## 1 Introduction  
Massive multiple-input and multiple-output (MIMO) is one of the most promising technologies for the next-generation wireless mobile communication systems [1; 2; 3; 4]. The large-scale antennas equipped at a base station (BS) can considerably improve the data rate, energy efficiency, and reliability; the theoretical results show that it perfectly cancels out inter-user interference with simple linear transceivers. In this case, the accurate knowledge of channel state information (CSI) is crucial at the BS to achieve the potential gain from large-scale antennas [1; 2; 3; 4]. In time division duplex (TDD) systems, the BS can acquire CSI during uplink training thanks to channel reciprocity [4; 5], and, hence, many existing works on massive MIMO systems consider TDD systems. However, many of current wireless mobile communication systems are frequency division duplexing (FDD) systems [6; 7], whose uplink and downlink channels are independent from each other, which motivates research to achieve the potential gains also in FDD systems [8; 9; 10; 11; 12; 13]. In FDD systems, a BS obtains the CSI from users' channel feedback due to the lack of reciprocity between the uplink and the downlink channels [6; 7; 8; 9; 10; 11; 12; 13]. In massive MIMO systems, the CSI feedback overhead problem becomes more severe because of the large number of antennas; it was already shown that the feedback size should linearly scale with the number of antennas to fully obtain the multiplexing gain [6; 7]. To resolve the feedback overhead problem, a two-stage precoder was widely used for FDD massive MIMO systems, which consists of the outer and the inner precoders [10; 11; 12; 13]. In the two-stage precoder, the outer precoder projects the original channel space of large dimensions onto a smaller dimensional subspace, and then the inner precoder controls the inter-user interference as multiuser MIMO precoding. There are various types of two-stage precoder designs in massive MIMO systems, and, among them, the hybrid architecture is widely considered for mmWave bands due to the limitation of hardware implementation [14; 15; 16; 17]. As a full digital architecture has many benefits in sub-6 GHz bands, there are also many papers that consider the full digital architecture in massive MIMO systems [10; 11; 12; 13]. Thus, we mainly consider a joint spatial division and multiplexing (JSDM) [10] with the full digital architecture in massive MIMO systems with a correlated channel environment. The key idea of JSDM is to divide whole users into multiple user groups according to the channel covariance matrices, and then the outer precoder and the inner precoder sequentially mitigate inter-group and inter-user interference, respectively [10; 18]. At the first stage, the outer precoder mitigates the inter-group interference (IGI) by projecting the original channel into a smaller dimensional subspace. Then, the inner precoder cancels out the same-group interference (SGI) with the dimension-reduced effective channels produced by the outer precoder; in this case, the BS exploits the quantized versions of dimension-reduced effective channels obtained via limited feedback. Therefore, the outer precoder design is important as it leverages many performance achieving factors such as the inter-group interference, the intra-group interference, and the channel quantization error. This motivates the sophisticated outer precoder design taking into account all of these factors. In this context, we optimized the dimension of outer precoder in [19] for a downlink massive MIMO system with limited feedback based on the lower bound-based analysis. Meanwhile, machine learning recently has attracted a lot of attention in wireless communication systems [20; 21]. The machine learning technique has shown good performance in many applications from image processing to economics [20; 21; 22].