{"paper":{"title":"Optimal Online Discrepancy Minimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Janardhan Kulkarni, Thomas Rothvoss, Victor Reis","submitted_at":"2023-08-02T20:01:11Z","abstract_excerpt":"We prove that there exists an online algorithm that for any sequence of vectors $v_1,\\ldots,v_T \\in \\mathbb{R}^n$ with $\\|v_i\\|_2 \\leq 1$, arriving one at a time, decides random signs $x_1,\\ldots,x_T \\in \\{ -1,1\\}$ so that for every $t \\le T$, the prefix sum $\\sum_{i=1}^t x_iv_i$ is $10$-subgaussian. This improves over the work of Alweiss, Liu and Sawhney who kept prefix sums $O(\\sqrt{\\log (nT)})$-subgaussian, and gives a $O(\\sqrt{\\log T})$ bound on the discrepancy $\\max_{t \\in T} \\|\\sum_{i=1}^t x_i v_i\\|_\\infty$. Our proof combines a generalization of Banaszczyk's prefix balancing result to t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.01406","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2308.01406/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"}