# A Data and Model-Driven Deep Learning Approach to Robust Downlink
Beamforming Optimization

Kai Liang, Gan Zheng, Zan Li, ,
Kai-Kit Wong, , and Chan-Byoung Chae K. Liang, and Z. Li are with the State
Key Laboratory of Integrated Service Networks, the School of
Telecommunications Engineering, Xidian University, Xi’an, 710071, China
(email<EMAIL_ADDRESS>zanli@xidian. edu.cn).G. Zheng is with the
School of Engineering, University of Warwick, Coventry, CV4 7AL, UK (email:
gan.zheng@warwick.ac.uk).K.-K. Wong is with the Department of Electronic and
Electrical Engineering, University College London, London, WC1E 6BT, UK
(email: kai-kit.wong@ucl.ac.uk). He is also affiliated with Yonsei Frontier
Lab., Yonseu University, Seoul, 03722, Korea.C.-B. Chae is with School of
Integrated Technology, Yonsei University, Seoul, 03722, Korea (email:
cbchae@yonsei.ac.kr).

###### Abstract

This paper investigates the optimization of the probabilistically robust
transmit beamforming problem with channel uncertainties in the multiuser
multiple-input single-output (MISO) downlink transmission. This problem poses
significant analytical and computational challenges. Currently, the state-of-
the-art optimization method relies on convex restrictions as tractable
approximations to ensure robustness against Gaussian channel uncertainties.
However, this method not only exhibits high computational complexity and
suffers from the rank relaxation issue but also yields conservative solutions.
In this paper, we propose an unsupervised deep learning-based approach that
incorporates the sampling of channel uncertainties in the training process to
optimize the probabilistic system performance. We introduce a model-driven
learning approach that defines a new beamforming structure with trainable
parameters to account for channel uncertainties. Additionally, we employ a
graph neural network to efficiently infer the key beamforming parameters. We
successfully apply this approach to the minimum rate quantile maximization
problem subject to outage and total power constraints. Furthermore, we propose
a bisection search method to address the more challenging power minimization
problem with probabilistic rate constraints by leveraging the aforementioned
approach. Numerical results confirm that our approach achieves non-
conservative robust performance, higher data rates, greater power efficiency,
and faster execution compared to state-of-the-art optimization methods.

###### Index Terms:

Robust beamforming, deep learning, outage probability constraint, multiuser
MISO.

##  I Introduction

Beamforming techniques in multi-user and multi-antenna systems can effectively
improve the throughput and spectral efficiency of the wireless communication
system, and have received long-term attention from both academia and industry
[1, 2, 3, 4]. Existing studies utilize beamforming algorithms with known
channel state information (CSI) at the base station (BS) and perform
interference cancellation and resource allocation among multiple users.
However, perfect CSI in practical scenarios is difficult to obtain, due to the
presence of channel estimation errors, the quantization errors from the user
side feedback, and the time-varying nature of wireless channels.

Since the imperfect CSI can worsen the performance of multi-user multi-antenna
system, this has motivated researchers to carry out the robust design of
beamforming, i.e., the optimal beamforming design based on predefined CSI
error model to mitigate the negative impact of imperfect CSI. Existing works
can usually be divided into three main categories, namely, the worst-case
robust approach [5, 6, 7], the average robust approach [8, 9], and the outage-
constrained approach [10, 11, 12]. The worst-case robust approach aims to
guarantee the worst-case performance over all the potential channel errors
which are assumed to be bounded within some closed sets. Since the worst-case
scenario is a rarely happening event in a multi-antenna system, the solutions
obtained by this approach are usually very conservative and requires high
resource consumptions. The average robust approach aims to optimizing the
average performance, but fails to guarantee the outage requirement, which is
vital for delay- or reliability- sensitive applications in the next generation
communication system. Thus, in this paper, we focus on the outage-constrained
approach, because it is able to guarantee the probabilistic quality of service
(QoS) requirements of users. This approach focuses on the QoS outage
probability constraint based on a probabilistic CSI error model (such as the
Gaussian model).

