{"id":"bf0ded43-3cfb-4414-8438-99b4b644670f","arxiv_id":"2506.17214","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Regularized HAL-TMLEs target within the selected HAL working model and, in simulations, improve stability, bias, and coverage over relaxed HAL for ATE and survival-curve estimation.","lead":"Two new estimation routines apply ridge or lasso regularization when solving the targeted maximum likelihood estimating equation inside the working model chosen by the Highly Adaptive Lasso. In simulations for treatment effects and survival curves, the projection-based variant reduces bias and improves coverage relative to relaxed HAL.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.5's verification of condition (C.3) relies on an unproved sup-norm rate for the HAL sieve; the advertised first-order bias removal for the original target is therefore not established.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing gap: Appendix D.4 verifies condition (C.3) using a sup-norm rate that Assumption 5.6 does not provide. My reading of the proof confirms this is not a stylistic omission but a substantive missing ingredient in Theorem 5.5, the theorem that carries the paper's advertised guarantee of first-order bias removal for the original target. The secondary issue about the regularized EIC approximation is also present in the algorithm but is secondary because even if the EIC were exact, condition (C.3) would still be unproved. I do not call for rejection: the simulations are extensive, code is provided, and the gap may be patchable by adding a suitable assumption or a separate sup-norm rate theorem for the HAL sieve. The correct status remains conditional, and my read does not change the reader's verdict, hence UNCHANGED.","tokens_in":27659,"tokens_out":5742,"duration_ms":58319,"concrete_test":"Independently re-derive Appendix D.4's verification of (C.3) using only Assumption 5.6 and the cited HAL rate theorems (Bibaut and van der Laan 2019; van der Laan 2023). If ∥Q0,Mn − Q0∥∞ = o(n^{−1/2}) does not follow from any stated theorem, the proof has a missing assumption. As a complementary numerical check, simulate a càdlàg Q0 with bounded sectional variation, compute the HAL sieve projection sup-norm error for n = 500, 1000, 2000, 4000, 8000, and estimate its decay rate; an empirical rate around n^{−1/3} rather than n^{−1/2} would confirm that the asserted rate is not a routine HAL property.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical guarantee is Theorem 5.5, whose proof (Appendix D.3) decomposes Ψ_Mn(P0) − Ψ(P0) and requires condition (C.3), P0 D*_{Mn}(P0,Mn) = o_p(n^{−1/2}). Appendix D.4 verifies (C.3) by asserting ∥Q0,Mn − Q0∥∞ = o(n^{−1/2}) 'by the HAL-sieve rate'. But Assumption 5.6 states only sup-norm o_P(1) and L2 O_P(n^{−r}) with r > 1/4; it does not state, and no theorem in the paper proves, a sup-norm rate of order o(n^{−1/2}). The HAL class is infinite-dimensional, and such a fast uniform rate is not a standard consequence of the cited HAL convergence results; it requires a separate argument that is absent. Thus condition (C.3) is not established, and Theorem 5.5 does not deliver the claimed first-order bias removal for the original target in the stated generality. A secondary, related gap is that Algorithm 1 solves the EIC equation only for the regularized/approximate D* and stops at a nonzero tolerance; no theorem shows that the approximation error between the projected/regularized EIC and the true working-model EIC is small enough to preserve the empirical-mean equation assumed in Theorem 5.5.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two regularized Targeted Maximum Likelihood Estimation procedures that operate inside the finite-dimensional working model implied by a Highly Adaptive Lasso fit: a delta-method regHAL-TMLE that uses a ridge-stabilized inverse of the empirical information matrix, and a projection-based regHAL-TMLE that uses a lasso-regularized projection of an influence function onto the HAL score space. The main theoretical claim, Theorem 5.5, asserts that the working-model TMLE is asymptotically linear for the original target parameter with influence curve equal to the EIC in a limiting oracle model. The paper also presents an adaptive extension with a plateau selector, and supports the methods with simulations for average treatment effects, survival curves, and adaptive model selection. The empirical section is extensive and consistently favors the projection-based estimator over relaxed HAL and delta-method