{"id":"0da461f4-c1f7-4012-b434-b037b0b7f9d6","arxiv_id":"2504.19924","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A communication-efficient score-type test for high-dimensional linear hypotheses on non-shared data provably matches the limiting power of the pooled-data score test under regularity conditions.","lead":"This paper proposes a collaborative score-type test for testing linear hypotheses when data are spread across sites that cannot share raw data. It shows the test can match the power of the pooled-data score test while only exchanging gradients, even with many sites and high-dimensional nuisance parameters.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposed communication-efficient protocol in §2.4 does not compute the CST statistic: the asserted identity T_S = Σ n_k ‖Ω̂_k ∇_{bb} L̂(β̂)‖² is false for matrix-valued Ω̂_k, so the scalar-sum algorithm implements a different statistic not covered by Theorem 2.","rationale":"Good-faith reading: the paper's theoretical core is a conditional theorem: if a consistent Ω̂ exists, CST has oracle power. The simulations appear consistent with that. The weakest point I find is not the master sample size per se but the link between the theorem and the algorithm. Condition 2(v) is an assumption, and the only concrete communication-efficient construction offered for Ω̂—the balanced local-variance scalar-sum scheme—uses a false algebraic identity. Consequently the quantity computed in the distributed protocol is not the T_S of Theorem 2. This is an internal gap, not a disagreement with prior consensus. It can likely be repaired by proving T_comm = T_S + o_p(1) under balanced sampling, or by using master-local variance estimators when n1 is large; either route preserves the main contribution. For this reason the verdict should remain CONDITIONAL (UNCHANGED relative to the reader), not ACCEPT, since the paper as written does not fully connect its theorem to its flagship communication protocol. I partially agree with the reader's weakest assumption: the master-sample scaling is also a genuine limitation, but the false identity is more directly falsifiable and more immediately blocks implementation.","tokens_in":20810,"tokens_out":12077,"duration_ms":130842,"concrete_test":"Analytic check of the claimed identity: take r=d=1, m=2, n1=n2=1, g=e1, Ω̂_1=[1,0], Ω̂_2=[0,1]. Then N‖(Ω̂_1+Ω̂_2)/2 g‖² = 2·(1/2)² = 0.5, while Σ n_k‖Ω̂_k g‖² = 1+0 = 1, so the equality in §2.4 fails. To check whether the protocol can still be saved, derive the null limit of T_comm = Σ n_k‖Ω̂_k ∇_{bb} L̂(β̂)‖² under the paper's conditions and run the balanced linear-regression simulation of §4.1 with n=200, m=20, p=1000 under H0: β1=0, computing T_comm exactly as described. If the empirical rejection rate is close to 0.05 and the QQ plot is diagonal, the protocol needs only an added equivalence proof; if not, the proposed communication-efficient test is not validated by Theorem 2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the CST statistic T_S in (9) has the oracle chi-square limit and can be built by aggregating local gradients. The communication-efficiency part is what distinguishes CST from shared-data tests. In §2.4, for balanced samples, the paper sets Ω̂ = Σ n_k Ω̂_k/N and states 'Since T_S = Σ n_k ‖Ω̂_k ∇_{bb} L̂(β̂)‖² in this case'. With matrix-valued Ω̂_k this equality is false. The left side is N‖Σ (n_k/N) Ω̂_k g‖² = (1/N)‖Σ n_k Ω̂_k g‖², which contains all cross terms; the right side Σ n_k ‖Ω̂_k g‖² does not. They differ by terms such as N Σ_{i<j} (n_i n_j/N²) g^T(Ω̂_i − Ω̂_j)^T(Ω̂_i − Ω̂_j)g, which do not vanish without further argument. Hence the advertised protocol, in which each site returns the scalar n_k‖Ω̂_k g‖², computes a different statistic T_comm. Theorem 2 is stated for T_S under Condition 2(v); no theorem or proof covers T_comm. Consistency of each local Ω̂_k alone is not obviously sufficient, because the discrepancy is multiplied by N and involves the global gradient. This is load-bearing: the actual distributed algorithm's validity is not established unless T_comm is shown asymptotically equivalent to T_S or a different communication-efficient estimator of Ω is provided. A master-only variance estimator could rescue the theory when n1 is large, but the paper instead recommends local variance estimators and the false identity. Moreover, in the unbalanced case Condition 2(v) is simply assumed with no explicit estimator, so this is the least secure link in the implementation chain.