{"id":"c07b78ff-959a-4eab-8a15-4357e0fb8007","arxiv_id":"1908.08791","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"SLOPE with Benjamini-Hochberg-type penalties scaled by 1+delta has asymptotic false discovery rate zero and power one for Gaussian random designs when the true model is sufficiently sparse and signals are sufficiently large.","lead":"This paper proves that a variant of the SLOPE regression method can, under certain sparsity and signal-strength conditions, drive false discoveries to zero while finding all true predictors, even when the number of predictors is much larger than the sample size. It matters because SLOPE is a widely used high-dimensional tool and its false-discovery behavior under random designs was previously guaranteed only for orthogonal designs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.9's proof uses a false uniform limit: log(p/r)/log p need not tend to 1 uniformly over r ≤ k*, so the FDR bound as written does not follow.","rationale":"The paper's central result is plausible and is supported by a detailed proof structure and simulations, but the proof contains a concrete gap in Lemma 3.9 that the reader's verdict did not identify. The reader's weakest assumption concerned the imported high-probability support containment and omitted uniqueness assumptions; those are genuine issues but are more plausibly patchable by citing Su-Candes and adding a general-position uniqueness remark. The Lemma 3.9 gap is internal and mathematical: the paper asserts a uniform limit that is false for parameter sequences allowed by Theorem 2.1, and the affected inequality is the one that makes the FDR tail sum vanish. This strengthens the case for a conditional verdict rather than acceptance. The gap appears repairable with r-dependent tail bounds, so rejection would be too harsh, especially given that the simulations and the heuristic roadmap in Section 3 are consistent with the stated theorem. The verdict should remain CONDITIONAL, which is what the reader already recommended, so verdict_should_be is UNCHANGED relative to the reader.","tokens_in":26477,"tokens_out":16097,"duration_ms":161298,"concrete_test":"Independently re-derive the proof of Lemma 3.9 for the explicit sequence p = n^1.2, k = n^0.4, k* = 1.25k, delta = 0.1, q = 0.2. Check whether (lambda_r - gamma)/(sigma(1 + 1/k*)) >= (1 + delta/2) sqrt(2 log p) holds at r = k* for large n; it fails by a constant factor approximately sqrt(1 - log k*/log p)^(-1). Then verify whether replacing the r-independent s by a_r = (lambda_r - gamma)/(sigma(1 + 1/k*)) still gives sum_{r=k+2}^{k*} P(|Z|_(r-k-1) > a_r) -> 0 under the theorem's assumptions. If the r-dependent sum converges, Lemma 3.9 can be repaired and Theorem 2.1 stands; if not, the FDR conclusion is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Lemma 3.9 needs to show that the sum over r > k of P(M^(1) in H^gamma_r, Q3) vanishes. In the proof of (6.12), the paper claims that for all r ≤ k*, the threshold (lambda_r - gamma)/(sigma(1 + 1/k*)) is bounded below by (1 + delta/2) sqrt(2 log p), and justifies this by saying that log(p/r)/log p = 1 - log r/log p -> 1 for r ≤ k*. This uniformity assertion is false under the assumptions available. Corollary 3.5 only gives k*/p -> 0 and (k*)^2 log p / n -> 0; it does not give log k*/log p -> 0. For example, take p = n^1.2 and k = n^0.4, so k/p -> 0 and k^2 log p/n -> 0. Corollary 3.5/Su-Candes Lemma 7.6 permits k* approximately (1+c)k/(1-q), so log k*/log p is about 1/3. Then for r = k*, log(p/r)/log p is about 2/3, not 1, and lambda_r is smaller than (1 + delta/2) sqrt(2 log p) by a constant factor sqrt(1 - log k*/log p). Thus the displayed inequality in the Lemma 3.9 proof fails near r = k*. Consequently the Beta-tail bound P(|Z|_(r-k-1) > s) with the r-independent s is applied with a threshold larger than the true threshold, so the claimed bound on P(M^(1) in H^gamma_r, Q3) is not established. Since Lemma 3.9 provides the final vanishing term in the FDR decomposition leading to (3.22), Theorem 2.1's FDR conclusion depends on this invalid step. The gap looks repairable: one should use the r-dependent threshold a_r = (lambda_r - gamma)/(sigma(1 + 1/k*)) and sum (q r/p)^((1+delta/2)^2) P(M^(1) in H^gamma_r), giving a factor (k*/p)^((1+delta/2)^2 - 1) -> 0. But the proof as written does not take this route.