{"id":"a4717523-de5a-46a8-97c8-f2dad30c69a3","arxiv_id":"2504.18212","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"PTL-SI computes valid post-selection p-values for features selected by TransFusion and Oracle Trans-Lasso in high-dimensional transfer learning, controlling the false positive rate at the user's alpha.","lead":"This paper gives a statistical test for features chosen by transfer-learning high-dimensional regression, so researchers can tell which selected features are real and which are noise. It builds on selective inference and shows that the method keeps false discoveries at the chosen rate in simulations and real data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 2 may not return the exact truncation set used in Lemma 1, so the implemented p-values do not inherit the claimed exact FPR control.","rationale":"The reader correctly identified the missing full-rank/general-position assumption as a real gap in the KKT-based truncation-region characterization. That concern is genuine and would break Lemmas 3-5 in degenerate designs. However, I judge the more directly load-bearing issue to be the mismatch between the exact truncation event assumed in Lemma 1 and the region actually enumerated by Algorithm 2. The finite z-window is acknowledged in a footnote, but for an exact FPR-control claim it is not a negligible detail: excluding portions of Z changes the conditioning event and hence the null distribution of the p-value. The breakpoint-stepping issue is more serious because it can cause whole intervals to be missed even inside the window, so the p-value may be evaluated against a truncation set that is not merely slightly smaller but structurally different from Z. This is a correctness risk for the central claim as implemented, though it is plausibly fixable by using a proper homotopy/continuation method to enumerate intervals and by stating and verifying general position. Since the paper's experiments show FPR near the target level and the theoretical architecture is otherwise standard selective inference, the reader's CONDITIONAL verdict remains appropriate; the authors should be required to prove that Algorithm 2 returns the exact truncation region (or state and analyze the approximation) before the exact-validity claim is accepted.","tokens_in":22959,"tokens_out":23823,"duration_ms":269585,"concrete_test":"On one synthetic instance from the Fig. 3 setup (e.g., nT=50, K=5, p=300), instrument Algorithm 2 to output all interval endpoints and compare the union of returned intervals with a brute-force evaluation of M(z) on a dense grid over [-50σ,50σ], using bisection to refine near breakpoints. Check: (1) every z in [zmin,zmax] with M(z)=Mobs is covered by the returned union; (2) the loop advances cleanly past each breakpoint; (3) recomputing the p-value with the grid-identified exact truncation set reproduces the Algorithm-3 p-value to numerical tolerance. If any covered point is missed, any interval overlaps inconsistently, or the two p-values differ materially, the implemented p-value is not the exact selective p-value of Lemma 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central validity claim rests on Lemma 1, which requires the exact truncation region Z={z : M(z)=M_obs} on the whole real line, as defined in Eq. (17). The p-value in Eq. (18) is exact only when Z is this set. In the implementation, Algorithm 2 constructs Z by (i) restricting attention to [z_min,z_max]=[-20σ,20σ] and (ii) enumerating intervals by solving the Lasso at the current z and jumping to the computed right endpoint. The paper gives no proof that the union of intervals produced by Algorithm 2 equals Z∩[z_min,z_max], nor that [z_min,z_max] contains all of Z. At a Lasso breakpoint, the active set obtained from a solver at that exact z can have zero-length KKT intervals; the algorithm then sets z to that endpoint and may fail to advance into the next interval, so intervals where M(z)=M_obs can be missed. Conditioning on a smaller truncation set changes the null distribution of Z, so the p-value is no longer exactly uniform. In addition, the KKT interval formulas in Lemmas 3-5 require (X_Ou^T X_Ou)^{-1} and (X_Lv^T X_Lv)^{-1} to exist; the paper never states the general-position/full-rank condition under which the active sets and signs are well-defined and the linear-inequality characterizations hold. Thus the implemented p-values are, at best, approximations to the exact selective p-values, and the statement P(p_selective_j ≤ α)=α in Lemma 1 has not been established for the p-values that Algorithm 3 actually outputs.