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a collaborative score-type test (CST) for testing linear hypotheses H0: Cθ* = t in high-dimensional sparse models when data are stored in m non-shared sites. The construction is based on a Kiefer-Bahadur decomposition of the pooled-data score statistic and a two-stage partial penalization scheme on a surrogate loss that combines the master site's loss with aggregated local gradients. The main theoretical results (Theorems 1 and 2) show that the CST statistic TS in (9) converges in distribution to a (possibly noncentral) chi-square with r degrees of freedom under the null and local alternatives, matching the oracle score test with known nuisance support. Corollaries 2 and 3 specialize the conditions to GLMs. Simulations for linear and logistic regression examine Type I error and power, and a comparison with divide-and-conquer testing is included, along with a real taxi-data application.","tokens_in":21150,"tokens_out":10026,"duration_ms":98096,"significance":"If the results are correct, the paper makes a useful contribution to distributed high-dimensional inference by providing a test for general linear hypotheses with a chi-square limiting distribution that matches the pooled-data oracle test, while using only gradient communication. The theoretical framework appears internally consistent, and the numerical results support the oracle property in the tested settings. The paper also correctly relaxes the balanced-sample-size requirement of earlier divide-and-conquer tests, at least when the master site is sufficiently large. However, the communication-efficient implementation described in Section 2.4 relies on an algebraic identity that is false for matrix-valued local variance estimators; the actual scalar-aggregated statistic is not covered by the theorems. Because communication efficiency is a central advertised feature, this issue is load-bearing.","major_comments":[{"comment":"The claim that T_S = Σ_{k=1}^m n_k ‖Ω̂_k ∇_{bb} L̂(β̂)‖_2^2 is algebraically false for matrix-valued Ω̂_k. With bΩ = Σ n_k Ω̂_k / N, the left side equals N ‖ Σ (n_k/N) Ω̂_k g ‖^2, whereas the right side is Σ n_k ‖ Ω̂_k g ‖^2 for g = ∇_{bb} L̂(β̂); these differ by N times the weighted variance of Ω̂_k g. If any site with small n_k is included in the local variance estimators, this discrepancy is O_p((d+s)/n_min) and need not vanish, so the scalar-sum protocol implements a statistic T_comm whose limiting distribution is not covered by Theorem 2. The paper should either prove an asymptotic equivalence theorem for the actual protocol under explicit conditions on the minimum site size used in variance estimation, or modify the protocol (e.g., compute bΩ on the master using transmitted local Hessian and covariance matrices, or use a master-only estimator) and state the resulting communication cost.","section":"Section 2.4, Eq. (9)"},{"comment":"The key condition (s+d)^3(log p + t_N) = o(n1) in Corollary 2 requires the master site's sample size to dominate the cube of the effective dimension. This is restrictive when data are partitioned across many sites with comparable sizes: if n1 is not sufficiently large, Assumption 4 fails and the remainder term R3 in (3) cannot be controlled, so the chi-square limit is not guaranteed. Since the paper motivates its method by allowing flexible site sizes (including 'arbitrarily small' non-master sites, Section 2.2), the master-site scaling should be presented as an explicit limitation, with a discussion of alternatives (e.g., aggregating local Hessians when communication allows) or an analysis of the unbalanced case without this scaling.","section":"Assumption 4 / Corollary 2"}],"minor_comments":[{"comment":"The phrase 'the line segment jointing' should read 'joining'; similar typos appear elsewhere (e.g., 'corvariates' in Table 4 and 'shrinkage the approximation error' in Remark 6).","section":"Section 2.2"},{"comment":"The main text contains no proof sketches for Theorems 1 and 2; given the complexity of Conditions 1 and 2, a short proof outline or a table mapping conditions to lemmas in the Supplementary Material would aid verification.","section":"Section 3"},{"comment":"The entries 'DC % %' for the multivariate hypotheses should be explained, since the DC test is only defined for univariate