Learning to Beamform in Heterogeneous Massive MIMO Networks

Beamforming optimization has been an active research area in the past two decades [5]. The power minimization based beamforming problems can be well-solved [5] or well-approximated by various convex optimization techniques [6]. On the contrary, the weighted sum rate maximization (WSRM) based beamforming problem is difficult to solve and in fact NP-hard in general [7, 8]. Many suboptimal but computationally efficient beamforming algorithms have been proposed in the literature. For example, the paper [9] proposed the zero-forcing based beamforming based on the generalized matrix inverse theory. Approximation algorithms based on successive convex approximation techniques are proposed in [10, 11] for efficient resource allocation and multiuser beamforming optimization. The inexact cyclic coordinate descent algorithm proposed in [8] relies on the block coordinate descent (BCD) and gradient projection (GP), and can achieve good performance with a low complexity. The celebrated weighted minimum mean square error (WMMSE) algorithm proposed in [12, 13] is based on the equivalence between signal-to-interference-plus-noise ratio (SINR) and MSE, which then is solved by the BCD method. The WMMSE algorithm provides the state-of-the-art performance and therefore is widely benchmarked in the literature. However, all these existing algorithms are iterative in nature, and their complexities quickly increase with the antenna number and network size.

In recent years, machine learning based approaches that depend on the deep neural network (DNN) have been considered in a range of wireless communication applications [14]. For instance, in [15], a parallel structured convolutional neural network (CNN) is trained with only geographical location information of transmitters and receivers to learn the optimal scheduling in dense device-to-device wireless networks. A black-box DNN is trained to approximate the WMMSE algorithm to learn the optimal power control strategy for WSRM in the interference channel [16, 17]. The paper [18] demonstrates the potential of multi-agent deep reinforcement learning techniques to develop a dynamic transmit power allocation scheme in wireless communication systems.

DNN based beamforming schemes have also been proposed for alleviating the computational issues faced in massive MIMO communications. For example, by considering the multiple-input single-output (MISO) broadcast channel, the work [19] proposed a beamforming neural network (BNN) that learns virtual uplink power variables based on the well-known uplink-downlink duality [5] for the power minimization problem and the SINR balancing problem. For the WSRM problem, they proposed a BNN to learn the power variables and Lagrange dual variables based on the optimal beamforming structure. By considering a coordinated beamforming scenario with multiple BSs serving one receiver, the work [20] proposed a black-box DNN to learn the radio-frequency (RF) downlink beamformers directly from the signals received at the distributed BSs during the uplink transmission.

Different from the black-box DNN approach, the deep unfolding technique [21, 22] can build a learning network based on approximating a known iterative algorithm with finite iterations. For example, the works [23], [24] and [25] respectively unfold the GP algorithm, alternating direction method of multipliers (ADMM) and gradicent descent algorithm to build learning networks for MIMO detection. For a single-cell multiuser beamforming problem, the authors in [26] proposed a learning network by unfolding the WMMSE algorithm. To overcome the difficulty of matrix inversion involved in the WMMSE algorithm, they approximate the matrix inversion by its first-order Taylor expansion. Another recent work [27] considered to unfold the WMMSE algorithm to solve the coordinated beamforming problem in MISO interference channels. They chose to avoid the matrix inversion by unfolding a GP algorithm to solve the beamforming subproblem. While these existing works have shown good performance, their learned networks cannot be easily used to optimize a new scenario in which network parameters such as the number of antennas, the number of BSs or the network size are different from the scenarios when the neural network is trained. This shortcoming makes the current designs not suitable for heterogeneous wireless environments.

In this paper, we consider learning-based beamforming designs for WSRM in the MISO interference channels as well as the cooperative multicell networks (see Fig. 1). We propose a beamforming learning network by unfolding the simple parallel GP (PGP) algorithm [28]. The proposed learning network has a recurrent neural network (RNN) structure and is referred to as “RNN-PGP”. The first key advantage of the proposed RNN-PGP is that it has a parallel structure and the deployed neural network is identical for all BSs. The second advantage is that the complexity of the neural network can be made independent of both the number of BSs and the number of BS antennas (which is usually large in massive MIMO communications), and thus the dimension of learnable parameters does not increase with the network size and antenna size. Consequently, the proposed RNN-PGP has the third advantage that it has good generalization capability with respect to the cell radius, the number of antennas, and the number of BSs, which means that the proposed RNN-PGP can be easily deployed in heterogeneous networks where the BSs may have different number of antennas and non-uniform inter-BS distances. Our specific technical contributions are summarized as follows.

