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Equation 31 · Part 3 · How a Model Actually Gets Small Enough to Run on a Phone

Symbol hatx

E ⁣[(x−x^)2]≈s212\mathbb{E}\!\left[(x - \hat{x})^2\right] \approx \frac{s^2}{12}
x^\hat{x}

What this part means

the dequantised approximation actually used in arithmetic.

Its job in the formula

hatx is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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 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 E ⁣[(x−x^)2]≈s212\mathbb{E}\!\left[(x - \hat{x})^2\right] \approx \frac{s^2}{12}. so halving the number of bits, which roughly doubles s at fixed range, roughly quadruples the expected squared error per weight. That is why INT8 is usually described as close to free and INT4 is not: the error a network has to absorb does not grow gently as bits are removed, it grows quadratically in the step size, and every bit…

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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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