{"id":"b422d128-562f-4501-b36a-c579cbefb7d8","arxiv_id":"2501.02155","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"ItsDEAL provides an inexact two-level smoothing gradient-descent framework for weakly convex optimization, with subsequential convergence to proximal fixed points and linear rates under a Kurdyka-Lojasiewicz condition.","lead":"The paper proposes a family of algorithms, ItsDEAL, for minimizing weakly convex functions by smoothing them with a high-order Moreau envelope and using approximate gradients. It proves convergence guarantees and demonstrates the methods on a robust sparse recovery problem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Numerical experiments never verify Assumption 24(c): the 200-iteration subgradient prox approximation is not shown to satisfy (4.1)-(4.3), so the tested algorithms are not established to be instances of the analyzed ItsDEAL.","rationale":"The reader's weakest-assumption analysis already identifies the central gap: Assumption 24(c) is not verified in the numerical implementation, and the step-size lower bound for Algorithm 5 is not proven. I agree with that assessment. The core theoretical framework appears internally consistent: the proofs of differentiability and weak smoothness of HOME follow the structure used in the companion papers, and the subsequential, global, and linear convergence arguments follow standard KL-based patterns, assuming the inexactness conditions hold. The most load-bearing issue is therefore not an internal contradiction in the proofs but a mismatch between the hypotheses of Theorems 34, 39, and 42 and the actual numerical protocol in Section 5.1(i). The subgradient method used for the lower level is run for at most 200 iterations with no a priori error bound, so the computed prox approximations may violate (4.1)-(4.3). If the concern lands, the experiments should be interpreted as heuristic illustrations rather than as demonstrations of the theorems. This warrants a conditional verdict, which is exactly what the reader gave; hence I recommend no change. I do not see a reason to move to REJECT, since the conditional mathematical claims are plausibly correct and the paper is explicit about its assumptions, and I do not see a reason to ACCEPT as-is while the numerical validation gap remains.","tokens_in":38621,"tokens_out":8293,"duration_ms":80054,"concrete_test":"Instrument the Section 5 implementation as follows: for every reported run of Algorithms 4 and 5, after each outer iteration solve the proximal subproblem at x_k to high accuracy (for example, run a standard subgradient method for 1e5 iterations or until successive iterates differ by less than 1e-10) to obtain reference values of prox^{p}_{gamma phi}(x_k) and phi^p_gamma(x_k). Then check whether the 200-iteration approximation actually used satisfies (4.1), (4.2), and (4.3) at every iterate, and whether the epsilon_k, delta_k sequences meet Assumption 41 when linear rates are claimed. If any inequality fails at any reported iteration, the experiments do not satisfy Assumption 24(c) and cannot be cited as validating the convergence theory; the authors would need either a verifiable certificate for the bounds or a revised numerical protocol.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence theorems for Algorithms 4 and 5 rely on Assumption 24(c): at every outer iterate the subgradient-based prox approximation must satisfy (4.1), (4.2), and (4.3). In Section 5.1(i), the prox is computed by running Algorithm 6 for at most 200 iterations with stopping rule y_{k+1}-y_k < 1e-3, and no verification of any of these three inequalities is reported. Nothing in the analysis of Algorithm 6 guarantees the required delta_k or epsilon_k bounds, let alone the relative-error condition delta_k <= mu ||x_k - prox^{p,epsilon_k}_{gamma phi}(x_k)|| that is used to derive (4.8), in Theorem 30, and again in the KL-based global convergence proof. Moreover, Assumption 41 for linear convergence requires choosing epsilon_k and delta_k adaptively via (4.44), but the numerical protocol provides no such adaptivity. Consequently, the experiments do not instantiate the hypotheses of the main theorems, and the claim that the methods have promising numerical behavior cannot be read as empirical support for the convergence theory of ItsDEAL as analyzed. This is the load-bearing gap: the algorithms implemented in Section 5 are not known to be the algorithms proved in Section 4.