## 1 Introduction
More recently, the SMO algorithm for solving QP problems is further improved by the studies [27; 28; 29]. In several real-world problems, the noise exhibits different distributions rather than a specific distribution such as Gaussian. Thus, beyond traditional \(l_{1}\) and \(l_{2}\) loss functions, a diverse landscape of alternatives has emerged, each tailored to specific noise distributions [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. In [30], a novel variant of SVM was introduced, where the traditional hinge loss in SVM was replaced with the pinball loss. The dual QP problem with box constraints was subsequently solved using the SMO algorithm in [31]. Ref. [32] extended it with the squared pinball loss, resulting in an asymmetric least squares SVM, and [33] employs this loss function for SVR with a SMO-based solver. In [34], SVR models with asymmetric Huber and \(\epsilon\)-insensitive Huber loss functions were presented, leading to strongly convex minimization problems which are solved in the primal by a functional iterative method. The classical ridge regression assumes that the noise follows a Gaussian distribution. However, Ref. [35] revealed that, in certain practical applications, such as wind speed prediction, the noise models may not adhere to a Gaussian distribution. So, in [35], a nonlinear loss function optimal to Beta noise distribution in the maximum likelihood sense was employed for wind speed prediction and the kernel ridge regression with this nonlinear loss was solved by the Augmented Lagrangian Multiplier method. In [36], SVR was formulated with a loss function determined based on the noise distribution in such a way that the optimal loss functions were determined in the maximum likelihood sense for Laplace, Gaussian, Beta, Weibull and Marshall-Olkin generalized exponential distributions. A naive online R minimization algorithm was chosen as the optimization method to solve this dual nonlinear SVR and it was reported that SVR with a loss determined based on noise distribution performs better than classical \(\epsilon\)-SVR. Ref. [37] proposed a LSSVM and an extreme learning machine with a homotopy loss possessing two tunable parameters, which covers different loss functions such as \(l_{1}\)-norm loss, logarithmic loss, Geman-Reynolds loss, Geman-McClure loss and correntropy-based loss. Although the proposed loss covers these above-mentioned losses, only the problem of LSSVM with homotopy loss, which becomes equivalent to the reweighted LSSVM model for some specific values of one tun able parameter, was solved via reweighted least squares algorithm. Recently, a convex piecewise linear loss function, namely the \(\varepsilon\)-penalty loss function, with two tunable parameters, where the popular \(\varepsilon\)-insensitive \(l_{1}\) loss function and the Laplace loss function are particular cases of this loss function, was introduced in [38], and resulting QP and linear programming problems of the SVR models with this loss function were solved by the interior point algorithm. Another convex, continuous and differentiable loss function, namely \(l_{s}\) loss, was presented in [39] and used to construct two kernel-based regressors for improved noise robustness. The \(l_{s}\) loss was used in place of the traditional loss function in LSSVR and ELM. An iteratively reweighted least squares method was utilized to optimize these LSSVR and ELM problems. Another study [40] introduced an SVR model with a continuously differentiable convex loss function, namely Incosh loss, which is optimal in the maximum likelihood sense for the hyper-secant error distribution. This loss function has been applied in various fields [41; 42; 43; 44; 45; 46; 47; 48] and it was noted in the study [40] that SVR models generated using various parameter settings of the Incosh loss exhibit many of the favorable attributes found in well-known loss functions like Vapnik's loss, the squared loss and Huber's loss functions.