## 1 Introduction  
In the machine learning literature, one of the most important challenges involves estimating parameters accurately and selecting relevant variables from highly correlated, high-dimensional data. Researchers have noticed many highly correlated features in high-dimensional data [1]. Models often overfit or underfit high-dimensional data because they have a large number of variables but only a few of them are actually relevant; most others are irrelevant or redundant. An underfitting model contributes to estimation bias (i.e., high bias and low variance) in the model fitting because it keeps out relevant variables, whereas an overfitting raises estimation error (i.e., low bias and high variance) since it includes irrelevant variables in the model. To illustrate an application of our proposed method, consider a study of gene expression data. This is a high-dimensional dataset and contains highly correlated genes. The geneticist always likes to determine which variants/genes contribute to changes in biological phenomena (e.g., increases in blood cholesterol level, etc.) [2]. Therefore, the aim is to explicitly identify all relevant variants. The penalized regularization models such as \(\ell_{1}\), \(\ell_{2}\), and so forth have recently become a topic of great interest within machine learning, statistics [1], and optimization [3] communities as classic approaches to estimate parameters. The \(\ell_{1}\)-based method is not a preferred selection method for groups of variables among which pairwise correlations are significant because the lasso arbitrarily selects a single variable from the group without any consideration of which one to select [4]. Furthermore, if the selected value of a parameter is too small, the \(\ell_{1}\)-based method would select many irrelevant variables, thus degrading its performance. On the other hand, a large value of the parameter would yield a large bias [5]. Another point worth noting is that few \(\ell_{1}\) regularization methods are either adaptive, computationally tractable, or distributed, but no such method contains all three properties together. Therefore, the aim of this study is to develop a model for parameter estimation and to determine relevant variables in highly correlated, high-dimensional data based on the ordered \(\ell_{2}\). This model has the following three properties all together: adaptive (Our method is adaptive in the sense that it reduces the cost of including new relevant variables as more variables are added to the model due to rank-based penalization properties. ), tractable (A computationally intractable method is a computer algorithm that takes a very long time to execute a mathematical solution. A computationally tractable method is exactly the opposite of intractable method. ), and distributed. Several adaptive and nonadaptive methods have been proposed for parameter estimation and variable selection in large-scale datasets. Different principles are adopted in these procedures to estimate parameters. For example, an adaptive solution (i.e., an ordered \(\ell_{1}\)) [5] is a norm and, therefore, convex. Regularization parameters are sorted in non-increasing order in the ordered \(\ell_{1}\), in which the ordered regularization penalizes regression coefficients according to their order, with higher orders closer to the top and having larger penalties. [PERSON] et al. [6] proposed a partial sorted \(\ell_{p}\) norm, which is non-convex and non-smooth. In contrast, the ordered \(\ell_{2}\) regularization is convex and smooth, just as the standard \(\ell_{2}\) norm is convex and smooth in Reference [7]. [PERSON] et al. [6] considered p-values between \(0<p\leq 1\) that do not cover \(\ell_{2}\), \(\ell_{\infty}\) norms, and so forth. [PERSON] et al. [6] also did not provide details of other partially sorted norms when \(p\geq 2\) and used random projection and the partial sorted \(\ell_{p}\) norm to complete the parameter estimation, whereas we have used ADMM with the ordered \(\ell_{2}\).