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To use newer accelerators like this, I need to perform matmul in 4-bit precision. How do I tell whether this operation is stable? Wondering if there well common heuristics in terms of properties of matrices $\{A_i\}$.

For classification tasks, matrix multiplication is stable if $\langle y, y_{\text{4-bit}}\rangle>0$ with high probability for random $x$ and $y=A_d\ldots A_2 A_1 x$.

Orthogonal matrices seem more likely to work in 4-bits, but that's too hard of a restriction.

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    $\begingroup$ What is "4-bit precision"? Are these integers? I'm not aware of 4-bit floating point numbers. $\endgroup$ Feb 21 at 0:57
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    $\begingroup$ That's wild.... $\endgroup$ Feb 21 at 13:18
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    $\begingroup$ @WolfgangBangerth This is what majority of worlds compute will soon be used for (if not already) - super low precision matmuls for AI models $\endgroup$ Feb 21 at 18:45
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    $\begingroup$ Using 4-bit numbers you can either represent 16 non-negative numbers, or 8 positive and 8 non-positive numbers. The moment you multiply two number above 4, it will overflow. Matrix-matrix multiplication would be very unstable with these numbers $\endgroup$ Feb 25 at 23:17
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    $\begingroup$ That is a fairer question. If the input is normalized and convolution kernel is relatively small (or if you are working with finite fields -think of modular arithmetic-) then I would expect it to be reasonably stable. For example, if the input is the output of a logit layer and the convolution is 3x3 (as it is a classic scenario in image classifiers), this quantization would be sufficient. It is a very operator and domain dependent question. $\endgroup$ Feb 26 at 4:40

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