[python-package] allow custom weighing in fobj for scikit-learn API (closes #5027)

[jmoralez]

This removes the automatic weighing for grad and hess when using a custom objective in the scikit-learn API and makes the signature equivalent to the custom eval (y_true, y_pred, weight, group).

Also adds tests to check that the behavior using the built-in objectives and the custom ones is consistent, i.e. if we give the weights to the dataset/fit using the built-in objectives we get the same results as if we use custom objectives and do the weighing ourselves.

Closes #5027.

[StrikerRUS]

Thank you so much for picking up this issue!
Please consider checking some my minor comments below:

[StrikerRUS]

Nice! Thank you very much!

[StrikerRUS]

@shiyu1994 Would you like to take a look at this PR as you were a creator of #5027?

[StrikerRUS]

@shiyu1994

[StrikerRUS]

@shiyu1994

[StrikerRUS]

@jameslamb Please help with a second review.

[jameslamb]

Thanks for this! I believe I understand the changes, prior discussion on this PR, and previous discussion in #5207.

It's unfortunate that we have to make the breaking change where providing a callable that takes 3 arguments will now be treated as (y_true, y_pred, weight) when previous releases treated it as (y_true, y_pred, group), since the values in weight and group can be similar and therefore code that worked in lightgbm v3.3.2 could silently start producing incorrect results in v4.0.0.

But I agree that that's worth it in exchange for consistency with the existing interface for scikit-learn eval functions, as @StrikerRUS mentioned in

#5027 (comment)

Since v4.0.0 will contain many breaking changes intended to reduce inconsistencies, and since the project's current state doesn't really support doing a round of deprecation warnings first (e.g. #5133 (review)).

[github-actions]

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