FEAT - Double quadratic datafit (quad-quad)

[tomaszkacprzak]

Description of the feature

In many applications there is a need to have an asymmetric loss that weights negative and positive residuals differently. A good function for this is double quadratic loss (quad-quad):

$L (\epsilon, \alpha) = 2 · ( \alpha + (1 − 2\alpha) · 1[\epsilon > 0] ) · || \epsilon ||_2^2$

where $\epsilon = Xw - y$ is the residual, $\alpha$ controls the degree of asymmetry and $1[·]$ is the indicator function. For $\alpha=0.5$ this function is equivalent to Quadratic datafit.

Would it be possible to add DoubleQuadratic datafit and have it compatible with the Lasso and other penalties?

Considered alternatives

Another possibility is to use Pinball loss.