ or the value bound (4.3) while the algorithm still converges to a proximal fixed point, then the assumptions are stronger than needed; if such violations coincide with stalling or cycling, the relative-error condition is load-bearing.

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Extended reading notes

Core claim

The central claim is that the high-order Moreau envelope provides a usable smooth surrogate for weakly convex optimization even when the proximal operator is only computed approximately. For $p\in(1,2]$, under $\rho$-weak convexity and a small enough smoothing parameter $\gamma$, the envelope $\varphi^p_\gamma$ is differentiable on any bounded ball and its gradient is Hölder continuous with exponent $(p-1)/2$; its gradient is $\frac{1}{\gamma}\|x-\mathrm{prox}^p_{\gamma\varphi}(x)\|^{p-2}(x-\mathrm{prox}^p_{\gamma\varphi}(x))$. Replacing the exact proximal point by an inexact one gives an inexact gradient oracle whose error can be bounded by a multiple of the inexact gradient norm provided the prox error satisfies $\delta_k \le \mu\|x_k-\mathrm{prox}^{p,\varepsilon_k}_{\gamma\varphi}(x_k)\|$. With this oracle, search directions satisfying a Hölderian sufficient-descent condition are genuine descent directions, and the resulting Algorithms 4 and 5 produce iterates whose every cluster point is a proximal fixed point; under a KL property with quasi-additivity the iterates converge, and with KL exponent $\theta=1/(1+\vartheta)$ the tail sums of gradient norms converge linearly.

Load-bearing premise

The whole theory depends on the assumption that at every step the approximate proximal solve is accurate enough that its error stays below a small fixed fraction of the current step length it produces, with the objective-value error summable; the numerical experiments terminate the subproblem solver after 200 iterations without checking either condition, so the theorems are not verified in the experiments.

Editorial extensions

If this is right

  • Exact proximal evaluations are not needed: any inexact lower-level solver whose errors satisfy (4.1)–(4.3) can be plugged into the upper-level descent scheme and still yields subsequential convergence to a proximal fixed point.
  • The Hölder exponent $(p-1)/2$ is interior to the algorithm: the backtracking rules in Algorithms 4 and 5 are parameter-free in the sense that the user need not know the Hölder constants $L_p$ in advance.
  • Under the KL property with quasi-additivity, subsequential convergence upgrades to global convergence of the whole iterate sequence to a single proximal fixed point.
  • When the KL desingularizing function has exponent $\theta=1/(1+\vartheta)$, the algorithm converges Q-linearly, with the rate inherited from the tail sums $B_k=\sum_{i\ge k}\|\nabla\varphi^{p,\varepsilon_i}_\gamma(x_i)\|^\vartheta$.
  • On robust sparse recovery with $\ell^1$ fidelity and a weakly convex clipped-quadratic penalty, the new algorithms—especially IDEALS—recover sparse signals with higher success probability under a strict $10^{-3}$ error tolerance than the subgradient baselines tested.

Reading between the lines

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

  • A testable extension would be to implement a stopping rule for the lower-level subgradient method that checks or bounds the relative prox error $\delta_k$; the theory predicts convergence whenever the check passes, and the check would turn the heuristic 200-iteration cap into a certified stopping test.
  • Since $p=1.25$ outperforms $p=2$ on the sparse-recovery test problems, the paper leaves open that the smoothing order is a tunable shape parameter: one could test whether the optimal $p$ shifts with sparsity level, conditioning, or measurement noise.
  • The linear-rate theorem for Algorithm 5 assumes the backtracking step-sizes $\{\alpha_k\}$ stay bounded away from zero, which the backtracking rule alone does not guarantee; adding a safeguard that monitors accepted step-sizes would be a natural follow-up.
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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

3 major / 5 minor

Summary. The paper proposes an inexact two-level smoothing descent framework (ItsDEAL) for weakly convex optimization. It studies the high-order Moreau envelope (HOME) and establishes several new properties: boundedness of the high-order proximal operator, differentiability and Hölder smoothness of HOME on balls (Theorems 21–22 and Corollary 23), and descent conditions based on an inexact oracle (Lemma 26). Three algorithms are presented: a constant-step Hölderian inexact gradient descent (HiGDA, Algorithm 3), a parameter-free version with line-search-like backtracking (Algorithm 4), and an inexact Armijo line-search method (IDEALS, Algorithm 5). Subsequential convergence and convergence rates are proved (Theorems 34 and 37), and global and linear convergence are established under Kurdyka–Łojasiewicz assumptions with quasi-additivity (Theorems 39 and 42). Preliminary experiments on robust sparse recovery compare Algorithms 4 and 5 with subgradient methods.