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces PTL-SI, a selective-inference framework for post-selection inference after transfer-learning-based high-dimensional regression, specifically for the TransFusion algorithm and its extension to Oracle Trans-Lasso. The method conditions on the selection event that the active sets and signs of the intermediate and final estimators match the observed values, reduces the conditional data space to a one-dimensional line, characterizes the truncation region through KKT-based linear inequalities, and computes a selective p-value from a truncated normal distribution. The central theoretical claim is Lemma 1, which states that under the selected-model null hypothesis the selective p-value is exactly uniform, thus controlling the per-feature false positive rate at any significance level α. Experiments on synthetic and real data compare FPR and TPR against naive, data-splitting, Bonferroni, and over-conditioned baselines, and include a robustness check under non-Gaussian noise.","tokens_in":23317,"tokens_out":7371,"duration_ms":74631,"significance":"If the exactness of the enumerated truncation region can be established, PTL-SI would be the first method to provide valid post-selection inference for transfer learning in high-dimensional regression, extending classical Lasso selective inference to a multi-source transfer setting. The paper's KKT-based interval characterizations cover two representative TL procedures, the experiments are reasonably broad, and a code repository is provided. These are genuine strengths. However, the central validity claim currently outruns the implementation: the exactness of Algorithm 2's truncation set and the required general-position assumptions are not proven, so the uniform-p-value guarantee is not yet established for the p-values that the algorithm actually outputs.","major_comments":[{"comment":"The paper claims that Eq. (21) gives the exact truncation region Z, but no proof is provided that the union of intervals collected in A by Algorithm 2 equals Z ∩ [zmin, zmax]. The loop advances z to the computed right endpoint r_{u,v,t}; at a Lasso breakpoint the interval for the currently active set can have zero length, and nothing in the pseudocode or its description guarantees that the subsequent Lasso solve at that endpoint discovers the next interval. Since Lemma 1 is exact only for the set in Eq. (17), the p-values produced by Algorithm 3 are not shown to satisfy P(p_selective_j ≤ α) = α. The authors should either prove exact enumeration, for example by a parametric-programming or homotopy argument, or clearly label the implementation as approximate and quantify the error.","section":"Section 3.3, Algorithm 2, Eq. (21)"},{"comment":"The KKT characterizations invert matrices such as X_Ou^T X_Ou, (X_Lv^T X_Lv), and the corresponding matrices for the selected set M. The paper never states the general-position/full-column-rank assumption under which these inverses exist, the Lasso solution is unique, and the active set and signs are well defined. Without such an assumption, the linear-inequality descriptions of Z_u, Z_v, and Z_t in Lemmas 3-5 collapse, so the interval endpoints used by Algorithm 2 are not well defined. This assumption must be stated explicitly and its failure modes discussed.","section":"Section 3.3 and Appendices A.3-A.5, Eqs. (A3), (A8), (A12)"},{"comment":"Eq. (18) is exact only when Z is the full truncation region on the real line. Restricting the search to [zmin, zmax] = [-20σ, 20σ] changes the conditioning event; the statement that the probability mass outside the interval is 'negligibly small' is not a proof of exact FPR control. If the bounded search is intended, the paper should state that the p-values are approximate and provide a bound on the deviation from uniformity; if exactness is claimed, the algorithm must be shown to cover all of Z.","section":"Section 3.3, footnote 1, and Eq. (18)"},{"comment":"The displayed formula for ζ^{otl}_{uv} contains ρ^{otl}_u where the analogous expression for ζ_{uv} in Appendix A.5 contains ι_u. This is not a purely notational difference, because it changes the intercept in the linear inequality characterizing Z^{otl}_t. The expression should be corrected, and the 'analogous' proof for Z^{otl}_v and Z^{otl}_t should either be written out or the exact substitutions specified, since Lemma 6 underlies the Oracle Trans-Lasso extension and the results in Fig. 8.","section":"Appendix A.6, definition of ζ^{otl}_{uv}"}],"minor_comments":[{"comment":"The event {Q(Y) = Qobs} has probability zero under continuous Y, and the proof of Lemma 1 integrates over Qobs without defining a regular conditional distribution. This is standard in the selective-inference literature, so a short justification or a citation to the measure-theoretic treatment would suffice.","section":"Section 