A Deep Learning Framework for Optimization of MISO Downlink Beamforming

“roblems P1 and P2 has K neurons but that in the BNN for problem P3 has 2K neurons. The Glorot normal initializer [62] is used for weight initialization and biases are initialized to 0. Adam optimizer [63] is used with the MSE metric-based loss function. However, in the second stage of the BNN for problem P3, the metri…”

“Thus, we can use a neural network to approximate the mapping function from h to q [43].”

“Practical algorithms to find optimal solutions are available for P1 [8], [12], [47] and P2 [5], [7], [8], [12], [13], thus supervised learning can be adopted for those problems.”

“To evaluate the performance of the proposed BNNs, we carry out numerical simulations to compare the BNNs with several benchmark solutions (when available), including the optimal beamforming, the ZF beamforming [58], the RZF beamforming [59], and the WMMSE algorithm.”

“According to the Perron-Frobenius theory, for any nonnegative real matrix Ω with spectral radius χ ( Ω ) , there exist a vector δ ≥ 0 such that Ω δ = χ ( Ω ) δ [54].”

“P3 is a difficult nonconvex problem and is usually solved using the iterative WMMSE approach [9, 10], therefore, supervised learning alone is insufficient for this case.”

“…ratio (SINR) balancing problem (also known as interference balancing problem) under a total power constraint [2–4], as a power minimization problem under quality of service (QoS) constraints [5–8], or as a sum rate maximization problem under a total power constraint [2, 9–11].”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“Finally, [36, 37] provide an overview on the recent advances in DL-based physical layer communications and [38] suggests potential applications of DL to the physical layer.”

“Thus, the total time complexity of all convolutional layers measured by the number of multiplications is l c l − 1 c l [52].”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“Before we present the BNN for the SINR balancing problem P1, we first introduce the following lemma to describe the uplink-downlink duality of problem P1 [12].”

“Practical algorithms to find optimal solutions are available for P1 [8], [12], [47] and P2 [5], [7], [8], [12], [13], thus supervised learning can be adopted for those problems.”

“On the contrary, the MSE loss function is highly sensitive to outliers in the dataset because the MSE function tries to adjust the model according to these outlier values, at the expense of other samples [50].”

“The WMMSE algorithm is a good choice to find the locally optimal solutions [9], [10], and such an iterative algorithm ensures the continuity of the mapping from the channel to the solution, and can be learned by a neural network [27], [30].”

“…problems, instead of estimating the beamforming matrix with NK elements, where N is the number of the transmit antennas at the BS and K is the number of users, we exploit the uplink-downlink duality of solutions [5, 6, 12] and predict the virtual uplink power allocation vector with only K elements.”

“Among these efforts, the autoencoder based on unsupervised DL, investigated in [33], [34], is an ambitious attempt to learn an end-to-end communications system [35].”

“To evaluate the performance of the proposed BNNs, we carry out numerical simulations to compare the BNNs with several benchmark solutions (when available), including the optimal beamforming, the ZF beamforming [58], the RZF beamforming [59], and the WMMSE algorithm.”

“Finally, [36, 37] provide an overview on the recent advances in DL-based physical layer communications and [38] suggests potential applications of DL to the physical layer.”

“The power minimization problem can be reformulated as a second-order cone programming (SOCP) [7, 8] or semidefinite programming (SDP) problem [13, 14], which can be solved directly by an optimization software package such as CVX [15].”

“Motivated by the above facts and the universal approximation theorem [43, 44], we propose a general DL framework to achieve not only near-optimal beamforming matrix, but also reduce complexity and latency as compared to the iterative methods.”

“Its optimal solution can also be obtained using iterative algorithms such as Algorithm A of [16] and the dual algorithm of [5, 12].”

“The sum rate performance in [40] was restricted by the quantized codebook constraint.”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“There are K = 4 single-antenna users and these users are distributed uniformly within the coverage of the BS. Note that none of these users is closer to the BS than 100 m. √ The channel of user k is modelled as where ˜ h k ∼ CN ( 0 , I N ) is the small-scale fading [60] and d k = 128 .”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“Furthermore, [28] proposed a real-time CSI feedback architecture called CsiNet-long short-term memory for time-varying massive multiple-inputmultiple-output (MIMO) channels.”

“Another transformation, suggested by [48], is to map the complex channel vector h into two real vectors P(hk) and M(hk), where the former contains the phase information and the latter includes the magnitude information of h.”

“Such a hybrid learning method of the supervised and unsupervised learning can significantly improve the learning performance and also accelerate convergence [29].”

