Paper Citation Record · LEDGER

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks

As of 20 August 2026, this Paper Citation Record lists 17 of 17 outbound references and 0 inbound Pith citation observations for arXiv:2606.00340.

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

pith.paper-citation-record.v1
2606.00340 v1

Coverage vector

measured 17 of 17 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-07-11T11:50:26.030339Z

measured 17 of 17 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-20T06:33:59.587034+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

17 of 17 outbound references displayed

verified exact0
verified fuzzy0
unresolved13
parse uncertain1
malformed identifier2
metadata mismatch1

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 7e8891af-deba-4bfc-baf5-7898fb9b6ef6 · outbound

This paper cites SGDR: Stochastic Gradient Descent with Warm Restarts.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks SGDR: Stochastic Gradient Descent with Warm Restarts

Reference 1

Resolution: metadata mismatch; local_arxiv, observed 2026-06-28T23:12:46.362276Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation ff8932ad-57e2-43c2-9524-1ac25922e95d · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 2

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 89737936-0ea9-4731-ad59-2faeec2d59bd · outbound

This paper cites We have B0 1 = B0 1 F = q tr(B 0 1 ⊤B0.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks We have B0 1 = B0 1 F = q tr(B 0 1 ⊤B0

Reference 3

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f3b4cd13-bf19-4467-89c1-48235fed66fc · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 4

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a820afe4-d530-4fb9-8e15-81d761b489d1 · outbound

This paper cites Thus, we get that G0 1 −A 0 1 ≤ 1√ h−1 G0 1 , G0 2 −A 0 2 ≤ 1√ h−1 G0 2.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Thus, we get that G0 1 −A 0 1 ≤ 1√ h−1 G0 1 , G0 2 −A 0 2 ≤ 1√ h−1 G0 2

Reference 5

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5661474e-988a-47fe-8dca-21c839ec78b7 · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 6

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3cd8d4e4-e6eb-4447-88e7-476dbe6dad8d · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 7

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 147f80d7-aaf1-40f4-ad7c-36d1e61333de · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 8

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8b084bef-2f4d-4a24-85a5-7d9c65c6f9e1 · outbound

This paper cites Thus, we get that G0 1 −A 0 1 ≤ 1√ h−1 G0 1 , G0 2 −A 0 2 ≤ 1√ h−1 G0 2.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Thus, we get that G0 1 −A 0 1 ≤ 1√ h−1 G0 1 , G0 2 −A 0 2 ≤ 1√ h−1 G0 2

Reference 9

Resolution: malformed identifier; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a054caff-314c-4a33-9e9f-6d6958e51ddb · outbound

This paper cites 1 h W 1 1 W 1 2 −M ⊤ ˜x⊤ 0 ˜x0 1 h W 1 1 W 1 2 −M #! =tr EW 0 1 ,W 0 2 ,ξ, ˜x0,X.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks 1 h W 1 1 W 1 2 −M ⊤ ˜x⊤ 0 ˜x0 1 h W 1 1 W 1 2 −M #! =tr EW 0 1 ,W 0 2 ,ξ, ˜x0,X

Reference 10

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation eb3adf63-e769-4646-986f-563de304716c · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 11

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 86e12ed9-e35f-4cf7-bea7-874ae10dee40 · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 12

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4b6cc828-0838-4808-883f-b69a4f6bdda4 · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 13

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bc77ad06-d0d0-442c-b288-4a346a18d2e2 · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 14

Resolution: parse uncertain; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3b99c05f-cbdb-4eaa-8639-09b97a102ba9 · outbound

This paper cites 1√ h W 1 1 W 1 2 a−β ∗ ⊤ ˜x⊤ 0 ˜x0 1√ h W 1 1 W 1 2 a−β ∗ # =tr EW 0 1 ,W 0 2 ,a,ξ, ˜x0,X.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks 1√ h W 1 1 W 1 2 a−β ∗ ⊤ ˜x⊤ 0 ˜x0 1√ h W 1 1 W 1 2 a−β ∗ # =tr EW 0 1 ,W 0 2 ,a,ξ, ˜x0,X

Reference 15

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b6274469-c609-4974-ae33-fe4b79e0e092 · outbound

This paper cites an unresolved cited work.

Balancing Learning Rates Across Layers: Exact Two-Step Dynamics and Optimal Scaling in Linear Neural Networks Unresolved cited work

Reference 16

Resolution: unresolved; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 50908399-bfef-40ec-ba2d-e617b4f23b44 · outbound

This paper cites 1 h W 1 1 W 1 2 −M ⊤ ˜x⊤ 0 ˜x0 1 h W 1 1 W 1 2 −M #! =tr EW 0 1 ,W 0 2 ,ξ, ˜x0,X.

Reference 17

Resolution: malformed identifier; no resolver link, observed 2026-06-28T23:06:06.894036Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.