Paper Citation Record · LEDGER

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size

As of 15 August 2026, this Paper Citation Record lists 38 of 38 outbound references and 3 inbound Pith citation observations for arXiv:2506.15025.

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

pith.paper-citation-record.v1
2506.15025 v1

Coverage vector

measured 38 of 38 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-15T19:54:42.614806Z

measured 41 of 41 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-15T06:32:42.880941+00:00

measured 3 of 3 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-06-30T17:31:32.533941Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-06-30T17:34:57.593618Z

Reference resolution

38 of 38 outbound references displayed

verified exact0
verified fuzzy17
unresolved21
parse uncertain0
malformed identifier0
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External citation measurements

No source-named external measurement is stored.

Outbound references

Observation b3f8b846-6e14-4517-be05-f687fba975d8 · outbound

This paper cites u-$\mu$P: The Unit-Scaled Maximal Update Parametrization.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size u-$\mu$P: The Unit-Scaled Maximal Update Parametrization

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.444889Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b4fc21cb-ea38-4af4-afab-63a2f63c7b5e · outbound

This paper cites Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.450340Z

Source-reported events for the cited work

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Observation 4e9517dd-5fdd-4a8e-91e5-5d50efc79d17 · outbound

This paper cites On Lazy Training in Differentiable Programming.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size On Lazy Training in Differentiable Programming

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.454722Z

Source-reported events for the cited work

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Observation 2e495ec2-61aa-4a99-b5c0-c7a317109251 · outbound

This paper cites Infinite- width limit of deep linear neural networks.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Infinite- width limit of deep linear neural networks

Reference 4

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Source-reported events for the cited work

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Observation 52002095-c746-4854-844c-d90658e74767 · outbound

This paper cites Alemi, Roman Novak, Peter J.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Alemi, Roman Novak, Peter J

Reference 5

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Observation 82f197d9-4eb3-444a-83c2-647fe2f7f6c1 · outbound

This paper cites On the infinite-depth limit of finite-width neural networks.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size On the infinite-depth limit of finite-width neural networks

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:43.115696Z

Source-reported events for the cited work

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Observation 87e1e4de-8cd6-4997-a736-fe3820039dec · outbound

This paper cites On the impact of the ac- tivation function on deep neural networks training.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size On the impact of the ac- tivation function on deep neural networks training

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:43.102018Z

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Observation 2f285087-ba6c-4b17-946f-12894b8c479c · outbound

This paper cites Stable resnet.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Stable resnet

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:43.087750Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation b5cf8226-42e4-4f65-8ba2-6adf96c288c4 · outbound

This paper cites Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification

Reference 9

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Source-reported events for the cited work

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Observation fd4661cb-4347-492a-aca5-915c0d87ec2a · outbound

This paper cites Neural Tangent Kernel: Convergence and Generalization in Neural Networks.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Neural Tangent Kernel: Convergence and Generalization in Neural Networks

Reference 10

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Source-reported events for the cited work

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Observation 590b4620-eaa3-47ac-be14-15a5c3c5747e · outbound

This paper cites Muon: An optimizer for hidden layers in neural networks,.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Muon: An optimizer for hidden layers in neural networks,

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:43.070446Z

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No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 0f979c3d-4478-4ecb-953e-6ca8318b1ec3 · outbound

This paper cites Kingma and Jimmy Ba.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Kingma and Jimmy Ba

Reference 12

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Source-reported events for the cited work

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Observation d8abe900-b9de-495f-bf1e-7662454d1e7f · outbound

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Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Unresolved cited work

Reference 13

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Source-reported events for the cited work

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Observation 3075e622-a266-4e46-b146-b163ba8bfff7 · outbound

This paper cites The llama 3 herd of models, 2024.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size The llama 3 herd of models, 2024

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.516190Z

Source-reported events for the cited work

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Observation 230ffc16-b8eb-421f-8d85-49e6a96e7ec7 · outbound

