{"id":"e8774d61-7b42-4111-abfa-5a353732394b","arxiv_id":"2607.24358","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Fully non-adaptive public-coin 1-bit queries attain the order-optimal finite-moment mean-estimation rates previously known only for adaptive protocols.","lead":"A fully non-adaptive 1-bit protocol matches the best adaptive rates for mean estimation under only a finite central moment. It answers a COLT open problem by fixing every query up front and letting the decoder reinterpret stored bits after a coarse center is found.","discovery_kind":"new_method","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Theorem 2.1 is a clean reduction to Lau–Scarlett’s non-adaptive localization theorem; that imported guarantee is the one point whose failure would collapse the claim.","rationale":"The reader identified the same soft spot: the result stands on an external non-adaptive localization block and explicitly uses arbitrary measurable, generally nonlocal public-coin queries. That is an honest scope limitation, not an internal inconsistency. The main new mechanism appears sound in structure. Disjoint localization and refinement samples justify conditioning on the decoded center; the correlated-threshold factors in (10) and (14) correctly cancel the phase and scale probabilities; the adjacent residues telescope pointwise; the safe-phase condition confines scale-j variance to |X−c|≥Lj/4; and pj∝Lj^((2−k)/2) is the square-root allocation for the resulting moment envelope. The continuous construction supplies a largely independent reproduction identity with the same cutoff and moment calculations.\n\nAccordingly, I would not lower the verdict. The appropriate action is a dependency audit of the cited localization theorem rather than a conditional verdict based on an unsubstantiated suspicion. If that theorem is exactly as quoted, the paper’s central existence claim remains well supported as an arbitrary-measurable-query, public-coin result.","tokens_in":22546,"tokens_out":7040,"duration_ms":268105,"concrete_test":"Audit Lau–Scarlett (2026b, v2), Theorem 16, line by line and instantiate it with η=δ/3 and E|X−μ|≤σ from Lyapunov. Produce an explicit notation map verifying all five requirements: deterministic fully non-adaptive queries; independence from separate refinement samples; premises |μ|≤λ and first central moment ≤σ; success probability at least 1−η; interval length ≤Clocσ; and sample size O(ℓ(λ/σ)+log(1/η)). If any property differs, substitute the actual guarantee into Proposition 2.2 and recalculate Theorem 2.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The refinement constructions do not themselves find a coarse center. Every bias and variance calculation begins by conditioning on localization success, so that the decoded midpoint satisfies |c−μ|≤(Cloc/2)σ and hence E|X−c|ᵏ≤τᵏ in equations (3), (4), and (17). Lemma 2.3 then relies on that center being generated from samples independent of the refinement samples and seeds. Thus Theorem 2.1 requires Proposition 2.2 to provide, under exactly |EX|≤λ and E|X−EX|≤σ, a deterministic fully non-adaptive block using O(ℓ(λ/σ)+log(1/η)) samples and returning an O(σ) interval with probability 1−η.\n\nIf the cited Lau–Scarlett Theorem 16 has a different premise, requires adaptivity or private randomness, gives only expected interval length, or has a different confidence dependence, the present conditioning and sample-complexity argument does not go through as stated. I did not find an internal break in the decoder-side refinement: the safe-phase telescope, crossing-supported variance bound, and moment-matched scale allocation are mutually consistent. The concern is therefore dependency risk rather than an identified error.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper addresses the Lau–Scarlett open problem: whether fully non-adaptive arbitrary 1-bit queries can match the adaptive minimax rates for mean estimation over the class D(k,λ,σ) with |EX|≤λ and k-th central moment ≤σ^k, k>1 fixed. The main result (Theorem 2.1) constructs a fully non-adaptive public-coin protocol whose sample complexity matches the adaptive rate r_k(λ,σ,ε,δ) in (1), up to k-dependent constants, in the small-error regime ε≤c_kσ. The protocol combines an imported non-adaptive localization block (Proposition 2.2, Lau–Scarlett Theorem 16) with two universal refinement constructions: a dyadic scheme using safe periodic residues with an adjacent-scale telescope and moment-matched scale sampling