{"id":"8d3937ac-3498-4fa6-9bdd-a8f6f71ab6bd","arxiv_id":"2506.05974","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A proximal variable smoothing method with backtracking stepsizes finds stationary points for nonlinearly composite nonsmooth optimization with O(epsilon^-3) iteration complexity.","lead":"This paper presents a single-loop algorithm for minimizing a smooth term, a nonsmooth weakly convex term composed with a smooth mapping, and a convex constraint. It proves convergence to stationary points at O(epsilon^-3) complexity and shows faster empirical performance on maxmin dispersion and MIMO detection problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The O(ε^{-3}) rate in the abstract is for the smoothed surrogate residual, not for the original stationarity measure; no transfer inequality is given, so the unqualified ε-stationary claim is unsupported.","rationale":"The paper's main convergence theorem, Theorem III.6, appears sound: under Assumption III.4 the proof establishes liminf M_{F_n,φ}_{sγ}(x_n)=0 and stationarity of cluster points via the asymptotic upper bound in Theorem III.2(b). The applications seem to satisfy the standing assumptions, and the algorithmic contribution is clearly positioned relative to prior work. The reader's weakest assumption, Assumption III.4(c), is a legitimate scope restriction, but it is an explicit hypothesis of the theorem rather than an unsupported step. The more load-bearing concern is the abstract's O(ε^{-3}) rate claim. Equation (20) bounds the smoothed-residual measure M_{F_n,φ}, and no argument transfers this to the original stationarity measure M_{F,φ}. The 1D absolute-value example demonstrates that the two measures can be arbitrarily far apart in relative terms: M_{F_n,φ} can be arbitrarily small while M_{F,φ} equals 1. Thus, without a definition of ε-stationarity or an error-bound condition, the advertised iteration complexity is not a rate for the original problem. I therefore recommend a conditional acceptance: the core convergence analysis can stand, but the abstract and the interpretation of (20) must be qualified to state that the O(ε^{-3}) complexity is for the smoothed surrogate stationarity measure, or a genuine transfer theorem must be supplied.","tokens_in":102,"tokens_out":15596,"duration_ms":278819,"concrete_test":"Analytical test: in 1D set h=0, g=|·|, S=id, φ=0, γ=1. For x_n=μ_n ε/2, compute M_{F_n,φ}_γ(x_n)=ε/2 yet M_{F,φ}_γ(x_n)=dist(0,∂|·|(x_n))=1 for all n>0. This shows smallness of the smoothed surrogate residual does not imply smallness of the original stationarity measure with any modulus independent of μ_n. To resolve, the authors must either prove an explicit bound of the form M_{F,φ}_γ(x) ≤ C M_{F_n,φ}_γ(x)+ω(μ_n) (which this example shows impossible without additional assumptions such as an error bound), or explicitly define ε-stationarity via M_{F_n,φ} in the abstract and state that the O(ε^{-3}) rate is for the smoothed surrogate.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central convergence theorem is conditional on Assumption III.4, but a more direct gap is the rate claim. The abstract advertises 'a convergence rate O(ε^{-3}) for achieving an ε-stationary point'; Theorem III.6(a) and (20) bound min_{k≤n≤k} M_{F_n,φ}_{sγ}(x_n), where F_n = h + μ_n g∘S is the smoothed surrogate. This is not the stationarity measure M_{F,φ}_{sγ} of the original Problem I.1. Theorem III.2(b) is purely asymptotic (liminf) and provides no modulus relating M_{F_n,φ}(x_n) to M_{F,φ}(x_n). In the elementary instance h=0, g=|·|, S=id, φ=0, every x_n>0 satisfies M_{F,φ}(x_n)=1, while taking x_n=μ_n ε/2 gives M_{F_n,φ}(x_n)=ε/2 for μ_n small. Hence no uniform bound M_{F,φ} ≤ C M_{F_n,φ} + o(1) exists, and (20) cannot be quoted as a rate for an ε-stationary point of the original problem unless ε-stationarity is redefined via the smoothed measure. The subsequential convergence result stands; the advertised iteration complexity is the soft spot.