How can I model regression with an asymmetric loss function?

Question

Mosek provided a concrete example of using the Huber loss function, Huber loss, which is great!

One problem I am trying to tackle is to use asymmetric loss, as described in the answer of asymmetric loss.

Simply speaking, instead of using a classic quadratic square error loss, the loss function becomes:

\begin{align}e &= y-y_{\text{predicted}}\\\text{loss} &= x\cdot x \cdot (\operatorname{sgn}(x) + a) \cdot(\operatorname{sgn}(x) + a).\end{align}

As you can see, the loss is not quadratic any more. The advantage of using some loss function, is that we can penalize under-estimate and over-estimate differently.

Can we somehow restructure the problem and make it solvable by Mosek (or other optimization tools)?

Answer 1

I suspect you want to minimize $\sum\limits_i \left[k_1\max(e_i,0)^2 + k_2\max(-e_i,0)^2\right]$. For instance, $a=0.5$ would correspond to $k_1 = 0.25$ and $k_2 = 2.25$).

This can be formulated as a quadratic program by introducing new variables $u$ and $v$, using the objective $k_1 u^\top u + k_2 v^\top v$ with the constraints $u\geq 0, u \geq e, v\geq 0, v\geq -e$.