“According to [56], the optimal solution is achieved when the equality of the constraint in problem P1 holds.”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“For example, the DNN was used to predict transmit power [27, 28].”

“DL can also facilitate resource management [25, 26], including power allocation [27–31].”

“Among these efforts, the autoencoder based on unsupervised DL, investigated in [33], [34], is an ambitious attempt to learn an end-to-end communications system [35].”

“which predicted the beamforming matrix in the finite solution space, [41], [42] directly estimated the beamforming matrix; in that case the number of variables to predict increases significantly as the numbers of transmit antennas and users increase, leading to high training complexity of the neural networks.”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“DL can also facilitate resource management [25, 26], including power allocation [27–31].”

“DL can also facilitate resource management [25, 26], including power allocation [27–31].”

“Then given the optimal beamforming ˜ W ∗ , the optimal p ∗ is obtained as the first K components of the dominant eigenvector of the following matrix [53] Υ ( ˜ W ∗ , P max ) = DU D σ 1 Lemma 1 can be easily extended to the case with non-identical noise power levels.”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“…a signal-to-interference-plus-noise ratio (SINR) balancing problem (also known as interference balancing problem) under a total power constraint [2–4], as a power minimization problem under quality of service (QoS) constraints [5–8], or as a sum rate maximization problem under a total power…”

“An improved and faster branch-and-bound algorithm was developed in [37, 51] to find the globally optimal solution, MSE”

“Different from existing works where the CNN was applied to power control [29, 30], resource allocation [45], and wireless scheduling [46], the proposed framework combines a signal processing module with the neural network module by exploiting expert knowledge such as the uplink-downlink duality and…”

“A lot of research has attempted to use DL to address physical layer issues, including channel decoding [17, 18], detection [19–21], channel estimation [22–24], and resource management [25–32].”

“Different from [39, 40] which predicted the beamforming matrix in the finite solution space, [41, 42] directly estimated the beamforming matrix; in that case the number of variables to predict increases significantly as the numbers of transmit antennas and users increase, leading to high training…”

“However, with the exception of [39–42], there are no works focusing on beamforming design in multi-antenna communications based on DL.”

“This article was presented in part at the IEEE International Conference Communication (ICC), Shanghai, China, May 2019 [1].”

“Adaptive beamforming in a vehicular communication scenario [139] and [140] is possible, but frequent beam training may be needed.”

“In this context, a DL based framework for optimization of downlink beamforming is proposed in [92].”

“Most of the existing works adopt MLPs [7], [9], [11], [21] or convolutional neural networks (CNNs) [8], [12].”

“The performance of deep neural networks on beamforming was studied in [20] but without the use of reinforcement learning.”

“Hence, it is a natural way to adopt DL for fast beamforming design and the DL based beamforming design method has been studied in mmWave systems [24], [25] and MISO systems [26], [27].”

“In [11] and [12], deep neural networks (NNs) are used to design a downlink beamforming (BF) matrix.”

“Recent research on physical layer optimization that exploit ML includes a deep learning framework for optimization of multi-input multi-output downlink beamforming [19].”

“technologies [14]–[16], [18], including SEI [11], [19]–[30].”

“In [211], the supervised deep learning approach is extended from the power control problem to a multi-user beamforming problem by utilizing convolutional neural networks and expert knowledge such as uplink-downlink duality (which has been reviewed in Section III-E).”

“According to the uplink-downlink duality in [3] and [25], using the uplink power allocation vector to replace beamforming as the output of the neural network is possible because the same SINR region of the uplink and downlink problems can be achieved.”

“To cope with multi-user scenarios, [14, 15] presented a model using DNN and custom layers, achieving comparable performance with traditional schemes.”

“In particular, the authors of [22] considered a multi-input single-output system and solved three optimization problems.”

“Preliminary investigations have leveraged data-driven DL methods for the design of precoding and combining matrices [38], [39], [40], [41].”

“Although authors in [236], [237], [238], [239] considered deep learning or graph neural networks (GNNs) to estimate the beamforming vectors, the sum rate performance of these methods was not higher than that of the WMMSE algorithm.”

“Different from existing network structures [6]–[8], we use three subnets to extract the features from complex channel transformations and association matrices respectively.”

“In the BF design problem, spectral efficiency is widely used to assess the BF performance as in [14], [33], [35].”

“To maximize the Weighted Sum Rate (WSR) in a MISO channel, the authors of [166] develop a model based on deep learning named the beamforming neural network.”

“Moveover, compared with DL [9], [10], it utilizes the samples generated previously to train the agent and does not need large amount data for off-line training.”