3.1, Eq. (13)"},{"comment":"The method 'No inference' is listed and appears in the FPR plots, but no description is given of how an FPR is computed when no p-values are produced; please clarify.","section":"Section 5.1"},{"comment":"The axis label '# true beta' should be 'true coefficient Γ' to match the text, and 'FNR' should be defined where first used in the captions.","section":"Figures 3-8"},{"comment":"The GitHub link in the introduction contains a space ('PTL SI') and should be a properly URL-encoded repository link.","section":"Section 1, code link"},{"comment":"The operators ⊘ and ◦ and the dimensions of the vectors ψ, γ, ν, κ, ω, ρ should be defined explicitly for readers not familiar with the notation.","section":"Appendix A.3"},{"comment":"The sentence 'Although in certain cases the p-values obtained from PTL-SI-oc are smaller than those from PTL-SI, overall, PTL-SI consistently yields smaller p-values' is confusing because smaller p-values indicate more significance; please rephrase to state which method is more powerful.","section":"Section 5.3"}],"recommendation":"major_revision","confidential_remarks":"The main uncertainty is whether the enumeration gap can be closed by a homotopy-based exact algorithm. If the authors can supply a proof that Algorithm 2, or a corrected version of it, returns Z exactly under the stated general-position assumption, the paper will be acceptable. I see no concerns about novelty or citation pattern; the self-citations are to generic selective-inference tools and the central claim does not rest on them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is the first to give selective-inference p-values for features selected by a transfer-learning high-dimensional regression method (TransFusion, and also Oracle Trans-Lasso). That is genuinely new. The SI logic is textbook Lee et al. — condition on the selection event, reduce to a line, compute a truncated normal p-value — but the transfer-learning event is structurally harder because it involves two nested Lasso problems and a summation. The authors derive KKT-based truncation intervals for all three stages and assemble them with a divide-and-conquer scheme. The empirical FPR plots are consistent with the claim, and they ship code and run real-data experiments. Credit where due: this is a useful contribution to a niche that had no exact inference method.\n\nNow the soft spots, in order of severity.\n\nFirst, the gap between the stated lemma and the implementation. Lemma 1 promises exact uniform p-values using the exact truncation set Z = {z : M(z)=M_obs} on the whole real line. Algorithm 2 restricts to [-20σ,20σ]. The paper calls the outside mass negligible, but gives no bound; it also gives no proof that the interval enumeration covers Z. The p-values actually output by Algorithm 3 are therefore approximations, not the exact object in Lemma 1. For most Gaussian problems the error is tiny, but the claim as written is stronger than what is implemented.\n\nSecond, the algorithm can in principle stall at a Lasso breakpoint. If the active set returned at a breakpoint has a zero-length KKT interval, the right endpoint equals the current z, and the while loop doesn't advance. The paper does not analyze this case or provide an epsilon-perturbation safeguard. It's a fixable implementation detail, but it means the current code may miss intervals and condition on a smaller set.\n\nThird — and this is the one a referee will insist on — Lemmas 3-5 require (X_Ou^T X_Ou)^{-1} and (X_Lv^T X_Lv)^{-1} to exist, i.e., full column rank of the selected design submatrices. The paper never states the general-position assumption. In the target regression, X_M has only n_T rows; if |M| > n_T, the inverse fails. In typical runs |M| is small, but the theory needs the condition on the table. It's a missing assumption, not a wrong derivation.\n\nThe central argument holds up. The missing pieces are all repairable: add the full-rank assumption, handle breakpoints, and explicitly state the finite-window approximation with an error bound. Who benefits: anyone doing inference after transfer learning in genomics or other data-scarce settings. The paper deserves a serious referee — I would not desk reject it — but I would send it back for revision before accepting. I'd bring it to reading group, mostly to argue about the implementation gap.","headline":"First real attempt at selective inference for transfer-learned high-dimensional regression; the theory is standard and mostly sound, but the implementation's finite z-window and unstated full-rank assumptions mean the advertised exact FPR control has not actually been established.","tokens_in":23794,"tokens_out":3869,"would_cite":false,"duration_ms":37805,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"PTL-SI