  Ridge regression (RR) is a regularization technique that penalizes the
L2-norm of the coefficients in linear regression. One of the challenges of
using RR is the need to set a hyperparameter ($\alpha$) that controls the
amount of regularization. Cross-validation is typically used to select the best
$\alpha$ from a set of candidates. However, efficient and appropriate selection
of $\alpha$ can be challenging, particularly where large amounts of data are
analyzed. Because the selected $\alpha$ depends on the scale of the data and
predictors, it is not straightforwardly interpretable. Here, we propose to
reparameterize RR in terms of the ratio $\gamma$ between the L2-norms of the
regularized and unregularized coefficients. This approach, called fractional RR
(FRR), has several benefits: the solutions obtained for different $\gamma$ are
guaranteed to vary, guarding against wasted calculations, and automatically
span the relevant range of regularization, avoiding the need for arduous manual
exploration. We provide an algorithm to solve FRR, as well as open-source
software implementations in Python and MATLAB
(https://github.com/nrdg/fracridge). We show that the proposed method is fast
and scalable for large-scale data problems, and delivers results that are
straightforward to interpret and compare across models and datasets.
