Add normalization to HessianDiagonalAggregator

[ashelkovnykov]

From a comment by @XianXing on PR #131:

"The computation and approximation for Hessian diagonal was designed and implemented independent of Photon's normalization, because previously we didn't enable normalization for GLMix.

But it will be helpful if we take the normalization into account as well here for Hessian diagonal, in order to make it consistent with the rest of code."

[fastier-li]

Assigning to myself as I am working on normalization right now.

@XianXing - can you review this PR, and let me know if it is actually incorrect from an algorithm point of view, if we don't normalize the Hessian diagonal when normalization is turned on?

[XianXing]

The PR looks good to me in general, but I haven't got a chance to look at it carefully.

Can you have other folks to take a look at the PR as well, for example @ankans or Ajith, so that you guys get the help needed and won't get blocked?

[fastier-li]

Yes, we are having it reviewed by Ankan too. What it does is only setup normalization contexts in GAME and then call the same machinery as photon. If it was correct in photon after the normalization contexts were set, it should be correct in GAME, that hasn't changed at all. What happens to the Hessian diagonal in photon, in case of normalization?

[fastier-li]

@XianXing - what is the real impact of not having this? Is it correctness? Is it performance? Is it required to compute variances correctly?

[XianXing]

Correctness. The impact would be downstream applications. For example, I don't know if we are using the estimated variance for multi-armed bandit and explore-exploit tradeoffs at LNKD or not, but based on our experience the quality of estimated variance would have direct impact on those applications (can't be more specific than this without getting into trouble :) ).

Having said that, if there are no such use cases at LNKD, please feel free to de-prioritize it.

@pogil @beechung

[fastier-li]

So, it is correctness of variances, only. The correctness of coefficient means themselves does not depend on that, right?