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Equation 26 · Part 1 · How a Model Actually Gets Small Enough to Run on a Phone

Symbol s

s=max⁡(∣x∣)2 b−1−1s = \frac{\max(|x|)}{2^{\,b-1} - 1}
ss

What this part means

the scale.

Its job in the formula

s is the quantity selected or evaluated by the optimization written on the right.

Where the article explains it

The standard affine mapping takes a real value x , a scale s and a zero-point z , and produces an integer xqx_q = clip\mathrm{clip}(⌊xs⌉+z, qmin⁡, qmax⁡)\left(\left\lfloor \frac{x}{s} \right\rceil + z,\ q_{\min},\ q_{\max}\right), \qquad x^\hat{x} = s(xq−z)\left(x_q - z\right) where ⌊\lfloor ⋅\cdot ⌉\rceil is round-to-nearest and x^\hat{x} is the dequantised approximation actually used in arithmetic.

The passage around this formula

For weights, a common simplification is symmetric quantization around zero, with z=0 and the scale set directly by the largest magnitude present: s=max⁡(∣x∣)2 b−1−1s = \frac{\max(|x|)}{2^{\,b-1} - 1}. for a signed integer of b bits. An 8-bit integer has 2^8=256 representable levels; a 4-bit integer has 2^4=16 . That sixteen-fold reduction in available codes is the entire cost of quantization in one number. Treating the rounding error as approximately uniform over one quantization step s , its expected squared magnitude is the classical result

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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