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> and standard ML theory works over real numbers.

This paper uses binary numbers only, even for training, with a solid theoretical foundation: https://proceedings.neurips.cc/paper_files/paper/2024/file/7...

TL;DR: They invent a concept called "Boolean variation" which is the binary analog to the Newton/Leibniz derivative. They are then able to do backpropagation directly in binary.



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