Pith Number

pith:ZS2SKAOO

pith:2026:ZS2SKAOOPR7PRYBOJLEYQNVX2I

not attested not anchored not stored refs pending

Zeroth-Order Optimization at the Edge of Stability

Bingcong Li, Liang Zhang, Michael Muehlebach, Minhak Song, Niao He, Sewoong Oh

Zeroth-order methods remain mean-square stable only when their step size satisfies a bound that depends on the full Hessian spectrum rather than its largest eigenvalue alone.

arxiv:2604.14669 v2 · 2026-04-16 · cs.LG · math.DS · math.OC · stat.ML

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\pithnumber{ZS2SKAOOPR7PRYBOJLEYQNVX2I}

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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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

we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator... mean-square stability of ZO methods depends on the entire Hessian spectrum.

C2weakest assumption

The linear stability analysis assumes the loss can be locally approximated by a quadratic form whose Hessian is constant over the relevant trajectory, and that the two-point estimator is used without additional smoothing or variance reduction.

C3one line summary

Zeroth-order methods achieve mean-square stability when the step size satisfies a condition involving the entire Hessian spectrum, with full-batch ZO optimizers operating at the edge of stability and large steps regularizing the Hessian trace.

Receipt and verification

First computed	2026-07-02T01:17:31.259570Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ccb52501ce7c7ef8e02e4ac98836b7d209df2144765ef2366c12dc9fb3ae21d1

Aliases

arxiv: 2604.14669 · arxiv_version: 2604.14669v2 · doi: 10.48550/arxiv.2604.14669 · pith_short_12: ZS2SKAOOPR7P · pith_short_16: ZS2SKAOOPR7PRYBO · pith_short_8: ZS2SKAOO

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZS2SKAOOPR7PRYBOJLEYQNVX2I \
  | 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: ccb52501ce7c7ef8e02e4ac98836b7d209df2144765ef2366c12dc9fb3ae21d1

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "1e3b5b5f0d85cc6d5838d159f8430a92acd19e59916cfe0e1b08dc4e9ac8fc9f",
    "cross_cats_sorted": [
      "math.DS",
      "math.OC",
      "stat.ML"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-04-16T06:23:18Z",
    "title_canon_sha256": "579a22f9f3d9dd0870be92dac9c81190f376974fd9f771bf37ac8126983d8a4f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.14669",
    "kind": "arxiv",
    "version": 2
  }
}