Pith Number

pith:IRZQA5WS

pith:2026:IRZQA5WS332PYAELVP6XMRQNEO

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A Quadratic-Approximation-Based Stochastic Approximation Method for Weakly Convex Stochastic Programming

Benqi Liu, Liwei Zhang, Xiantao Xiao, Yule Zhang

PMQSopt integrates proximal multipliers with quadratic approximations to reach O(T^{-1/4}) expected convergence on KKT metrics for weakly convex stochastic programs.

arxiv:2605.03400 v2 · 2026-05-05 · math.OC

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Claims

C1strongest claim

For each of these metrics, we establish an expected convergence rate of O(T^{-1/4}) after T iterations. Furthermore, we show that with probability at least 1-1/T^{2/3}, the gradient of the Lagrangian satisfies an O(T^{-1/8}) bound; with probability at least 1-2/T^{2/3}, the constraint violation achieves an O(T^{-1/4}) bound; and with probability at least 1-3/T^{2/3}, the complementarity violation attains an O(T^{-1/4}) bound.

C2weakest assumption

All results are established under two mild conditions: (i) weak convexity of all problem functions, and (ii) the existence of a strictly feasible point.

C3one line summary

PMQSopt achieves expected O(T^{-1/4}) convergence rates for three epsilon-KKT metrics after T iterations in weakly convex stochastic programming under weak convexity and strict feasibility assumptions.

Receipt and verification

First computed	2026-06-29T01:14:32.771397Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

44730076d2def4fc008babfd76460d239e336100378a6d3e722f4670db71bb75

Aliases

arxiv: 2605.03400 · arxiv_version: 2605.03400v2 · doi: 10.48550/arxiv.2605.03400 · pith_short_12: IRZQA5WS332P · pith_short_16: IRZQA5WS332PYAEL · pith_short_8: IRZQA5WS

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/IRZQA5WS332PYAELVP6XMRQNEO \
  | 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: 44730076d2def4fc008babfd76460d239e336100378a6d3e722f4670db71bb75

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "95b4bb54c30c63ed070050bd0465a424503cf7df016f25aef3088f6c71e416a4",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-05T06:21:12Z",
    "title_canon_sha256": "c07ed026070189a7302cab001d80ceb30e4a1946da94c0c4239408b92a3a2e31"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.03400",
    "kind": "arxiv",
    "version": 2
  }
}