Permutation feature importance on Train vs Validation set When I compare on Permutation Feature Importance (PFI) on Train vs Validation set, some features has high values (of PFI) for train but the low values (PFI) for validation.
One the conclusion, for me, that features : c, d, f, g and i seams to be kind of noise or "label leakage".
Is there any additional conclusions?
The bar plots of permutation feature importance:
Permutation Feature Importance on Train set

Permutation Feature Importance on Validation set

N.B. The task is Regression (Forecasting) which is done by Random Forest Regression.
 A: The difference in the observed importance of some features when running the feature importance algorithm on Train and Test sets might indicate a tendency of the model to overfit using these features. This is indeed closely related to your intuition on the noise issue.
In other words, your model is over-tuned w.r.t features c,d,f,g,I.
Running feature importance on train or test sets hasn't been addressed enough in literature. However, a good, albeit simple, analysis of this issue is provided in this blogpost.
A: First: ignore the results you have for the training set, they are worthless. Who cares how good a feature is at predicting for records that built the model?
Second: At this point you can’t do anything with features c,d,f,g. You’ve already built your model, yes these features are not useful in predicting the values for your test set, but if you were to remove them now, your test set would become part of the training process and you would be without a test set. All you can use this for at this point is to know that these features are uninformative for your model. To remove them and retrain the model would be akin to stepwise regression which we all know is bad.
If you wanted to use this measure in order to select features and improve your model I believe something like this would work: split your data into train/validation/test. Then split train into train/test, call it train2 and test2 I guess. Build a model on train2 and test the feature importance on test2. You could then remove any irrelevant features. Now continue as usual on train/test/validation as usual.
A: Well, let's think at what those numbers actually mean. 
If we take feature g for example, we know that our model relies on it a bit. So much that if you shuffle its values when making predictions on train data, your model performance drops by around $0.002$ (if I am correct). 
However, when we do the same thing and shuffle it before predicting unseen data, the model performace is on average unchanged, which means that the feature has no predictive power on your target, and that the importance it has with training data comes from using some pattern of your training data that does not generalize (aka, you are overfitting). 
Random forests tend to build very deep trees (possibly, up to a point where no split is possible). This means that even if you have a feature that is just white noise some trees will end up using it for splits at some point because they will see some pattern in it. This means that overall, it is likely even for noise features to have a positive permutation importance on the training data - and this is why the permutation importance you should really care about is on you validation set!

Now, what should you do next? Random Forests are somewhat resistant to this kind of overfitting, and having a few variables that contain only noise is not too detrimental to the overall performance, as long as their relative importance (on the training data) is not excessive, and there is not too many of them. 
However, as you now know that those features are useless for your regression (and in general as good practice), the best option would be to remove them and retrain your model.
