{"paper":{"title":"Order-Optimal Sequential 1-Bit Mean Estimation in General Tail Regimes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"An adaptive estimator using sequential randomized 1-bit threshold queries achieves order-optimal sample complexity for mean estimation under any fixed moment bound k greater than 1.","cross_cats":["cs.IT","cs.LG","math.IT","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Ivan Lau, Jonathan Scarlett","submitted_at":"2026-04-09T04:49:21Z","abstract_excerpt":"In this paper, we study the problem of mean estimation under 1-bit communication constraints. We propose a novel adaptive mean estimator based solely on randomized threshold queries, where each 1-bit outcome indicates whether a given sample exceeds a sequentially chosen threshold. Our estimator is $(\\epsilon, \\delta)$-PAC for any distribution with a bounded mean $\\mu \\in [-\\lambda, \\lambda]$ and a bounded $k$-th central moment $\\mathbb{E}[|X-\\mu|^k] \\le \\sigma^k$ for any fixed $k > 1$. Moreover, our sample complexity is order-optimal in all such tail regimes, i.e., for every such $k$ value. Fo"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our estimator is (ε, δ)-PAC for any distribution with a bounded mean μ ∈ [−λ, λ] and a bounded k-th central moment E[|X−μ|^k] ≤ σ^k for any fixed k > 1. Crucially, our sample complexity is order-optimal in all such tail regimes, i.e., for every such k value.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The distribution belongs to the class with bounded mean in [−λ, λ] and bounded k-th central moment for some fixed k>1; the analysis assumes randomized threshold queries can be chosen sequentially and adaptively without additional constraints on query implementation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An adaptive 1-bit mean estimator using sequential threshold queries achieves order-optimal sample complexity for any fixed k-th moment bound, with a necessary logarithmic penalty only when variance is finite.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"An adaptive estimator using sequential randomized 1-bit threshold queries achieves order-optimal sample complexity for mean estimation under any fixed moment bound k greater than 1.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"eea7821cf94e7a13a62a803dcd390eb15fd11f9a9f12c6a7f520cbbfb677a3f2"},"source":{"id":"2604.07796","kind":"arxiv","version":2},"verdict":{"id":"c855c06a-a33e-4cf2-9476-6267d70f7ca3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T18:14:26.529999Z","strongest_claim":"Our estimator is (ε, δ)-PAC for any distribution with a bounded mean μ ∈ [−λ, λ] and a bounded k-th central moment E[|X−μ|^k] ≤ σ^k for any fixed k > 1. Crucially, our sample complexity is order-optimal in all such tail regimes, i.e., for every such k value.","one_line_summary":"An adaptive 1-bit mean estimator using sequential threshold queries achieves order-optimal sample complexity for any fixed k-th moment bound, with a necessary logarithmic penalty only when variance is finite.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The distribution belongs to the class with bounded mean in [−λ, λ] and bounded k-th central moment for some fixed k>1; the analysis assumes randomized threshold queries can be chosen sequentially and adaptively without additional constraints on query implementation.","pith_extraction_headline":"An adaptive estimator using sequential randomized 1-bit threshold queries achieves order-optimal sample complexity for mean estimation under any fixed moment bound k greater than 1."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.07796/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"605df4a1d9a8ce75b573094bec70e9110994df13ff6e75b522448760d1bea886"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}