Does gradient boosting algorithm error always decrease faster and lower on training data?
You should worry about overfitting when the test error rate starts to go up again. Until then I would set it aside. Overfitting is rather about the number of parameters, e.g. When two models with the same performance have different number of parameters you would prefer the one with less parameters to preserve the generalisation power of the model.
Generally the training error will be lower than the test error for obvious reasons (the model specifically sees the training data but not the test data), but even this can be reversed in some cases. For example when there is very little test data, or the test data has a different distribution than the training data.
A &quot;perfect&quot; learning curve doesn't exist, the final performance is what counts.