{"id":"25f9bc56-8240-487c-9505-12380100171a","arxiv_id":"1908.01253","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"MOCE constructs simultaneous confidence regions after LASSO by debiasing on an expanded model with a ridge-type precision matrix, claiming validity under a=o(n/log p) sparsity.","lead":"This paper proposes MOCE, a debiased LASSO procedure that expands the selected model and uses a ridge approximation of the precision matrix to build simultaneous confidence regions in high-dimensional regression. The authors claim weaker sparsity requirements and much lower computational cost than existing debiasing methods.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.1's own bound for the remainder I11 is only o_p(1), not o_p(1/√n), under the stated sparsity and τ_a conditions, so the asymptotic normality in Theorem 5.1 is not established.","rationale":"The reader's weakest assumption is exactly the load-bearing issue: Lemma 4.1's own rate for I11 is incompatible with the claimed o_p(1/√n) remainder under the advertised sparsity a = o(n/log p) and the permitted τ_a = o(√(log p/n)). This is not a disagreement with the surrounding literature; it is an internal rate mismatch in the proof. Because Lemma 4.1 feeds directly into Theorem 5.1's expansion, and Theorem 5.2, Corollary 5.2, and the simultaneous test all depend on the same distributional result, the central theoretical guarantee is not established as written. The simulation evidence is genuinely useful and the computational speed advantage is credible, but the simulations use a = 3 and do not cover the claimed boundary regime, so they cannot repair the derivation. The paper also leaves Theorem 5.3's proof dependent on an unverified appeal to Bai and Saranadasa conditions, but the I11 rate gap is already decisive. A revised version could fix the claim by either strengthening the sparsity condition or imposing a much smaller τ_a and proving the expansion procedure satisfies Assumptions 3 and 4; until then, rejection is appropriate.","tokens_in":21984,"tokens_out":6738,"duration_ms":64858,"concrete_test":"Analytically re-derive the I11 bound in Lemma 4.1 while tracking √n. Set a = n/(log p · log log n), τ_a = (log log n)^{-3/4} √(log p/n), λ ≍ √(log p/n), and p growing with n. Apply the proof's own inequalities: ‖I11‖_2 ≤ |Σ̂_{ÃÃ}^{-1} τ_a|_∞ ‖β̂_Ã − β*_Ã‖_1. The displayed right-hand side is O_p((log log n)^{-7/4}), so √n times it diverges, while Lemma 4.1 and Theorem 5.1 require √n I11 = o_p(1). If no sharper bound can be derived from Assumptions 1–5 alone, the theorem fails as stated; if a sharper bound exists, it must identify the missing condition on τ_a or a.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.1 is the gate to Theorem 5.1: the expansion (4.4)/(4.7) has remainder r_a = I11 + I12 + I13 (Eq. 4.6), and the normality proof requires √n r_a = o_p(1). In the proof of Lemma 4.1, I11 = Σ̂_{ÃÃ}^{-1} τ_a (β̂_Ã − β*_Ã) is bounded by |Σ̂_{ÃÃ}^{-1} τ_a|_∞ · ‖β̂_Ã − β*_Ã‖_1 = O_p(τ_a · a · √(log p / n)), using Lemma 3.1 with ã < n and the standard LASSO ℓ1 rate. With the theorem's τ_a = o(√(log p/n)), this is o_p(a log p / n). The stated sparsity a = o(n/log p) makes this only o_p(1), not o_p(1/√n). Concretely, choose a = n/(log p · log log n) and τ_a = (log log n)^{-3/4} √(log p/n); both satisfy the theorem's conditions, yet the proof's displayed bound is (log log n)^{-7/4}, so √n times it diverges. Thus the debiasing expansion leaves a remainder of larger order than the claimed N(0,1) leading term, and the central distributional result does not follow from the stated assumptions. The gap propagates through I22 r_a to r_c and hence to Theorem 5.1. The simulations use a = 3 and small n, so they do not probe the advertised a = o(n/log p) relaxation; no independent verification or machine-checked proof is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes MOCE, a post-model-selection inference method for high-dimensional linear regression. After a LASSO 'contraction' step selects an initial model, an 'expansion' step constructs a larger model, and a ridge-type approximation to the precision matrix is used to debias the LASSO estimator on the expanded model and its complement. The main theoretical results are Lemma 4.1, claiming that the debiasing remainders are o_p(n^{-1/2}), and Theorems 5.1-5.2, claiming asymptotic normality of any bounded-support linear functional of the MOCE estimator under sparsity a = o(n/log p). The paper also proposes a Wald-type simultaneous test W_bs in Theorem 5.3 and reports simulations comparing MOCE with the LDP method. The central selling point is that