1. 1.

   We propose a RNN-PGP beamforming learning network by unfolding the PGP algorithm, which is a simple but effective method to handle the WSRM problem. In order to accelerate the algorithm convergence, we employ a multi-layer perception (MLP) to predict the gradient vector with respect to the beamforming vectors. By showing that the gradient vector lies in a low-dimensional subspace, the MLP simply learns the coefficients required to construct the gradient vector. Since the MLP is identical across the RNN iterations and for all BSs, the parameter space to be learned can be small.

2. 2.

   We make two key steps to improve the generalization capability of RNN-PGP with respect to the number of transmit antennas and the number of BSs. Firstly, the low-dimensional structure of the optimal beamforming solution is exploited to transform the target WSRM problem into a dimension-reduced problem. This ensures that the proposed RNN-PGP actually solves a problem whose dimension is independent of the number of transmit antennas. Secondly, instead of considering the interference from the whole network, the MLP only considers the signals and interference that are sufficiently strong to predict the beamforming gradient vector. This makes the MLP dimension independent of the whole network size, and thus the RNN-PGP can be robust against the number of BSs.

3. 3.

   The design of the RNN-PGP is further extended to the cooperative multicell beamforming problem, which is a joint transmission scheme with BS cooperation. The performance of the proposed RNN-PGP is examined by both synthetic channel dataset and ray-tracing based DeepMIMO dataset [30]. The results show that the proposed RNN-PGP can well approximate the WMMSE solution. More importantly, the RNN-PGP shows promising generalization capability with respect to the number of BSs, number of antennas and inter-BS distance.

Synopsis: Section II presents the system model of the MISO interference channel and formulates the WSRM problem. The existing beamforming algorithms are also briefly reviewed. Section III presents the main design of the proposed RNN-PGP, including introduction of the approach to improving its generalization capability. In Section IV, we extend our RNN-PGP to solve the more complex cooperative multicell beamforming problem. The simulation results are given in Section V and the paper is concluded in Section VI.

Notations: Column vectors and matrices are respectively written in boldfaced lower-case and upper-case letters, e.g., ${\bm{a}}$ and ${\bf A}$, respectively. The superscripts $(\cdot)^{{\mathsf{T}}}$, $(\cdot)^{*}$ and $(\cdot)^{{\mathsf{H}}}$ represent the transpose, conjugate and hermitian transpose respectively. ${\bm{I}}_{K}$ is the $K\times K$ identity matrix; $\|{\bm{a}}\|$ denotes the Euclidean norm of vector ${\bm{a}}$. $\Im(\cdot)$ and $\Re(\cdot)$ represent the imaginary and real part of a complex value respectively. $\{a_{jk}\}$ denotes the set of all $a_{jk}$ with subscripts $j,k$ covering all the admissible intergers, $\{a_{jk}\}_{k}$ denotes the set of all $a_{jk}$ with the first subscript equal to $j$.

The WMMSE algorithm [13] is one of the most popular algorithms for handling the WSRM problem in (4). It reformulates (4) as an equivalent weighted MSE minimization problem by the MMSE-SINR equality [32], followed by solving the problem with the BCD method [33]. The iterative steps of WMMSE are given by: for iteration $r=1,\ldots,$ perform

It is worth mentioning that the WMMSE algorithm developed in [13] was for a more general multi-user scenario where one BS could serve multiple UEs, and it can be extended to network MIMO scenarios [2]. Theoretically, it has been shown that the WMMSE algorithm can converge to a stationary solution of (4), and practically performs well.

Since the power budget constraints of problem (4) have a simple structure, gradient ascent based methods, such as the gradient projection (GP) method [28], can be applied. For example, the inexact cyclic coordinate descent (ICCD) algorithm proposed in [8] deals with problem (4) by applying gradient projection update for each beamformer ${\bm{v}}_{k}$ in a sequential and cyclic fashion. Specifically, the ICCD algorithm has the following steps: for iteration $r=1,2,\ldots,$ perform for $k=1,\ldots,K$ sequentially

where $s_{k}^{r}>0$ is the step size, and the step in (II-B) is projection onto the power budget constraint in (4). A key advantage of the gradient ascent based methods is that they involve simple computation steps. However, when comparing to the WMMSE algorithm, the gradient ascent based methods usually require a larger number of iterations to converge to a solution as good as the WMMSE solution.

