Equation 26 · Dense, Sparse, and Distilled: Comparing Approaches to Frontier Model Capacity
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 b_max
ax is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
Symbol M_device
evice occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol M_weights
eights occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol M_kv
v occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol s
s is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
subtraction
Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.
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.
Denominator: M_kv(s)
The complete quantity below the fraction bar; it must be nonzero for this division.
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
Trace one design decision through that structure to see why the tiers behave as they do. Suppose the middle tier’s key–value cache is halved by moving from multi-head to grouped-query attention. Achievable batch size is roughly . so halving the denominator roughly doubles concurrency at a given context length, which roughly halves cost per request at fixed hardware. The quality cost of that change, per the uptraining results, is small and concentrated on tasks that stress long-range attention [ 8 ] . A buyer sees a price cut. What actually happened is a targeted quality trade whose effects appear on a specific class of input.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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