{"id":"5462e95d-c4c3-4718-87db-97e1da6c6d55","arxiv_id":"2411.14875","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A generalized elastic-net sparse regression model with a nonconvex penalty is solved with two reweighted algorithms, with convergence guarantees and promising numerical results.","lead":"This paper develops two optimization algorithms for sparse regression with a flexible penalty that can handle different kinds of noise. Readers may care because such tools are used in genomics, finance, and signal processing, where the number of predictors far exceeds the number of observations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convergence guarantee for the implemented algorithms is not connected to the experiments because the epsilon threshold in (2.8) is never reported or verified.","rationale":"The reader's weakest assumption identified the same load-bearing issue: the epsilon threshold in (2.8) is required by Theorems 3 and 4, and the paper never reports how epsilon is set in the experiments. This is indeed the most consequential gap because it directly controls whether accumulation points of the implemented reweighted sequence are stationary for the original model (1.3) or only for the approximation. Without this verification, the central convergence claim is not tied to the numerical results. I also considered the unproven Corollary 2.1 and the inexact inner solves, but those are secondary: Corollary 2.1 is not used in the main convergence argument, and the inexactness could be patched by a perturbation analysis, whereas a wrong epsilon cannot be patched without changing the experiments. The paper has a reasonable theoretical core, so the appropriate verdict remains CONDITIONAL; my read does not change the reader's verdict. The proposed concrete test would settle the issue by checking whether the experiments respect (2.8) or by rerunning with controlled epsilon values.","tokens_in":21164,"tokens_out":30948,"duration_ms":237251,"concrete_test":"Request or infer the epsilon used in each numerical experiment (Tables 1-3 and Figures 1-2). For the stated designs (e.g., n=300, p=1000, kappa=0.2, q=0.5, lambda1=0.2, r=1/2/infinity), compute the right-hand side of (2.8) for all columns and verify the inequality. If epsilon is not available, rerun the experiments with epsilon set to, say, 0.1 times the minimum over i of p*lambda1*(||X_i||_r/(q*lambda1))^{q/(q-1)}, and also with a value 10 times larger that violates (2.8); compare RE, MSE, and the KKT residual of the original stationarity condition (2.1). If the violating case yields materially different solutions or fails the original stationarity test, the concern lands; if all tested epsilon satisfying (2.8) give the same results and the residual is small, the claim is empirically supported despite the reporting gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the iterative reweighted ADMM and PMM-SSN algorithms converge to generalized first-order stationary points of the original model (1.3). Theorem 4 proves this only for the exact reweighted iteration (3.2), and its final step invokes Theorem 3, which requires the approximation parameter epsilon to satisfy (2.8): 0 < epsilon < p*lambda1*(||X_i||_r/(q*lambda1))^{q/(q-1)} for every column i. The proof needs this condition to force every nonzero component of a stationary point of the epsilon-approximation out of the linear region of h_u^epsilon, so that partial h_u^epsilon coincides with the true lq derivative. If epsilon violates (2.8), stationary points of the approximate problem can have small nonzero coefficients in the linear region, and they need not be stationary for the original nonconvex model. Section 4 never states the value of epsilon used in any experiment, nor checks (2.8). Hence there is no evidence that the reported solutions are stationary points of (1.3); the claim is only supported for the epsilon-approximation. Additionally, the algorithms solve the inner problem (3.1) only inexactly (stopping at eta2 <= 1e-4), while Theorem 4 assumes exact minimization; this widens the gap between the theorem and the implemented method.