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pith:XYXST5G7

pith:2026:XYXST5G7ERBW3DN52M4PL55X4Z

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The Mean-Field Limit of Online Stochastic Vector Balancing

Christian Fiedler, Daniel Lacker, Joe Jackson, Jonathan Niles-Weed

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.

arxiv:2605.14149 v1 · 2026-05-13 · math.PR · math.OC

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

32 extracted · 32 resolved · 0 Pith anchors

[1] Altschuler and Konstantin Tikhomirov.A threshold for online balancing of sparse i.i.d 2025

[2] Discrepancy Minimization via a Self-Balancing Walk · doi:10.1137/21m1442450.url:https://doi.org/10.1137/21m1442450

[3] Paul Dupuis Amarjit Budhiraja.Analysis and Approximation of Rare Events. Springer, 2019 2019

[4] Constructive algorithms for discrepancy minimization 2010

[5] On-line balancing of random inputs 2020 · doi:10.1002/rsa

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:11.597751Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

be2f29f4df24436d8dbdd338f5f7b7e6657e9861bb5c14a57baf5337e403280d

Aliases

arxiv: 2605.14149 · arxiv_version: 2605.14149v1 · doi: 10.48550/arxiv.2605.14149 · pith_short_12: XYXST5G7ERBW · pith_short_16: XYXST5G7ERBW3DN5 · pith_short_8: XYXST5G7

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/XYXST5G7ERBW3DN52M4PL55X4Z \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: be2f29f4df24436d8dbdd338f5f7b7e6657e9861bb5c14a57baf5337e403280d

Canonical record JSON

{
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    "abstract_canon_sha256": "33543d947cf6ba05f035984e6b8bacfb50c4d0c3238d9a0b08f8a8471f1c07a9",
    "cross_cats_sorted": [
      "math.OC"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-13T22:00:17Z",
    "title_canon_sha256": "e45b7d59b1ba5059c71ce81e45be38166c29a5b520047e4ef02daecb5704bfc8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14149",
    "kind": "arxiv",
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}