p_j ∝ L_j^{(2−k)/2}, and a continuous-scale scheme using shifted random grids, Rademacher cell coloring, and a compactly supported kernel identity. All queries are fixed before communication; the decoded center affects only decoder-side weights. I verified the internal derivations in detail (safe-phase lemma, correlated-threshold identity, telescope and terminal bias (20)–(22), the three variance regimes (23), the k=2 pointwise summation (24), median-of-means concentration, and the continuous-scale analogues (48)–(59)) and found them correct.","tokens_in":22785,"tokens_out":7143,"duration_ms":697498,"significance":"If the imported localization theorem holds as quoted, this resolves a named COLT open problem and establishes a sharp separation: interaction is unnecessary for scalar finite-moment 1-bit mean estimation with arbitrary measurable queries, while it is provably necessary under threshold/interval restrictions. Strengths worth naming: two complete and internally verified constructions with full proofs (Appendices A–B), including a clean moment-matched scale allocation that yields all three tail regimes from one importance distribution; a careful k=2 boundary analysis that avoids an artificial extra logarithm via the pointwise bound (24), with honest disclosure of non-uniformity as k→2 (Remark 4.3); matching against an external minimax lower bound rather than a fitted benchmark; and a reproducible numerical appendix with Rao–Blackwellized evaluation, mechanism tests, and a public artifact.","major_comments":[{"comment":"The entire refinement analysis is conditioned on the localization guarantee: Lemma 2.3, Eq. (3), and the moment inflation in (17) all begin from Proposition 2.2, which imports Theorem 16 of Lau–Scarlett (2026b, version 2). If that theorem's guarantee differs in form — expected rather than high-probability interval length, an adaptive or private-coin model, a stronger moment premise, or a different confidence dependence — the conditional decoupling and the sample-complexity accounting in §A.6 do not go through as stated. Because this is the single point whose failure would collapse Theorem 2.1, I ask the authors to (i) restate the imported theorem in full (hypotheses, model, guarantee, confidence scaling) in an appendix, (ii) verify explicitly that each hypothesis holds on D(k,λ,σ) (the Lyapunov check in (2) covers only the first-moment premise), and (iii) pin the arXiv version number in","section":"§2, Proposition 2.2"},{"comment":"The corollary asserts minimax optimality of the fully non-adaptive class by combining Theorem 2.1 with Lau–Scarlett's Theorem 9. The argument takes c'_k = min{cup_k, clb_k} and δ < δ_0, which is fine, but it should also verify that the lower bound's localization term Ω(log(λ/σ)) is proved under the same premise (|EX| ≤ λ, k-th central moment ≤ σ^k) and the same query model, and that the refinement lower bounds hold for arbitrary measurable queries rather than only thresholds. A two-sentence verification would close this; as written the reader must trust that the ranges and models align.","section":"§4.4, Corollary 4.4"}],"minor_comments":[{"comment":"The affiliation/email line and the repository link run together ('miaoyc@mails.neu.edu.cn /githubhttps://...'); please repair the header formatting.","section":"Title page"},{"comment":"The notation R(B − B_c) for the correlated-threshold statistic is used before it is formally introduced in Lemma 3.2; a forward pointer would help first-time readers.","section":"§1.3, second paragraph"},{"comment":"The variance bound is stated with ℓ(τ/ϵ) in Proposition 5.1 (34) but proved as Cτ²log(eτ/ϵ) in Lemma B.3; since ℓ(x) = 1 + log x, these agree only up to the convention that the log is at least one — please use one notation or note the equivalence explicitly.","section":"§5, Eq. (34) vs. Appendix B.3, Eq. (58)"},{"comment":"The experiments supply the decoder with an oracle center c = 0, so only the refinement block is validated; the localization block and the end-to-end pipeline are not exercised. This is reasonable but should be stated in one sentence in the main text when the experiments are mentioned.","section":"Appendix C.1"},{"comment":"The ordinate of Figure 3 is defined only via v_k in Appendix C.3; adding the definition to the caption would make the figure self-contained. The color-independent hatching in Figure 4 is appreciated.","section":"Appendix C, Figures 3–4"},{"comment":"Remark 4.3's disclosure that constants are not uniform as k → 