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For example, given a face image, and you want to predict the gender. You also have age information for each person, should you feed the age information as input or should you use it as auxiliary output that the network needs to predict?

How do I know analytically (instead of experimentally) which approach will be better? What is the logic behind this?

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For extra input that does not matter, you should not input it to the network.

Feature selection, the process of finding and selecting the most useful features in a dataset, is a crucial step of the machine learning pipeline. Unnecessary features decrease training speed, decrease model interpretability, and, most importantly, decrease generalization performance on the test set.

Source: A Feature Selection Tool for Machine Learning in Python

As the source says, unnecessary features decreases accuracy and training speed. Moreover, they have no mapping to the labels, so they won't end up being used. They are unnecessary and adding them will only cause you trouble. Hope this helps you and have a nice day!

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