Solving the outage constrained beamforming optimization problem appears very
hard, and the key challenge is how to deal with the outage probability which
in general does not have a closed-form expression. Traditionally, the outage
constrained beamforming problem can be solved by convex optimization after
approximating the rate outage constraints. In [12], the authors provided two
approximation methods (i.e., Bernstein-type inequality (BTI) method and large
deviation inequality (LDI) method) together with semidefinite relaxation [13,
14] under Gaussian CSI uncertainties to solve the outage constrained robust
beamforming problem and they are still known to achieve the best solution so
far. These methods have been widely used in the studies of robust beamforming
design with CSI uncertainties, such as intelligent reflecting surface [15],
simultaneous wireless information and power transfer (SWIPT) [16],
heterogeneous networks [17], satellite-based Internet of Things [18], and
radar-communication systems [19]. Nonetheless, these methods have a high
computational complexity and algorithmic latency caused by, for example,
iterative algorithms and complex matrix computations. In addition, since these
methods attempt to approximate the outage constraint by finding the upper
bound of outage probability, they usually yield sub-optimal and conservative
solutions as well. Finally, for multi-user multi-input single-output (MU-MISO)
systems, these methods may obtain high-rank solutions due to semidefinite
relaxation, which cannot satisfy the rank-1 constraint and cannot be
decomposed into beamforming vectors. Although Gaussian randomization procedure
[13] may deal with this problem, it further weakens the optimality of the
solution, increases the computational and delay overhead, and even obtains
non-feasible solutions for the outage constraint.

In recent years, deep learning-based beamforming approaches have been
increasingly investigated. It uses deep neural networks (DNNs) to approximate
the mapping of inputs and optimal solutions of traditional optimization
algorithms, thus bypassing the tedious iterative solution process and handling
non-convex optimization problems with ease. Compared to purely data-driven
methods, the data- and model-driven learning method can effectively improve
the training efficiency and performance of neural networks, and has a wide
range of applications in the field of wireless communications, such as
transceiver design in over-the-air federated learning [20], and computation
offloading in mobile-edge computing networks [21]. As for beamforming
optimization, we proposed a model-driven learning approach for MU-MISO systems
with the aid of a convolutional neural network (CNN) in [22], where the CNN’s
output dimension is reduced by adopting an optimal beamforming structure [23].
We further used this method for the case of the absence of uplink and downlink
channel reciprocity as in [24]. This approach is then widely extended for
applications in DNN-based optimal beamforming designs such as universal power
MISO beamforming [25], scalable beamforming optimization though graph neural
network [26], and multi-user mutli-input mutli-output precoding [31]. In
addition to optimal beamforming structures, other types of mathematical model-
driven methods are also used in the design of beamforming based on perfect
CSI. In [27], a deep unfolding network was proposed, which unfolds the
weighted minimum mean-square error (WMMSE) algorithm [28] for solving problem
sum rate maximization problems and learns beamforming matrices through neural
networks. The method was further extended to the design of hybrid beamforming
in millimeter-wave systems [29]. In [30], a model GNN was proposed by using
first-order Taylor’s expansion to approximate inverse matrices and achieve
better performance. However, these methods are designed with perfect CSI at
the transmitters, and cannot directly applied for robust beamforming
optimization.

Meanwhile, many works have emerged for the deep learning based robust
beamforming optimization. The authors in [32] proposed a DNN-based robust
beamforming approach to solve average sum rate maximization problem in MU-MISO
system under Gaussian channel uncertainties. In [33], a graph neural network
(GNN) based robust beamforming method was proposed with Gaussian channel
uncertainties. Nevertheless, these works did not consider the outage
probability performance. In [34], we proposed a data augmentation based DNN
approach to obtain robust beamforming for the outage constrained power
minimization problem, which achieves better power efficiency and the execution
delay. However, this method directly learns beamforming vectors without
dimensional reduction through the optimal structure, making it hard to train
the neural network. A similar method was presented for MU-MISO to address the
outage-constrained rate quantile maximization problem in [35], where
beamforming vectors are recovered by the learned power features and the
regularized zero-forcing (RZF) [36, 37] beamforming structure. This scheme
reduces the dimensionality of the output and makes the neural network easy to
train, but sacrifices the transmission performance due to the inferior
performance of RZF beamforming.

