Approximate Leave-One-Out for High-Dimensional Non-Differentiable Learning Problems

Consider the following class of learning schemes: \begin{equation} \label{eq:main-problem1} \hat{\boldsymbol{\beta}} := \underset{\boldsymbol{\beta} \in \mathcal{C}}{\arg\min} \;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbol{\beta}; y_j) + \lambda R(\boldsymbol{\beta}), \qquad \qquad \qquad (1) \end{equation} where $\boldsymbol{x}_i \in \mathbb{R}^p$ and $y_i \in \mathbb{R}$ denote the $i^{\rm th}$ feature and response variable respectively. Let $\ell$ and $R$ be the convex loss function and regularizer, $\boldsymbol{\beta}$ denote the unknown weights, and $\lambda$ be a regularization parameter. $\mathcal{C} \subset \mathbb{R}^{p}$ is a closed convex set. Finding the optimal choice of $\lambda$ is a challenging problem in high-dimensional regimes where both $n$ and $p$ are large. We propose three frameworks to obtain a computationally efficient approximation of the leave-one-out cross validation (LOOCV) risk for nonsmooth losses and regularizers. Our three frameworks are based on the primal, dual, and proximal formulations of (1). Each framework shows its strength in certain types of problems. We prove the equivalence of the three approaches under smoothness conditions. This equivalence enables us to justify the accuracy of the three 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.