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the asymptotic model-selection properties of SLOPE in high-dimensional linear regression with i.i.d. Gaussian design. The main result, Theorem 2.1, states that if p tends to infinity, k/p tends to 0, k^2 log p / n tends to 0, and all nonzero true coefficients exceed 2 sigma (1+delta) sqrt(2 log p), then SLOPE with the tuning sequence lambda_i(q,delta) = sigma (1+delta) Phi^{-1}(1 - q i/(2p)) has FDR converging to 0 and power converging to 1. The proof uses a general characterization of the SLOPE support in terms of sets H_r (Theorems 3.1 and 3.2), expresses FDR as a sum over the support size r and null coordinates i, and then reduces the problem to Gaussian tail bounds for the vector M = X'epsilon + b0, using a resolvent set and bounds imported from Su and Candes (2016). The paper also reports extensive simulations comparing SLOPE, a heuristic SLOPE, and LASSO.","tokens_in":26958,"tokens_out":14853,"duration_ms":142005,"significance":"If fully established, the result would be a useful extension of the orthogonal-design FDR control of SLOPE to random Gaussian designs, and the general support characterization for convex differentiable losses is of independent interest. The paper builds on substantial external results rather than circularly fitting the target theorem, and the simulation study is extensive and clearly presented. However, the proof as written has a load-bearing uniformity gap in Lemma 3.9 and an unaddressed uniqueness issue in the general support theorems, so the main claims are not yet fully supported.","major_comments":[{"comment":"The claim immediately before (6.12) that log(p/r)/log p = 1 - log r / log p tends to 1 uniformly for r <= k* is false under the assumptions of Corollary 3.5. Corollary 3.5 only gives k*/p -> 0 and (k*)^2 log p / n -> 0; it does not give log k* / log p -> 0. For example, with p = n^1.2 and k = n^0.4, one has k/p -> 0 and k^2 log p / n -> 0, while Lemma 7.6 permits choosing k* of order k, so log k*/log p is about 1/3 and at r = k* one has log(p/r)/log p about 2/3, not 1. Consequently the displayed inequality (lambda_r - gamma)/(sigma(1 + 1/k*)) > (1 + delta/2) sqrt(2 log p) can fail by a constant factor, and the bound P(|Z|_(r-k-1) > s) is applied with a threshold larger than the true threshold. This invalidates the proof of (6.12), and since (6.12) supplies the vanishing tail term needed for Lemma 3.9 and hence for (3.22), the FDR conclusion of Theorem 2.1 is not established as written. The gap appears repairable by using the r-dependent threshold a_r = (lambda_r - gamma)/(sigma(1 + 1/k*)) and summing (q r/p)^((1+delta/2)^2) P(M^(i) in H^gamma_r), but the argument must be rewritten.","section":"Section 6, proof of Lemma 3.9 (around (6.12))"},{"comment":"The support characterization and the FDR formula (3.4) treat the SLOPE estimate as a well-defined random vector, but the optimization problem (3.1) is convex and need not have a unique minimizer when the loss is quadratic with p > n; the argmin set can contain vectors with different supports. No selection rule or uniqueness proof is provided. This matters because R and the indicators {hat b_i != 0}, and hence the FDR expression (3.4)/(3.10) and all subsequent reductions, are not well-defined unless a unique minimizer is specified. The issue is likely fixable, for example by proving uniqueness under the Gaussian design with strictly decreasing lambda or by working with the argmin set, but it is load-bearing and should be addressed before the central claim can be accepted.","section":"Section 3.1, Theorems 3.1 and 3.2 and Corollary 3.3"}],"minor_comments":[{"comment":"The sentence 'Due to conditions (7.5), (3.14) and (3.16)' should cite (3.12), (3.14), and (3.16); equation (7.5) appears only in the supplementary material and is not among the