computes valid p-values for features selected by transfer learning in high-dimensional regression, controlling the false positive rate at any chosen $\\alpha$.","keywords":["transfer learning","high-dimensional regression","post-selection inference","selective p-value","false positive rate","TransFusion","Oracle Trans-Lasso","Lasso"],"falsifier":"Simulate the null ($\\beta_0=0$) with exact duplicate columns among candidate features so the selected design matrix can be rank-deficient; if the PTL-SI p-values are not uniform under $H_{0,j}$ or the empirical FPR at $\\alpha=0.05$ departs from 0.05, the full-rank premise is shown to be load-bearing.","tokens_in":22764,"feed_emoji":"📊","tokens_out":11806,"duration_ms":99939,"temperature":0.7,"pith_summary":"Transfer learning for high-dimensional regression can borrow strength from source data, but the features it selects are chosen by the data, so ordinary p-values computed as if the features were fixed beforehand are invalid. This paper introduces PTL-SI, a conditional-inference method that computes a selective p-value for every feature selected by the two-stage TransFusion estimator (and by Oracle Trans-Lasso). The paper claims these p-values are exactly valid under the selected-model null: for each selected feature, the probability of a false positive equals the chosen significance level $\\alpha$ (e.g., 0.05) rather than being inflated by the selection step. A divide-and-conquer procedure makes the conditioning event tractable by characterizing, through systems of linear inequalities, the range of data perturbations that reproduce the same selected features. If the central claim holds, users of TL-HDR can attach error-controlled significance statements to individual selected features.","feed_headline":"PTL-SI p-values control false positives after transfer learning","feed_subtitle":"In high-dimensional regression, PTL-SI keeps each selected feature's false positive rate at the chosen alpha.","key_machinery":"The load-bearing object is the conditional-inference event, made concrete by a one-dimensional parametrization. For the test statistic $Z=\\eta_j^{\\top}Y$, the condition $Q(Y)=Q_{\\mathrm{obs}}$ forces $Y$ onto the line $Y(z)=a+bz$; Lemma 2 identifies the truncation region $\\mathcal{Z}$ as exactly the $z$-values for which TransFusion (with signs) returns the observed selected set. The divide-and-conquer step (Lemmas 3-5) rewrites $\\mathcal{Z}$ as a union of intersections $\\mathcal{Z}_u\\cap\\mathcal{Z}_v\\cap\\mathcal{Z}_t$, where the co-training active set and signs, the local-debias active set and signs, and the final coefficient active set and signs stay fixed; each such region is a system of linear inequalities obtained from Lasso KKT conditions. The selective p-value is then the two-sided tail probability of a truncated normal distribution over this region.","core_discovery":"On its own terms, the paper's discovery is that valid post-selection inference is possible for transfer-learning high-dimensional regression. Conditioning on the selection event $M(Y)=M_{\\mathrm{obs}}$ and on the sufficient statistic $Q(Y)=Q_{\\mathrm{obs}}$ reduces the conditional data space to the line $Y(z)=a+bz$, and the truncation region $\\mathcal{Z}=\\{z: M(z)=M_{\\mathrm{obs}}\\}$ is a finite union of intervals; Lemma 1 then gives $P_{H_{0,j}}(p_j^{\\mathrm{selective}}\\leq\\alpha)=\\alpha$ for every $\\alpha\\in[0,1]$. The proof identifies each interval through KKT conditions of the co-training weighted Lasso and the local-debias Lasso, and the same construction is extended to Oracle Trans-Lasso. Thus each selected feature receives a p-value that is exact for testing whether its coefficient in the projection onto the selected target design is zero.","pith_inferences":["Editorial inference: the same KKT-interval machinery should transfer to any two-stage transfer estimator that is piecewise linear in $Y$; a concrete test is to run PTL-SI on a different TL-HDR estimator and check that null p-values are uniform.","Editorial inference: PTL-SI controls the per-feature false positive rate, not the error rate across the whole selected set; applying it to many selected features simultaneously still needs a multiplicity adjustment before global claims.","Editorial inference: the search is run over $[-20\\sigma,20\\sigma]$, so a truncation region lying almost entirely beyond those bounds would be missed; adaptively widening the interval until the computed region stabilizes would make the procedure fully exact."],"forward_implications":["A user can threshold a selected feature's PTL-SI p-value at $\\alpha$ and know that, if the feature is truly null, it is declared relevant