“The common procedure in [147] to solve these problems was to relax them to virtual equivalent problems such that the solution, i.”

“Recently, due to its ability to provide fast and accurate prediction and decision making, DL has shown great promise in many practical applications [6], [9]–[11].”

“In [25], the authors proposed a neural network framework based on the convolutional neural network to address the optimization problems, i.”

“Beamforming, which exploits the degree of freedom (DoF) in the spatial domain, is a powerful technique for improving system capacity [1].”

“works on DL-based beamforming design [3], [4].”

“for the MU-MISO systems [9]–[11] and MU multiple-input”

“A. Simulation Settings 1) Network Settings: Unless stated otherwise, the network parameters are given as follows [22], [33]–[35].”

“Recently, deep learning (DL) has been applied to physical layer communications [15], [16], [23], [24], [25], [27], [28], [29] such as channel state information (CSI) feedback [25], signal detection [26], [27], channel estimation [15], [16], [28], precoder design [29], traffic analysis [30], [31], [32], and end-to-end…”

“MAB-based RL-aided solutions have already shown promising performance in such physical-layer-related problems [177]–[182].”

“In [93], a knowledge-driven DL framework for MISO downlink beamforming is proposed, which includes a neural network for feature extraction and a knowledge-embedded output layer for beamforming recovery.”

“The optimization of downlink beamforming via DL techniques is for the MISO system is done in [269,270].”

“The authors of [16] propose a beamforming neural network (BNN) for minimizing the transmit power of multiuser MISO systems, which employs convolutional neural networks (CNN) and a supervised-learning method to predict the magnitude and direction of the beamforming vectors.”

“Inspired by the recent advances in data-driven models, deep learning-based methods were proposed to solve the power allocation problems in wireless networks to combat the computation complexity [9], [10], [11], [12], [13].”

“In addition, the authors of [3], and [4] have also studied the ISAC systems under MIMO deployments, demonstrating the importance of optimizing beamforming.”

“However, most of these results [17]–[20], [25] consider deep learning models that are highly parametrizable (e.”

“…of the downlink data symbol. χ lk ∈ CN (0 , 1) denotes the data symbol intended for the k -th UE in the l -th cluster and satisfies [24], [38], [39] Following [4], η nlk denotes the corresponding power control coefficients, which is selected to satisfy the power constraint, i.e., By…”

“Similarly, CNNs have also been applied for power control [12] and beamforming design [13] in the multiple-input single-output downlink systems.”

“beamforming has been explored [13], [14], [15], [16], [17].”

“Furthermore, the authors in [37] exploited the structure of the optimal solutions to design DNN for fast beamforming design.”

“2) Network Backbone Design: With pre-processing and post-processing, we adopt the network architecture in [29] to”

“…can be viewed as an efficient tool in solving communication problems, such as channel estimation [18], [19], signal detection [20]–[22], channel modeling [23]–[25], beam selection [26], [27], resource allocations [28]–[30], indoor fingerprinting [31], [32], and smart congestion con-trol [33].”

“For pre-processing stage, EVD operation is used to acquire the eigenvalue, whose complexity is Then, for prediction stage, based on [28] and [29], the complexity for the forward calculation of the network can be expressed as where N L is the number of layers, C Fl , C Kl , F l − 1 , F l are the…”

“In the considered VANET, the VNFs connects the mobile edge hosts and applications within the data plane through the SFC [31], [33].”

“whereα as the portion of the channel estimation overhead in an coherent interval and therefore it satisfies α < 1, Re is the average sum rate for the estimated channels, which can be obtained using the existing neural network method such as that in [14] as if the CSI is perfect, while Rp is the average sum rate for th…”

“To address such a large complexity of the problems, deep learning methods have been widely applied [2]–[6] since they can deal with such a complex problem thanks to a large representational capability of deep neural networks (DNNs) [6].”

“Deep neural networks for downlink beamforming in [13] utilize the low-dimensional structure.”

“In [15], [16], and [17], the DL-based approaches exploiting expert knowledge, i.e., the known structure of optimal solutions, are studied for the beamforming optimization in a single-cell MU-MISO downlink system.”

“Deep learning has emerged as a new paradigm to learn the mapping from the channel state information to the beamforming solution offline, and then infer the near-optimal solution in real time with low complexity and computational latency [7]– [9].”

“Moreover, the works in [20], [21], [22], and [23] focused only on the transmitter optimization assuming perfect CSIT, and thus any joint design of the pilot sequences and the limited feedback along with the”

“When the variance of the received signal noise is equal, that is, 3 optimal active beamforming vector in (10) can be written as [37] 3 The assumption is reasonable when the users are in a similar environment and the user device type are the same.”

