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I am predicting a target $y$ using regression. In my application, the prediction $\hat{y}$ should be always no less than $y$. If $y>\hat{y}$, it is definitely a wrong prediction. On the $y<\hat{y}$ side, I want $\hat{y}$ is close to $y$.

In this case, I cannot use $(\hat{y}-y)^2$ to measure the error because it measures two sides. I guess it should be a common case in ML practices. Is there any commonly used continuous loss function that measures on-side error?

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How about something like: \begin{eqnarray} (y-\hat{y})^2 , y < \hat{y} \\ (y-\hat{y})^3 , y \geq \hat{y} \end{eqnarray}

This is both continuous and differentiable, and penalizes $y > \hat{y}$ more severely than $y < \hat{y}$. You can even control the severity of the penalization by changing the exponent from 3 to a higher odd number.

Here's how it looks ($x = y - \hat{y}$):

enter image description here

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