A robust hybrid of lasso and ridge regression

Ridge regression and the lasso are regularized versions of least squares regression using L2 and L1 penalties respectively, on the coefficient vector. To make these regressions more robust we may replace least squares with Huber’s criterion which is a hybrid of squared error (for relatively small errors) and absolute error (for relatively large ones). A reversed version of Huber’s criterion can be used as a hybrid penalty function. Relatively small coefficients contribute their L1 norm to this penalty while larger ones cause it to grow quadratically. This hybrid sets some coefficients to 0 (as lasso does) while shrinking the larger coefficients the way ridge regression does. Both the Huber and reversed Huber penalty functions employ a scale parameter. We provide an objective function that is jointly convex in the regression coefficient vector and these two scale parameters.