“Recently, machine learning [25] has demonstrated great potential in solving the communication problems, such as channel estimation [26], beamforming [27], [28], beam tracking [29] and so on.”

“tion [7]-[11], channel estimation [12]-[14], beamforming and precoding [15], [16], and autoencoder based transceiver designs for fading channels [17].”

“Without loss of generality, the communication channel and the target-free channel are generated using the method in [42].”

“In the literature, it has been reported that unsupervised training achieves higher sum rate [11] which”

“This makes it possible to obtain the precoder with low computational complexity and high realtime performance [29], [30].”

“Recently, few work has explored AI methods to compute an optimized solution for both the beam selection and power control [31], [32], [33].”

“sidered to solve problems whose conventional solutions become too computationally complex, or the search spaces are too vast for considering approaches like genetic algorithms [23], [24], [25], [26], [27].”

“Similar to [40], the channel model is generated using h k = √ m k ˜ h k ∈ C M × 1 , where ˜ h k ∼ CN ( 0 , I M ) is the small-scale fading and m k = 128 .”

“The authors of [43], [44], and [45] used convolutional neural networks (CNNs) with supervised training to approximate traditional iterative beamforming algorithms.”

“heorem in [50] to the continuous mapping in Algorithm 2. Based on results in Theorem 1, next we find solutions through designing the neural networks with the DL technique. Similar to our previous work [47], we introduce a general DL structure to approximate the mapping function from the channel coefficients to the beamf…”

“Conventional learning algorithms need to collect a large number of training samples from the devices and train the machine learning model in a central server [4], [5].”

“In recent years, Deep Learning (DL) has demonstrated strong capabilities in data analysis and feature extraction, and has been widely applied in wireless communication technologies [11].”

“Recently deep learning has been recognized as a new “learn to optimize” approach to design beamforming [3]–[7].”

“A similar conclusion has been demonstrated in [25], where DNNs are used for BF design of multiuser multiple-input single-output (MISO) downlink systems, but the size of the considered problem is rather small.”

“methods [12], [16], [20], [21], since the training process is unsupervised, and is essentially embedded with the solution”

“The authors in [26], [27], and [28] proposed a machine learning method to optimize performance of wireless communication system.”

“, channel estimation [16], joint channel estimation and signal detection [17], precoding [18], and semantic communication [19].”

“Current approaches to training large-scale models are usually directly implemented at the cloud server, which has a powerful computation capacity and collects training samples from widely distributed mobile devices [1, 2].”

“In this way, the proposed MLBF approach is less time-consuming and resource-dependent to be applied than the previous deeplearning based approaches [9]–[12], as the training process is essentially embedded in the solution process while the traditional deep-learning based approaches have to train the parameters of the…”

“With the optimal solution structure revealed in [31], the data driven method that learns the undetermined parameters in the solu-tion structure was proposed in [32], and further extended in [33].”

“In addition, [20], [21] propose an approach in which the channel state information is input to a CNN network to produce a prediction of key parameters in traditional beamforming algorithms.”

“Recently, deep learning (DL) has been applied to physical layer communications [7]–[9], such as channel state information (CSI) feedback [10], channel estimation [11] and precoder design [12], [13].”

“Especially in [25], the authors showed that their proposed scheme can achieve the higher sum rate than the existing algorithms.”

“As a benchmark, the conventional non-robust DL [6], [7] is considered which trains a DNN only with the perfect CSI.”

“There have been some existing works investigating the implementation of wireless FL, e.g., [1], [2], [3], [4].”

“Recently, Xia et al. [9] proposed a deep learning scheme for the downlink beamforming optimization to solve the signal-to-interference-plus-noise ratio (SINR) balancing and power minimization problems.”

“For example, in [7], a DL based framework was proposed for fast power-allocation design.”

“Simulation results of the power minimization problem [5]: a) power performance; b) execution time per sample averaged over 5000 samples under QoS constraints of 5 dB and N = 8, and testbed results of the SINR balancing problem with per-antenna power constraints; c) static channel condition; d) dynamic channel conditio…”

“Deep learning (DL) can always be used to solve optimization tasks while enforcing simple restrictions by applying generic equality and inequality constraints to optimization problems [38].”

“Next to black-box-based approaches [13], [14], [15], [16], [17], [18], [19], [20], [21], in which the cumbersome operations of matrix inverse, eigendecomposition, and bisection search are replaced by deep neural networks trained to either approximate the output of the WMMSE algorithm or to directly maximize the WSR, d…”