Pith Number

pith:Y5VVQRJ2

pith:2026:Y5VVQRJ27EHDSTDVRFSWWNC4VI

not attested not anchored not stored refs resolved

Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale

Han Shao, Shashaank Aiyer, Shay Moran, Tom Waknine, Yishay Mansour

For any bounded real-valued function class, uniform convergence at scale γ is equivalent to agnostic learnability at scale γ/2 and finite fat-shattering dimension at all scales above γ.

arxiv:2605.13684 v1 · 2026-05-13 · cs.LG · cs.IT · math.IT

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\usepackage{pith}
\pithnumber{Y5VVQRJ27EHDSTDVRFSWWNC4VI}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

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

for every bounded real-valued class and every γ>0, uniform convergence at scale γ, agnostic learnability at scale γ/2, and finiteness of the fat-shattering dimension at every scale γ'>γ are equivalent

C2weakest assumption

The real-valued function class must be bounded, which is required for the scale-sensitive notions of uniform convergence and learnability to be well-defined.

C3one line summary

For bounded real-valued function classes, uniform convergence at scale γ, agnostic learnability at γ/2, and finite fat-shattering dimension above γ are equivalent.

References

41 extracted · 41 resolved · 1 Pith anchors

[1] Long , editor = 2001 · doi:10.1007/3-540-44581-1

[2] Inventiones Mathematicae , volume = 2003

[3] IEEE Transactions on Information Theory , volume = 2002

[4] Bartlett and Sanjeev R 1997

[5] International Conference on Computational Learning Theory , pages= 2002

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T02:44:17.009589Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c76b58453af90e394c7589656b345caa1447f6be81b1b38a57220f8da413fb63

Aliases

arxiv: 2605.13684 · arxiv_version: 2605.13684v1 · doi: 10.48550/arxiv.2605.13684 · pith_short_12: Y5VVQRJ27EHD · pith_short_16: Y5VVQRJ27EHDSTDV · pith_short_8: Y5VVQRJ2

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI \
  | 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: c76b58453af90e394c7589656b345caa1447f6be81b1b38a57220f8da413fb63

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "738bd5a200a235f069c4876d6e7662332e05dc641469a3eee3c104736d18e6c8",
    "cross_cats_sorted": [
      "cs.IT",
      "math.IT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-13T15:41:30Z",
    "title_canon_sha256": "a24f49898933fac590f36df73ac706625010550d4db5b3a6713d6d6a14053f7c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13684",
    "kind": "arxiv",
    "version": 1
  }
}