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What is the proper technical name of the classification problem where each data sample can be classified according to two different criteria and each of them can have two or more classes?

For example age/gender estimation problem where age is one criterion, gender is second. Age can be divided into 4 age groups and gender into 2. And model should generated estimate of both age and gender for every sample.

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I think what you are looking for is multi-label classification. Multi-label classification can take care of both of your constraints -

each data sample can be classified according to two different criteria and each of them can have two or more classes

If you just had the latter constraint, I'd have suggested multi-class classification, but multi-label classifiers are a subset of multi-class classifiers.

See What is the difference between Multiclass and Multilabel Problem.

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This is called multi-task learning, as you have two independent classification tasks. Note that this is not the same as multi-label learning.

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