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Fast matrix multiplication saves multiplications through exact cancellation, but rounding sums that mix token rows can leave contributions from later tokens in earlier language model outputs. This threatens prefix invariance, which multiple-choice likelihood scoring relies on: a scored likelihood must depend only on it...
Low-precision training rounds tensors that the backward pass reads again, often for several gradients; each use can read the forward's rounded value, the original, or a new random rounding. This backward-state policy looks like a memory and precision detail, settled by copy accuracy and final loss. We argue that it is...
Algorithm realization becomes an independent low-precision axis with a global $\Phi$ optimum and measured fp8 relevance: an independent low-precision axis with a global $\Phi$ optimum and measured fp8 relevance.
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