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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) + \lambdaR(\boldsymbol{\beta}), \qquad \qquad \qquad (1) \end{equation} where$\boldsymbol{x}_i \in \mathbb{R}^p$ and $y_i \in \mathbb{R}$ denote the $i^{\rmth}$ feature and response variable respectively. Let $\ell$ and $R$ be theconvex loss function and regularizer, $\boldsymbol{\beta}$ denote the unknownweights, 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 $p$ arelarge. We propose three frameworks to obtain a computationally efficientapproximation of the leave-one-out cross validation (LOOCV) risk for nonsmoothlosses and regularizers. Our three frameworks are based on the primal, dual,and proximal formulations of (1). Each framework shows its strength in certaintypes of problems. We prove the equivalence of the three approaches undersmoothness conditions. This equivalence enables us to justify the accuracy ofthe three methods under such conditions. We use our approaches to obtain a riskestimate for several standard problems, including generalized LASSO, nuclearnorm regularization, and support vector machines. We empirically demonstratethe effectiveness of our results for non-differentiable cases.

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