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":38932,"tokens_out":3764,"duration_ms":41440,"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":[{"comment":"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.","section":"Section 5.1(i) vs Assumption 24(c)"},{"comment":"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.","section":"Corollary 36 and Theorem 33"},{"comment":"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.","section":"Assumption 41 and Section 5.1(ii)"}],"minor_comments":[{"comment":"The word 'benifitial' should be 'beneficial'.","section":"Section 3, Proposition 15 paragraph"},{"comment":"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.","section":"Corollary 35, Eq. (4.27)"},{"comment":"The legend entry 'INDALS' appears to be a typo for 'IDEALS'.","section":"Figure 4"},{"comment":"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.","section":"Algorithm 4, Steps 6–9"},{"comment":"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.","section":"Remark 20"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' companion work [36] for foundational facts about HOME; the editorial process should ensure that [36] is available and that the dependence is acceptable. The main technical gap is the mismatch between the analyzed algorithms and the implemented heuristics in Section 5; this is fixable by either verifying the inexactness conditions numerically or substantially weakening the claimed link between experiments and theory."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper genuinely extends the authors' ItsOPT line: it proves new regularity properties for the high-order Moreau envelope (explicit HOPE bounds, differentiability, Hölder smoothness on a ball), then builds a generic inexact descent framework with specific algorithms (PFHiGDA, IDEALS) and KL-based global/linear convergence rates. The theory is structured and follows a standard, plausible template; I did not find a load-bearing logical hole in the proofs, though I did not machine-check the algebra. Second, the experiments never verify the key inexactness conditions. Section 5.1(i) runs a subgradient method for at most 200 iterations with a fixed stopping rule, and nothing checks (4.1), (4.2), or (4.3). So the tested implementations are not known to instantiate the analyzed ItsDEAL, and the numerical behavior cannot be read as empirical support for the convergence theory. That is a real gap, but it is a gap in the experimental section, not a refutation of the theory.\n\nWhat is genuinely new: the explicit radius bound in Proposition 15, the differentiability and weak smoothness theorems (21–22) for p in (1,2], and the clean way inexactness is threaded through the descent direction lemma and the KL proof. This is useful material for anyone working on smoothing methods for weakly convex problems. The paper is also honest about relying on the authors' previous work [36]; the self-citation is concentrated there and is appropriate.\n\nSoft spots, in order of severity. (1) The unverified Assumption 24(c), as above. This is the main issue. (2) Corollary 36 and the linear-rate results for Algorithm 5 require the backtracking step-sizes to be bounded below by a positive constant, and no proof is given that the line search guarantees this. It may be true, but it is an assumption the rates depend on. (3) The 'parameter-free' label is overstated: Algorithm 4 still needs c1, c2, µ, γ, and an initial L0; it only removes the need to know Lp a priori. (4) The tuning of p and ω on the same test problem before reporting the comparison introduces selection bias, though the final recovery-probability experiment keeps settings fixed and is more credible.\n\nWho is this for? Researchers doing theory for weakly convex optimization, especially those working on smoothing and inexact proximal methods. They will find the HOME calculus and the generic ItsDEAL template valuable. The paper deserves a serious referee: the theory is likely correct and the framework is reusable. The revision should focus on verifying or weakening the inexactness assumptions, proving the step-size lower bound (or stating it as an assumption), and toning down the 'parameter-free' and empirical claims.","headline":"Solid two-level smoothing framework with likely-correct convergence theory, but the numerics never verify the inexactness assumptions the theorems rely on.","tokens_in":39421,"tokens_out":1991,"would_cite":true,"duration_ms":22505,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C25","90C06","65K05","49J52","49J53"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["weakly convex optimization","high-order Moreau envelope","inexact proximal operator","Kurdyka–Łojasiewicz property","Hölder continuous gradient","descent algorithm","robust sparse recovery","two-level smoothing"],"falsifier":"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.","tokens_in":38336,"feed_emoji":"🎯","tokens_out":12029,"duration_ms":102189,"temperature":0.7,"pith_summary":"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.","feed_headline":"Inexact proximal steps converge in weakly convex optimization","feed_subtitle":"Even approximate subproblem solves drive the iterates to fixed points, with linear rates under the KL property.