conditions established in the main text.","section":"Section 3.3"},{"comment":"The final displayed bound of (6.11) appears to contain an algebraic slip: the factor ek/p should be ek (without the extra 1/p), since the intended convergence uses k/p^(((1+delta/4)^2)-1) -> 0; please check the displayed inequality.","section":"Section 6, proof of Lemma 3.9, bound (6.11)"},{"comment":"The captions refer to the parameter alpha, but alpha is only introduced implicitly through k = round(n^alpha); please define alpha explicitly in the caption or in the surrounding text.","section":"Figure captions 2.3 and 2.4"},{"comment":"The notation P(A, Q3 | epsilon) is slightly abusive because Q3 is deterministic conditional on epsilon; writing P(A cap Q3 | epsilon) would be clearer.","section":"Equations (6.7)-(6.8)"},{"comment":"The line '|C_i| = |A_i - B_i| > ||A_i| - |B_i||' should have a weak inequality, since equality can occur when A_i and B_i have the same sign.","section":"Proof of Proposition 6.3"}],"recommendation":"major_revision","confidential_remarks":"The false-uniformity gap in Lemma 3.9 is the main barrier to accepting the paper; it is specific and appears repairable, so I recommend major revision rather than rejection. The authors should also clarify the uniqueness of the SLOPE minimizer or formulate the support results for a specified selection rule. The reliance on Su and Candes (2016) is legitimate and clearly attributed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper gives the first asymptotic FDR and power result for SLOPE with i.i.d. Gaussian design, and the general support characterization for convex differentiable losses is genuinely useful beyond the FDR application. The simulations are honest and match the stated regime. The proof builds carefully on Su and Candes, and the high-level roadmap is sound.\n\nThe problem is Lemma 3.9. The proof needs the sum over r>k of P(M^(1) in H^gamma_r, Q3) to vanish, and to get there it uses the claim that log(p/r)/log p -> 1 uniformly over r ≤ k*. That uniformity is false under the assumed scaling. Take p = n^1.2 and k = n^0.4; the resolvent-set lemma permits k* ~ (1+c)k/(1-q), so log k*/log p ≈ 1/3, and at r = k* the ratio log(p/r)/log p ≈ 2/3. Then the bracketing inequality (λ_r - γ)/(σ(1+1/k*)) > (1+δ/2)√(2log p) fails, and the subsequent Beta-tail bound with a fixed s overstates the threshold. The displayed tail bound for the H^gamma_r event is therefore not established, and since this is the vanishing term in the FDR decomposition, Theorem 2.1 is not proven as written.\n\nThat said, the gap looks genuinely repairable. Using the r-dependent threshold a_r and summing (qr/p)^((1+δ/2)^2) P(M^(1) in H^gamma_r) gives the needed (k*/p)^((1+δ/2)^2 - 1) -> 0 factor. My guess is the authors intended something like this and wrote a shortcut that doesn't work. The missing uniqueness assumption in Theorems 3.1 and 3.2 is a smaller but real issue; with Gaussian designs uniqueness fails in general, and the support characterization should either assume uniqueness or be stated for a particular minimizer. Neither problem makes me doubt the underlying claim; both are fixable in revision.\n\nWho is this for? Statisticians and machine learning people working on high-dimensional variable selection with FDR control, and anyone building on SLOPE's support behavior. The general-loss theorems alone are worth the read. I'd send it to referees, and I'd ask them to verify the corrected tail argument carefully. A serious referee can push this to a solid paper.","headline":"Real progress on SLOPE's FDR under random Gaussian designs, but the proof of the key vanishing lemma has a uniformity gap that needs repair before the theorem is established.","tokens_in":27485,"tokens_out":2752,"would_cite":true,"duration_ms":28378,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62J05","62J07","62F12"],"pacs":[],"model":"deepseek-v4-flash","headline":"SLOPE can be tuned to drive false discoveries to zero and power to one in high-dimensional Gaussian linear models.","keywords":["SLOPE","sorted L1 penalty","false discovery