Pinball loss

A loss useful for quantile regression. Penalize undershoot or overshoot one side than the other side.

$$L_{\tau }(y,f(x))=\{\begin{array}{ll}\tau \cdot |y-f(x)|& \text{if }y\ge f(x)\\ (1-\tau )\cdot |y-f(x)|& \text{if }y<f(x)\end{array}$$

So it may look like this:

For 𝜏=0.1, we would expect the model to underpredict 90% of the time and overpredict 10% of the time. And now we have already bridged the gap to distributions: This is equivalent to the 10% quantile.

Reference:

• This blog: How I made peace with quantile regression