hypotheses.","section":"Table 4"}],"recommendation":"major_revision","confidential_remarks":"The false identity in Section 2.4 is the main obstacle. If the authors can provide a valid communication-efficient implementation or prove the limit for the scalar-aggregated statistic under stated conditions, the paper's contribution would be solid. As currently written, the advertised protocol is not supported by the theorems. I recommend major revision rather than rejection because the core theoretical framework for the oracle statistic appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper has a genuinely new idea: a collaborative score-type test for general linear hypotheses in distributed high-dimensional data, allowing diverging number of constraints and target parameters, and it relaxes the balanced-sample restriction of divide-and-conquer tests. The second thing is that the communication-efficient protocol in §2.4 contains a false identity. The stress-test note is correct: with matrix-valued Ω̂_k, T_S = Σ n_k ‖Ω̂_k ∇_{bb} L̂(β̂)‖² does not hold. The left side has all cross terms; the right side does not. So the advertised scalar-sum protocol computes a different statistic, and no theorem in the paper covers it.\n\nWhat is genuinely good: the two-stage partial penalization with linear constraints, the oracle property result for the CST statistic, and the honest disclosure that the surrogate loss (5) coincides with the gradient-enhanced loss of Wang et al. (2017), Jordan et al. (2019), and Fan et al. (2023). The main theorems are internally consistent: the residual condition f*_II r*_II = o(N^{-1/2}) lines up with (s+d)^3(log p + t_N) = o(n1) in Corollary 2. Simulations report null behavior close to nominal and power increasing with effect size, in both linear and logistic regression.\n\nThe soft spots are mostly in the implementation chain. Beyond the false identity, the balanced-sample protocol is the only communication-efficient variant offered; the unbalanced case simply assumes Condition 2(v) with no explicit estimator. The chi-square convergence with diverging r has no stated rate, and the master site still needs (s+d)^3(log p + t_N) = o(n1), so the relaxation of balanced sizes is partial rather than complete. Proofs live entirely in a supplementary that is not part of the submission, and no code is provided.\n\nWho this is for: people working on distributed and federated inference will want to read the theory, but they should not adopt the protocol as written. The gap to close is to show that the computed T_comm is asymptotically equivalent to T_S, or to design a different communication-efficient variance estimator that admits a genuine sum-of-squares representation.\n\nMy recommendation: send it to serious peer review, but flag the §2.4 identity as the central technical issue. The paper deserves referee time; it is not ready to be used as-is.","headline":"Real new theory for a distributed score test, but the §2.4 communication-efficient protocol computes a different statistic than the one Theorem 2 covers.","tokens_in":21773,"tokens_out":3793,"would_cite":false,"duration_ms":33983,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F03","62F12","62J07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper demonstrates that a gradient-shifted collaborative score test reaches the oracle pooled-data chi-square limit in sparse high-dimensional models with non-shared data.","keywords":["collaborative score test","non-shared data","distributed inference","high-dimensional sparse models","linear hypothesis testing","oracle property","communication efficiency","two-stage penalization"],"falsifier":"Simulate a null logistic or linear regression with large $p$, sparse support of size $s$, and a master sample $n_1$ such that $(s+d)^3\\log p$ is comparable to $n_1$ rather than $o(n_1)$, then compare the empirical distribution of $T_S$ with $\\chi^2_r$ across many replications; if the distribution visibly drifts away from the chi-square reference as $p$ grows while other conditions hold, the master-sample scaling is doing the load-bearing work. The paper's own Corollary 2 makes the same prediction, since it states convergence requires $(s+d)^3(\\log p+t_N)=o(n_1)$.","tokens_in":20542,"feed_emoji":"📊","tokens_out":11117,"duration_ms":100422,"temperature":0.7,"pith_summary":"This paper proposes a collaborative score-type test (CST) for testing linear