2 is welcome; consider adding one line noting whether c_k, C_k could in principle be tracked explicitly from the proofs, to set reader expectations.","section":"§4.3, Remark 4.3"}],"recommendation":"minor_revision","confidential_remarks":"The result is a clean reduction onto Theorem 16 of Lau–Scarlett (arXiv:2604.07796, version 2), a very recent and possibly still-moving preprint by a different group; I could not independently verify the imported statement. The editor may wish to confirm that the pinned version is stable and that Proposition 2.2 quotes it faithfully, since the paper's central claim inherits any gap there. The numerical artifact is author-hosted; I did not run the code."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: for arbitrary measurable 1-bit queries, interaction is unnecessary at the order-optimal finite-moment rates. Miao fixes every query up front, then lets the decoded coarse center reweight already-stored bits. That closes the COLT open problem in the public-coin model they actually asked about.\n\nWhat is new is the universal refinement, not the rates or the localization. Two independent constructions do the same job—safe periodic residues with an adjacent-scale telescope, and a continuous random-grid identity with a compact kernel ψ_a and Rademacher cell colors. Both move location dependence entirely to the decoder, get crossing-supported variance, and allocate scales to the moment envelope so the three regimes (k>2, k=2, 1<k<2) fall out cleanly. Appendices A–B are complete; Appendix C has Rao–Blackwellized checks and code. Citation pattern is honest: rates and lower bound from Lau–Scarlett, correlated-threshold idea from Mayekar et al., and the paper states what it does not claim (threshold/interval queries still have a gap).\n\nThe soft spot the stress-test flags is real but proportionate: Theorem 2.1 is a reduction. Every bias/variance line conditions on a successful O(σ) center from their non-adaptive localization (Prop. 2.2 / their Thm 16), with independent samples and seeds (Lemma 2.3). If that imported block fails under the stated first-moment premise, the claim collapses. I do not see an internal break in the refinement algebra itself. Other scoped limits—nonlocal measurable sets, public coins, constants blowing up as k→2, optimality only where their lower bound applies—are written clearly in the text, not hidden.\n\nThis is for people in information-constrained estimation and distributed mean estimation who care about interaction vs. query class. The central argument holds as an existence result. I would send it to peer review and bring it to reading group; if you work near 1-bit or finite-moment distributed estimation, it is worth citing for the decoder-side separation and the two constructions.","headline":"Clean affirmative answer to the Lau–Scarlett non-adaptive open problem via decoder-side universal refinement; the math holds as a reduction on top of their localization block.","tokens_in":23805,"tokens_out":538,"would_cite":true,"duration_ms":19298,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G05","94A29","68Q32"],"pacs":[],"model":"grok-4.5","headline":"Interaction is unnecessary: fully non-adaptive 1-bit queries match adaptive order-optimal rates for mean estimation under finite central moments.","keywords":["1-bit mean estimation","non-adaptive protocols","finite central moments","distributed estimation","public-coin protocols","decoder-side refinement","periodic quantization"],"falsifier":"Exhibit a matching lower bound, or a concrete distribution family in D(k,λ,σ), showing that every fully non-adaptive protocol with arbitrary measurable 1-bit queries still needs asymptotically more samples than the stated rate r_k in the small-error, high-confidence regime where adaptive lower bounds already apply.","tokens_in":23686,"feed_emoji":"📡","tokens_out":944,"duration_ms":33061,"temperature":0.7,"pith_summary":"When each device can send only one bit about a fresh sample, adaptive threshold queries were known to achieve the best possible sample rates under a single central-moment bound. This paper shows those same rates are attainable with no interaction at all, provided the bits may answer arbitrary measurable questions fixed in advance. A public-coin protocol generates every localization and refinement query in one batch; after a coarse center is decoded, it changes only how the already-stored refinement bits are weighted, never which sets were queried. Two constructions—dyadic safe