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E.g. In detriment of a smaller mean error, I want to have fewer big mistakes

I'm working on a time series forecasting task and in some specific cases I don't need perfect accuracy, but the network cannot by any mean miss by a lot.

Any suggestions of loss functions or other methods to solve this issue?

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As you increase the harshness of big misses, you make the model less willing to miss big.

For instance, absolute loss considers missing by $2$ to be twice as bad as missing by $1$, but square loss considers missing by $2$ to be four times as bad as missing by $1$!

Imagine if you use cubic loss:

$$L_3(y,\hat{y}) = \sqrt[3]{\sum\big\vert y_i - \hat{y}_i \big\vert^3}$$

Imagine if you use quintic loss:

$$L_5(y,\hat{y}) = \sqrt[5]{\sum\big\vert y_i - \hat{y}_i \big\vert^5}$$

This gets at the general form of $L_p$ loss, which I'm guessing you can guess before reading what I type.

$$L_p(y,\hat{y}) = \sqrt[p]{\sum\big\vert y_i - \hat{y}_i \big\vert^p}$$

This relates to the idea of $L^p$ norms.

At the extreme of $L^p$ norms, you can set $p=\infty$, which is equivalent to just looking at the maximum. If you use $L_{\infty}$ loss, you minimize the greatest error.

I've wanted to play around with $L_{\infty}$ loss for a while and would be very curious what happens if you choose this loss function. I do, however, think this would be a very extreme loss function.

(Remember that $RMSE$, $MSE$, and $SSE$ all have the same $argmin$; by similar logic we are free to take the $p^{th}$ roots to get the cubic and qunitic loss functions I gave.)

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You could consider something like the relative error $$L(y,\hat{y}) = \sqrt[]{\sum\big\vert (y_i - \hat{y}_i) / y_i \big\vert^2}$$

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