alternatives.","tokens_in":27989,"tokens_out":3461,"duration_ms":35565,"significance":"If the theoretical guarantee were fully established, this would be a practically valuable contribution: it offers a computationally stable way to perform TMLE inside HAL-induced working models, avoiding ill-conditioned matrix inverses and expensive clever-covariate constructions, and the simulation evidence is strong and reproducible from the provided GitHub repository. The paper's central advertised property, however, is not currently backed by a complete proof. The gap in condition (C.3) and the mismatch between the EIC solved by the algorithm and the EIC used in the theorems are load-bearing, so the paper cannot yet be accepted as providing the claimed first-order bias-removal guarantee. The extensive simulations and reproducible code are genuine strengths and give reason to believe the practical claims, but the theoretical section needs substantial repair.","major_comments":[{"comment":"The verification of condition (C.3) in Appendix D.4 uses the assertion that ||Q_{0,M_n} - Q_0||_infty = o(n^{-1/2}) 'by the HAL-sieve rate', and the same sup-norm rate is invoked in Appendix D.3 for the remainder R(P_{0,M_n}, P_0, M_0). However, Assumption 5.6 only states sup-norm convergence o_P(1) and L2 convergence O_P(n^{-r}) with r > 1/4. A sup-norm rate of order o_P(n^{-1/2}) is not a consequence of the stated assumption and is not proved anywhere in the manuscript or in the cited results. Because (C.3) is exactly the term P_0 D*_{M_n}(P_{0,M_n}) used to show that the working-model target equals the original target up to o_p(n^{-1/2}), Theorem 5.5 does not establish the advertised first-order bias removal for the original target in the stated generality. The assumption r > 1/4 only yields O_P(n^{-1/2}) for the product term in (C.2), and the additional sup-norm rate in (C.3) needs to be either proved or added as an explicit assumption.","section":"Section 5.4, condition (C.3), and Appendix D.4"},{"comment":"Theorem 5.4 assumes that the regHAL-TMLE P*_{M_n} solves P_n D*_{M_n}(P*_{M_n}) = o_p(n^{-1/2}) for the working-model EIC D*_{M_n}. Algorithm 1, however, computes an approximated or regularized EIC D*_{n,beta,Q} and stops when |P_n D*_{n,beta,Q}| < se(D*_{n,beta,Q})/(sqrt(n) log n). No theorem or proof shows that the regularized/projected D*_{n,beta,Q} is close enough to the true working-model EIC D*_{M_n} that the empirical-mean equation assumed in Theorem 5.4 is satisfied, nor is there a bound on the difference between the two influence curves. In particular, the lasso projection with penalty lambda and the ridge parameter eta are algorithmic choices, yet the asymptotic results are stated for the exact parametric EIC. The first-order bias-removal guarantee therefore remains conditional on an unverified relationship between the algorithm's stopping criterion and the theorem's EIC equation.","section":"Section 5.2 and Algorithm 1"},{"comment":"The proof of conditions (A.2) and (B.2) asserts that the map P -> D*_{M_n}(P) is Lipschitz in sup-norm because Psi_{M_n} is pathwise differentiable on the finite-dimensional model M_n. This Lipschitz property is not demonstrated. The working-model EIC has the form gamma(P)^T S_beta, where gamma involves the inverse Fisher information, so a uniform Lipschitz bound in sup-norm is not automatic and may fail as the information matrix becomes nearly singular. Since (A.2) and (B.2) are needed for the empirical-process equicontinuity steps in Theorems 5.4 and 5.5, this is another load-bearing point in the theoretical verification that should either be proved under explicit conditions on the HAL basis and target parameter or replaced by a weaker sufficient condition.","section":"Section 5.4, conditions (A.2) and (B.2)"}],"minor_comments":[{"comment":"The section header 'D.4 Proof for theorem 5.4' is a duplicate; the content proves the verification of the regularity conditions in Section 5.4 and should be labeled accordingly, e.g., 'Proof of Theorem 5.7'.","section":"Appendix D.4"},{"comment":"The introduction states that both updates 'guarantee first-order bias removal', but in light of the gaps in Section 5 this is stronger than what the theorems currently establish; the wording should be softened or conditioned on the additional sup-norm rate and EIC-equation assumptions.","section":"Section 1, bullet (a)"},{"comment":"There are minor typos: 'statstical' should be 'statistical' in Section 7.2, and 'collniearity' should be 'collinearity' in Section 10.1.","section":"Section 7.2 and Section 10.1"},{"comment":"The plateau selector in Algorithm 2 is described only by an intuitive rule and no formal guarantee is given for its selected model complexity; this is acceptable as a heuristic, but the text should clearly separate this algorithmic heuristic from the formal asymptotic claims in Section 5.","section":"Section 8.