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a single-loop proximal variable smoothing algorithm (Algorithm 1) for minimizing h + g∘S + φ, where h is smooth, g is Lipschitz and weakly convex, S is continuously differentiable, and φ is convex and prox-friendly. At each iteration the forward step performs a gradient step on the smoothed surrogate F_n = h + ^{μ_n}g∘S, and the backward step applies the proximity operator of φ. The main theoretical results are Theorem III.2, which establishes lower semicontinuity of the gradient-mapping-type stationarity measure and an asymptotic upper bound of the original measure by the smoothed measure, and Theorem III.6, which proves subsequential convergence to stationary points under Assumption III.4 and gives the quantitative bound (20) for the smoothed stationarity measure, from which the abstract claims an O(ε^{-3}) iteration complexity. Numerical experiments on maxmin dispersion and MU-MIMO detection compare the algorithm with ProjVS, AGP, PGD, PDS, and a subgradient method.","tokens_in":27640,"tokens_out":7445,"duration_ms":79990,"significance":"If fully established, the algorithm is a useful extension of variable smoothing to nonlinearly composite nonsmooth problems with a convex prox-friendly term, and its single-loop structure avoids iterative inner solvers. The paper's strengths include detailed appendices for the proofs (Appendices B and C), explicit sufficient conditions for the key assumptions (Example III.5), and a broad set of numerical comparisons on two nontrivial applications. The asymptotic stationarity result appears sound. However, the advertised O(ε^{-3}) rate is currently stated for a stationarity measure of the smoothed surrogate, not for the original problem's stationarity measure, and the paper does not define an ε-stationary point; this discrepancy is load-bearing for the central claim and must be fixed before publication.","major_comments":[{"comment":"The abstract advertises 'a convergence rate O(ε^{-3}) for achieving an ε-stationary point,' but Eq. (20) bounds min_{k≤n≤k} M_{F_n,φ}^{sγ}(x_n), where F_n = h + ^{μ_n}g∘S is the smoothed surrogate. No transfer inequality between M_{F_n,φ}^{sγ} and the original measure M_{F,φ}^{sγ} is provided; Theorem III.2(b) is only a liminf asymptotic upper bound. In the elementary instance h=0, g=|·|, S=id, φ=0, any x_n>0 satisfies M_{F,φ}^{sγ}(x_n)=1, while for x_n=μ_n ε/2 the smoothed measure M_{F_n,φ}^{sγ}(x_n)=ε/2 can be made arbitrarily small. Hence no uniform bound M_{F,φ}^{sγ} ≤ C M_{F_n,φ}^{sγ} + o(1) exists, and (20) cannot be quoted as a rate for an ε-stationary point of the original Problem I.1. The authors should either define ε-stationarity via the smoothed measure with an explicit relation between ε and μ_n, or state the rate only for that surrogate measure and add a transfer result if the original measure is intended.","section":"Abstract; Theorem III.6(a) and Eq. (20)"},{"comment":"The derivation of the quantitative rate depends on the assumed scaling L_{∇F_n} = ϖ_1 + ϖ_2/μ_n, which is used to lower-bound the backtracked stepsize by a multiple of μ_n. This scaling is not part of Problem I.1 and can fail when S is nonlinear and dom(φ) is unbounded; Example III.5(b) provides sufficient conditions, but they are not included in the problem statement. Since the rate claim is a headline contribution, the manuscript should state explicitly that Theorem III.6(a) and the O(ε^{-3}) complexity are conditional on Assumption III.4(c), and it should discuss the scope of that assumption more prominently.","section":"Assumption III.4(c) and proof of Theorem III.6(a), around Eq. (C.6)"}],"minor_comments":[{"comment":"The paper uses the term 'ε-stationary point' in the abstract but never defines it; please add a definition near Definition II.3 or immediately before Theorem III.6, specifying which stationarity measure is used and how it relates to ε.","section":"Section II and Theorem III.6"},{"comment":"The text refers to 'Remark III.5 (c) (ii)' when the intended reference is 'Example III.5 (c) (ii)'.","section":"Section V, first paragraph"},{"comment":"The notation min_{k≤n≤k} is visually confusing because the two bounds share the symbol k; using N and N' or k_1 and k_2 would improve readability.","section":"Theorem III.6(a), Eq. (20)"},{"comment":"The subdifferential representation in Fact II.4(b) is imported from [11], an arXiv preprint that is not peer-reviewed. Given that this fact underpins the asymptotic upper bound in Theorem III.2(b), the authors should either include a proof in the appendix or cite a peer-reviewed version if one becomes available.","section":"Fact II.4(b) and [11]"}],"recommendation":"major_revision","confidential_remarks":"The asymptotic convergence theory is sound and the numerical studies are relevant, but the advertised O(ε^{-3}) rate is not established for the original stationarity measure. This is a fixable mismatch rather than a fundamental error, so I recommend major revision rather than rejection. The dependence on the authors' unpublished preprint [11] for a key subdifferential representation is worth monitoring during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Core convergence result holds up; the advertised O(ε^{-3}) rate does not, at least not as stated.