GNN-Based Beamforming for Sum-Rate Maximization in MU-MISO Networks

Consider a downlink MU-MISO system as shown in Fig. 1, where one $N_{\rm T}$-antenna BS intends to serve $K$ single-antenna users over a common spectral band. We use $\mathcal{K}\triangleq\{1,2,...,K\}$ to denote the index set of the users.

Denote the symbol for the $k$-th user and the corresponding beamforming vector by $s_{k}$ and ${{{\bf{w}}_{k}}}$, respectively. The received signal at the $k$-th user is given by

where ${{{\bf{h}}_{k}}}\in{\mathbb{C}^{{N_{\rm{T}}}}}$ denotes the CSI between the BS and the $k$-th user, and $n_{k}\sim\mathcal{CN}({0,{\sigma_{k}^{2}}})$ denotes the additive white Gaussian noise (AWGN). Without loss of generality, it is assumed that ${{\mathbb{E}}}\{{{{|{s_{k}}|}^{2}}}\}=1$ ($\forall k\in\mathcal{K}$). Then, the achievable data rate at the $k$-th user is expressed as

The sum-rate maximization problem is formulated as

where ${R_{\rm Req}}$ denotes the information rate requirement of each user and ${P_{\rm Max}}$ denotes the power budget of the BS.

The problem (2) can be solved by conventional convex optimization, which however, requires to be implemented for every CSI realization. To extract the knowledge of the mapping from the CSI to the optimal beamforming vectors, the GNN-based approach is investigated by solving the following graph optimization problem.

The network topology nature of the considered MU-MISO system makes it suitable to be modeled as a graph. In particular, the considered MU-MISO system can be modeled as a fully connected directed graph denoted as ${\cal G}=({\cal V},{\cal E})$ as shown in Fig. 1, where $\cal V$ denotes the set of nodes and $\cal E$ denotes the set of edges. Each BS-user desired link is modeled as a node while each interference link is modeled as an edge. That is, there are $K$ nodes and $K(K-1)$ edges. The node feature of the $k$-th node is defined as the corresponding CSI, i.e., ${\bf h}_{k}$, while there is no edge feature as the inter-user interference is implicit.

The node feature matrix of the graph ${\cal G}$ is denoted as $\mathbf{H}\triangleq[{\bf{h}}_{1},\cdots,{\bf{h}}_{k}]^{T}\in{\mathbb{C}^{{K}\times{N_{\rm{T}}}}}$. By defining the beamforming matrix $\mathbf{W}\triangleq[{\bf{w}}_{1},\cdots,{\bf{w}}_{k}]^{T}\in{\mathbb{C}^{{K}\times{N_{\rm{T}}}}}$, the problem (2) can be rewritten as the following graph optimization problem:

where

and $\mathcal{N}(k)\triangleq{\cal K}/k$ denotes the set of neighbor nodes of the $k$-th node.

The CRGAL is the key component of the proposed CRGAT, and it updates the feature of each node by aggregating the features of its neighboring nodes. As there is no edge feature, each node can only learn its interference to/from other nodes via the node features. To facilitate the aggregating process, the multiple attention mechanisms are adopted to assign different weights to neighboring nodes¹¹1As the considered system is modelled as a fully connected graph, the neighboring nodes of a node are actually the other $(K-1)$ nodes., such that the central node pays more attention to its neighboring nodes with greater impact on it.

It is noted that the deeper CRGALs may induce the oversmoothing issue, which homogenizes the aggregated features of different nodes and degrades the availability of the aggregated features. Therefore, the residual structure is adopted to balance the trade-off between adopting deeper CRGALs and avoiding the oversmoothing issue.

As shown in Fig. 3, each CRGAL consists of two processes, i.e., the attention-enabled aggregation and the residual-assisted combination.

The CFCLs are utilized to map the embedded features, i.e., ${{\mathbf{H}}}^{\left(L+1\right)}$, to the learned beamforming matrix. Let ${\mathbf{H}}^{\left(c\right)}\in{\mathbb{C}^{{K}\times{G(c)}}}$ denote the input node feature matrix of the $c$-th CFCL ($c\in{\cal C}\triangleq\{1,...,C\}$), where ${{G}(c)}$ denotes the corresponding dimension of each node. In particular, for the $c$-th CFCL, the mapping from the input to the net activation, denoted by ${\widetilde{\mathbf{H}}^{(c+1)}}$, is realized by

where ${\bm{\Theta}}^{(c)}\in\mathbb{C}^{G(c)\times G(c+1)}$ denotes the learnable parameters of the $c$-th CFCL.