The POA algorithm [34] is a monotonic optimization method which can globally solve problem (4). Specifically, in the POA algorithm, a sequence of surrogate problems are systematically constructed, whose feasible set contains the feasible set of the original problem. The constructed feasible set will shrink iteratively and converge to the true feasible set of the original problem [35], while the objective values of the constructed problems will converge to the true optimal value from above asymptotically. Therefore, the POA algorithm provides an upper bound solution for the original optimization problem (4). However, the POA algorithm suffers from very high computation complexity, making it impractical to be used in real-time scenarios.

It is worth noting that all of the above algorithms are iterative in their original form. Therefore, all of them suffer from significant computational delays especially for massive MIMO scenarios where the numbers of cells and transmit antennas are large. To alleviate the computation issues, researchers have proposed the use of DNN [20] and the deep unfolding techniques [21] for approximating the WMMSE beamforming solution in a computationally efficient fashion. While one of the challenges of implementing the WMMSE algorithm by a deep-unfolding neural network lies in the matrix inversion in (6c), the recent work [27] proposed to employ a GP method to handle the update of ${\bm{v}}_{k}^{r}$ in order to avoid explicitly computing the matrix inversion. This results in a double-loop deep-unfolding structure. A related work [26] also proposed a deep unfolding based network for the WMMSE beamforming solution, where the authors approximate the matrix inversion by its first-order Taylor expansion. However, both beamforming designs in [27, 26] have limited generalization capability with respect to the number of transmit antennas or network size. In particular, the learning networks have to be retrained whenever they are deployed in a scenario with different numbers of BSs and transmit antennas.

In the next section, we present a new beamforming learning network, which is designed to improve the generalization capability of the existing DNN based approaches.

Since in massive MIMO scenarios the number of transmit antennas $N_{t}$ is large, it is desirable to avoid handling problem (4) in its original form. It has been shown in [36, Proposition 1] that the optimal beamforming vectors actually have a low-dimensional structure, as stated below.

By this property, the beamforming vector for each $\text{BS}_{k}$ lies in the low-dimensional subspace spanned by the channel vectors $\{{\bm{h}}_{kj}\}_{j}$. Let ${\bm{H}}_{k}=\begin{bmatrix}{\bm{h}}_{k1},\ldots,{\bm{h}}_{kK}\end{bmatrix}\in{\mathbb{C}}^{N_{t}\times K}$ and ${\bm{\xi}}_{k}=\begin{bmatrix}\xi_{k1},\ldots\,\xi_{kK}\end{bmatrix}^{\top}\in{\mathbb{C}}^{K}$. We can let ${\bm{v}}_{k}={\bm{H}}_{k}{\bm{\xi}}_{k}$ for all $k\in\mathcal{K}$, and rewrite problem (4) as

To avoid handling the ellipsoid constraint $\|{\bm{H}}_{k}{\bm{\xi}}_{k}\|^{2}\leq P_{k}$, we consider the eigen-decomposition of

where ${\bf U}_{k}\in\mathbb{C}^{K\times K}$ is the unitary eigen-matrix and ${\bf\Lambda}_{k}\in\mathbb{R}^{K\times K}$ is a diagonal eigenvalue matrix. By letting ${\bm{w}}_{k}={\bf\Lambda}_{k}^{1/2}{\bf U}_{k}^{\mathsf{H}}{\bm{\xi}}_{k}$ and ${\bm{g}}_{jk}={\bf\Lambda}_{j}^{-1/2}{\bf U}_{j}^{\mathsf{H}}{\bm{H}}_{j}^{\mathsf{H}}{\bm{h}}_{jk}$, we write (III-A) as

By comparing (III-A) with (4), the number of unknown parameters are reduced from $O(KN_{t})$ to $O(K^{2})$. Therefore, when $N_{t}\gg K$, it will be beneficial to deal with the dimension-reduced problem (III-A). In addition, problem (III-A) is independent of $N_{t}$, and therefore a learning network based on (III-A) inherently has a good generalization capability with respect to the number of transmit antennas.

While the WMMSE algorithm can provide state-of-the-art performance, the matrix inversion structure of the updating rule makes it difficult to build a learning network to learn the beamforming solution. In this section, in view of the separable power constraint, we consider the simple PGP method [28] to handle problem (III-A). Moreover, based on the deep unfolding technique, we show how an effective and computationally efficient beamforming learning network can be built.

