Pith Number

pith:IYHWGC3F

pith:2026:IYHWGC3FVITNJHE2HKKPOXTFX2

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Convergence Rate Analysis of the AdamW-Style Shampoo: Unifying One-Sided and Two-Sided Preconditioning

Huan Li, Yiming Dong, Zhouchen Lin

AdamW-style Shampoo unifies one-sided and two-sided preconditioning to achieve an SGD-like convergence rate in nuclear norm.

arxiv:2601.07326 v3 · 2026-01-12 · math.OC · cs.LG

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Claims

C1strongest claim

Our analysis unifies one-sided and two-sided preconditioning and establishes the convergence rate 1/K sum_{k=1}^K E[||nabla f(X_k)||_*] <= O(sqrt(m+n) C / K^{1/4}) measured by nuclear norm.

C2weakest assumption

The derivation relies on standard stochastic-optimization assumptions (smoothness, bounded variance or similar) together with the specific form of the AdamW-style preconditioner; the abstract does not list the precise conditions under which the nuclear-norm bound holds.

C3one line summary

AdamW-style Shampoo unifies one- and two-sided preconditioning and achieves convergence rate O(sqrt(m+n) C / K^{1/4}) in nuclear norm, analogous to SGD's optimal rate when nuclear norm scales as Theta(sqrt(m+n)) times Frobenius norm.

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Cited by

2 papers in Pith

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers

Convergence Rate Analysis of SOAP with Arbitrary Orthogonal Projection Matrices

Receipt and verification

First computed	2026-05-26T02:04:02.946717Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

460f630b65aa26d49c9a3a94f75e65bebbd95699c8da782d92748a3c0e6658d6

Aliases

arxiv: 2601.07326 · arxiv_version: 2601.07326v3 · doi: 10.48550/arxiv.2601.07326 · pith_short_12: IYHWGC3FVITN · pith_short_16: IYHWGC3FVITNJHE2 · pith_short_8: IYHWGC3F

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/IYHWGC3FVITNJHE2HKKPOXTFX2 \
  | 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: 460f630b65aa26d49c9a3a94f75e65bebbd95699c8da782d92748a3c0e6658d6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "acf196fa014f2c358fe7b80ca3cdbbfc37e6d17a27e0372a5eb3f25ab39b88c6",
    "cross_cats_sorted": [
      "cs.LG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-01-12T08:51:03Z",
    "title_canon_sha256": "69c3092ca2f2c829965d7a48a0aa50ef530a4f53d345f7b0a9bedd43b5260bb4"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.07326",
    "kind": "arxiv",
    "version": 3
  }
}