TABLE I.

Summary of loss functions $\mathcal{L}(·;\beta )$ employed with MPRidge.

Task	Loss Function	$\mathcal{L}\left(\left(y_i,x_i\right);\mathit{\beta }\right)$
Linear regression	Least-square loss	${\left(y_i-x_i^T\mathit{\beta }\right)}^2$
Regression with outliers	Huber loss	$\{\begin{array}{l}{\left(y_i-x_i^T\mathit{\beta }\right)}^2,\text{\hspace{0.17em}if\hspace{0.17em}}\left|y_i-x_i^T\mathit{\beta }\right|<\delta \hfill \\ 2\delta \left|y_i-x_i^T\mathit{\beta }\right|-{\delta }^2,\text{O}.\text{W}.\hfill \end{array}$
Classification	Logistic loss	$-y_i{x}_i^T\mathit{\beta }+\text{\hspace{0.17em}log\hspace{0.17em}}\left(1+\text{exp\hspace{0.17em}}\left(x_i^T\mathit{\beta }\right)\right)$
Classification	Hinge loss	$\text{max}\left\{0,1-y_i{x}_i^T\mathit{\beta }\right\}$