For the $c$-th CFCL with $c\in{\cal C}\setminus C$, the $\mathbb{C}\rm{BatchNorm}$ ($\mathbb{C}$BN) layer is added in order to prevent the proposed CRGAT from overfitting as well as enhancing the convergence behavior. The $\mathbb{C}$BN layer can be treated as an activation function, which is applied to the net activation after $\mathbb{C}\rm{SELU}()$ as follows:

where ${\rm BN}()$ denotes the batch normalization function.

As for the last CFCL, a new activation function is proposed to guarantee that the output beamforming matrix, denoted as $\widehat{\mathbf{W}}$, satisfies the sum-power constraint (2b), which is given by

Note that $\widehat{\mathbf{W}}$ also represents the output of the CRGAT, i.e., the learned beamforming matrix.

With the proposed activation function (11), the problem (2) is equivalently transformed into the following optimization problem:

where $\bm{\theta}\triangleq\{\mathbf{A}^{(l)},{\bm{\Theta}}^{(l)},\overline{\bm{\Theta}}^{(l)},\widetilde{\bm{\Theta}}^{(l)},{\bm{\Theta}}^{(c)}\}$ represents all trainable parameters in the proposed CRGAT.

The PM embeds the rate requirement constraint (15b) into the loss function as a penalty term to enforce the output of CRGAT to satisfy (15b). For the problem (15), the loss function based on the PM is given by (16), where $N$ denotes the batch size, $\operatorname{ReLU}(x)\triangleq\max(x,0)$ represents the rectified linear unit activation function, $\widehat{\mathbf{W}}_{\left(n\right)}$ represents the ouput of the CRGAT assosicated with the $n$-th sample, i.e., ${\mathbf{H}}_{\left(n\right)}$, and $\lambda>0$ denotes the penalty coefficient which is treated as a hyperparameter.

Particularly, the penalty term is imposed only if at least one rate requirement constraint cannot be satisfied. With a larger value of $\lambda$, the rate requirement constraints are more likely to be satisfied, which however, may degrade the impact of the objective function on the loss function. By carefully adjusting the value of $\lambda$, a tradeoff between maximizing the sum rates and satisfying the rate requirement constraints can be achieved.

The unsupervised training procedure with the loss function based on PM is shown in Algorithm 1 where $M$, $E$ and $B$ respectively denote the total numbers of samples, epochs and minibatchs in the training dataset, and $\omega$ denotes the learning rate.

Instead of using the constant penalty coefficient, the LDM adopts the learnable Lagrangian multipliers.

Let $\bm{\mu}\triangleq[\mu_{1},...,\mu_{K}]\in\mathbb{R}_{+}^{K}$ denote non-negative Lagrangian multipliers associated with (15b). The Lagrangian dual optimization problem of the problem (15) is given by

where ${{\cal L}\left(\bm{\theta},\bm{\mu}\right)}$ is given by (18).

The Lagrangian dual optimization problem can be solved by alternately optimizing $\bm{\theta}$ and $\bm{\mu}$. Similarly, the loss function based on the LDM is given by (21), and we alternately update $\bm{\theta}$ and $\bm{\mu}$ during the training phase. Then, the unsupervised training procedure with the loss function based on LDM is shown in Algorithm 2, where $\tau>0$ denotes the step size for updating $\bm{\mu}$.

Simulation scenario: The number of antennas is set as $N_{\rm T}\in\{8,16\}$. The number of users is set as $K\in\{3,4,5,6,7,8,9\}$. The total power budget is set as $P_{\rm Max}\in\{1,2,3\}$. The rate requirement is set as $R_{\rm Req}\in\{1,2,3\}$ bit/s/Hz. The CSI of each user includes both large-scale path loss and the small-scale fading. The path-loss model is based on $10^{-(140.7+36.7\log_{10}({\widetilde{d}}))/10}$ where ${\widetilde{d}}$ (in kilometer) denotes the distance between the BS and the user. Rayleigh fading is adopted for the considered small-scale fading. The noise power spectrum is $162$ dBm/Hz and the bandwidth is $10$ MHz. The average ratio between path loss and noise power is set as $10$ dB.

Computer configuration: The convex formulations and the NNs are respectively processed by convex solver SeDuMi under Mathworks MATLAB R2021b and Python 3.8.18 with Pytorch 1.10.0 on a computer with Intel(R) Core(TM) i9-12900K CPU and one NVIDIA RTX 3090 GPU (24 GB of memory).