rate","power","high-dimensional regression","Gaussian design","asymptotic properties","resolvent set"],"falsifier":"Run the model with $p=n^2$, $k=\\sqrt{n}/10$, and the smallest signal exactly at the boundary $2(1+\\delta)\\sqrt{2\\log p}$, and evaluate the FDR via the explicit formula (3.4) or simulation for increasing $n$ (e.g., 200, 400, 800, 1600). If the FDR does not decrease toward zero as $n$ grows while the resolvent-set containment holds, the claimed convergence is false.","tokens_in":26274,"feed_emoji":"📊","tokens_out":10899,"duration_ms":101009,"temperature":0.7,"pith_summary":"This paper proves that SLOPE, a convex regression estimator that replaces LASSO's single penalty with a decreasing list of penalties sorted by coefficient size, can be tuned so that in a high-dimensional linear model with independent Gaussian predictors it eventually selects every true predictor and no false ones. The central result gives explicit conditions: the number $k$ of nonzero coefficients must satisfy $k/p \\to 0$ and $k^2 \\log p / n \\to 0$, and the smallest true signal must exceed $2\\sigma(1+\\delta)\\sqrt{2\\log p}$ for some $\\delta>0$. Under those conditions, using the Benjamini–Hochberg threshold sequence scaled up by any factor $1+\\delta$, the false discovery rate converges to zero and the power converges to one. This matters because SLOPE is a computationally tractable procedure that adapts to unknown sparsity, and previous FDR guarantees were restricted to orthogonal designs.","feed_headline":"SLOPE's FDR goes to zero while power goes to one under Gaussian design","feed_subtitle":"A small inflation of the Benjamini–Hochberg thresholds does the trick, given sparse and strong signals.","key_machinery":"The argument rests on three pieces. First, the SLOPE estimator itself, defined by the sorted $\\ell^1$ penalty $\\sum_{i=1}^p \\lambda_i |b|_{(i)}$, which shrinks coefficients both toward zero and toward each other. Second, a support characterization: for any convex differentiable loss, the selected set is exactly described by the condition $T(a)\\in H_r$, where $T(a)=U(\\hat{b})+a\\hat{b}$, $U$ is the gradient of the negative loss, and $H_r$ is a set of vectors whose sorted partial sums compare in a specific way with the $\\lambda_i$. Third, a reduction to independence: on a high-probability event $Q = Q_1\\cap Q_2\\cap Q_3$, where $Q_1$ asserts the support is inside a resolvent set of size $k^*$ with $k^*/p\\to0$ and $(k^*)^2\\log p/n\\to0$, $Q_2$ bounds the $\\ell_\\infty$ norm of $\\Gamma$ by $\\gamma = C_q\\sqrt{(k^*)^2\\log p/n}\\,\\lambda_{k^*}^{BH}$, and $Q_3$ keeps $\\|\\epsilon\\|_2$ near $\\sigma\\sqrt{n}$, the event $\\{T\\in H_r, |T_i|>\\lambda_r\\}$ is contained in an event depending only on $M$, whose coordinates are conditionally independent Gaussians. This yields the product bound with the factor $(k^*/p)^\\delta$ that drives the FDR to zero.","core_discovery":"The main theorem states that for the linear model $Y = Xb_0 + \\epsilon$ with $X$ having iid $N(0,1/n)$ entries, if $p\\to\\infty$, $k/p\\to0$, $k^2\\log p/n\\to0$, and $\\min_{j:b_0^j\\neq0}|b_0^j| > 2\\sigma(1+\\delta)\\sqrt{2\\log p}$, then the SLOPE estimator with tuning parameters $\\lambda_i = \\sigma(1+\\delta)\\Phi^{-1}(1 - qi/(2p))$ satisfies $\\mathrm{FDR}\\to0$ and power $\\to1$ for any $q\\in(0,1)$. The factor $\\delta>0$—an arbitrarily small inflation of the BH thresholds—is what drives the FDR to zero; the signal-strength assumption secures power one. The proof reduces the problem to the vector $M = X'\\epsilon + b_0$ by showing that on a high-probability event the remainder $\\Gamma = (I - X'X)(\\hat{b}-b_0)$ is small in $\\ell_\\infty$ and that the support of the solution is contained in a small deterministic resolvent set, so that false-discovery probabilities factor into independent Gaussian tails.","pith_inferences":["A natural next step would be to use the exact FDR formula (3.4) to select $\\delta$ in practice, trading off a target FDR against power in finite samples; the simulations suggest a heuristic sequence already does this more evenly than