Backward-State Policy Is Part of the Learning Algorithm

Shuxiao Xie, Shuyang Xie, Dezhi Ran, Wei Yang, Tao Xie

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 part of the learning algorithm, and that neither check shows whether it is right. Copy accuracy does not decide the outcome: in three pairs of 390M runs with an emulated FP8 backward, training fails when attention's backward reuses the forward's rounded output and succeeds with a new rounding from the same distribution. Even the most accurate copy, the original itself, can be wrong by our reference: the gradient of the forward pass as it actually ran, with gradients passed through rounding unchanged. For example, a normalization output stored in low precision feeds two gradients: the gain's gradient needs the original, but the next layer's weight gradient needs the rounded value that layer multiplied. Final loss, the other check, does not rule out the error of reading the original for both: it persists in models trained with such a store, while planned loss…