with probability exactly $\\alpha$.","PTL-SI separates false positives from true positives in TL-HDR: null features receive large p-values while genuine signals receive small ones, which naive p-values fail to do.","The divide-and-conquer algorithm makes the truncation region computable by solving linear inequalities, so a valid p-value is obtained without re-running TransFusion over a continuum of datasets.","The same conditional-inference construction gives valid p-values for Oracle Trans-Lasso, not only for TransFusion.","Experiments on synthetic and real data indicate that FPR control survives non-Gaussian noise (Laplace, skew-normal, t20), while power remains higher than data splitting and Bonferroni baselines."],"supporting_citations":[{"why":"Supplies the truncated-normal conditional-inference construction and the KKT characterization of Lasso selection regions that Lemmas 2-5 adapt.","marker":"[6]"},{"why":"Defines the TransFusion two-stage TL-HDR estimator whose selected set and signs are the object of inference.","marker":"[2]"},{"why":"Defines Oracle Trans-Lasso, the second TL-HDR estimator to which Section 4 extends PTL-SI.","marker":"[1]"},{"why":"Provides the divide-and-conquer parametric-programming strategy for turning the conditioning event into linear-inequality subregions.","marker":"[13]"},{"why":"Establishes the precedent of selective inference after domain adaptation by decomposing the conditioning event into subproblems.","marker":"[16]"},{"why":"Makes explicit the one-dimensional line reduction of the conditional data space that Lemma 2 and Remark 2 rely on.","marker":"[15]"},{"why":"The closest prior work on post-adaptation feature-selection inference, which the paper contrasts with its multi-source high-dimensional setting.","marker":"[14]"}],"fun_headline_variants":["PTL-SI gives exact p-values for transfer-learning selection","Exact selective p-values after transfer learning","Post-transfer inference: valid p-values for selected features","Controlled false positives: PTL-SI p-values","Exact p-values for transfer-learning feature selection"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that among the features selected at each stage, no column is a linear combination of the others; if collinearity makes the selected design matrix rank-deficient, the Lasso solution can be non-unique and the truncation-region description breaks down.","fun_headline_variants_meta":{"raw":{"variants":["PTL-SI gives exact p-values for transfer-learning selection","Exact selective p-values after transfer learning","Post-transfer inference: valid p-values for selected features","Controlled false positives: PTL-SI p-values","Exact p-values for transfer-learning feature selection"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001022,"raw_usage":{"total_tokens":4296,"prompt_tokens":919,"completion_tokens":3377,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":3311}},"tokens_in":535,"tokens_out":3377,"duration_ms":25343,"temperature":1.0,"reasoning_tokens":3311,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:22:40.561300+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the null ($\\beta_0=0$) with exact duplicate columns among candidate features so the selected design matrix can be rank-deficient; if the PTL-SI p-values are not uniform under $H_{0,j}$ or the empirical FPR at $\\alpha=0.05$ departs from 0.05, the full-rank premise is shown to be load-bearing.","supporting_citations":[{"cited_title":"The Annals of Statistics 44(3), 907–927 (2016)","cited_arxiv_id":null,"evidence_quote":"Supplies the truncated-normal conditional-inference construction and the KKT characterization of Lasso selection regions that Lemmas 2-5 adapt."},{"cited_title":"In: International Conference on Artificial Intelligence and Statistics, pp","cited_arxiv_id":null,"evidence_quote":"Defines the TransFusion two-stage TL-HDR estimator whose selected set and signs are the object of inference."},{"cited_title":"Journal of the Royal Statistical Society Series B: Statistical Methodology 84(1), 149–173 (2022)","cited_arxiv_id":null,"evidence_quote":"Defines Oracle Trans-Lasso, the second TL-HDR estimator to which Section 4 extends PTL-SI."},{"cited_title":"The Journal of Machine Learning Research 23(1), 13544–13580 (2022)","cited_arxiv_id":null,"evidence_quote":"Provides the divide-and-conquer parametric-programming strategy for turning the conditioning event into linear-inequality subregions."},{"cited_title":"In: International Conference on Artificial Intelligence and Statistics, pp","cited_arxiv_id":null,"evidence_quote":"Establishes the precedent of selective inference after domain adaptation by decomposing the conditioning event into subproblems."}],"review_version":1}