The PGP method applied to (III-A) entails the following iterative steps: for iteration $r=1,\ldots,$ perform for $k=1,\ldots,K$ in parallel

where ${\bm{G}}=\{{\bm{g}}_{jk}\}_{j,k\in\mathcal{K}}$ contains all the transformed channel information. According to the deep unfolding idea, one can build an RNN with a finite iteration number to imitate the iterative updates of the PGP method (11)-(12). In the existing works such as [27], only the step sizes $\{s_{k}^{r-1}\}$ are set to the learnable parameters. Here, we attempt to learn both the step size $s_{k}^{r-1}$ and the gradient vector $\nabla_{{\bm{w}}_{k}}R(\{{\bm{w}}_{j}^{r-1}\},{\bm{G}}\})$, aiming to find good ascent directions that may expedite the algorithm convergence.

It is interesting to note that the gradient vector $\nabla_{{\bm{w}}_{k}}R(\{{\bm{w}}_{j}^{r-1}\},{\bm{G}})$ in fact lies in the range space of the channel vectors $\{{\bm{g}}_{kj}\}_{j}$. Specifically, one can have

​​ where

As shown in Fig. 2, we construct a beamforming learning network termed “RNN-PGP” based on the above ideas. The learning network has an RNN structure with $T$ iterations, each of which mimics the iterative PGP update (11)-(12) for solving problem (III-A). Specifically, in each iteration $r$, it contains $K$ identical function blocks that produce the beamforming solutions $\{{\bm{w}}_{k}^{r+1}\}$ in parallel. Here, we illustrate the detailed operations of the learning network.

Gradient prediction: In each iteration, a central preprocessor first gathers the channel information $\{{\bm{g}}_{kj}\}$, the noise variances $\{\sigma_{k}^{2}\}$ and the beamforming vectors $\{{\bm{w}}^{r}_{k}\}$ obtained in the previous iteration, and then calculates for each $\text{BS}_{k}$

and $\{{\bm{g}}_{kj}^{\mathsf{H}}{\bm{w}}_{k}\}_{j\in\mathcal{K}}$. The terms $D_{j}^{r}$ and $I_{j}^{r}$ stand for the information signal power and the suffered interference plus noise power of each $\text{UE}_{j}$, respectively; while ${\bm{g}}_{kj}^{\mathsf{H}}{\bm{w}}_{k}$ is the out-going interference of $\text{BS}_{k}$ to $\text{UE}_{j}$. The computed $\{D_{j}^{r},I_{j}^{r},{\bm{g}}_{kj}^{\mathsf{H}}{\bm{w}}_{k}\}_{j\in\mathcal{K}}$ is used as the input of a multilayer perception (MLP) block associated with each $\text{BS}_{k}$, which is trained to predict the complex coefficients $\{a_{kj}^{r}\}_{j}$ in (13) and the step size $s_{k}^{r}$. Note that the parallel MLPs for all $K$ BSs have an identical network structure with common parameters ${\bm{\theta}}\in\mathbb{R}^{M}$, where $M$ is the model size.

The predicted $\{a_{kj}^{r}\}_{j}$ are used to construct the gradient vector through the function block $\nabla$

Gradient ascent: With $\nabla_{{\bm{w}}_{k}}R^{r}$ and $s_{k}^{r}$, the gradient ascent update is performed as

This step is shown in the middle of the enlarged rectangle in Fig. 2, where $\otimes$ and $\oplus$ represent the multiplication and addition operations.

Projection: Following (12), the function block $\mathcal{P}$ projects the beamforming vector $\tilde{{\bm{w}}}_{k}^{r+1}$ onto the feasible set by computing

Phase rotation: It is worth noting that problem (III-A) does not have a unique solution. In fact, if $\{{\bm{w}}_{k}\}$ is an optimal solution of (III-A), then any phase rotated solution $\{{\bm{w}}_{k}e^{i\psi_{k}}\}$ is also an optimal solution, where $i=\sqrt{-1}$. In order to make sure that the RNN learns a one-to-one mapping, we rotate the phases of $\{\hat{}{\bm{w}}_{k}^{r+1}\}$ so that each rotated $\hat{}{\bm{w}}_{k}^{r+1}$, denoted by ${\bm{w}}_{k}^{r+1}$, aligns with ${\bm{g}}_{kk}$ (i.e., ${\bm{g}}_{kk}^{\mathsf{H}}{{\bm{w}}}_{k}^{r+1}$ is real). In particular, we perform via the function block $\angle_{\mathcal{F}}$

As seen in the previous subsection, the proposed RNN-PGP network would have good generalization capability with respect to the number of transmit antennas $N_{t}$ since both the MLPs and operations in each iteration do not depend on it. To further make the MLP easy to generalize to settings with different number of BSs $K$, we need the MLP to have its input $\{D_{j}^{r},I_{j}^{r},{\bm{g}}_{kj}^{\mathsf{H}}{\bm{w}}_{k}^{r}\}_{j\in\mathcal{K}}$ and output $\{a_{kj}^{r}\}_{j\in\mathcal{K}}$ to be independent of the BS number $K$.

