{"paper":{"title":"Function-free Optimization via Comparison Oracles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Introduces a comparison-oracle framework using preference level-set geometry to achieve near-optimal query complexity for normal direction estimation and descent-based optimization under regularity and convexity.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Katya Scheinberg, Zikai Xiong","submitted_at":"2026-04-29T16:34:11Z","abstract_excerpt":"In this work, we study optimization specified only through a comparison oracle: given two points, it reports which one is preferred. We call it function-free optimization because we do not assume access to, nor the existence of, a canonical application-given objective function. The goal is to find the most preferred feasible point, which we call the optimal solution. This model arises in preference- and ranking-based settings where objective values and derivatives are unavailable or meaningless. Even when a representative function exists, it may be nonsmooth, nonconvex, or discontinuous. We de"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"Under regularity of the preference relation in a d-dimensional Euclidean space, we estimate normal directions to accuracy ε using O(d log(d/ε)) comparisons, nearly matching a lower bound of Ω(d log(1/ε)). Under convexity, regularity, and a local growth condition on the regularity radius, the resulting normal direction descent method reaches an ε level-set optimality gap using at most Õ(d D²/ε²) comparisons over O(D²/ε²) normal direction estimation steps.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The regularity of the preference relation together with convexity and the local growth condition on the regularity radius; these are invoked to derive the normal-direction estimation and descent complexity bounds but are not guaranteed to hold for arbitrary comparison oracles.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces a comparison-oracle framework using preference level-set geometry to achieve near-optimal query complexity for normal direction estimation and descent-based optimization under regularity and convexity.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"0c20cfc3539c15e702f52ea80fc4acf033b7701cb868d565942f0b9aee04df84"},"source":{"id":"2604.26867","kind":"arxiv","version":2},"verdict":{"id":"b83445bf-2d25-4542-bffc-130f793fb5af","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T11:53:01.481806Z","strongest_claim":"Under regularity of the preference relation in a d-dimensional Euclidean space, we estimate normal directions to accuracy ε using O(d log(d/ε)) comparisons, nearly matching a lower bound of Ω(d log(1/ε)). Under convexity, regularity, and a local growth condition on the regularity radius, the resulting normal direction descent method reaches an ε level-set optimality gap using at most Õ(d D²/ε²) comparisons over O(D²/ε²) normal direction estimation steps.","one_line_summary":"Introduces a comparison-oracle framework using preference level-set geometry to achieve near-optimal query complexity for normal direction estimation and descent-based optimization under regularity and convexity.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The regularity of the preference relation together with convexity and the local growth condition on the regularity radius; these are invoked to derive the normal-direction estimation and descent complexity bounds but are not guaranteed to hold for arbitrary comparison oracles.","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.26867/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":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}