Huber cost function

As explained, mean squared error isn't robust to outliers, because it's always quadratic independently of the distance between actual value and prediction. To overcome this problem, it's possible to employ the Huber cost function, which is based on threshold t_H, so that for distances less than t_H, its behavior is quadratic, while for a distance greater than t_H, it becomes linear, reducing the entity of the error and, therefore, the relative importance of the outliers.

The analytical expression is: