{"paper":{"title":"Geometrizing rates of convergence under local differential privacy constraints","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Angelika Rohde, Lukas Steinberger","submitted_at":"2018-05-03T16:49:04Z","abstract_excerpt":"We study the problem of estimating a functional $\\theta(\\mathbb P)$ of an unknown probability distribution $\\mathbb P \\in\\mathcal P$ in which the original iid sample $X_1,\\dots, X_n$ is kept private even from the statistician via an $\\alpha$-local differential privacy constraint. Let $\\omega_{TV}$ denote the modulus of continuity of the functional $\\theta$ over $\\mathcal P$, with respect to total variation distance. For a large class of loss functions $l$ and a fixed privacy level $\\alpha$, we prove that the privatized minimax risk is equivalent to $l(\\omega_{TV}(n^{-1/2}))$ to within constant"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01422","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}