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Equation 26 · Part 10 · Dense, Sparse, and Distilled: Comparing Approaches to Frontier Model Capacity

Numerator: M_device - M_weights

bmax⁡≈Mdevice−MweightsMkv(s),b_{\max} \approx \frac{M_{\mathrm{device}} - M_{\mathrm{weights}}}{M_{\mathrm{kv}}(s)},
Mdevice−MweightsM_{\mathrm{device}} - M_{\mathrm{weights}}

What this part means

The complete quantity above the fraction bar.

Its job in the formula

MdM_device - MwM_weights occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

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 bmax⁡≈Mdevice−MweightsMkv(s)b_{\max} \approx \frac{M_{\mathrm{device}} - M_{\mathrm{weights}}}{M_{\mathrm{kv}}(s)}. 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.

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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