In this paper, we explore the application of data and model-driven deep
learning as a novel approach to address two robust beamforming optimization
problems in MU-MISO systems. Our proposed method exhibits improved rate and
power performance in the presence of channel uncertainty. Additionally, we
achieve lower output dimensions to enhance neural network training efficiency
and employ a more flexible neural network structure to accommodate universal
transmit power. To the best of our knowledge, our work is the first to surpass
the performance and computational efficiency of the traditional solution
presented in [12]. The contributions are summarized in the following:

  * •

We propose a data and model-driven deep learning approach to solve the outage-
constrained minimum rate quantile maximization problem, while adhering to the
rate outage probability constraint and the transmit power constraint,
considering Gaussian channel uncertainties. We first propose a modified
optimal beamforming structure that introduces new trainable interference
features arising from channel estimation errors. These features, along with
the power features, enhance system performance while simultaneously reducing
the neural network’s output dimension. Furthermore, we introduce a specialized
GNN, as described in [26], to train interference and power features for
improved rate performance, scalability and generalizability of the neural
network. Unlike the message passing GNN [39] that can only handle homogeneous
graphs, this GNN realizes the bipartite message-passing (BMP) inference, which
makes it capable of handling heterogeneous vertex types using different neural
networks. After implementing beamforming recovery using the modified optimal
structure, we estimate the rate quantile through data augmentation and Monte
Carlo (MC) sampling. MC sampling is unnecessary for the inference of robust
beamforming, ensuring that it does not increase the execution time or
complexity.

  * •

We further propose a bisection-based deep learning method to address the
robust beamforming design for the more challenging transmit power minimization
problem with the rate outage probability constraint. Existing neural network
methods, such as those discussed in [34], pose challenges in training as they
require simultaneous updating of the neural network parameters and Lagrange
multipliers. To overcome this limitation, we modify the neural network
framework designed for rate quantile maximization. We incorporate transmit
power as an additional input, enabling the neural network to feature universal
transmit power. The trained universal neural network, in conjunction with the
bisection method, allows us to infer the minimum transmit power that satisfies
the outage probability constraints. This leverages the observation that the
transmission rate increases with higher transmit power.

  * •

Numerical results show that our proposed robust beamforming method outperforms
state-of-the-art works, including the classical optimization method [12], the
RZF-model based learning method [35] and the DNN-based learning method without
assistance of model information, in terms of achievable robust rate, execution
time, power consumption and feasibility.

The rest of this paper is organized as follows. System model and problem
formulation are described in Section II, after which the model-driven learning
approach for the rate quantile maximization problem is provided in Section
III. Then, the bisection based deep learning approach for the power
minimization problem is given in Section IV. Numerical results and analysis
are provided in Section V, followed by which we conclude this paper in Section
VI.

Notation: Bold uppercase and lowercase letters (e.g., ${\bf A}$ and ${\bf a}$)
represent matrices and column vectors, respectively. ${\bf I}$ denotes the
identity matrix. The superscripts $(\cdot)^{T}$ and $(\cdot)^{\dagger}$ stand
for transpose and Hermite conjugate transpose of a matrix or vector,
respectively. We use $\mathbb{R}^{m\times n}$ and $\mathbb{C}^{m\times n}$ to
denote the real and complex space of m by n dimensions, respectively.
$Re(\cdot)$ and $Im(\cdot)$ represent the real and imaginary parts of a
complex variable, respectively. $|\cdot|$ and $\|\cdot\|$ refer to the
absolute value operator and the Euclidean norm operator, respectively.
$\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ denote the floor and ceil
functions (i.e. the downward and upward rounding operations), respectively.
$dim({\bf a})$ represents the dimension of a vector ${\bf a}$.

##  II System Model and Problem Formulation

We consider a MU-MISO downlink system in which a BS with $N$-antennas serves
$K$ single-antenna users. Let $\mathcal{N}=\\{1,\cdots,N\\}$ and
$\mathcal{K}=\\{1,\cdots,K\\}$ denote the antenna set and the user set,
respectively. Suppose $x_{k}$ is the transmit signal to the user $k$ with unit
power and the BS transmits with a total power $P$. The transmitted data symbol
$x_{k}$ is mapped onto the antenna array elements by multiplying the
beamforming vector $\mathbf{w}_{k}\in\mathbb{C}^{N\times 1}$. The received
signal at the user $k$ can be expressed as