As inspired by [18], for each ${\rm BS}_{k}$ we define two neighbor subsets. The first is the set of BSs that cause a relatively strong interference to ${\rm BS}_{k}$, i.e.,

where $\eta$ a preset threshold. We order $\ |{\bm{g}}_{jk}^{\mathsf{H}}{\bm{w}}_{j}^{r}|^{2}/\sigma_{k}^{2}$, $j\in\mathcal{I}_{k}^{r}$ in a decreasing fashion, and further select the first $c$ indices in $\mathcal{I}_{k}^{r}$. The selected subset is denoted by $\mathcal{I}_{k}^{r}(c)\subseteq\mathcal{I}_{k}^{r}$, which indicates the first $c$ most “interferering” BSs that have strong interference on ${\rm BS}_{k}$.

Analogously, for each ${\rm BS}_{k}$ we define in the following the set of UEs whom ${\rm BS}_{k}$ causes strong interference to

and further select a subset $\mathcal{O}_{k}^{r}(c)\subseteq\mathcal{O}_{k}^{r}$ which corresponds to the $c$ most “interfered” UEs by ${\rm BS}_{k}$. An example is shown in Fig. 3, where $c=3$, and the “interferering” neighbors of $\text{BS}_{5}$ are $\{\text{BS}_{1},\text{BS}_{2},\text{BS}_{6}\}$ and “interfered” UEs of it are $\{\text{UE}_{4},\text{UE}_{6},\text{UE}_{7}\}$.

Then, we consider only BSs in $\mathcal{I}_{k}^{r}(c)$ and UEs in $\mathcal{O}_{k}^{r}(c)$ for computing the input and output of the MLP associated with $\text{BS}_{k}$. Specifically, instead of $\{D_{j}^{r},I_{j}^{r},{\bm{g}}_{kj}^{\mathsf{H}}{\bm{w}}_{k}\}_{j\in\mathcal{K}}$, we let the input of the MLP associated with ${\rm BS}_{k}$ be $\{D_{j}^{r},I_{j}^{r}(c),{\bm{g}}_{kj}^{\mathsf{H}}{\bm{w}}_{k}^{r}\}_{j\in\{\mathcal{O}^{r}_{k}(c),k\}}$, where

In addition to the step size $s_{k}^{r}$, the output of the MLP associated with ${\rm BS}_{k}$ is changed to $\{a_{kj}^{r}\}_{j\in\{\mathcal{O}^{r}_{k}(c),k\}}$, which now is dependent on the set $\mathcal{O}^{r}_{k}(c)$ only. Therefore, the input and output of the MLP are independent of the whole network size $K$. The diagram of the revised RNN-PGP network based on $\mathcal{I}_{k}^{r}(c)$ and $\mathcal{O}_{k}^{r}(c)$ is illustrated in Fig. 4, where only the block associated with ${\rm BS}_{k}$ at iteration $r$ is plotted.

Suppose that a training data set of size $L$ is given, which contains the (dimension reduced) CSI $\{{\bm{G}}^{(\ell)}\}$, WSR coefficients $\{\alpha_{k}^{(\ell)}\}_{k\in\mathcal{K}}$, and the beamforming solutions $\{{\bm{w}}_{k}^{(\ell)}\}_{k\in\mathcal{K}}$ obtained by an existing algorithm, for $\ell=1,\ldots,L$. Suppose that the RNN-PGP has a total number of $T$ iterations (see Fig. 2), and denote $\{{\bm{w}}_{k}^{(\ell),r}\}_{k\in\mathcal{K}}$ as the output of RNN-PGP in the $r$th iteration for the $\ell$th data sample. The RNN-PGP network is trained by a two-stage approach. The first stage is based on supervised learning, using the following loss function

where $\gamma$ is the penalty parameter. Note that the second term in the right hand side of (23) encourages the output of earlier iterations of RNN-PGP to be close to $\{{\bm{w}}_{k}^{(\ell)}\}$, which may help speed up the convergence of RNN-PGP.