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the sparse linear regression model F(β)=||Xβ−y||_r+λ2||β||_2^2+λ1||β||_q^q with r≥1 and 0<q<1. It defines a generalized first-order stationarity condition, proves that local minimizers satisfy it, and derives a lower bound on the magnitude of nonzero entries of stationary points. It then replaces the ℓ_q penalty by Lu's locally Lipschitz ε-approximation, obtains a convex reweighted subproblem, and proposes ADMM and PMM-SSN algorithms for it. The main theoretical claim is that accumulation points of the exact reweighted iteration (3.2) are generalized stationary points of (1.3) when ε satisfies (2.8), and the numerical sections report experiments on simulated and real data.","tokens_in":21402,"tokens_out":8499,"duration_ms":80999,"significance":"If the convergence claims hold, the paper provides a useful extension of the iterative reweighted framework to ℓ_r-loss elastic-net problems with nonconvex ℓ_q penalties, and the lower-bound result is a concrete identifiability statement. The PMM-SSN dual approach, which exploits strongly semismooth proximal mappings, is a sensible and potentially efficient design, and the numerical study compares with a recent alternative. However, the significance is conditional: the implemented algorithms are not connected to the convergence theorem by the reported parameters, and one corollary asserts an unproved direction. No machine-checked proofs or code are provided, so verification rests on the written proofs and tables.","major_comments":[{"comment":"The statement that an optimal solution of (1.3) is also an optimal solution of (2.7) is not a consequence of Theorems 1 and 3. Theorem 3 only shows that stationary points of (2.7) are stationary for (1.3), and Theorem 1 only shows that local minimizers of (1.3) are stationary for (1.3); neither gives the asserted direction. Please correct the statement or provide a separate proof.","section":"§2.3, Corollary 2.1"},{"comment":"The convergence guarantee of Theorem 4 requires ε to satisfy the column-dependent threshold (2.8), but Section 4 never reports the value of ε used in any experiment and never checks (2.8) for the reported data and parameters. Without this information, the numerical solutions are only proved to be stationary points of the ε-approximation (2.7) under unverified conditions, not of the original model (1.3). The experiments should state ε and verify the condition, or the claims should be weakened accordingly.","section":"§3, Theorem 4; §4.1"},{"comment":"Theorem 4 assumes that each outer iterate β^{k+1} is an exact minimizer of the reweighted problem (3.1), while both algorithms stop the inner iterations at η2≤1e−4 (or η1≤1e−4 in §4.4). The theory therefore does not cover the implemented algorithms. Please add an inexact-version convergence theorem with explicit tolerance conditions, or justify that the chosen tolerances place the iterates in the theorem's regime.","section":"§3, Theorem 4 and Algorithms 1–2"},{"comment":"The convergence of the ADMM for (3.1) is attributed to references [33,34], but neither reference establishes the needed two-block ADMM convergence for this problem: [33] is a counterexample paper on multi-block ADMM, and [34] concerns Hankel matrix rank minimization. Please give a precise convergence statement and a correct citation, or prove the result.","section":"§3.1"}],"minor_comments":[{"comment":"In the first paragraph, the complement \\bar T is taken in {1,...,n}, but β∈R^p; it should be {1,...,p}.","section":"§2.1"},{"comment":"The fixed-data experiment sets q=1, which is the ordinary elastic-net penalty rather than the 0<q<1 generalized penalty studied in the paper; please explain why this case is included or move it to a baseline.","section":"§4.2.1"},{"comment":"In the UN, λ1=0.05 row, ADMM reports MSE=3.432e−01 and RE=1.129e−01, substantially worse than the neighboring rows; this apparent failure should be discussed rather than summarized as consistent performance.","section":"Table 1"},{"comment":"The displayed update for β^{k+1} is typeset ambiguously and should include the factor (2λ2)^{-1} explicitly, as in Prox_{(2λ2)^{-1}g_k(·)}(−(2λ2)^{-1}X^T \\bar u^{k+1}).","section":"Algorithm 2"},{"comment":"There are numerous typographical errors (e.g., \"miminizes\", \"resuts\", \"Thorem\", \"PMM-SSN is efficient than ADMM\"); a careful proofread is needed.","section":"Throughout"},{"comment":"The LIBSVM URL given for the regression datasets points to a binary classification page; please supply the correct URLs or dataset descriptions.","section":"§4.4"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends known iterative reweighted machinery to a generalized elastic net with an ℓr loss and an ℓq penalty, and the core theory is mostly standard and believable. The lower bound on nonzero coefficients (Theorem 2) is a useful, explicit extension of earlier results, and Theorem 4 correctly adapts Lu's accumulation-point argument to the ℓr loss. The two algorithms are natural and reasonably well motivated, and the experiments show PMM-SSN is consistently faster and more accurate than ADMM, with a moderate advantage over the one baseline they compare against.