{"id":"b06ec80d-c521-48f2-b944-98b45486403c","arxiv_id":"2608.04566","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Combining local-polynomial Lasso variable selection with projected gradient descent yields near-minimax rates for estimating the minimizer and minimum of a sparse nonparametric regression function.","lead":"Researchers propose a two-step method to estimate the minimum of a smooth function from noisy samples when only a few of many input variables actually matter. The method recovers the relevant variables and then runs projected gradient descent, achieving near-optimal accuracy as if the relevant variables were known in advance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Active-variable identification relies on (A6)(1), which requires every active partial derivative at the single point x0 to be bounded away from zero; without it, Lemma 5.1 can fail and the optimization runs on the wrong subspace, so the claimed rate is not established for the natural strongly…","rationale":"I read the construction and proofs carefully. Theorems 3.1-3.3 are internally coherent: the decomposition into variable-selection error and optimization error is standard, and the concentration lemmas, while partly deferred to Akhavan et al., are plausible. My stress-test did not uncover a contradiction inside (A1)-(A6). The load-bearing point is the 'unknown active variables can be identified' step. (A6)(1) is what makes Lemma 5.1 valid, and it is a real restriction: it requires each active direction to have a first-order effect at the chosen x0, which is not implied by the other assumptions and cannot be checked from the sample. The quadratic example shows the failure mode is not just a proof artefact: a missed active coordinate can be frozen at a non-optimal value, giving a constant error. This agrees with the reader's weakest assumption, so no verdict change is needed. I would keep the reader's conditional verdict: acceptance should require either a relaxation or replacement of (A6)(1) (for instance, multi-point or higher-order screening), or an explicit statement that the method's guarantees are conditional on this identifiability condition, together with a separate lower bound for the restricted class before using the word 'optimal'.","tokens_in":29257,"tokens_out":22738,"duration_ms":256511,"concrete_test":"Numerical/analytic check: set d = r = 2, Θ = [-2,2]^2, uniform design on a slightly larger box, Gaussian noise, and f(x1,x2) = 2 x1^2 + 2 (x2 - 1)^2 + x1 x2, with x0 = (0,0). This obeys (A1)-(A5) and has ∂1 f(x0) = 0, so (A6)(1) fails, while the true minimizer is x* ≈ (-4/15, 16/15). Run the estimator (8)-(10) with the bandwidth and λ specified in Lemma 5.1 for large n, and compute (i) the frequency with which 1 ∈ \\hat J and (ii) the empirical quadratic error E \\|x* - \\hat x\\|^2. If 1 is missed with probability bounded below and the error does not converge to zero at the rate in Theorem 3.1, the concern is confirmed: (A6)(1) is load-bearing and the central claim is limited to that identifiable subclass.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The rate theorems rest on Lemma 5.1, which guarantees that \\hat J = J with probability 1 - c0 exp(c0 d) exp(-c1 n h^{d+2}). That lemma requires the separation condition encoded in (A6)(1): |∂_j f(x0)| ≥ C for every j ∈ J. The first step (8)-(10) estimates the coefficients h ∂_j f(x0); if an active variable has ∂_j f(x0) = 0, that coefficient is exactly zero and the rule 'keep nonzero coefficients' cannot distinguish j from an inactive variable, so the variable-selection step breaks. Strong convexity plus β-Hölder smoothness does not prevent this: for example, f(x1,x2) = 2 x1^2 + 2 (x2 - 1)^2 + x1 x2 on Θ = [-2,2]^2 with x0 = (0,0) satisfies (A1)-(A5) (after scaling, α > 1) but ∂1 f(x0) = 0 while the minimizer is x* ≈ (-4/15, 16/15). If \\hat J misses variable 1, the projected gradient descent (13)-(18) fixes x1 = 0 and \\|W(x* - \\hat x)\\|^2 has a positive constant lower bound. Thus (A6)(1) is not a harmless regularity assumption; it is an unverifiable identifiability condition that excludes natural strongly convex functions. Moreover, the paper's optimality discussion cites lower bounds for the known-active-variable problem, so no lower bound is supplied for the class actually restricted by (A6).