# What's the pros and cons between Huber and Pseudo Huber Loss Functions?

The Huber Loss is: $$huber = \begin{cases} \frac{1}{2} t^2 & \quad\text{if}\quad |t|\le \beta \\ \beta |t| &\quad\text{else} \end{cases}$$ The pseudo huber is: $$pseudo = \delta^2\left(\sqrt{1+\left(\frac{t}{\delta}\right)^2}-1\right)$$

What are the pros and cons of using pseudo huber over huber? I don't really see much research using pseudo huber, so I wonder why?

For me, pseudo huber loss allows you to control the smoothness and therefore you can specifically decide how much you penalise outliers by, whereas huber loss is either MSE or MAE. Also, the huber loss does not have a continuous second derivative.

So, what exactly are the cons of pseudo if any?

1. You don't have to choose a $$\delta$$. (Of course you may like the freedom to "control" that comes with such a choice, but some would like to avoid choices without having some clear information and guidance how to make it.)
• Thanks, although i would say that 1 and 3 are not really advantages, i.e. we can make $\delta$ so it is the same curvature as MSE. And for point 2, is this applicable for loss functions in neural networks? I'm not sure Feb 14, 2021 at 16:59