$\displaystyle y_{k}=\mathbf{h}_{k}^{\dagger}\mathbf{w}_{k}x_{k}+\sum_{j\neq
k}\mathbf{h}_{k}^{\dagger}\mathbf{w}_{j}x_{j}+n_{k},$ (1)

where
$\mathbf{h}_{k}=[h_{1k},\cdots,h_{nk},\cdots,h_{Nk}]^{T}\in\mathbb{C}^{N\times
1}$, and $h_{nk}$ is the channel between the antenna $n$ at the BS and the
user $k$. Then, the overall channel matrix can be denoted as ${\bf H}=[{\bf
h}_{1},\cdots,{\bf h}_{K}]$. $n_{k}$ denotes the additive white Gaussian noise
(AWGN) component with zero mean and variance $\sigma^{2}_{k}$. Therefore, the
signal-to-interference-plus-noise ratio (SINR) that measures quality of the
data detection at the $k$-th user is given by

$\Gamma_{k}=\frac{|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{k}|^{2}}{\sum_{j\neq
k}|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{j}|^{2}+\sigma^{2}_{k}},$ (2)

and the corresponding achievable rate is formulated as

$R_{k}=B\log_{2}(1+\Gamma_{k}),$ (3)

where $B$ denotes the bandwidth.

Considering that it is hard to obtain perfect CSI at the BS in practice, the
channel error model is given as follows:

$\mathbf{h}_{k}=\mathbf{\tilde{h}}_{k}+\mathbf{e}_{k},$ (4)

where $\mathbf{\tilde{h}}_{k}$ is the estimated CSI known at the BS, and
$\mathbf{e}_{k}=[e_{1k},\cdots,e_{nk},\cdots,e_{Nk}]^{T}\in\mathbb{C}^{N\times
1}$ represents the channel error. The overall estimated channel matrix and
channel error matrix are denoted as $\tilde{{\bf H}}=[\tilde{\bf
h}_{1},\cdots,\tilde{\bf h}_{K}]$ and ${{\bf E}}=[{\bf e}_{1},\cdots,{\bf
e}_{K}]$, respectively. The commonly used Gaussian channel error is adopted to
model $\mathbf{e}_{k}$, where each element of $\mathbf{e}_{k}$ independently
identically distributed follows $\mathcal{CN}(0,\sigma_{e}^{2})$.

We first focus on the minimum rate quantile maximization problem, i.e.,
maximizing the corresponding quantile while satisfying the minimum rate outage
probability constraint and the BS transmit power constraint. Note that the
minimum rate (denoted as $R_{min}=\min_{k\in\mathcal{K}}R_{k}$) is the minimum
achievable rate among all users. This problem is given as follows:

P1: $\displaystyle\max_{\mathbf{w}_{k},\forall\mathcal{K}}r$ (5) s.t.
$\displaystyle\text{Prob}\big{(}R_{min}\leq r\big{)}\leq\rho$ (8)
$\displaystyle R_{min}\leq R_{k},\forall k\in\mathcal{K}$
$\displaystyle\sum_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}\leq P,$

where (8) refers to the outage constraint, and $\rho$ is the target outage
probability which often has a low value (e.g., 5%). (8) denotes the power
constraint at the BS and $P$ is the maximum transmit power of the BS.

Another conventional but more challenging problem considered in this paper is
the power minimization problem based on the minimum rate outage probability
constraint, which is given by

P2:
$\displaystyle\min_{\mathbf{w}_{k},\forall\mathcal{K}}\sum_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}$
(9) s.t. $\displaystyle\text{Prob}\big{(}R_{min}\leq r\big{)}\leq\rho$
$\displaystyle R_{min}\leq R_{k},\forall k\in\mathcal{K}.$

Obviously, the optimization problems P1 and P2 are both nonconvex and
difficult to solve. The traditional method solves such problems through
jointly using convex restriction approach and semi-positive definite
optimization (see [12]). However, it suffers from a long convergence time and
high-rank solutions. Although the Gaussian randomization procedure [13] claims
to be able to solve the high-rank matrix issue which is not designed for the
outage constrained problem, it is shown in our simulations that this method
can yield infeasible solutions to the above outage-constrained problems with a
higher probability. Therefore, we will present the proposed data and model-
driven deep learning approach to solve P1 and P2, achieving faster convergence
and better performance.

