Abstract

Consider the following class of leaning schemes: ,  where denote the feature and response variable respectively. Let and R be the loss function and regularizer, denote the unknown weights, and be a regularization parameter. Finding the optimal choice of is a challenging problem in high-dimensional regimes where both n and p are large. We propose two frameworks to obtain a computationally efficient approximation ALO of the leave-one-out cross validation (LOOCV) risk for nonsmooth losses and regularizers. Our two frameworks are based on the primal and dual formulations of (1). We prove the equivalence of the two approaches under smoothness conditions. This equivalence enables us to justify the accuracy of both methods under such conditions. We use our approaches to obtain a risk estimate for several standard problems, including generalized LASSO, nuclear norm regularization, and support vector machines. We empirically demonstrate the effectiveness of our results for non-differentiable cases.