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies estimation of the minimizer and minimum value of a regression function f on a compact convex set Θ, in a passive-design nonparametric model where f depends on an unknown subset J of r active variables among d. The proposed method first estimates J by a Lasso-type local polynomial fit at a reference point x0 (Section 2.1), then runs a projected gradient descent on the estimated active subspace using local polynomial gradient estimates (Section 2.2). The main results, Theorems 3.1 and 3.3, give non-asymptotic upper bounds of order (W(n)/n)^{2(β-1)/(2β+r)} for the squared error of the estimated minimizer and (W(n)/n)^{β/(2β+r)} for the absolute error of the estimated minimum, where W is the Lambert function. The proofs combine a variable-selection lemma from Bertin and Lecué (2008) with gradient/minimum estimation lemmas adapted from Akhavan et al. (2024).","tokens_in":29450,"tokens_out":8643,"duration_ms":80801,"significance":"If the results are correct, the paper makes a valuable contribution: it shows that in a passive design one can identify the active variables and optimize the function at rates that, up to a Lambert-function factor, match the minimax rates for the problem where the active subspace is known. The explicit choice of bandwidth and regularization in (26) and the detailed proofs of the main theorems are strengths, as is the use of the Lambert function to sharpen the logarithmic factor in prior bounds. However, the main optimality claim is not fully supported because the class of functions considered is restricted by a strong, unverifiable distinguishability condition, and the lower bounds cited do not cover that restricted class.","major_comments":[{"comment":"","section":"Assumption (A6)(1), Section 2.1; Lemma 5.1"},{"comment":"","section":"Section 3, optimality claims after Theorems 3.1 and 3.3"},{"comment":"","section":"Section 6, Lemmas 6.1-6.4, 6.5-6.6, 6.10-6.11"}],"minor_comments":[{"comment":"","section":"Section 2.1, text after (11)"},{"comment":"","section":"Lemmas 6.3 and 6.4"},{"comment":"","section":"Assumption (A6)(2) and Lemma 5.1"},{"comment":"","section":"Equation (11) and Lemma 5.1"},{"comment":"","section":"Theorem 3.1, condition on N"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically serious and the upper bounds appear to follow from the stated assumptions, but the central optimality claim is undermined by the strong and unverifiable condition (A6)(1). The counterexample in the report shows that the class of strongly convex functions satisfying (A1)-(A5) is much larger than the class covered by the theorem. I recommend major revision: the authors should either weaken or remove (A6)(1) (e.g., by using multiple reference points or a different variable-selection procedure), or explicitly restrict the theorem and provide a matching lower bound for the restricted class. The incomplete appendix proofs should also be completed. I do not recommend reject, as the method and rates are interesting and potentially fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper does something real. It combines Bertin-Lecué's Lasso-based variable selection with Akhavan et al.'s projected gradient descent and proves risk bounds for the unknown-active-set problem. The rates match the known-active-variable minimax rates up to a Lambert-function factor, which is a genuine extension. The main proofs are coherent; I checked the structure of Theorems 3.1–3.3 and the lemmas they rest on, and the bounds do follow from the stated assumptions. The Lambert refinement over the usual log factor is a nice touch. Credit where due: the variable-selection probability bound is imported correctly, and the decomposition in the proof of Theorem 3.1 is clean.\n\nThe soft spots are two. First, the \"optimal\" claim is not fully supported. The paper compares its upper bounds to lower bounds for the known-active-variable problem. That gives the right rate for the wider class, but the class actually restricted by (A6) — active partial derivatives at x0 bounded away from zero by C ≥ 1512 (p_M/p_m) L sqrt(r) — is smaller. No lower bound is proved for that restricted class, so \"optimal\" is an overreach, though a modest one; the rates are likely right, but that requires proof.\n\nSecond, (A6)(1) is load-bearing and not cosmetic. The stress-test example lands: f(x1,x2)=2x1^2+2(x2-1)^2+x1x2 at x0=(0,0) satisfies (A1)–(A5) after scaling, but the partial derivative in x1 vanishes at x0. Then the Lasso step in (8)–(10) cannot distinguish that variable from an inactive one, \\hat J misses it, and the minimization runs in the wrong subspace, giving a constant positive error. Strong convexity plus Hölder smoothness does not rule this out, and the paper gives no way to check (A6) from data. That is not fatal — the assumption is explicit — but it narrows the scope considerably, and the authors should present it that way.\n\nThird, several appendix lemmas are sketches referring to Akhavan et al. Full independent verification requires that paper at hand. Acceptable for a research article, but the authors should mark clearly which proofs are new and which are adaptations.