hypotheses about a high-dimensional sparse model when the data are split across sites that refuse to share raw information. The central claim is that the CST is oracle: its statistic converges in distribution to the same chi-square limit as a score test that knew the support of the nuisance parameters, so it inherits the pooled-data power function. If that claim holds, many sites with unbalanced sample sizes can cooperate at the full efficiency of a traditional shared-data score test while exchanging only gradients and estimated parameters. The paper also shows the method tolerates a diverging number of constraints and target parameters, and it relaxes the usual restriction that the number of sites stay negligible.","feed_headline":"Gradient-only collaboration reaches pooled-data test power","feed_subtitle":"A collaborative score-type test with two-stage penalization attains oracle chi-square limits without sharing raw data.","key_machinery":"The carrying identity is the Kiefer-Bahadur representation of the traditional score statistic, $\\nabla\\hat{L}(\\hat{\\beta}_{\\mathrm{global}}) \\approx C_a \\Psi^{-1}(C_a^\\top J^{-1}\\nabla\\hat{L}(\\beta^*) - h)$, which shows which gradient components a score test actually needs and motivates the gradient-shifted surrogate loss that approximates them without pooling data. The proof machinery then combines a two-stage partial penalization algorithm—an $\\ell_1$-penalized first stage followed by a reweighted folded-concave second stage under the null constraint—with restricted strong convexity and a weak support-stability bound ($\\max_{j\\notin S^{\\mathrm{a}}}\\|J_{j,S^{\\mathrm{a}}}(J_0)^{-1}\\|_1 \\le A_0$) to recover the nuisance support and control the three approximation errors $R_1, R_2, R_3$ in the Taylor expansion. The oracle property is obtained by coupling this estimator error control with a Lyapunov-type normality condition and consistent variance estimation.","core_discovery":"Under Conditions 1 and 2, the paper proves both the oracle test (OCST) and the feasible test (CST) satisfy $\\sup_x |P(T_S \\le x) - P(\\chi^2(r, N h^\\top V^{-1} h) \\le x)| \\to 0$, where $T_S = N\\|\\hat{\\Omega}\\nabla_{\\hat{b}}\\hat{L}(\\hat{\\beta})\\|_2^2$. This says that the feasible statistic, built without knowing the nuisance support, shares the limiting noncentral chi-square of the oracle shared-data score test and therefore has the same asymptotic power against local alternatives. The construction uses a surrogate loss $\\tilde{L}(\\beta;\\hat{\\beta}^{(0)}) = \\hat{L}_1(\\beta) + \\langle \\nabla\\hat{L}(\\hat{\\beta}^{(0)}) - \\nabla\\hat{L}_1(\\hat{\\beta}^{(0)}), \\beta\\rangle$, which approximates the global loss to first order using only local gradients, followed by a two-stage partial penalization that first drives an $\\ell_1$-penalized estimator into a neighborhood of the truth and then applies a reweighted folded-concave penalty under the null constraint to remove nuisance bias.","pith_inferences":["The master role need not be fixed; rotating the master or aggregating Hessian approximations from several larger sites could relax the master-sample condition of Assumption 4 in unbalanced federations, though the paper does not pursue this.","The same gradient-shifted surrogate loss could be inverted to build confidence regions for $\\theta$ by testing a grid of hypothesized values, extending the method from hypothesis testing to interval inference.","Because the approximation error $R_3$ is a product of the Hessian mismatch and the estimator error, one could build a finite-sample diagnostic that flags calibration as suspect whenever $\\|\\nabla^2\\hat{L}_1(\\beta^*) - J\\|_{\\max}\\cdot\\|\\hat{\\beta}-\\beta^*\\|_2$ is not $o(N^{-1/2})$.","Applying the same CST construction to smoothed quantile losses, which the paper notes are not covered by its current assumptions, is a natural next step and is already hinted at in the discussion of distributed quantile regression."],"forward_implications":["Sites that can never share raw data can run hypothesis tests whose asymptotic power equals the pooled-data score test, exchanging only gradients and iterates.","The number of data sources $m$ no longer needs to be negligible relative to the sample size, and local sample sizes can be unbalanced, removing a binding restriction of one-shot divide-and-conquer tests.","General linear hypotheses with multivariate constraints