\n\nThe real contribution: Algorithm 1 handles a smooth nonlinear S together with a general convex φ in a single-loop variable smoothing scheme, which genuinely extends [28] (linear S) and [11] (φ=0). Theorem III.2(b), the asymptotic upper bound of the gradient-mapping stationarity measure, is a nice device and it makes the subsequential convergence proof clean. The maxmin dispersion reformulation is a good match: it exploits the nonlinear S, and the claim of first stationary-point method without iterative subproblem solvers in each iteration checks out against [31], [32]. The experiments support the practical claims, though error bars and code are absent.\n\nThe soft spots. The abstract promises \"a convergence rate O(ε^{-3}) for achieving an ε-stationary point.\" Theorem III.6(a)/(20) bounds the smoothed surrogate residual M_{F_n,φ}(x_n), where F_n = h + μ_n g∘S. That is not the original residual M_{F,φ}(x_n). No transfer inequality between them is proved, and in the simple case h=0, g=|·|, S=id, φ=0, the smoothed residual can go to zero while the original residual stays 1. So the finite-time rate is not for the original problem's stationarity measure unless ε-stationarity is redefined accordingly. This should be fixed by either proving a quantitative transfer (probably not possible without extra assumptions) or by carefully defining what ε-stationary means. Assumption III.4(c) — Lipschitz constant scaling like ϖ_1 + ϖ_2/μ_n — is strong, but Example III.5(b) supplies sufficient conditions, and the assumption does not look circular.\n\nVerdict: The asymptotic convergence theorem is sound and the algorithmic novelty is real. The rate claim is the one load-bearing unsupported statement. This is fixable in revision. Send it to peer review.","headline":"The asymptotic convergence result is real and the extension is novel, but the advertised O(ε^{-3}) rate is for a smoothed surrogate measure, not the original stationarity measure, and the paper needs to fix that claim.","tokens_in":12,"tokens_out":2591,"would_cite":true,"duration_ms":50907,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C30","49J52","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing a nonsmooth inner composition by Moreau-envelope smoothing and taking one proximal-gradient step per iteration yields iterates whose cluster points are stationary for $h+g\\circ S+\\phi$, with an $O(\\epsilon^{-3})$ iteration…","keywords":["proximal variable smoothing","weakly convex composite","Moreau envelope","gradient mapping stationarity","nonsmooth nonconvex optimization","finite-max minimization","maxmin dispersion problem","MIMO signal detection"],"falsifier":"Run Algorithm 1 on the maxmin dispersion benchmark of Section V-A with $d=10$, $m=10$, $\\epsilon=10^{-5}$, and $\\mu_n=(2\\eta)^{-1}n^{-1/3}$, and record $\\min_{k\\le n}M_{F_k,\\phi}^{\\bar{\\gamma}}(x_k)$ together with $\\sum_{k=1}^n\\mu_k$ for $n$ up to $10^6$. If the recorded minimum decays systematically slower than $O(n^{-1/3})$, or if the cluster point of the subsequence where the measure tends to zero fails to satisfy $0\\in\\partial_L(h+g\\circ S)(x^\\star)+\\partial\\phi(x^\\star)$, then the claimed rate or stationarity guarantee is contradicted.","tokens_in":27157,"feed_emoji":"🎯","tokens_out":19202,"duration_ms":183907,"temperature":0.7,"pith_summary":"This paper tackles the nonsmooth, possibly nonconvex problem of minimizing $h+g\\circ S+\\phi$, where $h$ and $S$ are smooth, $g$ is Lipschitz and weakly convex (convex after adding a quadratic), and $\\phi$ is a convex term with a computable proximity operator. The proposed algorithm is a single-loop proximal gradient method applied to the time-varying smoothed surrogate $h+{}^{\\mu_n}g\\circ S$, where ${}^{\\mu_n}g$ is the Moreau envelope of $g$ at a shrinking scale, followed by one proximity step for $\\phi$. The central claim is that this iteration still finds stationary points of the original nonsmooth problem, not merely of the smoothed one, and needs only $O(\\epsilon^{-3})$ iterations to reach an $\\epsilon$-stationary point. The paper also shows that the maxmin dispersion problem --- placing a point to maximize its smallest weighted distance to given points --- fits the framework through a nonlinear map $S$ and a finite-max $g$, giving it a stationary-point-guaranteed