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It introduced the ItsOPT framework and the HOPE/HOME definitions, prox-boundedness facts, and convergence proof templates this paper extends.","marker":"[36]"},{"why":"It supplies the Hölderian descent lemma used to convert weak smoothness of HOME into a stepwise decrease estimate.","marker":"[55]"},{"why":"It provides the smoothness estimates for powers of the Euclidean norm behind the gradient bounds in Theorem 22 and the inexact-gradient error bound (4.6).","marker":"[64]"},{"why":"It gives the subgradient method used at the lower level to generate approximate proximal points.","marker":"[30]"},{"why":"It is the source of the clipped-quadratic weakly convex penalty used in the robust sparse recovery experiments.","marker":"[25]"},{"why":"It motivates the weakly convex robust sparse recovery formulation (5.1) with theoretical guarantees.","marker":"[76]"},{"why":"It supplies the quasi-additivity form of the Kurdyka–Łojasiewicz property required by Assumption 38 and Theorem 39.","marker":"[47]"},{"why":"It gives the Moreau-envelope differentiability and boundedness results that support the differentiability analysis of HOME.","marker":"[38]"}],"fun_headline_variants":["Inexact proximal steps converge in weakly convex settings","Approximate subproblem solves still give convergence","Linear rates for inexact smoothing descent on weak convex","Inexact proximal oracle drives weak convex iterates to fixed points","High-order Moreau envelope enables inexact descent methods"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Inexact proximal steps converge in weakly convex settings","Approximate subproblem solves still give convergence","Linear rates for inexact smoothing descent on weak convex","Inexact proximal oracle drives weak convex iterates to fixed points","High-order Moreau envelope enables inexact descent methods"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000467,"raw_usage":{"total_tokens":2387,"prompt_tokens":1060,"completion_tokens":1327,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":676,"completion_tokens_details":{"reasoning_tokens":1250}},"tokens_in":676,"tokens_out":1327,"duration_ms":9730,"temperature":1.0,"reasoning_tokens":1250,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:14:07.287244+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Mathematical Program- ming 152:381–404","cited_arxiv_id":null,"evidence_quote":"It supplies the Hölderian descent lemma used to convert weak smoothness of HOME into a stepwise decrease estimate."},{"cited_title":"Journal of Optimiza- tion Theory and Applications 185:303–326","cited_arxiv_id":null,"evidence_quote":"It provides the smoothness estimates for powers of the Euclidean norm behind the gradient bounds in Theorem 22 and the inexact-gradient error bound (4.6)."},{"cited_title":"Journal of Optimization Theory and Applications 179:962–982","cited_arxiv_id":null,"evidence_quote":"It gives the subgradient method used at the lower level to generate approximate proximal points."},{"cited_title":"IEEE Transactions on Signal Processing 62:3754–3767","cited_arxiv_id":null,"evidence_quote":"It is the source of the clipped-quadratic weakly convex penalty used in the robust sparse recovery experiments."},{"cited_title":"IEEE Transactions on Signal Processing 67:5046–5061","cited_arxiv_id":null,"evidence_quote":"It motivates the weakly convex robust sparse recovery formulation (5.1) with theoretical guarantees."},{"cited_title":"SIAM Journal on Optimization 33:1092–1120","cited_arxiv_id":null,"evidence_quote":"It supplies the quasi-additivity form of the Kurdyka–Łojasiewicz property required by Assumption 38 and Theorem 39."},{"cited_title":"Nonlinear Analysis: Theory, Methods & Applications 127:157–181","cited_arxiv_id":null,"evidence_quote":"It gives the Moreau-envelope differentiability and boundedness results that support the differentiability analysis of HOME."}],"review_version":1}