Paper Citation Record · LEDGER

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient

As of 21 August 2026, this Paper Citation Record lists 14 of 14 outbound references and 0 inbound Pith citation observations for arXiv:2602.11557.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2602.11557 v2

Coverage vector

measured 14 of 14 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-03T00:19:06.654629Z

measured 14 of 14 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-21T06:32:19.484+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

14 of 14 outbound references displayed

verified exact0
verified fuzzy0
unresolved14
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 544b21a7-3484-4651-ba80-48e5ecd12f74 · outbound

This paper cites an unresolved cited work.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Unresolved cited work

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:05.953554Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6e37de04-10d4-40e0-b62e-c9f83d498947 · outbound

This paper cites an unresolved cited work.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Unresolved cited work

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:05.735241Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 09573839-9346-4066-a0ab-6a64f0dafa4e · outbound

This paper cites an unresolved cited work.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Unresolved cited work

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.122583Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ade1eb50-cfef-4266-947c-e56e9e640c8d · outbound

This paper cites 20 •Case II: Entry-wise p-norms.Using the consistency of vector and matrix norms: ∥vv⊤∥=∥v∥ 2 p ≤(∥∆∥∥x∥ ∗)2 ≤R 2∥∆∥2.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient 20 •Case II: Entry-wise p-norms.Using the consistency of vector and matrix norms: ∥vv⊤∥=∥v∥ 2 p ≤(∥∆∥∥x∥ ∗)2 ≤R 2∥∆∥2

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:05.827367Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 59d5fd22-cabd-4cd3-8f97-448430b04710 · outbound

This paper cites Since t1 is a constant independent of n (determined only by ρ and curvature), for large n, the LHS is dominated by the sum up to t2.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Since t1 is a constant independent of n (determined only by ρ and curvature), for large n, the LHS is dominated by the sum up to t2

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.043765Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c63a7a71-f76e-4b18-a6f3-23e6efc7a166 · outbound

This paper cites Let Ss = Ps−1 τ=t 2 ητ.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Let Ss = Ps−1 τ=t 2 ητ

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.185300Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2b55e46b-f8b9-438c-89c9-6313ce4f7aa7 · outbound

This paper cites The contribu- tion to the sum is: t1,polyX s=0 ηs =O (ξ1/a)1−a =O ξ 1 a −1 .(Fora= 1 : logξ).

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient The contribu- tion to the sum is: t1,polyX s=0 ηs =O (ξ1/a)1−a =O ξ 1 a −1 .(Fora= 1 : logξ)

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.250556Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3c481eee-c4b2-49d8-8096-a378fb6765cd · outbound

This paper cites The contribution to the sum is: t1,expX s=0 ηs = Θ (log(1/β1))a−1 .(Fora= 1 : log( 1 log(1/β1) )).

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient The contribution to the sum is: t1,expX s=0 ηs = Θ (log(1/β1))a−1 .(Fora= 1 : log( 1 log(1/β1) ))

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.305190Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a01d3777-c756-4bc3-bd08-cbca11b03cea · outbound

This paper cites The order of t0 is given in Eq.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient The order of t0 is given in Eq

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.393864Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e9f37ca2-e709-4136-a693-5ae5a4a538db · outbound

This paper cites Since 1 − 1/a <0, the function y1−1/a logy is bounded on [1 ,∞ ), and hence there exists a constant Ca >0 such that L−1 log(1/L)≤C a L−1/a,∀L∈(0,1].

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Since 1 − 1/a <0, the function y1−1/a logy is bounded on [1 ,∞ ), and hence there exists a constant Ca >0 such that L−1 log(1/L)≤C a L−1/a,∀L∈(0,1]

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.485493Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2fec5291-c55f-43b0-8bdf-d2050ece7131 · outbound

This paper cites Then, the accumulated error in epochris bounded by: ∥Er∥F ≤ nX k=1 |ηrn+k−1 −η rn|∥Msign σr(k)∥F ≤n· can (rn)a+1 CM = caCM n2 (rn)a+1.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Then, the accumulated error in epochris bounded by: ∥Er∥F ≤ nX k=1 |ηrn+k−1 −η rn|∥Msign σr(k)∥F ≤n· can (rn)a+1 CM = caCM n2 (rn)a+1

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.574661Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 45361c72-a9ea-47a8-bcb2-65e5780b8ba2 · outbound

This paper cites 61 Proof.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient 61 Proof

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:06.654629Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation de8bbbba-21aa-46eb-9ad5-cb83fd061767 · outbound

This paper cites Muon is Scalable for LLM Training.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient Muon is Scalable for LLM Training

Reference 2014

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:05.635066Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 514e274e-7f32-4e7e-8f2f-7b8658b0027b · outbound

This paper cites On the Implicit Bias of Adam.

The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient On the Implicit Bias of Adam

Reference 2025

Resolution: unresolved; no resolver link, observed 2026-08-03T00:19:05.549126Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.