MOCE relaxes the super-sparsity assumption while retaining valid simultaneous inference at low computational cost.","tokens_in":22327,"tokens_out":15452,"duration_ms":160420,"significance":"If the theory were correct, the MOCE idea would be an attractive contribution: replacing node-wise LASSO precision estimation with a ridge-type inverse on an expanded model is computationally inexpensive, and the method explicitly allows different variances for signal and null parameters. The simulation study is detailed and documents favorable coverage and runtime comparisons against the LDP method. However, the central distributional result is not established as stated: the proof of Lemma 4.1 contains a rate error, Theorem 5.3 is proved only by invoking an inapplicable fixed-dimensional result and a sketchy appeal to Bai and Saranadasa, and the expansion algorithm is not shown to deliver the assumptions on which the theory rests. Because these issues affect the main claims of the paper, I cannot recommend acceptance.","major_comments":[{"comment":"The bound for the term I_11 has a rate error. The proof gives ||I_11||_2 <= |\\hat\\Sigma_{\\tilde A\\tilde A}^{-1} \\tau_a|_\\infty ||\\hat\\beta_{\\tilde A} - \\beta^*_{\\tilde A}||_1 = O_p(\\rho^+_{\\max}(\\tau_a) a \\sqrt{\\log p / n}). Under the stated condition \\rho^+_{\\max}(\\tau_a) = o(\\sqrt{\\log p / n}), this is o_p(a \\log p / n), and the stated sparsity a = o(n/\\log p) yields only o_p(1), not o_p(1/\\sqrt n). For example, taking a = n/(\\log p \\cdot \\log\\log n) and \\tau_a = (\\log\\log n)^{-3/4} \\sqrt{\\log p / n} satisfies the assumptions, but the displayed bound is (\\log\\log n)^{-7/4}, so \\sqrt n times it diverges. Since Lemma 4.1 supplies the remainders r_a and r_c used in Theorem 5.1, the central asymptotic normality result and the simultaneous confidence regions built on it are not established.","section":"§4.1, proof of Lemma 4.1, Eq. (4.6)"},{"comment":"The proof of Theorem 5.3 is not sufficient. It states that 'Theorem 5.1 implies \\sqrt n(\\hat\\beta_{\\tau G} - \\beta_{0,G}) \\to N(0, \\sigma^2 \\hat\\Sigma^*_{GG})', but Theorem 5.1 is proved only for a fixed projection vector d \\in M_m with bounded support size m. In Theorem 5.3 the group G has |G| = g with g/n \\to \\gamma \\in (0,1), so the subvector \\hat\\beta_{\\tau G} has growing dimension. The subsequent appeal to Bai and Saranadasa (1996) does not verify their conditions in this regression setting, and the displayed variance formula 'var(M2)' is asserted without derivation. Therefore the proposed test statistic W_bs is not shown to be asymptotically N(0,1).","section":"§5.5, Theorem 5.3"},{"comment":"The abstract claims that the expanded model 'can be selected with theoretical guarantees', but no theorem connects the proposed construction of \\tilde A (the threshold \\lambda_a and the noise-injection step) to Assumptions 3 and 4. Assumption 3 is a condition on the unknown true coefficients on the unobserved set \\tilde A^c \\cap A_s, and Assumption 4 is a condition on the LASSO estimator on \\tilde A^c \\cap \\hat A; both are used directly in the proof of Lemma 4.1. Moreover, because \\tilde A is data-dependent, the Gaussianity of the leading term in Theorem 5.1 requires either conditioning on \\tilde A or a uniformity argument, neither of which is supplied. Without a proof that the expansion algorithm produces an \\tilde A satisfying these assumptions, the theorems are conditional on properties that the procedure is not shown to deliver.","section":"§4.2 and Assumptions 3-4"}],"minor_comments":[{"comment":"The sentence 'If \\rho^+_{\\max}(\\tau_a) = o(\\sqrt{\\log p / n}) holds, Lemma 4.1 shows that \\|r_a\\|_2 = o_p(1/\\sqrt n)' appears before Lemma 4.1 is stated and uses a condition that is not sufficient for the claimed conclusion; please align the text with the actual hypothesis of the lemma.","section":"§4.1"},{"comment":"The expression 'var(M2)' appears to be a typo for 'var(M_n)', and the variance formula should be derived rather than asserted.","section":"§5.5, proof of Theorem 5.3"},{"comment":"The MSC codes are listed as 60K35 for both primary and secondary classifications, which is not appropriate for a statistics methodology paper; the authors should provide the correct subject classifications.","section":"Title page"},{"comment":"The simulation setup uses a = 3 and n = 200, far from the regime a = o(n/\\log p) featured in the theorems; an experiment with larger a would make the claimed relaxation of the sparsity assumption more credible.","section":"§6.1"},{"comment":"The proof of Lemma 4.1 attributes the LASSO \\ell_1 error