  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.
