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Let say I do the regression task(I am using the deep neural network) for some skewed distribution. Now I am using mean absolute error as loss function.

All typical approaches in machine learning are minimizing mean loss, but for skewed that is unappropriating. It is better from a practical point of view to minimize median loss. I think one way is to penalize big losses with some coefficient. And then mean will be close to the median. But how to calculate that coef for the unknown distribution type? What can you to advice?

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The weighted penalty you are thinking of is the idea behind Quantile Regression:

https://en.wikipedia.org/wiki/Quantile_regression

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