bound to 'Lemma 3.1 with \\tilde a < n', but Lemma 3.1 concerns only the size of \\hat\\Sigma_{\\tilde A\\tilde A}^{-1}\\tau_a; the standard LASSO bound for \\|\\hat\\beta_{\\tilde A} - \\beta^*_{\\tilde A}\\|_1 under Assumption 1 should be stated explicitly and proved or cited.","section":"§4.2 and proof of Lemma 4.1"}],"recommendation":"reject","confidential_remarks":"I agree with the main concern in the stress-test note: the proof of Lemma 4.1 fails to deliver the o_p(n^{-1/2}) remainder under the stated assumptions, and this is load-bearing for Theorems 5.1 and 5.2. The gap in Theorem 5.3 and the lack of a proof that the expansion algorithm satisfies Assumptions 3-4 compound the problem. In principle the rate issue might be repaired by imposing a stronger condition on \\tau_a or a stronger sparsity condition, but either repair would alter the paper's advertised contribution, and the missing selection guarantee and the incomplete proof of Theorem 5.3 would still need substantial new work. On balance I recommend rejection of the current manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know one thing before you read this paper: the MOCE idea is genuinely interesting and the computational speedups are real, but the main theorem—the asymptotic normality that carries all the confidence intervals and tests—has a proof gap that looks load-bearing. I checked the stress-test note against the text, and it holds up.\n\nWhat is new: MOCE expands the LASSO-selected model and then debiases with a block lower-triangular ridge inverse of the sample covariance. That block structure is a clever dodge: it lets the expanded-model coefficients be debiased without estimating a full precision matrix, and it naturally gives different interval lengths for signals and nulls. The Wbs test applied to this estimator is a new combination. The paper also reports large speedups over node-wise LASSO debiasing, and the simulation tables show coverage close to nominal in the small-a settings they tried. The MSE comparison to LDP is a reasonable contribution.\n\nWhere it goes wrong: Lemma 4.1 is the gate. The proof bounds the term I11 by O_p(τ_a · a · √(log p/n)). Under the theorem's τ_a = o(√(log p/n)), that is o_p(a log p/n). For a as large as n/log p, that's only o_p(1), not o_p(1/√n). The proof writes the equality to o_p(1/√n) as if it followed, and it doesn't. Pick a = n/(log p · log log n) and τ_a = (log log n)^{-3/4}√(log p/n), both within the stated conditions, and √n times the bound diverges. So the debiased expansion leaves a remainder of larger order than the claimed N(0,1) term. The gap propagates to Theorem 5.1 and to everything built on it. I also agree with the reader that Theorem 5.3's proof is thin—it defers to Bai and Saranadasa without checking that the debiased covariance satisfies their conditions.\n\nI want to be fair: this is not a crank paper. The estimator is well-defined, the simulations are honest, and the citation pattern is appropriate. The flaw is a missed rate, not a missing idea. Simulations with a=3 and n=200 do not probe the advertised a=o(n/log p) regime, so they don't close the gap.\n\nWho is this for? A theory-oriented reader interested in debiased inference might still find the construction worth understanding, but as written the main guarantee is unproven.\n\nMy recommendation: send it to referees, but ask them to focus on the remainder rate. The authors need to either strengthen the sparsity assumption to one that makes I11 o_p(1/√n), sharpen the τ_a condition, or prove the expansion algorithm gives the extra control. That is a revisable paper, not a degenerate one.","headline":"MOCE is a clever debiasing construction with real computational appeal, but the paper's central asymptotic normality claim is not established: Lemma 4.1's own bound only gives o_p(1), not o_p(1/√n), under the stated sparsity and τ_a conditions.","tokens_in":22927,"tokens_out":4021,"would_cite":false,"duration_ms":38506,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a contraction-then-expansion debiasing scheme gives valid simultaneous post-LASSO inference under weaker sparsity than standard debiased Lasso, with a block-ridge inverse replacing precision-matrix estimation.","keywords":["MOCE","post-model selection inference","high-dimensional linear regression","LASSO","debiasing","simultaneous confidence regions","ridge regularization","Wald-type test"],"falsifier":"Simulate Gaussian linear models at the boundary $a = c n/\\log p$ with weak signals of size $\\lambda_s\\sigma$, run MOCE with $\\tau_a = o(\\sqrt{\\log p/n})$, and check whether 95% confidence regions