Equation 11 · 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 L
L 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 N_c
occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol alpha_N
alph 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.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article.
What the article says around this equation
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 . has an irreducible floor and diminishing returns above it. Doubling N buys a fixed decrement in loss, not a fixed multiple of capability.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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