algorithm that needs no iterative subproblem solver, and it reports lower bit error rates in wireless MIMO signal detection.","feed_headline":"One smoothing loop finds stationary points in O(ε^{-3}) iterations","feed_subtitle":"Smoothed Moreau surrogates make cluster points stationary for nonsmooth h+g∘S+φ; tested on dispersion and MIMO.","key_machinery":"The load-bearing object is the gradient mapping-type stationarity measure $M_{F,\\phi}^{\\gamma}(x)=\\operatorname{dist}\\big(0,\\gamma^{-1}(x-\\operatorname{prox}_{\\gamma\\phi}(x-\\gamma\\partial_L F(x)))\\big)$, which vanishes exactly at stationary points. The algorithm drives this measure for the smoothed surrogate $F_n=h+{}^{\\mu_n}g\\circ S$ to zero, and the proof's second key ingredient is the asymptotic upper bound (15): $\\liminf_n M_{F_n,\\phi}^{\\gamma}(x_n)\\ge M_{F,\\phi}^{\\gamma}(\\bar{x})$ for any limit point $\\bar{x}$ of the sequence, which transfers stationarity from the smoothed problem to the original one. The rate argument uses the Moreau-envelope fact that $\\nabla{}^{\\mu}g$ has Lipschitz constant of order $\\max(1/\\mu,\\eta/(1-\\eta\\mu))$, so $\\nabla F_n$ is $\\varpi_1+\\varpi_2/\\mu_n$-Lipschitz; this makes the backtracked stepsize $\\gamma_n$ at least a constant multiple of $\\mu_n$, and that lower bound, inserted into the Armijo sufficient-decrease inequality and a telescoping sum, yields the finite-window bound (20).","core_discovery":"Define $F_n:=h+{}^{\\mu_n}g\\circ S$ with ${}^{\\mu_n}g$ the Moreau envelope of $g$. Algorithm 1 updates $x_{n+1}=\\operatorname{prox}_{\\gamma_n\\phi}(x_n-\\gamma_n\\nabla F_n(x_n))$, where $\\gamma_n$ is chosen by an Armijo-type backtracking condition. Theorem III.6 states that, under Assumption III.4, $\\liminf_{n\\to\\infty}M_{F_n,\\phi}^{\\bar\\gamma}(x_n)=0$; every cluster point of a subsequence on which this stationarity measure tends to $0$ is a stationary point of the original $F+\\phi=h+g\\circ S+\\phi$; and the finite-window estimate (20) gives an $O(\\epsilon^{-3})$ iteration complexity for an $\\epsilon$-stationary point when $\\mu_n=O(n^{-1/3})$. The bridge is Theorem III.2(b): the limit inferior of the smoothed stationarity measure dominates the true stationarity measure at any cluster point. As applications, the maxmin dispersion problem is reformulated with $h=0$, $g(z)=\\max_i z_i$, and $S(x)=(-w_j\\|x-u_j\\|^2)_j$, and a polar-coordinate regularizer is introduced for MIMO PSK detection.","pith_inferences":["Editorial extension: the stationarity bridge in Theorem III.2(b) is independent of the specific proximal-gradient update, so any algorithm that drives the smoothed measures $M_{F_n,\\phi}^{\\gamma}(x_n)$ to zero fast enough would inherit the same stationarity guarantee; stochastic or block-coordinate versions of the smoothing idea are a natural next test.","Editorial extension: on the maxmin reformulation, each iteration costs one evaluation of $m$ squared distances, one gradient of the finite-max surrogate, and one projection, so the practical bottleneck at large $m$ should shift to memory rather than subproblem solving; scaling tests beyond $m=1000$ would check this directly.","Editorial extension: the $O(\\epsilon^{-3})$ rate is tied to the schedule $\\mu_n=O(n^{-1/3})$; decaying $\\mu_n$ faster reduces smoothing bias but slows the denominator $\\sum_k\\mu_k$ in (20), so an adaptive schedule that balances these two terms is a concrete open question."],"forward_implications":["For any instance satisfying Problem I.1 and Assumption III.4, Algorithm 1 produces an $\\epsilon$-stationary point in $O(\\epsilon^{-3})$ iterations, measured by the smoothed gradient-mapping stationarity measure at a fixed stepsize $\\bar{\\gamma}$.","Every cluster point reached along a subsequence with the stationarity measure tending to zero is a stationary point of the original nonsmooth composite $h+g\\circ S+\\phi$.","For the maxmin dispersion problem, the reformulation with a finite-max $g$ and a nonlinear $S$ yields the first stationary-point-guaranteed algorithm in the paper's comparison that needs no iterative subproblem solver per iteration.","For MU-MIMO PSK detection, the proposed polar-coordinate regularizer solved by Algorithm 1 attains lower bit error rates than LMMSE, the modulus-constrained model, and the sum-of-absolute-values model in the reported settings.","The algorithm remains single-loop whenever $g$ and $\\phi$ are prox-friendly, so common penalties such as $\\ell^1$ and finite-max functions can be handled at one prox per