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This question is very general in the sense that the reason may differ depending on the area of ML you are considering. Below are two different areas of ML where the KL-divergence is a natural consequence: Classification: maximizing the log-likelihood (or minimizing the negative log-likelihood) is equivalent to minimizing KL divergence as typical used in DL-...

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In ML we always deal with unknown probability distributions from which the data comes. The most common way to calculate the distance between real and model distribution is $KL$ divergence. Why Kullback–Leibler divergence? Although there are other loss functions (e.g. MSE, MAE), $KL$ divergence is natural when we are dealing with probability distributions. It ...

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To add to nbro's answer, I'd say also that much of the time the distance measure isn't simply a design decision, rather it comes up naturally from the model of the problem. For instance, minimizing the KL divergence between your policy and the softmax of the Q values at a given state is equivalent to policy optimization where the optimality at a given state ...

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I did not read those two specified linked/cited papers and I am not currently familiar with the total variation distance, but I think I can answer some of your questions, given that I am reasonably familiar with the KL divergence. When you compute the $D_{KL}$ between two polices, what does that tell you about them The KL divergence is a measure of "...

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