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In machine learning and applied fields of statistics, receiver operating characterization (ROC) analysis is commonly used to select optimal algorithms/models. However, at a lecture I once attended on mathematical optimization, I remember the lecturer saying that ROC analysis just wasn't considered a useful approach to optimization. No justification was offered. As a non-mathematician, I want to know (1) is this true that ROC analysis is not used in optimization problems and (2) if so, why?

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The lecturer was wrong. It is now an active area of research (though outside my speciality). For instance, in a paper at NIPS in 2003, they discuss using ROC AUC instead of error as an objective function in training neural networks.

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