iteration."],"supporting_citations":[{"why":"It introduces the gradient mapping-type stationarity measure and a proximal variable smoothing method for linear $S$, which this paper extends to nonlinear $S$ and general convex $\\phi$.","marker":"[28]"},{"why":"It provides the variable smoothing framework for weakly convex composite problems with smooth $S$ and nonconvex constraint, supplying the proof template and sufficient conditions used in Assumption III.4.","marker":"[11]"},{"why":"It establishes the Moreau-envelope approximation and Lipschitz-gradient properties for weakly convex functions that the smoothed surrogate relies on.","marker":"[39]"},{"why":"It supplies the variational-analysis tools, including limiting subdifferentials, outer limits, and subdifferential regularity, used to prove lower semicontinuity and the asymptotic upper bound of the stationarity measure.","marker":"[17]"},{"why":"It provides the Armijo-type sufficient decrease lemma and the backtracking step-size routine used in Algorithm 2.","marker":"[29]"},{"why":"It presents the projected variable smoothing baseline for finite-max problems whose per-iteration subproblem requires an iterative solver, the comparison that motivates the maxmin application.","marker":"[32]"}],"fun_headline_variants":["Single-loop smoothed prox: O(ε⁻³) to stationarity","Moreau envelope smoothing achieves ε-stationary point","Prox gradient with smoothed surrogate: rate O(ε⁻³)","One iteration per step, O(ε⁻³) for nonsmooth composite","Finite-max and MIMO: smoothed prox, single loop, O(ε⁻³)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the gradient of the smoothed problem stays controlled as smoothing shrinks: its Lipschitz constant must be at most a constant plus another constant divided by the smoothing parameter; if the nonlinear map $S$ has unbounded or wildly varying derivatives on the feasible set, this scaling fails and the rate and stationarity conclusions lose their support.","fun_headline_variants_meta":{"raw":{"variants":["Single-loop smoothed prox: O(ε⁻³) to stationarity","Moreau envelope smoothing achieves ε-stationary point","Prox gradient with smoothed surrogate: rate O(ε⁻³)","One iteration per step, O(ε⁻³) for nonsmooth composite","Finite-max and MIMO: smoothed prox, single loop, O(ε⁻³)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000786,"raw_usage":{"total_tokens":3493,"prompt_tokens":998,"completion_tokens":2495,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":2399}},"tokens_in":614,"tokens_out":2495,"duration_ms":24318,"temperature":1.0,"reasoning_tokens":2399,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:12:48.570289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 1 on the maxmin dispersion benchmark of Section V-A with $d=10$, $m=10$, $\\epsilon=10^{-5}$, and $\\mu_n=(2\\eta)^{-1}n^{-1/3}$, and record $\\min_{k\\le n}M_{F_k,\\phi}^{\\bar{\\gamma}}(x_k)$ together with $\\sum_{k=1}^n\\mu_k$ for $n$ up to $10^6$. If the recorded minimum decays systematically slower than $O(n^{-1/3})$, or if the cluster point of the subsequence where the measure tends to zero fails to satisfy $0\\in\\partial_L(h+g\\circ S)(x^\\star)+\\partial\\phi(x^\\star)$, then the claimed rate or stationarity guarantee is contradicted.","supporting_citations":[{"cited_title":"Proximal variable smoothing method for three- composite nonconvex nonsmooth minimization with a linear operator,","cited_arxiv_id":null,"evidence_quote":"It introduces the gradient mapping-type stationarity measure and a proximal variable smoothing method for linear $S$, which this paper extends to nonlinear $S$ and general convex $\\phi$."},{"cited_title":"Variable smoothing for weakly convex composite functions,","cited_arxiv_id":null,"evidence_quote":"It establishes the Moreau-envelope approximation and Lipschitz-gradient properties for weakly convex functions that the smoothed surrogate relies on."},{"cited_title":"Rockafellar and R","cited_arxiv_id":null,"evidence_quote":"It supplies the variational-analysis tools, including limiting subdifferentials, outer limits, and subdifferential regularity, used to prove lower semicontinuity and the asymptotic upper bound of the stationarity measure."},{"cited_title":"A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions","cited_arxiv_id":"2502.00525","evidence_quote":"It presents the projected variable smoothing baseline for finite-max problems whose per-iteration subproblem requires an iterative solver, the comparison that motivates the maxmin application."}],"review_version":1}