Bowl-Shaped Loss

Bowl-shaped loss functions are an important class of loss functions in Statistics and Machine Learning that are both common and well-behaved. A loss function $L:{\Theta }^2\to {\mathbb{R}}_{\ge 0}$ is bowl-shaped, if there exists some function $\ell$ with convex and symmetric level sets, such that

$$L(\hat{\theta },\theta )=\ell (\hat{\theta }-\theta ).$$

• 📗 Example: $\ell (x)=\Vert x{\Vert }^p$ for any norm and $p$.