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].”

“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].”

“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 .”

“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].”

“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].”

“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 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…”

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

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

“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].”