Significance. If the theoretical claims hold, the paper makes a useful contribution by extending high-order Moreau smoothing to weakly convex problems with inexact oracles and by providing parameter-free variants with explicit rates. The framework is versatile, and the explicit constants and Hölderian descent inequalities are potentially valuable. The main strength is the combination of a generic two-level scheme with tractable instances and convergence theorems under KL-type conditions. However, the numerical experiments do not verify several assumptions that are load-bearing for the convergence theory, and one convergence result for Algorithm 5 relies on a step-size lower bound that is not proved. These gaps do not invalidate the theory as stated, but they currently prevent the experiments from being read as empirical confirmation of the analyzed algorithms.

major comments (3)
  1. [Section 5.1(i) vs Assumption 24(c)] The implementation computes the prox approximation by running Algorithm 6 for at most 200 iterations or until ||y_{k+1}-y_k||<1e-3, but it never checks the three inequalities in Assumption 24(c), namely (4.1), (4.2), and (4.3). In particular, the relative-error condition (4.2), i.e., δ_k ≤ μ||x_k - prox^{p,ε_k}_{γφ}(x_k)||, is used to derive (4.8), in Lemma 26, and in the convergence proofs. Since the stopping rule of Algorithm 6 gives no bound on δ_k or ε_k and no verification of (4.2) is reported, the implemented Algorithms 4 and 5 are not established to satisfy the assumptions of the theorems that are claimed to support them. The numerical results therefore cannot be interpreted as evidence for the convergence theory of ItsDEAL as analyzed; at most they show that the heuristic implementations behave well on the tested instances.
  2. [Corollary 36 and Theorem 33] Corollary 36 and Corollary 43(b) for Algorithm 5 assume that the step-size sequence {α_k} is bounded away from zero by some α>0. Theorem 33 only proves that the backtracking loop in Steps 7–9 terminates after finitely many iterations; it does not prove a uniform positive lower bound on the accepted step-sizes. Since the line-search condition is tested only at the current iterate and the error terms ε_{k+1} are not controlled in a way that prevents α_k from tending to zero, the linear convergence claim for Algorithm 5 rests on an unverified structural property. The authors should either prove a lower bound under their assumptions or explicitly state that the linear rate is conditional on this additional property and indicate how it could be checked.
  3. [Assumption 41 and Section 5.1(ii)] Assumption 41 requires choosing ε_k and δ_k adaptively so that (4.44) holds for all k, with the right-hand side depending on the trajectory through min_{0≤i≤j≤k} β_j ||x_i - prox^{p,ε_i}_{γφ}(x_i)||^{ϑ(p-1)}. The numerical protocol in Section 5.1(ii) sets ε_k = 1/(k+1)^2 and does not specify δ_k at all, and no attempt is made to enforce (4.44). Consequently, Theorem 42's linear convergence result is not instantiated by the experiments. This is a separate gap from the Assumption 24(c) issue and should be addressed explicitly in the revised version.
minor comments (5)
  1. [Section 3, Proposition 15 paragraph] The word 'benifitial' should be 'beneficial'.
  2. [Corollary 35, Eq. (4.27)] The numerator of the first rate bound is written as φ^{p,ε_k}_γ(x_0); the superscript ε_k appears to be a typo for ε_0, since the initial inexact value is taken at x_0.
  3. [Figure 4] The legend entry 'INDALS' appears to be a typo for 'IDEALS'.
  4. [Algorithm 4, Steps 6–9] The description of the inner loop would benefit from specifying how i is reset and how L_{k+1} is initialized at the start of each outer iteration; the text says 'Steps 6-9 of the algorithm facilitate the inner iterations' but the algorithm box itself does not explicitly show the initialization and update of i.
  5. [Remark 20] There is a typo: 'γmax is is sufficiently far' should read 'γmax is sufficiently far'. Also, the dependence of γmax on bγ is not made explicit.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation found: convergence is derived from explicit inexactness and KL assumptions, not from fitted parameters or self-referential definitions.