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's Section 5 explicitly says the theorems and proofs 'closely followed' reference [10], and the verification section does not fully connect those abstract theorems to the specific regularized algorithms. The authors should be asked to either prove the missing sup-norm rate, add it as a formal assumption, or restate the main theorem as conditional on it. The simulation study is strong and the code availability is a plus, so the paper is worth a revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2506.17214. First, the practical contribution is real: the projection-based regHAL-TMLE is a genuinely new algorithmic variant, and the simulations show it consistently beats relaxed HAL and the delta-method variant for ATE and survival-curve targets, especially under positivity violations and in the survival setting where relaxed HAL collapses. Second, the advertised theoretical guarantee—that both updates \"guarantee first-order bias removal\" for the original target—is not established as stated. Theorem 5.5's verification of condition (C.3) relies on an unproved sup-norm rate o(n^{-1/2}) for the HAL sieve; Assumption 5.6 only gives sup-norm o_P(1) and L2 O_P(n^{-r}) with r>1/4. That is a real gap, and it is load-bearing for the asymptotic linearity claim.\n\nWhat the paper does well: it identifies a genuine practical bottleneck (collinear HAL basis and costly clever-covariate construction), proposes two concrete regularized updates that stay inside the working model, and evaluates them with extensive simulations, including a high-dimensional survival target. Code is available. The paper is also honest: Section 5 states it closely follows the theorems and proofs in [10] (same group), and Section 10.5 flags incomplete theory for data-driven penalty selection.\n\nThe soft spots, in proportion. The theoretical scaffolding is adapted, not new; the only genuinely new content is algorithmic. The proof gap around (C.3) is the main issue; a separate argument for a uniform rate of order n^{-1/2} would be needed, or the guarantee should be downgraded to a conditional statement. A secondary gap: Algorithm 1 solves the EIC equation only for the regularized EIC and stops at a tolerance; no theorem quantifies the bias from this approximation. The plateau selector in Section 8 is heuristic, and the simulations, while extensive, are limited to three DGPs. None of this undermines the empirical case, which is strong.\n\nBottom line: this is a useful methods paper with an honest empirical evaluation and a theory section that overreaches. A serious referee should engage with it and push for either a proof of the sup-norm rate or a restated theorem with stronger assumptions. For practitioners who want to get HAL-TMLE working in survival and mediation settings, the projection-based variant is worth trying; the simulations back it up. I'd accept it for peer review with major revision, not desk reject.","headline":"Projection-based regHAL-TMLE is a real algorithmic improvement with strong simulation support, but the paper's central guarantee of first-order bias removal rests on an unproved sup-norm rate.","tokens_in":28494,"tokens_out":2382,"would_cite":true,"duration_ms":22149,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G05","62G20","62J07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two regularized updates let HAL-based TMLE remove first-order bias while staying inside the working model, even when p exceeds n.","keywords":["Highly Adaptive Lasso","targeted maximum likelihood estimation","working model","efficient influence curve","lasso projection","survival curve estimation","average treatment effect","adaptive model selection"],"falsifier":"Construct a data-generating process where the true regression function has a sharp discontinuity that the HAL sieve approximates only slowly; for increasing n, measure the sup-norm distance between the best approximation in the active HAL basis and the truth. If that distance does not decay faster than $n^{-1/2}$, the remainder term used to prove Theorem 5.5 is not controlled, and confidence intervals built from the projected EIC should fail to reach nominal coverage.","tokens_in":27448,"feed_emoji":"🎯","tokens_out":8265,"duration_ms":79134,"temperature":0.7,"pith_summary":"This