{"paper":{"title":"The Mean-Field Limit of Online Stochastic Vector Balancing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The optimal value for online vector balancing converges exactly to the value of a mean-field problem for steering Brownian motion under an L2 drift constraint.","cross_cats":["math.OC"],"primary_cat":"math.PR","authors_text":"Christian Fiedler, Daniel Lacker, Joe Jackson, Jonathan Niles-Weed","submitted_at":"2026-05-13T22:00:17Z","abstract_excerpt":"We study an online vector balancing problem, in which $n$ independent Gaussian random vectors $\\boldsymbol{\\zeta}(1),\\dots,\\boldsymbol{\\zeta}(n) \\sim \\mathcal{N}(0, I_n)$, each of dimension $n$, arrive one at a time. The goal is to choose signs $\\varepsilon(1),\\dots,\\varepsilon(n) \\in \\{\\pm 1\\}$ with $\\varepsilon(k)$ depending only on $\\boldsymbol{\\zeta}(1),\\dots,\\boldsymbol{\\zeta}(k)$, so as to minimize the expected $\\ell^{\\infty}$ norm of the signed sum $\\frac{1}{\\sqrt{n}}\\sum_{k = 1}^n \\varepsilon(k) \\boldsymbol{\\zeta}(k)$. Prior work showed that the optimal value $V^n$ is $O(1)$, at least "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our main contribution is to determine the exact limit V^∞ = lim V^n as the value of a nonstandard stochastic control problem of mean-field type: find the narrowest terminal interval into which a Brownian motion can be adaptively steered under a uniform-in-time L² constraint on the drift.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The upper bound relies on the Gaussian structure via a coupling procedure involving the Föllmer drift and dynamic programming; the lower bound holds more generally under i.i.d. mean-zero, variance-one, finite-fourth-moment entries using probabilistic compactness.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The optimal value V^n for online vector balancing with Gaussian vectors converges to the value of a mean-field stochastic control problem that finds the narrowest terminal interval for a Brownian motion under uniform L2 drift constraint.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The optimal value for online vector balancing converges exactly to the value of a mean-field problem for steering Brownian motion under an L2 drift constraint.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"dec6afb69460e417ed49e75f48cce54cb7aa0135bcbb98a8b11477806fa8690b"},"source":{"id":"2605.14149","kind":"arxiv","version":1},"verdict":{"id":"da94b48b-e24d-4db0-a8c2-3970047dc09f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:56:54.790325Z","strongest_claim":"Our main contribution is to determine the exact limit V^∞ = lim V^n as the value of a nonstandard stochastic control problem of mean-field type: find the narrowest terminal interval into which a Brownian motion can be adaptively steered under a uniform-in-time L² constraint on the drift.","one_line_summary":"The optimal value V^n for online vector balancing with Gaussian vectors converges to the value of a mean-field stochastic control problem that finds the narrowest terminal interval for a Brownian motion under uniform L2 drift constraint.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The upper bound relies on the Gaussian structure via a coupling procedure involving the Föllmer drift and dynamic programming; the lower bound holds more generally under i.i.d. mean-zero, variance-one, finite-fourth-moment entries using probabilistic compactness.","pith_extraction_headline":"The optimal value for online vector balancing converges exactly to the value of a mean-field problem for steering Brownian motion under an L2 drift constraint."},"references":{"count":32,"sample":[{"doi":"","year":2025,"title":"Altschuler and Konstantin Tikhomirov.A threshold for online balancing of sparse i.i.d","work_id":"f77b68f7-1a71-461f-ac93-a67b5bf54d08","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/21m1442450.url:https://doi.org/10.1137/21m1442450","year":null,"title":"Discrepancy Minimization via a Self-Balancing Walk","work_id":"f2dba25c-da21-4f30-97ea-3ae90f8fd31f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"Paul Dupuis Amarjit Budhiraja.Analysis and Approximation of Rare Events. Springer, 2019","work_id":"3fe44615-bd4f-40ab-bdfc-985d7950927e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Constructive algorithms for discrepancy minimization","work_id":"3496d8d3-73bc-47e6-8569-9c43ba959776","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1002/rsa","year":2020,"title":"On-line balancing of random inputs","work_id":"1d66be86-0b5c-430b-81b7-bcbc57b68f84","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"211d418e209c01163891b689c418bae25f7f0b9d34c5600e1658c3876e57bdc3","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"6caa6d10ec4a1c0e97762ce3deae8d7f1be478ab0d9f5e5539f148d58a7214a1"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}