Equation 8 · OpenAI Model Systems from First Principles: Weights, Post-Training, and Inference Compute
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol C_pre
re is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
Symbol N
N is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
Symbol D
D is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
How to interpret it
Its accuracy depends on the assumptions and range of use described in the article.
What the article says around this equation
Pretraining compute for a dense transformer is well approximated by . with N parameters and D training tokens, the factor of six counting the forward and backward passes. Empirically, test loss falls as a power law in each of parameters, data, and compute over many orders of magnitude, a relationship first characterised systematically by Kaplan and colleagues [ 5 ] . The important structural feature is the functional form: a term of the shape
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