paper tries to make Targeted Maximum Likelihood Estimation (TMLE) work after an initial Highly Adaptive Lasso (HAL) fit, without leaving the finite-dimensional working model that HAL selects and without inverting unstable information matrices. It proposes two updates: one that regularizes the inverse Fisher information with a small ridge term, and one that lasso-regresses an initial influence function onto the HAL score space. The paper claims that both updates stay inside the working model, remove first-order bias, and remain numerically stable even when the number of basis functions exceeds the sample size. The central theoretical result is that the projection-based estimator is asymptotically linear for the original target parameter with the full-model efficient influence curve, despite every computation happening inside the smaller working model. This matters for practice because existing relaxed-HAL updates are unstable under collinearity, while full nonparametric TMLE for survival or mediation targets is computationally heavy.","feed_headline":"Regularized HAL-TMLE cuts bias by up to 98 percent","feed_subtitle":"Two updates inside the HAL working model remove first-order bias and keep coverage near nominal, even under positivity violations.","key_machinery":"The load-bearing mechanism is the working-model efficient influence curve, $D^*_{M_n}(P) = \\gamma^T S_\\beta$, where $S_\\beta$ is the vector of HAL score functions and $\\gamma$ solves $\\gamma^T I_P(\\beta) = \\partial_\\beta \\Psi(Q_\\beta)$. The delta-method variant estimates $\\gamma$ by replacing the inverse Fisher information with a ridge-regularized inverse, $(I_{P_n} + \\eta I)^{-1}$, while the projection variant estimates $\\gamma$ by a lasso regression of an initial influence function on the score columns, avoiding the inverse entirely. The identity that carries the argument is Theorem 5.2, the coincidence of the projected parameter's influence curve with the working model's parametric influence curve inside $M_n$. The algorithm then updates the HAL coefficients in the direction of the approximated influence curve until the empirical mean of the EIC falls below a tolerance set by its standard error, turning targeting into a sequence of coefficient updates.","core_discovery":"On its own terms, the paper establishes that the finite-dimensional working model selected by HAL is enough for valid TMLE, provided that the target parameter's influence curve is approximated inside that model with regularization. Theorem 5.2 shows that inside the working model the parametric efficient influence curve coincides with the full-model efficient influence curve, and Theorem 5.5 gives the asymptotic linearity statement $\\Psi_{M_n}(P^*_{M_n}) - \\Psi(P_0) = P_n D^*_{M_0}(P_0) + o_p(n^{-1/2})$ for the original target parameter. The lasso-projected influence curve is the engine: projecting an initial gradient onto the HAL score space with an $\\ell^1$ penalty produces a stable update direction that removes first-order bias without requiring a matrix inverse. In the paper's simulations, the projection update reduces absolute bias by 50-85 percent for average treatment effects and by roughly 98 percent for survival curves compared with relaxed HAL, while keeping confidence interval coverage near nominal. The ridge-regularized delta-method variant is presented as the alternative when projection is not practical.","pith_inferences":["The same regularized projection could treat arbitrary initial estimators, such as random forests or neural networks, as high-dimensional parametric fits in their implicit feature spaces; the paper sketches this as a general TMLE blueprint but does not develop it.","Viewing one-step debiased lasso as an unbounded linear special case suggests that regularized targeting could replace debiasing corrections in collinear designs, trading a little bias for more stable confidence intervals; the paper identifies this connection but does not test it.","A testable extension is to tune the projection penalty by the stability of the resulting EIC rather than fixing a tiny lambda, since the paper acknowledges that data-driven penalty selection lacks theoretical guarantees.","The idea of targeting in one working model and inferring in a richer one could improve coverage under misspecification; the paper lists this as future work rather than a demonstrated result."],"forward_implications":["Projection-based regHAL-TMLE is reported to cut absolute bias by 50-85 percent for average treatment effect estimation compared with relaxed HAL, and by roughly 