for random sparse contrasts $d$ keep empirical coverage near 0.95 and the standardized statistic tracks $N(0,1)$; systematic undercoverage would show the missing $o_p(1/\\sqrt n)$ condition is essential.","tokens_in":21662,"feed_emoji":"📊","tokens_out":4553,"duration_ms":46729,"temperature":0.7,"pith_summary":"The paper proposes MOCE, a method for simultaneous inference on a set of regression coefficients in high-dimensional linear models after LASSO model selection. Instead of estimating the precision matrix, MOCE first contracts the model with LASSO, then expands the selected set to include weak signals, and debiases using a block ridge inverse. The paper claims this yields asymptotic normality under sparsity $a = o(n/\\log p)$, which is weaker than the super-sparsity $a = o(\\sqrt n/\\log p)$ often required, and gives lower mean squared error and far lower computational cost than existing low-dimensional projection methods. A new Wald-type test statistic is introduced for testing a group of parameters, with type I error controlled even when the group size grows proportionally with $n$.","feed_headline":"A two-step debiaser widens post-LASSO inference to weaker sparsity","feed_subtitle":"MOCE replaces node-wise LASSO precision estimates with a block ridge inverse, cutting computation and controlling group-test errors.","key_machinery":"The central object is the MOCE estimator $\\hat\\beta_\\tau = \\hat\\beta_\\lambda + \\lambda L_\\tau^{-1}\\kappa$, where $\\kappa$ is the subgradient from the LASSO KKT conditions and $L_\\tau$ is the lower block-triangular matrix $\\begin{pmatrix} \\hat\\Sigma_{\\tilde A\\tilde A} & 0 \\\\ S_{\\tilde A^c\\tilde A} & \\hat\\Sigma_{\\tilde A^c\\tilde A} \\end{pmatrix}$, built from ridge-adjusted sample covariance blocks rather than a full precision-matrix estimate. The expansion step chooses a larger model $\\tilde A$ so that the bias remainder terms $r_a$ and $r_c$ are $o_p(1/\\sqrt n)$, and the triangular form means the estimate on $\\tilde A^c$ does not feed back into the estimate on $\\tilde A$.","core_discovery":"Under Gaussian or suitably regular errors, and for a size-$\\tilde a$ expanded model satisfying assumptions on the design matrix and on the strength of missed signals, the MOCE estimator $\\hat\\beta_\\tau = \\hat\\beta_\\lambda + \\lambda L_\\tau^{-1}\\kappa$ satisfies $\\sqrt n\\, v^{-1} d^T(\\hat\\beta_\\tau - \\beta^*) = n^{-1/2} v^{-1} d^T L_\\tau^{-1} X^T\\epsilon + o_p(1)$, where the leading term is standard normal. This distributional result is used to construct simultaneous confidence regions for sparse contrasts and to justify the proposed Wald-type test. The paper also establishes that the variances of estimators for parameters inside the expanded model are separated from those outside it, so confidence intervals for signals are wider than those for nulls, a feature the paper argues improves coverage compared with low-dimensional projection methods.","pith_inferences":["If the missing $o_p(1/\\sqrt n)$ condition is genuinely needed, the practical relaxation of super-sparsity may only hold for models substantially sparser than $a = n/\\log p$, and the advertised boundary would need re-stating.","The lower block-triangular structure suggests a sequential debiasing recursion that might extend to multi-stage model building or to grouped inference beyond a single expanded model.","The variance separation between expanded and non-expanded parameters suggests a principled way to weight parameters in group tests, though the paper does not develop that weighting rule.","The expansion step is not tied to LASSO in principle, so the same contraction-expansion scheme could be tested with other selection rules that provide a ranking of predictors."],"forward_implications":["Simultaneous confidence regions can be built without estimating the precision matrix, under sparsity $a = o(n/\\log p)$ instead of $a = o(\\sqrt n/\\log p)$.","Confidence intervals for nonzero signals can be wider than those for zero signals, which the paper argues improves coverage and type I error relative to low-dimensional projection methods.","Computational cost is $O(n(p-\\tilde a)^2)$, versus $O(2np^2)$ for node-wise LASSO debiasing, with simulations showing hundreds-fold speedups.","The proposed test $W_{bs}$ controls type I error when the tested group size $g$ satisfies $g/n \\to \\gamma \\in (0,1)$, a regime where the classical Wald statistic fails.","The $\\ell_2$-norm error bound of the MOCE estimator on the expanded model is $O_p(\\sqrt{\\tilde a\\log\\tilde a/n})$, lower than the $O_p(\\sqrt{ap/n})$ bound of low-dimensional