Pith Number

pith:45F5CUW3

pith:2025:45F5CUW3IMYQOQIDNSAHHNCXXO

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Optimal Rates for Generalization of Gradient Descent for Deep ReLU Classification

Yiming Ying, Yuanfan Li, Yunwen Lei, Zheng-Chu Guo

Under NTK separability with margin γ, gradient descent on deep ReLU networks attains an excess risk of Õ(L^6/(n γ²)).

arxiv:2510.02779 v4 · 2025-10-03 · cs.LG

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\pithnumber{45F5CUW3IMYQOQIDNSAHHNCXXO}

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

under the assumption that the data are NTK separable from the margin γ, we prove an excess risk rate of Õ(L^6 / (n γ²))

C2weakest assumption

the assumption that the data are NTK separable from the margin γ (stated in the abstract as the condition under which the rate holds)

C3one line summary

Gradient descent on deep ReLU networks achieves excess risk rate of order L^6/(n gamma^2) under NTK margin separability via activation pattern control.

Receipt and verification

First computed	2026-06-03T01:05:45.517619Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e74bd152db43310741036c8073b457bbae4282571238749770d7854d1ea081f4

Aliases

arxiv: 2510.02779 · arxiv_version: 2510.02779v4 · doi: 10.48550/arxiv.2510.02779 · pith_short_12: 45F5CUW3IMYQ · pith_short_16: 45F5CUW3IMYQOQID · pith_short_8: 45F5CUW3

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO \
  | 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: e74bd152db43310741036c8073b457bbae4282571238749770d7854d1ea081f4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d9dbf981bef549d6867176946de85939839ae7d66dff89e44ec69cc0326de424",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2025-10-03T07:22:36Z",
    "title_canon_sha256": "4bc1d06d560a9fc43b13cd540ea8c56d0986ecce85e5c3fe2893cbb0477bcf0e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2510.02779",
    "kind": "arxiv",
    "version": 4
  }
}