or contrasts can be tested while both the number of constraints $r$ and the number of target parameters $d$ diverge.","With local variance estimators, the CST can be assembled at a communication cost of $\\mathcal{O}(mp)$ bits per round, with iteration counts growing logarithmically in $m$ and doubly logarithmically in $p$ for GLMs."],"supporting_citations":[{"why":"Supplies the divide-and-conquer test that the CST is compared against and whose restrictions on balanced local sizes and the number of data sources the paper aims to remove.","marker":"Battey et al. (2018)"},{"why":"Establishes the gradient-enhanced surrogate loss whose mathematical form coincides with the collaborative loss used here in the unconstrained case.","marker":"Jordan et al. (2019)"},{"why":"Provides the communication-efficient estimation framework and the homogeneity-type assumptions (their Assumption 3.2) that the paper adapts to its testing setting.","marker":"Fan et al. (2023)"},{"why":"Supplies Assumption 3's concentration-of-score framework, a standard building block for high-dimensional inference.","marker":"Ning and Liu (2017)"},{"why":"Introduces one-step weighted $\\ell_1$ estimates for folded-concave penalization, the template for the reweighted second stage.","marker":"Zou and Li (2008)"},{"why":"Provides strong oracle optimality of folded-concave penalized estimation, used to justify support recovery and the beta-min condition.","marker":"Fan et al. (2014)"},{"why":"Gives the unified $\\ell_1$-penalized M-estimator error bounds and cone conditions used for the initial estimator and stage-one rates.","marker":"Negahban et al. (2012)"},{"why":"Supplies restricted strong convexity and support-recovery machinery that motivates Assumptions 5 and 6 and the weak bound on $A_0$.","marker":"Loh and Wainwright (2017)"}],"fun_headline_variants":["Gradient-only stats match shared-data test power","Collaborative test hits oracle power without raw data","Score test with gradients reaches oracle chi-square","Two-stage penalized test attains pooled-data power","Local gradients yield global test efficiency"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire chi-square calibration rests on the condition that the master site's local Hessian is uniformly close to the population Hessian (Assumption 4); for generalized linear models this becomes the requirement that the master sample size satisfy $(s+d)^3(\\log p + t_N)=o(n_1)$, so if no single site is large enough, the Hessian approximation error $R_3$ does not vanish and the claimed limiting distribution can fail.","fun_headline_variants_meta":{"raw":{"variants":["Gradient-only stats match shared-data test power","Collaborative test hits oracle power without raw data","Score test with gradients reaches oracle chi-square","Two-stage penalized test attains pooled-data power","Local gradients yield global test efficiency"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000467,"raw_usage":{"total_tokens":2351,"prompt_tokens":990,"completion_tokens":1361,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":606,"completion_tokens_details":{"reasoning_tokens":1293}},"tokens_in":606,"tokens_out":1361,"duration_ms":10649,"temperature":1.0,"reasoning_tokens":1293,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:40:38.966163+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a null logistic or linear regression with large $p$, sparse support of size $s$, and a master sample $n_1$ such that $(s+d)^3\\log p$ is comparable to $n_1$ rather than $o(n_1)$, then compare the empirical distribution of $T_S$ with $\\chi^2_r$ across many replications; if the distribution visibly drifts away from the chi-square reference as $p$ grows while other conditions hold, the master-sample scaling is doing the load-bearing work. The paper's own Corollary 2 makes the same prediction, since it states convergence requires $(s+d)^3(\\log p+t_N)=o(n_1)$.","supporting_citations":[{"cited_title":"Strong oracle optimality of folded concave penalized estimation","cited_arxiv_id":null,"evidence_quote":"Provides strong oracle optimality of folded-concave penalized estimation, used to justify support recovery and the beta-min condition."},{"cited_title":"Assumption","cited_arxiv_id":null,"evidence_quote":"Supplies restricted strong convexity and support-recovery machinery that motivates Assumptions 5 and 6 and the weak bound on $A_0$."}],"review_version":1}