\n\nThe soft spots are real but not fatal. First, Corollary 2.1 as stated claims that an optimal solution of the original problem (1.3) is also optimal for the epsilon-approximation (2.7). That does not follow from Theorems 3 and 1; being a stationary point of the approximation does not imply global optimality in the original problem. I expect this is a typo or a misstated direction, but as written it is wrong. Second, the ADMM convergence is dismissed with references [33,34], which are about multi-block nonconvergence and matrix rank minimization, not this setting; that needs a proper two-block ADMM citation. Third, and most important for the paper's own claims: Theorem 4's accumulation-point result is conditional on the epsilon threshold in (2.8), but the experiments never report epsilon or verify that the chosen values satisfy the bound. Without that, the reported solutions are only supported as stationary points of the approximation, not of (1.3). The authors also solve the inner subproblem inexactly (stopping at eta2 ≤ 1e-4), while the theorem assumes exact solves; common in practice, but worth a remark. These gaps are fixable and do not undermine the practical value of the method, but they should be closed.\n\nWho is this for? Practitioners needing a robust ℓq elastic net with non-Gaussian noise will get a workable algorithm with reasonable guidance on parameter choices, provided they accept the gap between theory and implementation. The paper is incremental, and no code is released, which limits reproducibility, but the derivations are transparent enough to reimplement.\n\nRecommendation: send it to peer review. It deserves a serious referee, but expect major revision to correct the corollary, replace the ADMM citation, and add a paragraph on how epsilon is set and whether the condition is checked.","headline":"Solid incremental methods paper whose main theorems are fine but whose stated optimality corollary and experiment-theory connection both need fixing.","tokens_in":21980,"tokens_out":3678,"would_cite":false,"duration_ms":38875,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62J07","90C26","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the generalized elastic net with an ℓr-norm loss and an ℓq-norm penalty (0<q<1) is solvable by two iterative reweighted algorithms whose accumulation points are generalized first-order stationary points.","keywords":["sparse linear regression","generalized elastic net penalty","iterative reweighted framework","alternating direction method of multipliers","proximal majorization-minimization method","semismooth Newton method","ℓq-norm penalty","ℓr-norm loss"],"falsifier":"Run the one-predictor case ($p=1$, $q=1/2$, fixed $\\lambda_1$) with random noise, compute the threshold (2.8) from $\\|X_1\\|_r$, and run the reweighted algorithm with $\\varepsilon$ set above that threshold. If the accumulation point violates the lower bound $|\\bar{\\beta}_1|>(q\\lambda_1/\\|X_1\\|_r)^{1/(1-q)}$, or fails the stationarity condition (2.1), the claim that the implemented algorithm is covered by Theorem 4 would be falsified.","tokens_in":20921,"feed_emoji":"🎯","tokens_out":10840,"duration_ms":91726,"temperature":0.7,"pith_summary":"The paper is trying to establish that the generalized elastic-net model $\\min_{\\beta\\in\\mathbb{R}^p}\\{ \\|X\\beta-y\\|_r + \\lambda_2\\|\\beta\\|_2^2 + \\lambda_1\\|\\beta\\|_q^q \\}$, with $r\\ge 1$ and $0<q<1$, is numerically tractable despite the non-Lipschitz $\\ell_q$ penalty and the nonsmooth $\\ell_r$ loss. Its main theoretical result is that every accumulation point of an iterative reweighted sequence is a generalized first-order stationary point of the original model, provided the smoothing parameter $\\varepsilon$ stays below a computable threshold that depends on the column norms and tuning parameters. Building on that, it develops two solvers: an easy-to-implement ADMM and a faster proximal majorization-minimization method with semismooth Newton steps (PMM-SSN). If the claim is right, the practical payoff is that one penalty family covers heavy-tailed, Gaussian, and uniform noise by choosing $r=1$, $2$, or $\\infty$, while the $q<1$ penalty can deliver sparser solutions than the $\\ell_1$ elastic net. The reported experiments on simulated and real data support both algorithms, with PMM-SSN reaching higher accuracy in fewer iterations.","feed_headline":"Two reweighted solvers crack the ℓq generalized elastic net","feed_subtitle":"Iterative reweighting turns the non-Lipschitz ℓq penalty into convex steps with provable stationary points.","key_machinery":"The load-bearing object is the $\\varepsilon$-approximation $h_{u_\\varepsilon}(\\beta_i)=\\min_{0\\le s\\le u_\\varepsilon} q\\left(|\\beta_i|s - \\frac{q-1}{q}s^{q/(q-1)}\\right)$ with $u_\\varepsilon=(\\varepsilon/(p\\lambda_1))^{(q-1)/q}$; it is Lipschitz, has a computable Clarke subdifferential, and converts the $\\ell_q^q$ term into an iteratively reweighted $\\ell_1$ term with weights $w_i^k=\\min\\{u_\\varepsilon,|\\beta_i^k|^{q-1}\\}$. Around it sit the proximal mappings of the $\\ell_r$ norms for $r=1,2,\\infty$, which are strongly semismooth and give closed-form updates in ADMM and a well-defined generalized Hessian in the semismooth Newton step. The threshold (2.8), involving $\\|X_i\\|_r$, $\\lambda_1$, and $q$, is what makes a stationary point of the approximation a stationary point of the original model.","core_discovery":"The paper's central claim is that the nonconvex, non-Lipschitz problem (1.3) can be approximated by a convex reweighted $\\ell_1$ problem at each outer iteration, and that the approximation is faithful in a precise sense. Replacing $\\|\\beta\\|_q^q$ with the Lipschitz $\\varepsilon$-approximation $h_{u_\\varepsilon}$ turns the objective into the convex problem (2.7); the iterative reweighted scheme then solves (3.2), whose objective is the $\\ell_r$-loss plus a weighted $\\ell_1$ term. Theorem 4 states that if $\\varepsilon$ satisfies (2.8), any accumulation point $\\bar{\\beta}$ of the sequence is a generalized first-order stationary point of the original model in the sense of (2.1), and its nonzero entries satisfy $|\\bar{\\beta}_i|>(q\\lambda_1/\\|X_i\\|_r)^{1/(1-q)}$. The same framework underlies both algorithms: ADMM solves the convex subproblem with closed-form proximal updates for $\\ell_r$ norms, while PMM-SSN solves the subproblem through its dual with a semismooth Newton method. The numerical section claims that both are competitive with an existing generalized-elastic-net solver and that PMM-SSN is more efficient than ADMM.","pith_inferences":["The paper gives no default value for $\\varepsilon$; a natural extension is a data-dependent rule that sets $\\varepsilon$ as a fixed small fraction of the right-hand side of (2.8), making the guarantee hold by construction for every problem instance.","The lower bound (2.4) can be used as a post-hoc diagnostic: any reported coefficient below the bound is a warning that the solver stopped outside the stationary regime or that the model parameters are mis-specified.","If the stationarity argument only needs the loss's proximal map to be strongly semismooth, the same reweighted outer loop should extend to other losses such as quantile or Huber, as long as their proximal operators are available."],"forward_implications":["Every solution returned by either algorithm has no nonzero coefficient smaller than $(q\\lambda_1/\\|X_i\\|_r)^{1/(1-q)}$, giving a direct threshold for screening variables out of the support.","With $r=1$, $r=2$, and $r=\\infty$ the same algorithmic pair covers least-absolute-deviation, square-root, and uniform-noise losses without changing the stationarity guarantees.","The outer sequence decreases the $\\varepsilon$-approximation objective monotonically, so early termination based on relative change preserves the stationarity statement.","On the tested instances PMM-SSN reaches the stopping tolerance in fewer iterations than ADMM, and both outperform the IAGENR-Lq baseline in mean squared error and relative error."],"supporting_citations":[{"why":"It