In which case one would estimate the performance of, let's say, regression, using $L_1$ loss function and in which case $L_{2}$ is better?

I know that $L_{1}$ is more robust in a sense that it is less sensitive to outliers.

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    $\begingroup$ If conditional median rather then mean prediction is what you need, than $L1$ is your choice. For example, if one predicts stock price returns, which are heavy tailed and so influenced by "outliers" an $L1$ could be more stable, but the forecast will also be "median", not mean. $\endgroup$ – Alexey Burnakov Oct 30 '19 at 14:47

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