Dataset: The ratio between the training set, validation set and test set in the dataset is set as $9:1:1$. Since the unsupervised learning is adopted, the training set does not include labels. The labels of the validation set and test set are generated by an SCA-based approach (presented in Appendix A) with the convergence precision of $10^{-4}$. All the datasets prepared for the simulation experiments are shown in Table I.

GNN architecture: The detailed architecture of the CRGAT is shown in Table II where $4$ CGCLs and $3$ CFCLs are included.

Basic Metrics: The basic metrics are used to evaluate the learning-based approaches on a given test set in terms of the optimality performance, feasible rate and inference time.

• 1)

  Optimality performance: The ratio of the average achievable sum rates (with feasible solutions) by the learning-based approach to the average optimal sum rates.

• 2)

  Feasibility rate: The percentage of the feasible solutions to the considered problem by the learning-based approach.

• 3)

  Inference time: The average running time required to calculate the feasible beamforming vectors with the given CSI by the learning-based approach.

Advanced Metrics: Based on the basic metrics, we define the following three advanced metrics to evaluate the learning-based approaches. The detailed is described as follows.

• 1)

  Model effectiveness: Evaluate the efficacy of the model architecture based on the basic metrics, where the settings of the test set and the training set are identical.³³3The identical setting means that the scalar variables are the same while the random variables are i.i.d.

• 2)

  Scalability: Evaluate the efficacy of the learning-based approach on the test set with different settings from the training set.

• 3)

  Adaptability: Evaluate the efficacy of the activation function and the loss function with different constraints.

This subsection shows the performance of the proposed CRGAT based on the basic metrics. Two scenarios are considered, i.e., $(N_{\rm T},K,P_{\rm Max},R_{\rm Req})=(8,4,1,1)$ (Dataset No. 2) and $(N_{\rm T},K,P_{\rm Max},R_{\rm Req})=(16,8,1,1)$ (Dataset No. 5). To fully evaluate the proposed CRGAT, several convex optimization and learning based approaches are adopted as the benchmarks⁴⁴4The learning-based approaches are according to existing works and extended to solve the considered problem..

• 1)

  SCA: The SCA-based algorithm serves as a benchmark to evaluate the optimality and the computational efficiency of the learning-based approaches;

• 2)

  MRT: The SCA-based algorithm based on the maximum ratio transmission (MRT) scheme (given in Appendix B);

• 3)

  ZF: The SCA-based algorithm based on the zero-forceing (ZF) scheme (given in Appendix B);

• 4)

  CMLP[7]: Direct learning of the mapping from CSI, i.e., $\{{\bf h}_{i}\}$, to beamforming vectors, i.e., $\{{\bf w}_{i}\}$ via CFCLs;

• 5)

  CGCN[23]: Topological information among CSI is utilized through graph convolution;

• 6)

  CGAT[15]: Additive attention mechanism is used in the process of graph convolution;

• 7)

  CTGCN[28]: Key-value attention mechanism is used in the process of graph convolution.

For fair comparison, all learning-based approaches adopt the activation function (11) and trained via Algorithm 1. The numerical results are shown in Table III.

Compared with the benchmark learning-based approaches, the proposed CRGAT achieves the best optimality performance, and optimality performance gap to the SCA-based algorithm is less than $5\%$. The increment of antennas and users has an insignificant impact on the optimality performance of the CRGAT, which however, greatly degrades the optimality performance of other learning-based approaches. On the one hand, the attention-enabled aggregation adopted in the CRGAT is able to explore the hidden features due to the inter-user interference, which makes it more suitable to the considered problem than the CMLP and the CGCN. On the other hand, the residual-assisted combination adopted in the CRGAT avoids the oversmoothing issue, which enhances its feature embedding capability compared with the CGAT and the CTGCN. Besides, the feasible solution by the CRGAT is guaranteed.

Compared with traditional convex optimization based approaches, the inference time is greatly reduced by the learning-based approaches. Besides, the increment number of antennas and users has limited impact on the inference time of the learning-based approaches while greatly increasing the inference time of the convex optimization based approaches. That is, with limited optimality performance loss, the proposed CRGAT is able to make a real-time response to the time-varying CSI, e.g., several microseconds with our computer configuration. The following comparisons are among the learning-based approaches, and the inference time is omitted.

