Pith Number

pith:LCPWCFYM

pith:2026:LCPWCFYMFSWJOTK3N5IK5SFCBU

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Stochastic Mirror Descent under Iterate-Dependent Markov Noise: Analysis in the Asymptotic and Finite Time Regimes

Anik Kumar Paul, Shalabh Bhatnagar

Stochastic mirror descent converges almost surely under iterate-dependent Markov noise for both convex and non-convex problems.

arxiv:2605.15538 v1 · 2026-05-15 · math.OC · cs.SY · eess.SY

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Claims

C1strongest claim

We first establish almost sure convergence for both convex and non-convex problems under the mild assumption of Lipschitz continuity of the objective function, without requiring differentiability. We then derive finite-time concentration bounds for smooth objectives. In the convex setting, the resulting sample complexity matches the classical rate of stochastic mirror descent under i.i.d. noise.

C2weakest assumption

The Markov chain generated by the iterate-dependent sampling distribution satisfies sufficient mixing or ergodicity conditions that allow the bias and temporal dependence to be controlled; this property is invoked to justify the almost-sure convergence but is not stated explicitly in the abstract.

C3one line summary

Proves almost sure convergence and finite-time sample complexity bounds for stochastic mirror descent under iterate-dependent Markov noise for both convex and non-convex objectives.

References

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[1] A. AGARWAL, S. M. KAKADE, J. D. LEE,ANDG. MAHAJAN,On the theory of policy gradient methods: Optimality, approximation, and distribution shift, Journal of Machine Learning Research, 22 (2021), pp. 1–76 2021

[2] V. S. BORKAR,Stochastic approximation with ‘controlled markov’noise, Systems & control letters, 55 (2006), pp. 139–145 2006

[3] BOUMAL,An Introduction to Optimization on Smooth Manifolds, Cambridge University Press, 2023 2023

[4] BUBECK,Introduction to Online Optimization, Lecture notes, Princeton University, 2012 2012

[5] E. CHE, J. DONG,ANDX. T. TONG,Stochastic gradient descent with adaptive data, Operations Research, (2026) 2026

Formal links

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Receipt and verification

First computed	2026-05-20T00:01:04.165630Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

589f61170c2cac974d5b6f50aec8a20d133365bebeda45a712f2ec62217aa58b

Aliases

arxiv: 2605.15538 · arxiv_version: 2605.15538v1 · doi: 10.48550/arxiv.2605.15538 · pith_short_12: LCPWCFYMFSWJ · pith_short_16: LCPWCFYMFSWJOTK3 · pith_short_8: LCPWCFYM

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LCPWCFYMFSWJOTK3N5IK5SFCBU \
  | 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: 589f61170c2cac974d5b6f50aec8a20d133365bebeda45a712f2ec62217aa58b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a5f02d6570eff0977d4f0b36095950c65fbb18585a93a126fee1ccb2a51900db",
    "cross_cats_sorted": [
      "cs.SY",
      "eess.SY"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-15T02:19:06Z",
    "title_canon_sha256": "96621135131e6eeca8fdf69e3579ccaa2aed421dbe318c36a1753dc0674417e6"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15538",
    "kind": "arxiv",
    "version": 1
  }
}