periodic residues and continuous shifted random grids—deliver the standard three refinement regimes (light, critical, and heavy tails) plus an additive localization cost. In the small-error, high-confidence range where matching lower bounds already exist, the resulting sample complexity is minimax optimal.","feed_headline":"Non-adaptive 1-bit queries match adaptive mean rates","feed_subtitle":"All queries can be fixed in one batch; the coarse center only reweights stored bits.","key_machinery":"Universal decoder-side refinement: all measurable 1-bit queries are fixed before any message is seen; a later-decoded coarse center only reinterprets stored bits. The dyadic construction uses safe half-period residues, correlated thresholding, and an adjacent-scale telescope whose variance is supported only on boundary crossings; the continuous construction integrates a compactly supported kernel over random grid widths with adjacent-cell gradients. Moment-matched scale sampling yields the three optimal tail regimes.","core_discovery":"For every fixed moment order k>1, a fully non-adaptive public-coin 1-bit protocol is (ε,δ)-accurate over the class of distributions with mean bounded by λ and k-th central moment bounded by σ^k, using a sample size of the same order as the best adaptive protocols: an additive localization term 1+log(λ/σ) plus the usual refinement cost in σ/ε and log(1/δ) that depends on whether k is above, equal to, or below 2.","pith_inferences":["The same separate-query-from-decoder pattern may transfer to other one-dimensional distributed tasks where a coarse location is easy to code but fine residuals are location-dependent.","Because both a discrete telescope and a continuous integral identity work, the phenomenon is about decoder-side recentering rather than one algebraic representation.","Extending the idea past the scalar setting will need new geometry: coordinate-wise application need not preserve optimal dimension dependence, as the paper notes remains open."],"forward_implications":["Zero adaptive rounds suffice for order-optimal scalar 1-bit mean estimation once arbitrary measurable queries are allowed.","All localization and refinement bits can be issued in a single parallel batch, removing sequential latency.","The known adaptive–non-adaptive gap for this task is an artifact of restricting to threshold or interval queries, not of the 1-bit budget itself.","In the parameter range of existing high-confidence lower bounds, the non-adaptive sample complexity is minimax optimal up to k-dependent constants."],"fun_headline_variants":["Non-adaptive 1-bit protocol matches adaptive mean rates","Interaction unnecessary for order-optimal 1-bit mean estimation","One-batch 1-bit queries achieve adaptive-optimal mean rates","Decoder-side refinement makes non-adaptive 1-bit estimation optimal","Fully non-adaptive public-coin 1-bit mean estimation is order-optimal"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The argument needs arbitrary measurable query sets (generally nonlocal unions of intervals) and shared public randomness, and it sits on top of an existing non-adaptive localization block; it does not claim the same rates for ordinary threshold or interval queries alone.","fun_headline_variants_meta":{"raw":{"variants":["Non-adaptive 1-bit protocol matches adaptive mean rates","Interaction unnecessary for order-optimal 1-bit mean estimation","One-batch 1-bit queries achieve adaptive-optimal mean rates","Decoder-side refinement makes non-adaptive 1-bit estimation optimal","Fully non-adaptive public-coin 1-bit mean estimation is order-optimal"]},"model":"grok-4.5","effort":"low","cost_usd":0.00498,"raw_usage":{"total_tokens":1431,"prompt_tokens":847,"num_sources_used":0,"completion_tokens":97,"cost_in_usd_ticks":49804000,"prompt_tokens_details":{"text_tokens":847,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":487,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":847,"tokens_out":97,"duration_ms":8078,"temperature":1.0,"reasoning_tokens":487,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T17:05:28.859314+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a matching lower bound, or a concrete distribution family in D(k,λ,σ), showing that every fully non-adaptive protocol with arbitrary measurable 1-bit queries still needs asymptotically more samples than the stated rate r_k in the small-error, high-confidence regime where adaptive lower bounds already apply.","supporting_citations":[],"review_version":1}