##  III Rate Quantile Maximization

In this section, we provide a deep leaning approach for the rate quantile
maximization problem (i.e., P1) via an unsupervised learning manner. Suppose
there exists a function $\mathbf{\Psi}_{1}(\cdot)$ with its input being the
estimated channel information and its output being an optimal robust
beamforming solution that satisfies the constraints in problem P1. Then, the
problem P1 can be rewritten as follows:

$\\{{\bf w}_{1},\cdots,{\bf w}_{K}\\}=\mathbf{\Psi}_{1}(\tilde{{\bf
h}}_{1},\cdots,\tilde{{\bf h}}_{K}).$ (10)

Normally, the closed-form expression for function $\mathbf{\Psi}_{1}(\cdot)$
does not exist. Fortunately, the universal approximation theorem [38] has
verified that any continuous-valued functions can be approximated with very
small approximation error through a well-designed DNN. Thus, the original
problem can be turned into a trainable problem. Assume there is a DNN
$\mathbf{\Psi}_{1}(\cdot;\mathbf{\Theta})$ with trainable parameter set
$\mathbf{\Theta}$, which has the ability to approximate (10) as follows:

$\\{{\bf w}_{1},\cdots,{\bf w}_{K}\\}=\mathbf{\Psi}_{1}(\tilde{{\bf
h}}_{1},\cdots,\tilde{{\bf h}}_{K};\mathbf{\Theta}).$ (11)

This allows us to learn the parameter set $\mathbf{\Theta}$ to obtain robust
solutions for the problem P1, which is more tractable than the original
problem.

Different from commonly used data driven learning, we propose a model-based
beamforming structure to deal with channel uncertainties, which together with
data samples speeds up training the neural network. Then, we employ a unique
GNN combined with MC sampling to realize a robust beamforming optimization
while satisfying the outage probability constraint and the power constraint.

###  III-A Model-based Beamforming Structure

With the known CSI at the BS, we have previously proposed a hybrid supervised
and model-based deep learning method in [24] with the aid of the optimal
beamforming structure [23], which is given below

$\mathbf{w}_{k}=\sqrt{p_{k}}\frac{\big{(}\mathbf{I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\mathbf{h}_{j}\mathbf{h}_{j}^{\dagger}\big{)}^{-1}\mathbf{h}_{k}}{\bigg{\|}\big{(}\mathbf{I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\mathbf{h}_{j}\mathbf{h}_{j}^{\dagger}\big{)}^{-1}\mathbf{h}_{k}\bigg{\|}},$
(12)

where $p_{k}$, $q_{k}$ are non-negative trainable parameters and satisfy
$\sum_{k=1}^{K}p_{k}=\sum_{k=1}^{K}q_{k}=P$.
$\mathbf{p}=[p_{1},\cdots,p_{K}]^{T}\in\mathbb{R}^{K\times 1}$ denotes the
downlink power allocation vector, and
$\mathbf{q}=[q_{1},\cdots,q_{K}]^{T}\in\mathbb{R}^{K\times 1}$ is an auxiliary
variable used to determine the direction of optimal beamforming.

This practice reduces the neural network output dimension and thus improves
the training efficiency and accuracy. Unfortunately, this optimal structure
cannot be directly applied to the case of channel uncertainty due to the fact
that the neural network needs to reconstruct the beamforming vectors based on
the known CSI and the learned power parameters. To this end, we modify the
optimal beamforming structure in order to make it suitable for the case of
imperfect CSI. Specifically, we introduce a key feature implying the extra
interference arising from channel estimation errors into the optimal
beamforming structure (12). The modified robust beamforming structure is given
as follows:

$\hat{\bf
w}_{k}=\sqrt{p_{k}}\frac{\bigg{(}(1+s_{k})\mathbf{I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf
h}_{j}\tilde{\bf h}_{j}^{\dagger}\bigg{)}^{-1}\tilde{\bf
h}_{k}}{\bigg{\|}\bigg{(}(1+s_{k})\mathbf{I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf
h}_{j}\tilde{\bf h}_{j}^{\dagger}\bigg{)}^{-1}\tilde{\bf h}_{k}\bigg{\|}},$
(13)