defines the elastic net model (1.1) that the generalized model (1.3) extends.","marker":"[5]"},{"why":"It supplies the Lipschitz ε-approximation h and the iterative reweighted framework whose convergence proof is extended to ℓr-norm losses.","marker":"[27]"},{"why":"It provides the proximal majorization-minimization with semismooth Newton template used for the PMM-SSN subproblem.","marker":"[17]"},{"why":"It is the earlier generalized elastic net solver (IAGENR-Lq) used as the state-of-the-art comparison baseline in Section 4.3.","marker":"[8]"},{"why":"It gives the explicit proximal mappings and generalized Jacobians for ℓr norms with r=1,2,∞ that make ADMM and SSN updates closed-form.","marker":"[18]"},{"why":"It motivates the r=2 loss as a way to avoid knowing the noise standard deviation.","marker":"[14]"},{"why":"It is cited for the convergence of the ADMM inner solver.","marker":"[34]"}],"fun_headline_variants":["Iterative reweighting tames the ℓ_q elastic net","Two reweighted solvers for generalized elastic net","PMM-SSN outperforms ADMM on ℓ_q elastic net","Reweighted framework yields provable stationary points for ℓ_q"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The convergence and equivalence theorems hold only when the smoothing parameter $\\varepsilon$ is below the threshold in (2.8), which depends on the column norms $\\|X_i\\|_r$ and on $\\lambda_1$ and $q$; the numerical experiments never report how $\\varepsilon$ is set, so the implemented algorithms may be running outside the regime where their accumulation points are provably stationary for the original model.","fun_headline_variants_meta":{"raw":{"variants":["Iterative reweighting tames the ℓ_q elastic net","Two reweighted solvers for generalized elastic net","PMM-SSN outperforms ADMM on ℓ_q elastic net","Reweighted framework yields provable stationary points for ℓ_q"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000418,"raw_usage":{"total_tokens":2232,"prompt_tokens":1105,"completion_tokens":1127,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":721,"completion_tokens_details":{"reasoning_tokens":1054}},"tokens_in":721,"tokens_out":1127,"duration_ms":11068,"temperature":1.0,"reasoning_tokens":1054,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:46:11.273172+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the one-predictor case ($p=1$, $q=1/2$, fixed $\\lambda_1$) with random noise, compute the threshold (2.8) from $\\|X_1\\|_r$, and run the reweighted algorithm with $\\varepsilon$ set above that threshold. If the accumulation point violates the lower bound $|\\bar{\\beta}_1|>(q\\lambda_1/\\|X_1\\|_r)^{1/(1-q)}$, or fails the stationarity condition (2.1), the claim that the implemented algorithm is covered by Theorem 4 would be falsified.","supporting_citations":[{"cited_title":"SIAM Journal o n Matrix Analysis and Applications 34(3), 946–977 (2013)","cited_arxiv_id":null,"evidence_quote":"It is cited for the convergence of the ADMM inner solver."},{"cited_title":"Biometrika 98(4), 791–806 (2011)","cited_arxiv_id":null,"evidence_quote":"It motivates the r=2 loss as a way to avoid knowing the noise standard deviation."},{"cited_title":"Journal of the Royal Statistical Society Series B 67(2), 301–320 (2005)","cited_arxiv_id":null,"evidence_quote":"It defines the elastic net model (1.1) that the generalized model (1.3) extends."},{"cited_title":"Mathematical Programming 147(1–2), 277–307 (2014)","cited_arxiv_id":null,"evidence_quote":"It supplies the Lipschitz ε-approximation h and the iterative reweighted framework whose convergence proof is extended to ℓr-norm losses."},{"cited_title":"Journal of Machine Learning Research 21(226), 1–38 (2020)","cited_arxiv_id":null,"evidence_quote":"It provides the proximal majorization-minimization with semismooth Newton template used for the PMM-SSN subproblem."},{"cited_title":"Advances in Pure Mathematics 7(1), 66–74 (2017)","cited_arxiv_id":null,"evidence_quote":"It is the earlier generalized elastic net solver (IAGENR-Lq) used as the state-of-the-art comparison baseline in Section 4.3."},{"cited_title":"Optimization Methods a nd Software 38(2), 262–288 (2023)","cited_arxiv_id":null,"evidence_quote":"It gives the explicit proximal mappings and generalized Jacobians for ℓr norms with r=1,2,∞ that make ADMM and SSN updates closed-form."}],"review_version":1}