projection."],"supporting_citations":[{"why":"Provides the low-dimensional projection debiasing baseline that MOCE extends and compares against.","marker":"[27]"},{"why":"Represents node-wise LASSO debiasing methods whose precision-matrix estimation MOCE avoids.","marker":"[24]"},{"why":"Supplies the restricted eigenvalue and sparse eigenvalue conditions used for LASSO rates.","marker":"[4]"},{"why":"Gives the threshold order $\\lambda_{s+} = \\sqrt{2\\log p/n}$ that MOCE uses to set the expansion scale.","marker":"[9]"},{"why":"Establishes minimax rates and adaptivity limits for confidence intervals, framing the sparsity relaxation claim.","marker":"[7]"},{"why":"Provides the test statistic $W_{bs}$ that MOCE adapts for simultaneous group testing.","marker":"[1]"},{"why":"Supplies the bootstrap-based simultaneous inference approach that MOCE compares against computationally.","marker":"[26]"},{"why":"Raises the post-selection ambiguity that MOCE's expansion step aims to reduce.","marker":"[3]"}],"fun_headline_variants":["MOCE: contraction-expansion makes post-LASSO inference faster and wider","Contraction-expansion debiasing speeds up simultaneous post-LASSO tests","MOCE relaxes super-sparsity for simultaneous inference with less compute","Block ridge inverse in MOCE shrinks computation for post-selection CIs","Simultaneous intervals after LASSO without super-sparsity, via MOCE"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the debiasing remainder vanishes at $o_p(1/\\sqrt n)$ uses a bound that only yields $o_p(1)$ when the sparsity $a$ is as large as $n/\\log p$; asymptotic normality therefore rests on an unstated stronger sparsity condition or a smaller ridge tuning than Theorem 5.1 explicitly allows.","fun_headline_variants_meta":{"raw":{"variants":["MOCE: contraction-expansion makes post-LASSO inference faster and wider","Contraction-expansion debiasing speeds up simultaneous post-LASSO tests","MOCE relaxes super-sparsity for simultaneous inference with less compute","Block ridge inverse in MOCE shrinks computation for post-selection CIs","Simultaneous intervals after LASSO without super-sparsity, via MOCE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000692,"raw_usage":{"total_tokens":3105,"prompt_tokens":893,"completion_tokens":2212,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":2109}},"tokens_in":509,"tokens_out":2212,"duration_ms":13725,"temperature":1.0,"reasoning_tokens":2109,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:19:24.413985+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate Gaussian linear models at the boundary $a = c n/\\log p$ with weak signals of size $\\lambda_s\\sigma$, run MOCE with $\\tau_a = o(\\sqrt{\\log p/n})$, and check whether 95% confidence regions for random sparse contrasts $d$ keep empirical coverage near 0.95 and the standardized statistic tracks $N(0,1)$; systematic undercoverage would show the missing $o_p(1/\\sqrt n)$ condition is essential.","supporting_citations":[{"cited_title":"and Yu, B","cited_arxiv_id":null,"evidence_quote":"Provides the low-dimensional projection debiasing baseline that MOCE extends and compares against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Represents node-wise LASSO debiasing methods whose precision-matrix estimation MOCE avoids."},{"cited_title":"We first introduce MOCE and then discuss its computational complexity","cited_arxiv_id":null,"evidence_quote":"Supplies the restricted eigenvalue and sparse eigenvalue conditions used for LASSO rates."},{"cited_title":"and Saranadasa, H","cited_arxiv_id":null,"evidence_quote":"Gives the threshold order $\\lambda_{s+} = \\sqrt{2\\log p/n}$ that MOCE uses to set the expansion scale."},{"cited_title":"We developed a new method of contraction and expan- sion (MOCE) for simultaneous inference in the high-dimensional linear mod- els","cited_arxiv_id":null,"evidence_quote":"Establishes minimax rates and adaptivity limits for confidence intervals, framing the sparsity relaxation claim."},{"cited_title":"Method of Contraction-Expansion (MOCE) for Simultaneous Inference in Linear Models","cited_arxiv_id":"1908.01253","evidence_quote":"Provides the test statistic $W_{bs}$ that MOCE adapts for simultaneous group testing."},{"cited_title":", Meier, L","cited_arxiv_id":null,"evidence_quote":"Supplies the bootstrap-based simultaneous inference approach that MOCE compares against computationally."},{"cited_title":"The first regularity condition on the design matrix X is given as follows","cited_arxiv_id":null,"evidence_quote":"Raises the post-selection ambiguity that MOCE's expansion step aims to reduce."}],"review_version":1}