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Why does my manual derivative of Layer Normalization imply no gradient flow?

I recently tried computing the derivative of the layer norm function (https://arxiv.org/abs/1607.06450), an essential component of transformers, but the result suggests that no gradient flows through the … \\ \quad=\frac{-y_i}{\sigma}\mu_y\\ \textrm{BUT by properties of data following a standard normal distribution $\mu_y=0$, so}\\ \frac{dy_i}{dX}=\frac{-y_i}{\sigma}0\\ \quad=0\\ \textrm{Which means no gradient
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