\n\nBottom line: a solid, useful paper for people working on sparse nonparametric estimation and black-box optimization in passive designs. It deserves a serious referee. I would send it to review and ask for a revision that either relaxes (A6), proves a lower bound for the restricted class, or at least removes the unqualified \"optimal\" and adds an identifiability discussion. The math on its own terms is coherent.","headline":"A real two-step combination with clean rates for the unknown-active-set problem, but the 'optimal' claim and the (A6) identifiability condition need referee scrutiny before the paper's scope is accepted as stated.","tokens_in":30170,"tokens_out":1917,"would_cite":true,"duration_ms":21260,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G08","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Lasso-based variable selection plus projected gradient descent on the estimated subspace estimates the minimum and minimizer of a smooth noisy function at the same nonparametric rates as if the active variables were…","keywords":["multivariate regression","variable selection","dimension reduction","function minimization","local polynomial estimation","projected gradient descent","Hölder smoothness","Lambert function"],"falsifier":"Simulate the model with $d=2$, $r=2$, $f(x) = (x_1 - a)^2 + \\epsilon (x_2 - b)^2$, with $\\epsilon$ chosen so that $|\\partial_2 f(x_0)|$ is below the threshold $C$ at the chosen reference point, and compute the empirical risk of the two-step estimator over many replications. If the risk does not decay at the claimed rate once $\\hat J$ starts omitting variable 2, the theorem's validity is exactly limited to the range where (A6)(1) holds.","tokens_in":28929,"feed_emoji":"🎯","tokens_out":10942,"duration_ms":101898,"temperature":0.7,"pith_summary":"The paper studies passive-design nonparametric regression where the unknown regression function depends on $d$ coordinates but only on $r < d$ unknown 'active' coordinates. It proposes a two-step estimator: first locate the active coordinates with a Lasso on local polynomial coefficients at a reference point; then run projected gradient descent on the estimated active subspace using local polynomial estimates of the gradient, and finally evaluate the local-polynomial estimator at the resulting point. The central claim is that the expected squared error of the minimizer estimate and the absolute error of the minimum estimate match, up to a factor smaller than a power of a logarithm, the minimax rates one would get if the active set were known: $(W(n)/n)^{2(\\beta-1)/(2\\beta+r)}$ and $(W(n)/n)^{\\beta/(2\\beta+r)}$. A sympathetic reader should care because the method achieves dimension reduction automatically, without being told which coordinates matter.","feed_headline":"Two-step method matches known-variable rates without knowing them","feed_subtitle":"Lasso picks the active coordinates, gradient descent minimizes; errors match oracle rates up to a Lambert-function factor.","key_machinery":"The method is carried by two objects. First, the selection step solves a Lasso on the coefficients of a degree-one local polynomial at a fixed reference point $x_0$ with a uniform kernel; the nonzero coefficient of $\\partial_j f(x_0)$ flags variable $j$, and Lemma 5.1 (drawn from the paper's cited selection theorem) bounds the mis-selection probability. Second, minimization uses projected gradient descent with step size $\\eta_k = 1/k$ on the estimated subspace, where the gradient is replaced by a local polynomial regression estimator; the bandwidth and regularization are chosen as $h_n = (W(n)/n)^{1/(2\\beta+r)}$ and $\\lambda_n = (W(n)/n)^{\\beta/(2\\beta+r)}$. The Lambert-function inequalities of Lemma 5.4 convert the usual $\\log n$ factors into $W(n)$, tightening the bounds below any power of a logarithm while remaining within a logarithmic order.","core_discovery":"The paper's central claim is that unknown active variables can be identified and the function minimized at the same statistical rate as if those variables were known beforehand. Concretely, the set estimator $\\hat J$ from the Lasso step satisfies $\\mathbb{P}(\\hat J \\neq J) \\le c_0 \\exp(c_0 d) \\exp(-c_1 n h^{d+2})$, so with overwhelming probability the optimization runs on the correct subspace. On that event, projected gradient descent with local polynomial gradient estimates yields $E\\|W(x^*-\\hat x)\\|^2 \\le B (W(n)/n)^{2(\\beta-1)/(2\\beta+r)}$ and $E|f(x^*)-\\hat f^*| \\le B' (W(n)/n)^{\\beta/(2\\beta+r)}$, matching the known-active-variable minimax rates up to the Lambert factor $W(n)$, which is smaller than $\\log n$ at the first order. The paper states this as an improvement over the case where the active variables are unknown.","pith_inferences":["Beyond the paper's own claims, the distinguishability condition means the method selects variables from first-order behavior at a single reference point; a natural testable