full rationale

The central convergence claims—Theorem 37 (every cluster point is a proximal fixed point), Theorem 39 (global convergence under KL), and Theorem 42 (linear convergence under KL with a specific exponent)—are derived from Assumption 24's explicit oracle inequalities (4.1)–(4.3), the descent-direction conditions in Lemma 26, and concrete step-size rules in Algorithms 3–5. No equation in the proof is equivalent to a fitted value or to the theorem it is supposed to establish. In particular, the inexact prox approximation is not defined as the object that makes the conclusions true; it is an input controlled by δ_k and ε_k, and the analysis shows how those inputs propagate to the output. The main new HOME properties (differentiability and weak smoothness, Theorems 21–22 and Corollary 23) are proved in the paper itself, not merely imported. The cited results from the authors' companion paper [36]—Fact 1, Fact 12, Fact 13, Fact 14, Fact 18, and the uniformized KL property—are external lemmas with stated assumptions; none asserts the paper's iterative convergence theorems, so under the review rules they count as independent support rather than circular self-citation. The numerical experiments are not used as evidence inside the proof chain; the implementation in Section 5.1(i) computes prox approximations with a fixed 200-iteration subgradient procedure without verifying Assumption 24(c), and Corollary 36 assumes the step sizes are bounded away from zero. These are verification gaps between the analyzed algorithms and the implemented ones, and would be correctness or reproducibility concerns, but they are not instances of a prediction being forced by construction or of a theorem reducing to its own hypothesis. Consequently, the paper does not exhibit self-definitional, fitted-input, uniqueness-imported, ansatz-smuggled, or renamed-known-result circularity.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central results rest on a chain of assumptions: the weak convexity and coercivity of φ, the existence of inexact prox oracles satisfying (4.1)-(4.3), and for global/linear rates, KL-type properties with quasi-additivity and technical summability conditions. No new physical entities are introduced. The algorithmic constants (p, μ, c1, c2, γ, L0, etc.) are hand-tuned in the experiments, and the error sequences are chosen ad hoc; the theory provides no guidance for satisfying Assumption 24(c) or Assumption 41 in practice.

free parameters (8)
  • p = 1.25 (tuned in experiments)
    Order of the high-order Moreau envelope; no universal rule is given. In Section 5.2 the authors select p=1.25 after comparing p∈{1.25,1.5,1.75,2} on the test problem.
  • mu = 0.9*(1/2^{2-p})^{1/(p-1)} in experiments
    Hand-chosen relative error tolerance in Assumption 24(c).
  • c1, c2 = 1, 1
    Constants in descent conditions (4.14)/(4.11), set to 1 in Section 5.1.
  • gamma = 0.9
    Smoothing parameter, fixed in experiments.
  • L0 = 1e-3
    Initial value in line search (Algorithm 4).
  • upsilon (Algorithm 4) = 3
    Multiplier for increasing L_{k+1}.
  • lambda, upsilon (Algorithm 5) = 0.5, 0.4
    Armijo parameters.
  • epsilon_k, delta_k = epsilon_k = 1/(k+1)^2 in experiments; delta_k not specified
    Error sequences are assumed summable/non-increasing in theory, but no practical construction is given to satisfy Assumption 24(c).
assumptions (6)
  • standard math Facts from prior work [36] (Fact 1 basic inequality, Fact 12 HOME properties, Fact 13-14 coercivity/sublevel, Fact 18 differentiability characterization) are used throughout.
    These are not reproved here; they come from the authors' own previous paper.
  • standard math Hölderian descent lemma (Fact 2) and KL property (Definition 3) are assumed from the literature.
    Standard tools in nonconvex optimization.
  • domain assumption Assumption 24(a): φ is proper, lsc, coercive, ρ-weakly convex with at least one minimizer.
    Defines the class of problems addressed.
  • ad hoc to paper Assumption 24(b)-(c): error sequences ε_k summable, δ_k ↓0, and prox approximations satisfy (4.1)-(4.3).
    Strong conditions on the inexact oracle; not verified in the numerical implementation.
  • ad hoc to paper Assumption 38(b): summability of [(φ'(w_k))^{-1}]^{2/(p-1)}.
    Technical condition for global convergence under KL; difficult to check.
  • ad hoc to paper Assumption 41: error coupling condition ε_k^{θ/(1+θ)}, δ_k^{θ(p-1)} ≤ min_{0≤i≤j≤k} {β_j||x_i - prox^{p,ε_i}||^{θ(p-1)}}.
    Couples errors to past iterates; appears only for linear convergence.

