Pith Number

pith:HL3PWH3B

pith:2026:HL3PWH3BO7UNJFKMWZ4TDPZGZY

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Propagation of Chaos in Contextual Flow Maps

Kaizhao Liu, Philippe Rigollet, Shi Chen, Zhengjiang Lin

Finite-context models converge to infinite-context versions uniformly in depth and training steps at optimal Wasserstein rates.

arxiv:2605.16747 v1 · 2026-05-16 · cs.LG · math.AP · math.OC · math.PR · math.ST · stat.TH

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Claims

C1strongest claim

We establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate n^{-1/d} for general CFMs and parametric rate n^{-1/2} for a restricted class of CFMs that includes transformers as a special case.

C2weakest assumption

The dynamics of the attention blocks admit a McKean-Vlasov structure so that the finite-context empirical measure converges to a population measure in the large-n limit; this is invoked to apply classical propagation-of-chaos machinery (abstract, paragraph on McKean-Vlasov structure).

C3one line summary

Derives forward and backward propagation-of-chaos bounds for finite vs. infinite-context transformers modeled as contextual flow maps, achieving Wasserstein rate n^{-1/d} generally and n^{-1/2} for transformer-like cases.

References

21 extracted · 21 resolved · 6 Pith anchors

[1] [ÁLGRB26] Antonio Álvarez-López, Borjan Geshkovski, and Domènec Ruiz-Balet

[2] Perceptrons and localization of attention’s mean-field landscape · arXiv:2601.21366

[3] [BCL+26] Giuseppe Bruno, Shi Chen, Zhengjiang Lin, Yury Polyanskiy, and Philippe Rigollet

[4] Scaling Limits of Long-Context Transformers · arXiv:2605.08505

[5] [BO20] Tom B

Formal links

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Cited by

1 paper in Pith

Uniform-in-Time Weak Propagation-of-Chaos in Shallow Neural Networks

Receipt and verification

First computed	2026-05-20T00:03:19.480239Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3af6fb1f6177e8d4954cb67931bf26ce00271e6f1c3115f59895937bf63972e3

Aliases

arxiv: 2605.16747 · arxiv_version: 2605.16747v1 · doi: 10.48550/arxiv.2605.16747 · pith_short_12: HL3PWH3BO7UN · pith_short_16: HL3PWH3BO7UNJFKM · pith_short_8: HL3PWH3B

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/HL3PWH3BO7UNJFKMWZ4TDPZGZY \
  | 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: 3af6fb1f6177e8d4954cb67931bf26ce00271e6f1c3115f59895937bf63972e3

Canonical record JSON

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      "math.PR",
      "math.ST",
      "stat.TH"
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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-16T02:03:20Z",
    "title_canon_sha256": "5fd3a623f34f363407e3d4d72356d76f357b62346441e5963b0f3794533f43c6"
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