and modern deep models with three of the major benchmark datasets on image classification.

Preliminary {#sec:prelimminary}
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We consider an energy optimization problem based on a given set of training data in a supervised machine learning framework. Let $\chi = \{ (x_i, y_i) \}_{i = 1}^n$ be a set of training data where $x_i \in \X \subset \mathbb{R}^{N}$ is the $i$-th input and $y_i \in \Y \subset \mathbb{R}^{M}$ is its desired output. Let $h_w \colon \X \to \Y$ be a prediction function that is associated with its model parameters $w = ( w_1, w_2, \cdots, w_m ) \in \R^m$ where $m$ denotes the dimension of the feature space. The objective of the supervised learning problem under consideration is to find an optimal set of parameters $w^*$ by minimizing the empirical loss $\L(w)$ that typically consists of a data fidelity term $\rho(w)$ and a regularization term $\gamma(w)$ as follows: $$\begin{aligned}
    w^* = \arg\min_w \mathcal{L}(w), \quad
    \mathcal{L}(w) = \rho( w ) + \lambda \, \gamma( w ), \label{eq:loss}
    \end{aligned}$$ where $\lambda > 0$ is a control parameter, called weight-decay coefficient, that determines the trade-off between the data fidelity term $\rho(w)$ and the regularization $\gamma(w)$. The data fidelity term $\rho(w)$ is of the 