This paper cites Pointer sen- tinel mixture models, 2016.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Pointer sen- tinel mixture models, 2016

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:43.028154Z

Source-reported events for the cited work

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Observation 4976277b-3b6d-46c9-ada3-c88d92d87a5f · outbound

This paper cites An Empirical Study of $\mu$P Learning Rate Transfer.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size An Empirical Study of $\mu$P Learning Rate Transfer

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.511780Z

Source-reported events for the cited work

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Observation a6df0e87-6b3a-459e-9930-6d91c199d3ae · outbound

This paper cites Poole, S.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Poole, S

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:43.015329Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 07b7ee97-7df3-4b56-9441-1ac76142e1c4 · outbound

This paper cites Schoenholz, J.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Schoenholz, J

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:43.001690Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 19eda179-91e6-419b-b004-cbde3f0c47a5 · outbound

This paper cites an unresolved cited work.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Unresolved cited work

Reference 19

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Source-reported events for the cited work

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Observation 60dcbd05-82f5-42d8-827a-7668986fbb75 · outbound

This paper cites The Falcon Series of Open Language Models.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size The Falcon Series of Open Language Models

Reference 20

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Source-reported events for the cited work

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Observation 2d3fc3b8-693c-4e3e-9bc2-7b06c8d81e88 · outbound

This paper cites Gemma 3 technical report, 2025.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Gemma 3 technical report, 2025

Reference 21

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Source-reported events for the cited work

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Observation f736ab14-e453-4cf0-811d-2a28c79e8fe8 · outbound

This paper cites Scaling Laws with Vocabulary: Larger Models Deserve Larger Vocabularies.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Scaling Laws with Vocabulary: Larger Models Deserve Larger Vocabularies

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.539112Z

Source-reported events for the cited work

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Observation ea63cfd8-d429-48e1-af50-f8099316e3ec · outbound

This paper cites Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.557483Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9469a6ce-f8dc-40e0-b5ca-7656daaba731 · outbound

This paper cites Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.562346Z

Source-reported events for the cited work

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Observation ec43db71-939e-494a-b99c-f35d63d9c7a3 · outbound

This paper cites Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone

Reference 25

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.552969Z

Source-reported events for the cited work

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Observation 1d57d817-22eb-4cb8-a62c-a0f2eb0de283 · outbound

This paper cites How does critical batch size scale in pre-training?,.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size How does critical batch size scale in pre-training?,

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.981917Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 23af0345-ba61-4677-a09e-a194ec2b363b · outbound

This paper cites Selected Studies of the Principle of Relative Frequency in Language.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Selected Studies of the Principle of Relative Frequency in Language

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.969342Z

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No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 301056cd-5a09-4409-b131-1c21484cc905 · outbound

This paper cites Tensor Programs VI: Feature Learning in Infinite-Depth Neural Networks.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Tensor Programs VI: Feature Learning in Infinite-Depth Neural Networks

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.567261Z

Source-reported events for the cited work

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Observation 69665084-8ddd-457a-aa2c-ee73b03ebf63 · outbound

This paper cites Because Wj∼N (0,Im) and Mj =⟨v,Wj⟩, for anyi∈ [m] the pair (Mj,Wji) is jointly Gaussian with correlationρ := vi ∥v∥.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Because Wj∼N (0,Im) and Mj =⟨v,Wj⟩, for anyi∈ [m] the pair (Mj,Wji) is jointly Gaussian with correlationρ := vi ∥v∥

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.956995Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation f171a23a-e471-4f34-8e9d-ba48be49c6c9 · outbound

This paper cites Still conditional on v, sign(Mj) is±1 with equal probability, independent of the magnitude of Wj.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Still conditional on v, sign(Mj) is±1 with equal probability, independent of the magnitude of Wj

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.943251Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation a4358c98-7dd9-49d2-b588-4b01e82d6cf0 · outbound

