Equation 6 · The Hardest Unsolved Problems in Small and On-Device AI

What does this equation mean?

\Delta_{\max} = \max_{t}\left(\mathrm{Acc}_t^{FP} - \mathrm{Acc}_t^{Q}\right).

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Inputs and operationsmax_t(Acc_t^FP - Acc_t^Q)

Result or conditionDelta_max

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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\Delta_{\max}

Symbol Delta_max

Delt $a_m$ ax is part of the quantity the equation computes from the expression on the right.

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t

Symbol t

t is one of the signed contributions combined to compute the quantity on the left.

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t^{FP}

Symbol t^FP

$t^F$ P is one of the signed contributions combined to compute the quantity on the left.

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t^{Q}

Symbol t^Q

$t^Q$ is one of the signed contributions combined to compute the quantity on the left.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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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.

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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.

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How to interpret it

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What the article says around this equation

while what governs whether any individual deployment is safe to ship is closer to the worst-case regression $\Delta_{\max} = \max_{t}\left(\mathrm{Acc}_t^{FP} - \mathrm{Acc}_t^{Q}\right)$ . A quantization scheme can post a $\bar{\Delta}$ close to zero while $\Delta_{\max}$ is large, provided the loss concentrates on one or a few tasks that are a small share of the suite. Nothing about $\bar{\Delta}$ being small implies $\Delta_{\max}$ is small; the two only converge if degradation is spread evenly, and the studies below find that it is not.

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Sources cited in the article section

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