where $\mathbf{s}=[s_{1},\cdots,s_{K}]^{T}\in\mathbb{R}^{K\times 1}$ can be
viewed as the extra interference caused by the channel estimation error. It is
very difficult to rigorously prove the robust beamforming structure with
outage probabilistic constraints, so a mathematical interpretation of (13) is
provided in the Appendix at the end of this paper. We denote robust
beamforming matrix as $\hat{\bf W}=[\hat{\bf w}_{1},\cdots,\hat{\bf
w}_{K}]^{T}$. It is worth noting that (13), although not rigorously
mathematically derived, is able to approximate the value of ${\bf s}_{k}$
through a neural network description without the need for its exact
expression. This structure enables us to reconstruct the robust beamforming
matrix $\hat{\bf W}$ from the output $[{\bf p},{\bf q},{\bf s}]$ of the neural
network when only the estimated CSI is known, and the output dimension of the
neural network is reduced from $2NK$111Since neural network frameworks such as
PyTorch or TensorFlow have difficulty dealing with complex numbers, the
complex channel must be divided into real and imaginary parts. to $3K$.

###  III-B Bipartite GNN for Feature Learning

In our previous works [22, 34], power features $[{\bf p},{\bf q}]$ can be
learned by a fully connected neural network (FCNN) or CNN. Considering that
GNNs can achieve state of the art performance, and the potential scalability
as well as generalizability due to their permutation equivariant and
permutation invariant properties [26, 40, 41, 42], this paper employs a
special bipartite GNN (BGNN) as in [26] to realize the BMP inference and thus
to learn the features in (13). Different with [26], the weights of the edges
in the BGNN are estimated channel information $\tilde{{\bf H}}$ rather than
perfect channel information ${\bf H}$. Besides, we need not only to learn the
power feature $[{\bf p},{\bf q}]$ with a BGNN, but also to learn the
interference feature ${\bf s}$ with another BGNN considering the effect of
channel errors. The reason of using two BGNNs to learn the interference
feature ${\bf s}$ and power features $[{\bf p},{\bf q}]$ separately is due to
the difference in their ranges of values. For narrative simplicity, in this
subsection we use the notation ${\bf g}_{k}$ to denote the features of user
$k$ learned by the BGNN, which either stands for being ${\bf
s}_{k}\in\mathcal{R}$ or $[{\bf p}_{k},{\bf q}_{k}]\in\mathcal{R}^{2}$.

Figure 1: Graph representation and deep learning based BMP inference with
$N=2,K=2$.

The MU-MISO system can be represented as a weighted bipartite graph
$\mathcal{G}=(\mathcal{N},\mathcal{K},\tilde{\bf H})$, where the $\mathcal{N}$
and $\mathcal{K}$ respectively stand for disjoint vertex sets of antennas and
users, and estimated channel information $\tilde{\bf H}$ refers to the weight
matrix of all edges, as shown in Fig. 1. Each antenna-user pair $(n,k),\forall
n\in\mathcal{N},\forall k\in\mathcal{K}$ is connected through an edge with
weight $\tilde{h}_{nk}$. At this point, we realize the division of the basic
network entity for the MU-MISO system by considering each antenna of the BS
and the user as a vertex. In this paper, we adopt a deep learning method based
on vertices parameter sharing [26, 43], which facilitates the GNN to satisfy
the desired permutation equivariance properties and is irrelevant to how the
graph is constructed. The basic idea is that each vertex can complete its own
computation independently based on part of the estimated channel information,
and then transfer the relevant statistics (named as messages) between the
vertices to avoid the lack of overall channel information. By iterating the
this process, the overall computation of the system can be formed, and finally
the optimal beamforming of the MU-MISO system can be realized.

The iteration process of vertex parameter sharing for beamforming feature
calculation can be achieved through the BMP inference from the literature
[26], which defines three types of vertex operators, i.e., user message
generators $\mathcal{C}(\cdot)$, antenna message generators
$\mathcal{M}(\cdot)$, and feature decision makers $\mathcal{D}(\cdot)$. We use
an $L$-layer BGNN to realize the BMP inference, where each layer of the BGNN
implements an iteration in the BMP inference, as shown in Fig. 1. In the
following, we will briefly introduce the deep learning based BMP inference,
see [26] for more details.