Furthermore, we evaluate the robustness of the proposed CRGAT with $(N_{\rm T},K)=(16,8)$ by adding channel estimation errors to the original input, i.e, $\widehat{\textbf{H}}={\textbf{H}}+{\textbf{E}}$ ($\forall k\in\mathcal{K}$), where H represents the original input and E denotes the CSI error with ${\textbf{E}}_{(k,:)}\sim\mathcal{C}\mathcal{N}(0,\sigma^{2}_{{\rm E},k}{\bf I}_{N_{\rm T}})$ ($\forall k\in\mathcal{K}$). The numerical results are shown in Table IV. It is observed that the adding noise to the original input has limited impact on the optimality performance of the CRGAT.

The adaptability is to evaluate the performance of the learning-based approaches under different scenarios. The adaptability to different $R_{\rm Req}$ and ${P_{\rm Max}}$ is respectively evaluated as follows.

Table VII shows the results with different values of $R_{\rm Req}$. It is observed that with an increase of $R_{\rm Req}$, the optimality performance of each method almost keeps unchanged while the feasible rate decreases in general. The reason is that with larger value of $R_{\rm Req}$, the feasible set of problem (15) shrinks. Besides, the proposed methods are superior to the benchmark ones, especially in terms of the feasible rate. As it is of high significance for learning-based approaches to yield feasible solutions, the effectiveness of the proposed approach is validated.

Table VIII shows the results with different values of ${P_{\rm Max}}$. It is interesting to see that with an increment of ${P_{\rm Max}}$, the optimality performance of all methods is slightly degraded. This might be due to the fact that with larger ${P_{\rm Max}}$, there are more allocation schemes of the power budget, and the learning task becomes more complicated. Nevertheless, the feasibility rates of the proposed methods keeps $100\%$, which outperforms the benchmark methods.

The effectiveness of the attention-enabled aggregation has been analyzed in our previous work [15], which is also validated in Table III and Table V. For a deeper understanding of the residual-assisted combination, this subsection shows the performance of CGAT and CRGAT with different depths. The numerical results are shown in Table IX.

It is observed that with the increment of depths, the performance of the CGAT is degraded. In particular, a sharp decay is induced with $4$ depths. This is due to the oversmoothing issue as analyzed in Remark 1. However, the oversmoothing issue is avoided by the CRGAT, as the expression capacity of the CRGAT is enhanced with the increment of depths. Such a result validates the effectiveness and the importance of the residual-assisted combination.

Furthermore, we adopt the mean average distance (MAD) [26] among the node features to quantify oversmoothing issue, which is calculated by

where $\mathbf{D}\in{\mathbb{R}^{{K}\times{K}}}$ denotes the distance matrix which is calculated by

where $\overline{\mathbf{H}}^{(l)}={\rm Com}\left({\rm Re}(\mathbf{H}^{(l)}),{\rm Im}(\mathbf{H}^{(l)})\right)\in{\mathbb{C}^{{K}\times{2{{\widetilde{F}}(l)}}}}$. The MADs of different layers in CGAT and CRGAT are shown in Table X.

It is observed that the MAD of the CGAT converges to $0$ with a growth of the depths, which indicates that the node features intend to be indistinguishable. However, a larger MAD is obtained in the CRGAT. As the learning capability is roughly positive correlation to the depths, the residual-assisted combination in fact balances the trade-off between enhancing the learning capability and avoiding the oversmoothing issue.

As analysed before, the scalability to different numbers of users of the CRGAT is effected by $|K_{\rm Tr}-K_{\rm Te}|/K_{\rm Tr}$. To improve the optimality performance, instead of generalizing, one can also reuse of the pre-trained CRGAT on a new test set in a transfer learning manner.

Fig. 4 show the transfer learning performance of the CRGAT, where the learnable parameters of the CRGAT trained on the dataset with $K_{\rm Tr}=4$ (Dataset No. 2) is leveraged as the initial parameters for fine-tuning the CRGAT on the dataset with $K_{\rm Tr}=5$ (Dataset No. 3). For comparison, the convergence behavior of retraining the CRGAT on the dataset with $K_{\rm Tr}=5$ (Dataset No. 3) is also plotted. It is observed that with the transfer learning, the CRGAT is able to achieve about $95\%$ optimality performance after $4$ epochs while the generalization performance is about $92\%$ (in Table V). Besides, the convergence speed of the transfer learning is much faster than that of the retraining with a slight optimality performance gain.