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pith:6NXGUGCS

pith:2026:6NXGUGCSA35FUSGJFWK7ULPXGQ

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Beyond Bounded Variance: Variance-Reduced Normalized Methods for Nonconvex Optimization under Blum-Gladyshev Noise

Abolfazl Hashemi, Antesh Upadhyay, Arda Fazla

Normalized stochastic gradient descent with momentum converges under BG-0 noise with O(ε^{-6}) oracle complexity using one gradient per step.

arxiv:2605.15314 v1 · 2026-05-14 · cs.LG · math.OC

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Claims

C1strongest claim

We prove that normalized stochastic gradient descent with momentum, using only one stochastic gradient per iteration, converges under BG-0 noise with oracle complexity O(ε^{-6}). This rate holds both for standard smoothness and for α-symmetric generalized smoothness.

C2weakest assumption

The stochastic gradients satisfy the Blum-Gladyshev (BG-0) noise model in which the variance grows quadratically with the distance from the initialization point (stated in the problem setup and used throughout the convergence analysis).

C3one line summary

Normalized momentum SGD and variance-reduced STORM achieve O(ε^{-6}) and O(ε^{-4}) oracle complexities respectively under quadratic distance-dependent noise in nonconvex stochastic optimization.

References

46 extracted · 46 resolved · 1 Pith anchors

[1] Towards weaker variance assumptions for stochastic optimization.arXiv preprint arXiv:2504.09951, 2025 2025

[2] Lower bounds for non-convex stochastic optimization.Mathematical Programming, 199(1):165–214, 2023 2023

[3] Non-asymptotic analysis of stochastic approximation algorithms for machine learning 2011

[4] Approximation methods which converge with probability one.The Annals of Mathematical Statistics, pages 382–386, 1954 1954

[5] Optimization methods for large-scale learning 2018

Formal links

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Receipt and verification

First computed	2026-05-20T00:00:52.215898Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f36e6a185206fa5a48c92d95fa2df7341bed05422255e4acbb72b2772686cdfb

Aliases

arxiv: 2605.15314 · arxiv_version: 2605.15314v1 · doi: 10.48550/arxiv.2605.15314 · pith_short_12: 6NXGUGCSA35F · pith_short_16: 6NXGUGCSA35FUSGJ · pith_short_8: 6NXGUGCS

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/6NXGUGCSA35FUSGJFWK7ULPXGQ \
  | 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: f36e6a185206fa5a48c92d95fa2df7341bed05422255e4acbb72b2772686cdfb

Canonical record JSON

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    "abstract_canon_sha256": "8700fb82324411455c599c030787a900ff5fa50281c64229e7b4db6ac98b0b61",
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      "math.OC"
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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-14T18:27:49Z",
    "title_canon_sha256": "3c0127a2c571a8529af0bc48ed5b4c03a7fcfa562d9a39c424e731dc371f20ed"
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