Denote the BGNN as $\mathbf{\Phi}(\tilde{{\bf H}};\mathbf{\Theta})$ with the
trainable parameter set $\mathbf{\Theta}$. In $l+1$-th layer of the BGNN,
vertex operators $\mathcal{C}(\cdot)$, $\mathcal{M}(\cdot)$, and
$\mathcal{D}(\cdot)$ are respectively implemented by three FCNNs
$\mathbf{\Phi}_{\mathcal{C}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{C}_{l+1}})$,
$\mathbf{\Phi}_{\mathcal{M}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{M}_{l+1}})$
and
$\mathbf{\Phi}_{\mathcal{D}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{D}_{l+1}})$
as follows:

${\bf c}_{kn}^{(l+1)}=\mathcal{C}\bigg{(}{\bf g}_{k}^{(l)},{\bf
b}_{k}^{(l)},\tilde{h}_{ki}\bigg{)}=\mathbf{\Phi}_{\mathcal{C}_{l+1}}\bigg{(}{\bf
g}_{k}^{(l)},{\bf
b}_{k}^{(l)},\tilde{h}_{kn};\mathbf{\Theta}_{\mathcal{C}_{l+1}}\bigg{)},$ (14)
$\displaystyle{\bf b}_{nk}^{(l+1)}$ $\displaystyle=\mathcal{M}\bigg{(}{\bf
m}_{nk}^{(l)},{\bf c}_{n}^{(l+1)},\tilde{h}_{kn}\bigg{)}$ (15)
$\displaystyle=\mathbf{\Phi}_{\mathcal{M}_{l+1}}\bigg{(}{\bf
m}_{nk}^{(l)},{\bf
c}_{n}^{(l+1)},\tilde{h}_{kn};\mathbf{\Theta}_{\mathcal{M}_{l+1}}\bigg{)},$
${\bf g}_{k}^{(l+1)}=\mathcal{D}\bigg{(}{\bf
m}_{k}^{(l+1)}\bigg{)}=\mathbf{\Phi}_{\mathcal{D}_{l+1}}\bigg{(}{\bf
m}_{k}^{(l+1)};\Theta_{\mathcal{D}_{l+1}}\bigg{)},$ (16)

where the superscript $(\cdot)^{(l)}$ represents the value of $l$-th layer. In
the $l+1$-th layer, each user vertex has a copy of sub neural networks
$\mathbf{\Phi}_{\mathcal{C}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{C}_{l+1}})$
and
$\mathbf{\Phi}_{\mathcal{D}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{D}_{l+1}})$,
while each antenna vertex shares the same sub neural network
$\mathbf{\Phi}_{\mathcal{M}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{M}_{l+1}})$.
Therefore, the trainable parameter set of the BGNN can be denoted as

$\mathbf{\Theta}=\big{\\{}\mathbf{\Theta}_{\mathcal{C}_{l}},\mathbf{\Theta}_{\mathcal{M}_{l}},\mathbf{\Theta}_{\mathcal{D}_{l}},l=1,\cdots,L\big{\\}}.$
(17)

${\bf b}_{k}^{(l+1)},{\bf m}_{nk}^{(l+1)},{\bf c}_{n}^{(l+1)},$ and ${\bf
m}_{k}^{(l+1)}$ are given below

$\displaystyle{\bf b}_{k}^{(l+1)}=\mathcal{P}(\\{{\bf b}_{nk}^{(l+1)},\forall
n\in\mathcal{N}\\}),$ (18) $\displaystyle{\bf m}_{nk}^{(l+1)}=\\{{\bf
b}_{nk}^{(l+1)},\mathcal{P}(\\{{\bf b}_{nj}^{(l+1)},\forall j\neq k\\})\\},$
(19) $\displaystyle{\bf c}_{n}^{(l+1)}=\mathcal{P}(\\{{\bf
c}_{kn}^{(l+1)},\forall n\in\mathcal{K}\\}),$ (20) $\displaystyle{\bf
m}_{k}^{(l+1)}=\mathcal{P}(\\{{\bf m}_{nk}^{(l+1)},\forall
n\in\mathcal{N}\\}),$ (21)

where $\mathcal{P}(\cdot)$ stands for the well-known sum pooling operator,
which is widely used in existing works [44, 45]. Specifically, let $\mu_{a}$
be a variable of vertex $a\in\mathcal{A}$, and we use a simplified pooling
operator as $\mathcal{P}(\\{\mu_{a},\forall
a\in\mathcal{A}\\})=\sum_{a\in\mathcal{A}}\mu_{a}$.