98 percent for survival curves, while keeping coverage near nominal.","Because each update modifies only the coefficient vector inside the working model, survival-curve estimation no longer requires recursive clever-covariate integrations, and the per-iteration cost is $O(p)$.","For distributions inside the working model, targeting the projected parameter with the within-model EIC is equivalent to targeting the original parameter with the full-model EIC.","The plateau-selector extension provides a data-driven way to grow the working model, selecting simpler models under positivity violations and more complex models as the sample size grows."],"supporting_citations":[{"why":"Introduces the Highly Adaptive Lasso estimator and its working model, which is the starting point for the proposed targeting updates.","marker":"[1]"},{"why":"Supplies the bracketing-entropy and fast-rate results for cadlag functions used to verify Donsker and rate conditions in the appendix.","marker":"[2]"},{"why":"Provides the parametric EIC formula and TMLE framework that the paper adapts to the HAL working model.","marker":"[6]"},{"why":"Documents the recursive clever-covariate burden for survival-curve TMLE that the within-model updates are designed to avoid.","marker":"[7]"},{"why":"The adaptive debiased machine learning framework whose theorem structure and plateau selector are adapted for the main results and regHAL-ATMLE.","marker":"[10]"},{"why":"Used to justify the sup-norm and L2 convergence rates of HAL nuisance estimators that underpin condition (C.3).","marker":"[12]"},{"why":"The lasso-projection approach to approximating the efficient influence curve, which the projection step regularizes inside the working model.","marker":"[14]"}],"fun_headline_variants":["HAL-TMLE with lasso projection slashes bias up to 98%","Projection-based regHAL-TMLE cuts first-order bias by 85%","Stable HAL-TMLE: lasso projection kills bias, keeps coverage","New regularized TMLE cuts bias up to 98% without matrix inverses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on the HAL-selected working model being so close to the true regression function that the largest pointwise error shrinks faster than one over the square root of the sample size, a rate the paper invokes from the HAL sieve but does not derive from its own stated assumptions.","fun_headline_variants_meta":{"raw":{"variants":["HAL-TMLE with lasso projection slashes bias up to 98%","Projection-based regHAL-TMLE cuts first-order bias by 85%","Stable HAL-TMLE: lasso projection kills bias, keeps coverage","New regularized TMLE cuts bias up to 98% without matrix inverses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000929,"raw_usage":{"total_tokens":3987,"prompt_tokens":962,"completion_tokens":3025,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":2949}},"tokens_in":578,"tokens_out":3025,"duration_ms":22170,"temperature":1.0,"reasoning_tokens":2949,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:09:57.118631+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a data-generating process where the true regression function has a sharp discontinuity that the HAL sieve approximates only slowly; for increasing n, measure the sup-norm distance between the best approximation in the active HAL basis and the truth. If that distance does not decay faster than $n^{-1/2}$, the remainder term used to prove Theorem 5.5 is not controlled, and confidence intervals built from the projected EIC should fail to reach nominal coverage.","supporting_citations":[{"cited_title":"The highly adaptive lasso estimator","cited_arxiv_id":null,"evidence_quote":"Introduces the Highly Adaptive Lasso estimator and its working model, which is the starting point for the proposed targeting updates."},{"cited_title":"Springer International Publishing,, 2018","cited_arxiv_id":null,"evidence_quote":"Provides the parametric EIC formula and TMLE framework that the paper adapts to the HAL working model."},{"cited_title":"One-step TMLE for targeting cause-specific absolute risks and survival curves","cited_arxiv_id":"2107.01537","evidence_quote":"Documents the recursive clever-covariate burden for survival-curve TMLE that the within-model updates are designed to avoid."},{"cited_title":"A generally efficient hal-tmle.Targeted Learning in Data Science: Causal Inference for Complex Longitudinal Studies, pages 95–102, 2018","cited_arxiv_id":null,"evidence_quote":"The lasso-projection approach to approximating the efficient influence curve, which the projection step regularizes inside the working model."}],"review_version":2}