the asymptotic choice.","Because the proof only uses Gaussian concentration of $X'\\epsilon$ and singular values of submatrices, the same argument likely extends to independent sub-Gaussian columns; a direct extension would need only to verify the resolvent-set and remainder bounds for such entries.","The resolvent-set bound is the main bottleneck: if a sharper support-containment result were available, the sparsity condition $k^2\\log p/n\\to0$ might be relaxed, as the heuristic SLOPE simulations indicate.","The proof's decomposition into $M$ and $\\Gamma$ suggests that the same factorization of FDR into independent Gaussian tails could be used to design data-dependent threshold sequences that control FDR at a prespecified level with high probability rather than only asymptotically."],"forward_implications":["SLOPE can simultaneously control false discoveries and achieve full power in sparse high-dimensional regression with random Gaussian designs, without knowing the sparsity level in advance.","The signal-strength requirement is on the same $\\sqrt{2\\log p}$ scale as LASSO's support-recovery bound, so SLOPE gains adaptivity in sparsity at no asymptotically higher cost in signal magnitude.","The FDR bound contains the factor $(k^*/p)^\\delta$, which shows any $\\delta>0$, however small, suffices to drive FDR to zero; the paper leaves it open whether $\\delta$ can be chosen to hit exactly the nominal level $q$.","The support characterization for a general convex differentiable loss opens a route to analogous FDR and power statements for SLOPE in generalized linear models or Gaussian graphical models."],"supporting_citations":[{"why":"Supplies the resolvent-set lemma and singular-value bounds (Lemmas 4.4, A.11, A.12) that yield the high-probability support containment and the smallness of Gamma.","marker":"Su and Candès (2016)"},{"why":"Introduced SLOPE and its FDR guarantee under orthogonal design; the current paper extends that guarantee to Gaussian random designs.","marker":"Bogdan et al. (2015)"},{"why":"Provides the BH threshold sequence on which the tuning parameters lambda_i are based.","marker":"Benjamini and Hochberg (1995)"}],"fun_headline_variants":["Under Gaussian design, SLOPE's FDR converges to zero and power to one","Inflated BH thresholds let SLOPE achieve asymptotic perfection","SLOPE with BH thresholds: false discoveries vanish, power reaches one","Gaussian SLOPE asymptotics: FDR zero, power one with slight threshold boost","SLOPE's asymptotic selection: zero FDR, full power via BH inflation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire argument depends on a known high-probability bound that puts the support of the SLOPE solution inside a small deterministic set of size $k^*$, where $k^*/p\\to0$ and $(k^*)^2\\log p/n\\to0$; if that containment fails, both the FDR and power conclusions collapse.","fun_headline_variants_meta":{"raw":{"variants":["Under Gaussian design, SLOPE's FDR converges to zero and power to one","Inflated BH thresholds let SLOPE achieve asymptotic perfection","SLOPE with BH thresholds: false discoveries vanish, power reaches one","Gaussian SLOPE asymptotics: FDR zero, power one with slight threshold boost","SLOPE's asymptotic selection: zero FDR, full power via BH inflation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000743,"raw_usage":{"total_tokens":3342,"prompt_tokens":1001,"completion_tokens":2341,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":2244}},"tokens_in":617,"tokens_out":2341,"duration_ms":19212,"temperature":1.0,"reasoning_tokens":2244,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:30:32.889199+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the model with $p=n^2$, $k=\\sqrt{n}/10$, and the smallest signal exactly at the boundary $2(1+\\delta)\\sqrt{2\\log p}$, and evaluate the FDR via the explicit formula (3.4) or simulation for increasing $n$ (e.g., 200, 400, 800, 1600). If the FDR does not decrease toward zero as $n$ grows while the resolvent-set containment holds, the claimed convergence is false.","supporting_citations":[],"review_version":1}