This paper cites Because the Yj’s are conditionally independent, Ev Cov(X|v) = Ev h dX j=1 Cov(Yj|v) i =d Im− Ev µ(v)µ(v)⊤ | {z } = 2 πmIm = d− 2d πm Im.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Because the Yj’s are conditionally independent, Ev Cov(X|v) = Ev h dX j=1 Cov(Yj|v) i =d Im− Ev µ(v)µ(v)⊤ | {z } = 2 πmIm = d− 2d πm Im

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.930270Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 31f911b0-7376-4e10-a45a-49d7fdd94672 · outbound

This paper cites From Step1, E[X|v] =dµ (v), so Covv E[X|v] =d2 Covv µ(v) =d2 2 π Covv v ∥v∥ = 2d2 πmIm.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size From Step1, E[X|v] =dµ (v), so Covv E[X|v] =d2 Covv µ(v) =d2 2 π Covv v ∥v∥ = 2d2 πmIm

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.917670Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 99f7d6eb-a9a8-4b1c-8f78-015dd947a521 · outbound

This paper cites Conditioning on E, each entry of E⊤M a centered Gaussian variable, hence E[S(E⊤M)|E] = 0.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Conditioning on E, each entry of E⊤M a centered Gaussian variable, hence E[S(E⊤M)|E] = 0

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.903671Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation e1760d3f-2b56-4eaa-be4c-e3aca666982c · outbound

This paper cites Fix a column index k.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Fix a column index k

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.888341Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation ef577664-d8da-4277-9f3e-194bb3efff91 · outbound

This paper cites Different columns ofM (differentk) are independent, so Cov(X) is diagonal and each coordinate variance is the same as above.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Different columns ofM (differentk) are independent, so Cov(X) is diagonal and each coordinate variance is the same as above

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T19:54:42.871604Z

Source-reported events for the cited work

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Observation 52994ea5-ed58-4778-9f43-0c7fd4e17cd0 · outbound

This paper cites Adam: A Method for Stochastic Optimization.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Adam: A Method for Stochastic Optimization

Reference 2017

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.507645Z

Source-reported events for the cited work

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Observation 3c8dd48f-a720-4c59-839c-43c18097175c · outbound

This paper cites Scaling Exponents Across Parameterizations and Optimizers.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size Scaling Exponents Across Parameterizations and Optimizers

Reference 2024

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.469149Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 08d43654-9d72-4310-8e60-28d09a1f0868 · outbound

This paper cites How Does Critical Batch Size Scale in Pre-training?.

Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size How Does Critical Batch Size Scale in Pre-training?

Reference 2025

Resolution: unresolved; no resolver link, observed 2026-08-15T19:54:42.575268Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

Observation b12fee4f-c0fe-4629-9cfd-5c377a29cff1 · inbound

Quantifying Hyperparameter Transfer and the Importance of Embedding Layer Learning Rate cites this paper.

Quantifying Hyperparameter Transfer and the Importance of Embedding Layer Learning Rate Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size

Reference 17

Resolution: verified exact; arxiv_id, observed 2026-05-21T05:03:57.956920Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 00fd0d3b-1820-4c62-ad19-72b38dc51853 · inbound

One LR Doesn't Fit All: Heavy-Tail Guided Layerwise Learning Rates for LLMs cites this paper.

One LR Doesn't Fit All: Heavy-Tail Guided Layerwise Learning Rates for LLMs Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size

Reference 3

Resolution: verified exact; arxiv_id, observed 2026-05-22T07:44:42.593237Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation bc462715-4679-4150-bef0-3cee690c782f · inbound

One LR Doesn't Fit All: Heavy-Tail Guided Layerwise Learning Rates for LLMs cites this paper.

One LR Doesn't Fit All: Heavy-Tail Guided Layerwise Learning Rates for LLMs Optimal Embedding Learning Rate in LLMs: The Effect of Vocabulary Size

Reference 3

Resolution: verified exact; arxiv_id, observed 2026-06-30T17:34:57.595450Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.