$L_1$ vs $L_2$ loses in cross-validation

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.

• 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. – Alexey Burnakov Oct 30 '19 at 14:47