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Cite this review

Pith. "Pith review of ItsDEAL: Inexact two-level smoothing descent algorithms for weakly convex optimization." pith.science (2026). https://pith.science/paper/CTLJQRGW

@misc{pith2026250102155,
  author       = {Pith},
  title        = {Pith review of: ItsDEAL: Inexact two-level smoothing descent algorithms for weakly convex optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CTLJQRGW}},
  note         = {Machine review of arXiv:2501.02155}
}
read the original abstract

This paper deals with nonconvex optimization problems via a two-level smoothing framework in which the high-order Moreau envelope (HOME) is applied to generate a smooth approximation of weakly convex cost functions. As such, the differentiability and weak smoothness of HOME are further studied, as is necessary for developing inexact first-order methods for finding its critical points. Building on the concept of the inexact two-level smoothing optimization (ItsOPT), the proposed scheme offers a versatile setting, called Inexact two-level smoothing DEscent ALgorithm (ItsDEAL), for developing inexact first-order methods: (i) solving the proximal subproblem approximately to provide an inexact first-order oracle of HOME at the lower-level; (ii) developing an upper inexact first-order method at the upper-level. In particular, parameter-free inexact descent methods (i.e., dynamic step-sizes and an inexact nonmonotone Armijo line search) are studied that effectively leverage the weak smooth property of HOME. Although the subsequential convergence of these methods is investigated under some mild inexactness assumptions, the global convergence and the linear rates are studied under the extra Kurdyka-\L{}ojasiewicz (KL) property. In order to validate the theoretical foundation, preliminary numerical experiments for robust sparse recovery problems are provided which reveal a promising behavior of the proposed methods.

Figures

Figures reproduced from arXiv: 2501.02155 by the authors.

Figure 1
Figure 1. Plot of the function Φ in Example 19 for various values of p. On the basis of Example 19, it is clear that for p > 2 we cannot be hopeful about the differentiability of HOME. As such, in the rest of this study, we restrict our attention to the case p ∈ (1, 2]. Remark 20 As described in Remark 16 (c), to impose an upper bound on the radius τ obtained in Proposition 15, we consider an upper bound for picking γ, denote… view at source ↗
Figure 2
Figure 2. The relative errors, and function values versus iterations for various values of [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. The relative errors and Function values versus iterations for various values of [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Subfigures (a)-(b) stand for the relative errors, and function values versus iterations for considered [PITH_FULL_IMAGE:figures/full_fig_p030_4.png]

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On fundamental properties of high-order forward-backward envelope

    math.OC 2025-11 conditional novelty 6.0 of 10

    Under weak smoothness of f and prox-regularity of g, the high-order forward-backward envelope is differentiable and its gradient is Hölder continuous near p-calm points of the composite objective.

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

    math.OC 2025-11 conditional novelty 5.0 of 10

    Descent methods obeying f(x_{k+1}) ≤ f(x_k) − ρ‖∇f(x_k)‖^θ converge linearly when θ equals the inverse KL exponent, with a unified rate/complexity analysis.

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