This is the paper we all cite for Adam. I've done it many times in various papers (mostly unpublished class reports) but haven't actually read it in detail. I know how Adam works at a high level, though.
It's also worth noting that there seems to be some stuff to fix:
Fixing Weight Decay Regularization in Adam, ICLR 2018. https://openreview.net/forum?id=rk6qdGgCZ