By treating the supervised training in the first stage as a pre-training, we can further refine the network in an unsupervised fashion in the second stage. Specifically, we can directly train the network so that the WSR function of (4) is maximized; for example, we consider the following loss function

As will be shown in Section V, the two-stage training approach can outperform those that solely use supervised training or unsupervised training [19].

We first consider the MISO interference channel model in Section II and test the performance of the proposed RNN-PGP network in Fig. 2 under synthetic Rayleigh channel data. Like Fig. 3, we assume that each ${\rm BS}_{k}$ is located at the center of cell $k$ and ${\rm BS}_{k}$ is located randomly according to a uniform distribution within the cell. The half BS-to-BS distance is denoted as $d$ and it is chosen between $100$ and $1000$ meters. We set $P_{max}$, i.e., the maximum transmit power level of ${\rm BS}_{k}$, to be $38$ dBm over a $10$ MHz frequency band. The path loss between the UE and its associated BS is set as $128.1+37.6\log_{10}(s)$ (dB) where $s$ (km) is the distance between the UE and BS. The channel coefficients of $\{{\bm{h}}_{kj}\}$ are generated following the independent and identically distributed complex Gaussian distribution with mean equal to the pathloss and variance equal to one. The noise power spectral density of all UEs are set the same and equal to $\sigma^{2}=-174$ dBm/Hz. A total of $5000$ training samples ($L=5000$) are generated which include CSI $\{{\bm{h}}_{kj}^{(\ell)}\}$, $\ell=1,\ldots,L$, rate weights $\{\alpha_{k}^{(\ell)}\}$, $\ell=1,\ldots,L$, and beamforming solutions $\{{\bm{w}}_{k}^{(\ell)}\}$, $\ell=1,\ldots,L$, obtained either by the PGP method or the POA algorithm. Another $1000$ testing samples are also generated in the same way.

To train the RNN-PGP network in Fig. 2, we set the parameter $\eta$ in (20) and (21) to be $5$, and the parameter $\gamma$ in (23) and (IV) to be 0.95. The function $\tanh$ is used as the activation function in the MLP. The iteration number of RNN-PGP is set to $T=20$, and the Adam optimizer is used for training the RNN-PGP network. The simulation environment is based on Python 3.6.9 with TensorFlow 1.14.0 on a desktop computer with Intel i7-9800X CPU Core, one NVIDIA RTX 2080Ti GPU, and 64GB of RAM. The GPU is used during the training stage but not used in the test stage.

In the presented results, we benchmark the proposed RNN-PGP network with the existing beamforming algorithms such as the PGP method, WMMSE algorithm and the POA algorithm, as well as black-box based DNN approach. Specifically, they include

• •

  PGP, WMMSE and POA: Performance results obtained by applying the three methods to the dimension reduced problem (III-A), respectively.

• •

  RNN-PGP (PGP) and RNN-PGP (POA): The RNN-PGP networks in Fig. 2 trained by the hybrid strategy, and the beamforming solutions used for supervised training are obtained by the PGP method and the POA method, respectively.

• •

  DNN (PGP) and DNN (POA): The black-box DNNs, which have $5$ layers with the concatenated CSI as the input and the concatenated beamforming solutions as the output, are trained end-to-end by the hybrid training strategy, and the beamforming solutions used for supervised training are obtained by the PGP method and the POA method, respectively.

• •

  RNN-PGP (Unsuper): The RNN-PGP network in Fig. 2 trained solely by the unsupervised cost.

• •

  RNN-PGP (POA, Stepsize): The MLPs in the RNN-PGP only predicts the step size $s_{k}^{r}$, and the gradient vector $\nabla_{{\bm{w}}_{k}}R^{r}$ is computed explicitly by (13) and (14). The network is trained by the hybrid strategy and the beamforming solutions obtained by the POA method is used during the supervised training.

For the presented test results, if not mentioned specifically, the beamforming solutions of RNN-PGP are obtained by unfolding the network for $T=20$ iterations. Except for the (weighted) sum rate, we also show the “accuracy” (%) which is the ratio of the (weighted) sum rate achieved by the RNN-PGP and that achieved by the WMMSE solution.

In this subsection, we evaluate the sum-rate performance of different schemes by setting the weights $\alpha_{k}$ of all users to be $1$. The results are shown in Fig. 6(a), Fig. 6(b) and Tables I, II, III, respectively.