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A short discussion on Regression and Classification

On April 24th Deep Learning for Sciences, Engineering, and Arts Meetup, the following problem was discussed: “Why for binary classification don’t we just pick up some values to represent the two possible outcomes (e.g. 0 and 1) and use regression with a linear output and a MSE loss?”. I had the impression that the provided answers were not totally clear for everybody. I am therefore writing this short note, hoping that the arguments presented below will help for a better understanding.