extension is to use a degree-two local polynomial in the selection step to catch variables that enter only through second-order terms.","Because the mis-selection probability contains $\\exp(c_0 d)$, the method is best suited to moderate ambient dimension; for very high-dimensional problems, the same two-step logic would need an additional screening or sparsity assumption to control the $d$-dependence.","The risk bound in Theorem 3.2 holds explicitly for every number of iterations $N$, which suggests one could tune $N$ by monitoring the estimated gradient rather than using the theoretical $N_{n,\\beta}$."],"forward_implications":["The minimizer estimator reaches the rate $(W(n)/n)^{2(\\beta-1)/(2\\beta+r)}$ regardless of the ambient dimension $d$, so the curse of dimensionality is confined to the exponentially small probability of selecting the wrong active set.","The minimum estimator reaches $(W(n)/n)^{\\beta/(2\\beta+r)}$, the same rate as if the active set were known.","The required number of gradient iterations $N$ only needs to exceed $N_{n,\\beta}$ times a constant depending on $|\\Theta|^2$ and the strong-convexity ratio, so the computational cost grows only polynomially in $n$.","If the Hölder exponent $\\beta$ is unknown, taking $\\beta=2$ in the bandwidth choice yields a valid procedure with rates in powers of $(W(n)/n)$, per Remark 1 of the paper."],"supporting_citations":[{"why":"Supplies Lemma 5.1, the local-polynomial Lasso variable-selection bound guaranteeing $\\hat J = J$ with exponentially small failure probability.","marker":"Bertin and Lecué (2008)"},{"why":"Provides the projected gradient-descent procedure, the local polynomial gradient and function estimators, and the risk recursion the paper adapts to the unknown-active-set case.","marker":"Akhavan et al. (2024)"},{"why":"Establishes the passive-design minimax rate for minimizer estimation that the paper matches up to the Lambert factor.","marker":"Tsybakov (1990)"},{"why":"Used in Lemma 6.9 for the covering-number bound that controls the supremum of the empirical process in the variance estimates.","marker":"Vershynin (2018)"}],"fun_headline_variants":["Identify hidden variables, then minimize: optimal rates","Optimal minimization without knowing active variables","Two-step method matches known-variable rates","Find active inputs, then minimize: oracle rates","Unknown which inputs matter? Still get optimal rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption (A6)(1): at the reference point $x_0$, every active variable must satisfy $|\\partial_j f(x_0)| \\ge C$ with $C \\ge 1512\\,(p_M/p_m)\\,L\\sqrt{r}$, and there is no data-dependent way to verify it. If a truly active variable has a smaller or vanishing first derivative at $x_0$, the Lasso step can omit it, and the minimization is performed on a wrong subspace, so the claimed rates no longer apply.","fun_headline_variants_meta":{"raw":{"variants":["Identify hidden variables, then minimize: optimal rates","Optimal minimization without knowing active variables","Two-step method matches known-variable rates","Find active inputs, then minimize: oracle rates","Unknown which inputs matter? Still get optimal rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000217,"raw_usage":{"total_tokens":1439,"prompt_tokens":955,"completion_tokens":484,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":416}},"tokens_in":571,"tokens_out":484,"duration_ms":6513,"temperature":1.0,"reasoning_tokens":416,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:54:39.805051+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the model with $d=2$, $r=2$, $f(x) = (x_1 - a)^2 + \\epsilon (x_2 - b)^2$, with $\\epsilon$ chosen so that $|\\partial_2 f(x_0)|$ is below the threshold $C$ at the chosen reference point, and compute the empirical risk of the two-step estimator over many replications. If the risk does not decay at the claimed rate once $\\hat J$ starts omitting variable 2, the theorem's validity is exactly limited to the range where (A6)(1) holds.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 5.1, the local-polynomial Lasso variable-selection bound guaranteeing $\\hat J = J$ with exponentially small failure probability."},{"cited_title":"Gogolashvili, and A","cited_arxiv_id":null,"evidence_quote":"Provides the projected gradient-descent procedure, the local polynomial gradient and function estimators, and the risk recursion the paper adapts to the unknown-active-set case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the passive-design minimax rate for minimizer estimation that the paper matches up to the Lambert factor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used in Lemma 6.9 for the covering-number bound that controls the supremum of the empirical process in the variance estimates."}],"review_version":1}