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Scaling Laws for Neural Language Models

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Critical Batch Size and Training Dynamics

Critical Batch Size Power Law

Critical Batch Size Power Law

Source equation

Bcrit(L)=B∗L1/αB,B∗∼2⋅108 tokens,  αB∼0.21B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21

The critical batch size Bcrit(L)B_{\rm crit}(L) determines the optimal batch size for training neural language models. It represents the threshold where further increasing the batch size yields diminishing returns in terms of training speedup per step. This relationship is modeled as a power law of the cross-entropy loss LL.

Sources

S1.E4

Bcrit​(L)=B∗L1/αB,B∗∼2⋅108​ tokens,αB∼0.21formulae-sequencesubscript𝐵crit𝐿subscript𝐵∗superscript𝐿1subscript𝛼𝐵formulae-sequencesimilar-tosubscript𝐵∗⋅2superscript108 tokenssimilar-tosubscript𝛼𝐵0.21B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21 (1.4)
B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21

Bcrit(L)=B∗L1/αB,B∗∼2⋅108 tokens,  αB∼0.21B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21

Sources

S1.E4

Bcrit​(L)=B∗L1/αB,B∗∼2⋅108​ tokens,αB∼0.21formulae-sequencesubscript𝐵crit𝐿subscript𝐵∗superscript𝐿1subscript𝛼𝐵formulae-sequencesimilar-tosubscript𝐵∗⋅2superscript108 tokenssimilar-tosubscript𝛼𝐵0.21B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21 (1.4)
B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21

Illustrative Calculation

Let's calculate the critical batch size Bcrit(L)B_{\rm crit}(L) for a target cross-entropy loss of L=2.0L = 2.0:

  1. Given Parameters:

    • B∗=2⋅108B_{\ast} = 2 \cdot 10^8 tokens
    • αB=0.21\alpha_B = 0.21
    • L=2.0L = 2.0

  2. Compute the Exponent:

    • 1/αB=1/0.21≈4.76191 / \alpha_B = 1 / 0.21 \approx 4.7619

  3. Compute the Denominator:

    • L1/αB=2.04.7619≈27.135L^{1/\alpha_B} = 2.0^{4.7619} \approx 27.135

  4. Calculate BcritB_{\rm crit}:

    • Bcrit(2.0)=2⋅10827.135≈7.37⋅106B_{\rm crit}(2.0) = \frac{2 \cdot 10^8}{27.135} \approx 7.37 \cdot 10^6 tokens
Sources

S1.E4

Bcrit​(L)=B∗L1/αB,B∗∼2⋅108​ tokens,αB∼0.21formulae-sequencesubscript𝐵crit𝐿subscript𝐵∗superscript𝐿1subscript𝛼𝐵formulae-sequencesimilar-tosubscript𝐵∗⋅2superscript108 tokenssimilar-tosubscript𝛼𝐵0.21B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21 (1.4)
B_{\rm crit}\left(L\right)=\frac{B_{\ast}}{L^{1/\alpha_{B}}},\qquad B_{\ast}\sim 2\cdot 10^{8}\text{ tokens},\ \ \alpha_{B}\sim 0.21
Bcrit(L)B_{\rm crit}(L)
Critical batch size measured in tokens · scalar
LL
Cross-entropy loss value · scalar
B∗B_{\ast}
Scale parameter for the critical batch size · scalar
αB\alpha_{B}
Power-law scaling exponent · scalar