Pith Number

pith:6S4RW55B

pith:2026:6S4RW55B5YXNRQZUCML4XRYCP3

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A Single-Loop Penalty-based Algorithm for Stochastic Minimax Optimization with Nonlinear Coupled Constraints

Jin Zhang, Qichao Cao, Shangzhi Zeng, Yuxuan Zhou

SPACO is a single-loop stochastic gradient algorithm that solves nonconvex-concave minimax problems with nonlinear convex coupled constraints via penalty-based smoothing.

arxiv:2605.01246 v2 · 2026-05-02 · math.OC

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\usepackage{pith}
\pithnumber{6S4RW55B5YXNRQZUCML4XRYCP3}

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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 propose SPACO, a single-loop stochastic gradient algorithm built upon a penalty-based smooth approximation framework for MCC, and establish non-asymptotic complexity bounds and asymptotic stationarity of accumulation points.

C2weakest assumption

The quadratic penalty scheme with regularization produces a continuously differentiable approximation of the MCC problem while preserving enough structure for the nonconvex-concave convergence analysis to apply.

C3one line summary

SPACO is a new single-loop stochastic algorithm for stochastic nonconvex-concave minimax problems with nonlinear convex coupled constraints that uses penalty smoothing and provides non-asymptotic complexity bounds plus stationarity analysis.

Receipt and verification

First computed	2026-06-08T01:04:06.372102Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f4b91b77a1ee2ed8c3341317cbc7027ec359fcd54fd44fba11a1d9f1f5c181e9

Aliases

arxiv: 2605.01246 · arxiv_version: 2605.01246v2 · doi: 10.48550/arxiv.2605.01246 · pith_short_12: 6S4RW55B5YXN · pith_short_16: 6S4RW55B5YXNRQZU · pith_short_8: 6S4RW55B

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/6S4RW55B5YXNRQZUCML4XRYCP3 \
  | 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: f4b91b77a1ee2ed8c3341317cbc7027ec359fcd54fd44fba11a1d9f1f5c181e9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e29ad2a0206a5bed6f1c315bd9ba7a47aa41e6ce1917ac35fa09f7071013a7b7",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-02T04:57:02Z",
    "title_canon_sha256": "a5d153bdbe82aa7e975418aeb0621d08d90f6438be4f0b8b4fc4c0b2c43000ee"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.01246",
    "kind": "arxiv",
    "version": 2
  }
}