## 1 Introduction  
Optical fibers are widely deployed in modern telecommunication systems and networks [1, 2]. They have been used in several short-reach applications [3]. Metro and media access networks, data center networks (DCNs), cloud radio access networks (C-RANs), passive optical networks (PONs), and mobile front-haul networks (MFHNs) are a few examples among many other potential applications of short-haul fiber-optic communications [4, 5, 6]. In short-reach applications, intensity modulation and direct detection (IMDD) is typically used because of its low cost, simple implementation, and robustness [7]. The nonlinear Kerr effect is negligible in the short-haul fiber-optic transmission systems due to small launch power and short transmission range. However, linear optical distortions caused by the unavoidable chromatic dispersion phenomenon of the fiber turn into nonlinear impairments in the electrical domain due to the square-law detection of the IMDD system. Therefore, linear electrical equalizers, e.g., feed-forward equalizer (FFE) [8] and decision feedback equalizer (DFE) [9], fail to mitigate such nonlinear signal distortions at the receiver. Rather, a nonlinear equalizer has to be employed at the receiver side to perfectly compensate for the nonlinear distortions. Various nonlinear electronic equalization techniques have been proposed in the literature for mitigating the fiber distortions of the IMDD fiber communication systems [10]. For instance, maximum likelihood sequence estimator (MLSE) is the classical technique that provides the traditional benchmark compensation performance [11, 12]. However, it suffers from its high computational cost that prohibits its practical implementation. The MLSE computational cost per symbol is exponentially proportional to the length of the intersymbol interference (ISI) span, i.e., the required channel memory size. The high MLSEcomputational cost can though be reduced at the expense of lower equalization performance using a Volterra-series-based equalizer (VSE) [13; 14]. However, the computational cost of VSEs is still unreasonably high because of the enormous amount of calculations required, especially when a high transmission bit rate is needed. A powerful nonlinear equalization technique that can potentially reduce the computational complexity required by the VSE is a neural-network-based equalizer (NNE) [15]. While the computational cost of the NNEs increases in proportion to the number of neurons in each layer, the VSE cost increases exponentially as the number of tapped delays and the order of the Volterra series increase [16; 17; 18]. Particularly, in order to compensate for the nonlinear impairments of the short-reach fiber transmission system, a computational cost per symbol of as low as \(O\big{(}N^{2}\big{)}\) is required by the NNE, where \(N\) is the ISI span length [16; 17; 18]. Contrarily, the minimum computational complexity of the VSE is of the order \(O\big{(}N^{3}\big{)}\) per symbol [16; 17; 18]. Several types of NNEs have therefore been exploited for the compensation of the electrical nonlinear impairments in the IMDD short-haul fiber-optic communication systems. These include equalizers that are based on single-layer artificial neural networks (ANNs) [19; 20; 21], two-layer ANNs [22; 23; 24], radial basis function neural networks [25], convolutional neural networks [26; 23; 27], recurrent neural networks [28; 29], and multi-layer deep neural networks [6; 30]. Although the single-layer-ANN-based equalizers are the most basic option among all other types of NNEs, they are more popular to be practicably utilized in the short-reach fiber-optic applications because of their superiority from the computational cost point of view. In [21], we proposed an extensive training scheme that enhances the compensation performance of the single-layer-ANN-based equalizer, achieving a slightly lower compensation performance compared to that of the MLSE but with much lower computational cost.