{"id":"d9c07e7d-0bbd-41c0-a6db-5f31041b5522","arxiv_id":"2607.02896","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Is interaction necessary for order-optimal 1-bit mean estimation over nonparametric finite-moment classes, or can fully non-adaptive general quantizers match the adaptive rate?","lead":"This open-problem note asks whether fully non-adaptive 1-bit quantizers can match the known adaptive minimax rate for nonparametric mean estimation under moment bounds. A yes or no answer would settle whether one adaptive transition is necessary and sufficient for order-optimal 1-bit estimation.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the external dependence on the adaptive rate, but that dependence does not undermine the note: Open Problem 1 is still a precise, interesting question at the adaptivity boundary, and the surrounding discussion of barriers (tail aliasing under moment assumptions, insufficiency of packing for nonlocal quantizers) is accurate and self-contained. No load-bearing flaw appears in the problem formulation, the known partial results, or the technical discussion. The CONDITIONAL verdict already reflects that this is an open-problem note rather than a resolved result; no further adjustment is warranted.","tokens_in":6874,"tokens_out":396,"duration_ms":3434,"concrete_test":"Spot-check the two-stage construction cited in §3 (Lau & Scarlett 2026b, §4.3): verify that the first-stage coding-based localization uses only non-adaptive general 1-bit queries and that the second-stage refinement, once the O(σ) interval is known, recovers the refinement term of (1) for each k-regime. If that construction holds, the open problem is correctly framed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The note is a clean open-problem statement, not a theorem paper. Its central claim is the well-posedness of Open Problem 1: whether fully non-adaptive arbitrary 1-bit quantizers can match the adaptive rate r_k of equation (1). The only potential soft spot is that r_k is imported from the authors' concurrent works (Lau & Scarlett 2026a,b) without re-proof. That dependence is standard and transparent for COLT open-problem notes; the problem remains well-posed even if those rates later need minor constant adjustments, and the note itself contains no internal inconsistency or unsupported assertion about non-adaptive protocols.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript poses a clean open problem in communication-constrained nonparametric statistics: whether fully non-adaptive arbitrary 1-bit quantizers can achieve the order-optimal adaptive 1-bit minimax rate for mean estimation over the finite-moment class D(k, λ, σ). It recalls that adaptive threshold queries attain the rate r_k(λ, σ, ε, δ) of equation (1), that the same rate is achievable with general 1-bit queries using only one adaptive transition (two stages), and that non-adaptive threshold and interval queries are known to be highly suboptimal. Open Problem 1 asks whether there exist constants c_k, C_k such that a fully non-adaptive protocol matches this rate for all λ ≥ σ > 0, sufficiently small ε, and δ ∈ (0, 1/2). The note carefully separates localization from refinement costs, surveys parametric positive results that do not extend, and discusses technical barriers (tail aliasing under moment assumptions, the insufficiency of locality-based packing arguments for general measurable quantizers).","tokens_in":6994,"tokens_out":802,"duration_ms":7456,"significance":"If resolved either way, the problem would settle a genuine adaptivity gap for a fundamental primitive under the strictest communication constraint. A positive answer would yield an order-optimal one-shot protocol; a negative answer would certify that the known two-stage estimator is stage-optimal. The formulation is precise, the rate formula is stated explicitly, and the discussion of barriers (universal refinement without prior localization, simultaneous localization/refinement/tail control) is technically useful. As a COLT-style open-problem note it is well-scoped and does not overclaim; its value lies in isolating the zero-adaptivity, general-query regime that prior work leaves open.","major_comments":[],"minor_comments":[{"comment":"Section 3, Fourier-feature example: the O((σ/ε)^8 (log(λ/ε) + log(1/δ))) bound is useful for illustration, but a one-sentence remark on whether the exponent 8 is an artifact of the second-order bias analysis or inherent would help readers gauge how far the construction is from optimality.","section":null},{"comment":"Equation (1) and Open Problem 1: the dependence of the constants c_k, C_k only on k is stated clearly; it would be slightly cleaner to note explicitly that they may also hide absolute numerical factors independent of all parameters.","section":null},{"comment":"References: the concurrent works Lau & Scarlett (2026a,b) are cited as arXiv/AISTATS; once final versions or DOIs are available, updating the bibliography would improve long-term citability.","section":null},{"comment":"Section 4, last paragraph: the phrase 'standard packing arguments are insufficient' is accurate; a brief pointer to why a mutual-information or Assouad-style argument would also need new ingredients (because a single query can touch many locations) would make the barrier discussion even sharper.","section":null}],"recommendation":"accept","confidential_remarks":"This is a short, carefully written open-problem note that meets the usual COLT open-problem standard. The only soft dependence is that the rate r_k is imported from the authors' concurrent papers; that is transparent and ordinary for this format. I see no reason to request major changes or to treat the note as incomplete. Accept as is (or with the minor polish listed)."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a short open-problem note, not a theorem paper. What it does well is isolate one precise remaining question: can fully non-adaptive arbitrary 1-bit quantizers match the adaptive rate r_k for the nonparametric class D(k, λ, σ), or is the known two-stage protocol already stage-optimal?