To fix variance computation for complex Adam (#62946)
Summary:
Fixes https://github.com/pytorch/pytorch/issues/59998
It has been discussed in the issue that the variance term of Adam optimizer currently doesn't compute correctly for complex domain. As it has been stated in the Generalization to Complex numbers section in https://en.wikipedia.org/wiki/Variance variance is computed as E[(X - mu)(X-mu)*] (where mu = E[X] and * stands for conjugate) for complex random variable X.
However, currently the computation method in implementation of Adam is via E[(X - mu)(X-mu)] which doesn't return right variance value, in particular it returns complex number. Variance is defined to be real number even though underlying random variable is complex.
We fix this issue here, and testing that resulting variance is indeed real number.
Pull Request resolved: https://github.com/pytorch/pytorch/pull/62946
Reviewed By: albanD
Differential Revision: D30196038
Pulled By: iramazanli
fbshipit-source-id: ab0a6f31658aeb56bdcb211ff86eaa29f3f0d718