\n\nThe landscape is laid out cleanly. Adaptive thresholds achieve the rate; general queries need only one adaptive transition; non-adaptive thresholds and intervals are known to be highly suboptimal. The open case is therefore exactly the zero-adaptivity, general-measurable-quantizer regime. Equation (1) and Open Problem 1 are stated carefully, the technical barriers (tail aliasing under pure moment assumptions, the need for universal refinement that still controls far-tail samples) are named without overclaiming, and the citations to the parametric literature and to the authors’ own concurrent adaptive/two-stage results are transparent.\n\nThe only soft spot is the expected one for this genre: r_k is imported from Lau & Scarlett 2026a,b and not re-proved here. That does not make the problem ill-posed; even if those rates later need constant-factor polishing, the qualitative question remains well-defined. There are no new estimators, no new lower bounds, and no circular reasoning inside the note itself. Self-citation is ordinary when the cited results are the immediate prior art.\n\nThis is for people already working on communication-constrained estimation, distributed mean estimation, or adaptivity gaps. It is not a general-audience paper and it does not claim a resolved result. For the COLT open-problems track it is exactly the right length and focus. I would send it to referees without hesitation; the question is sharp enough to be worth community attention.","headline":"Clean COLT-style open problem that isolates the remaining non-adaptive gap for nonparametric 1-bit mean estimation; no new theorems, but well-posed and useful.","tokens_in":7571,"tokens_out":466,"would_cite":false,"duration_ms":4807,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G05","68Q32","94A29"],"pacs":[],"model":"grok-4.5","headline":"The paper asks whether any non-adaptive 1-bit quantizers can match the adaptive order-optimal rate for nonparametric mean estimation under moment bounds.","keywords":["1-bit mean estimation","communication constraints","adaptivity","non-interactive protocols","nonparametric estimation","finite-moment classes","open problem"],"falsifier":"Either exhibit a fully non-adaptive family of measurable 1-bit queries whose sample complexity is at most a constant multiple of r_k for every λ≥σ and every sufficiently small ε, or prove a lower bound showing that every non-adaptive protocol requires asymptotically more samples than r_k on some sequence of instances in D(k,λ,σ).","tokens_in":7758,"feed_emoji":"❓","tokens_out":744,"duration_ms":5978,"temperature":0.7,"pith_summary":"This open-problem paper asks whether interaction is truly required to achieve the best possible sample complexity for 1-bit mean estimation when the underlying distribution is known only through a bound on its k-th moment. Adaptive 1-bit protocols already match the unquantized minimax rate up to at most one unavoidable logarithmic factor, and the same rate is known to be achievable with only a single adaptive transition (two stages). Fully non-adaptive threshold and interval queries are known to be badly suboptimal, but the paper leaves open whether completely general non-adaptive 1-bit quantizers could still match the adaptive rate. A positive answer would give an optimal one-shot protocol that devices can run without feedback; a negative answer would prove that one adaptive transition is both necessary and sufficient. Either resolution would pin down the precise role of interaction in the nonparametric 1-bit setting.","feed_headline":"Is one adaptive bit enough for optimal mean estimation?","feed_subtitle":"Open problem: can non-adaptive 1-bit queries match the known adaptive rate under only moment bounds?","key_machinery":"The adaptive rate r_k(λ,σ,ε,δ) itself, written explicitly as localization cost plus refinement cost; the open problem is whether this same expression remains attainable when every binary query set must be fixed before any bits are observed.","core_discovery":"The paper formulates Open Problem 1: whether there exist constants depending only on the moment order k such that fully non-adaptive arbitrary 1-bit quantizers can achieve the known adaptive 1-bit minimax sample complexity r_k(λ,σ,ε,δ) for every distribution class D(k,λ,σ). That rate consists of a logarithmic localization term log(λ/σ) plus a refinement term that matches the classical unquantized rate (with an extra log(σ/ε) factor only when k=2).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Is interaction needed for optimal 1-bit mean estimation","Can non-adaptive 1-bit quantizers match the adaptive rate","Is one adaptive transition enough for order-optimal rates","Do non-adaptive queries fail for moment-bounded mean estimation","Must 1-bit mean estimation use adaptive stages to be optimal"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The entire problem rests on earlier claims that the stated adaptive rate is both achievable and minimax-optimal among all 1-bit protocols; if those rates are incomplete, the open question is ill-posed.","fun_headline_variants_meta":{"raw":{"variants":["Is interaction needed for optimal 1-bit mean estimation","Can non-adaptive 1-bit quantizers match the adaptive rate","Is one adaptive transition enough for order-optimal rates","Do non-adaptive queries fail for moment-bounded mean estimation","Must 1-bit mean estimation use adaptive stages to be optimal"]},"model":"grok-4.5","effort":"low","cost_usd":0.005532,"raw_usage":{"total_tokens":1454,"prompt_tokens":702,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":55320000,"prompt_tokens_details":{"text_tokens":702,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":683,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":702,"tokens_out":69,"duration_ms":5168,"temperature":1.0,"reasoning_tokens":683,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T06:17:39.117291+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Either exhibit a fully non-adaptive family of measurable 1-bit queries whose sample complexity is at most a constant multiple of r_k for every λ≥σ and every sufficiently small ε, or prove a lower bound showing that every non-adaptive protocol requires asymptotically more samples than r_k on some sequence of instances in D(k,λ,σ).","supporting_citations":[],"review_version":1}