Figure 2: The proposed overall neural network framework for
$\mathbf{\Psi}_{1}(\hat{\bf H};\mathbf{\Theta})$.

There are three main steps in $l+1$-th layer of the BGNN, which are detailed
as follows:

(1) User message generation. The user message ${\bf
c}_{kn}^{(l+1)}\in\mathbb{R}^{M}$ is generated at user vertex $k$ by (14) and
will be forwarded to the antenna $n$, where the hyperparameter $M$ is the
dimension of this message. The neural network
$\mathbf{\Phi}_{\mathcal{C}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{C}_{l+1}})$
realizes a combine operation of the previous feature decision ${\bf
g}_{k}^{(l)}$, antenna message ${\bf b}_{k}^{(l)}\in\mathbb{R}^{M}$ and
$\tilde{h}_{kn}$. Input and output dimensions of the neural network
$\mathbf{\Phi}_{\mathcal{C}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{C}_{l+1}})$
are $M+2+dim({\bf g}_{k}^{(l)})$ and $M$, where $dim({\bf g}_{k}^{(l)})=2$ for
power features $[{\bf p}_{k},{\bf q}_{k}]$ and $dim({\bf g}_{k}^{(l)})=1$ for
the interference feature ${\bf s}_{k}$. Therefore, the size of neural network
$\mathbf{\Phi}_{\mathcal{C}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{C}_{l+1}})$
is decoupled with the network size $N$ and $K$.

(2) Antenna message generation. After receiving user messages $\\{{\bf
c}_{kn},\forall k\in\mathcal{K}\\}$, the antenna vertex $n$ adopts the
aggregation operation to obtain ${\bf c}_{n}^{(l+1)}\in\mathbb{R}^{M}$ via
(20). Then, it generates the antenna message ${\bf
b}_{nk}^{(l+1)}\in\mathbb{R}^{M}$ by (15), where the neural network
$\mathbf{\Phi}_{\mathcal{M}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{M}_{l+1}})$
performs a combination operation on the previous antenna information ${\bf
m}_{nk}^{(l)}\in\mathbb{R}^{2M}$, ${\bf c}_{n}^{(l+1)}$ and $\tilde{h}_{kn}$.
The input and output dimensions of
$\mathbf{\Phi}_{\mathcal{M}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{M}_{l+1}})$
are $3M+2$ and $M$, respectively. Subsequently, the antenna vertex $n$ updates
${\bf m}_{nk}^{(l+1)}$ by (19) and forwards ${\bf b}_{nk}^{(l+1)}$ as well as
${\bf m}_{nk}^{(l+1)}$ to the user vertex $k$.

(3) Feature decision. The user $k$ aggregates all the received ${\bf
m}_{nk}^{(l+1)},\forall n\in\mathcal{N}$ to obtain ${\bf
m}_{k}^{(l+1)}\in\mathbb{R}^{2M}$ by (21), and then generates the feature
decision ${\bf g}_{k}^{(l+1)}$ through
$\mathbf{\Phi}_{\mathcal{D}_{l+1}}(\cdot;\mathbf{\Theta}_{\mathcal{D}_{l+1}})$
as in (16), without considering the network size $N$ and $K$. Besides, it
updates ${\bf b}_{k}^{(l+1)}$ by the aggregation operation according to (18).

Since each layer of the BGNN realizes one iteration of the BMP inference,
through several times of iterations (e.g., $L$), user vertices are able to
finally obtain the converged ${\bf g}_{k},\forall k\in\mathcal{K}$, which are
used to recover the robust beamforming matrix (13).

###  III-C Overall Neural Network framework

After introducing the BGNN , we provide the overall neural network framework
to address the outage constrained rate quantile maximization problem P1, as
shown in Fig. 2. Specifically, we provide a data augmentation method to
estimate the rate quantile and to gain robust beamforming solutions. The input
of the proposed network is the estimated CSI $\mathbf{\tilde{H}}$ with a
dimension of $2NK$ by splitting the real part $Re(\mathbf{\tilde{H}})$ and the
imaginary part $Im(\mathbf{\tilde{H}})$, and the outputs are the key features
of robust beamforming $[{\bf s},{\bf p},{\bf q}]$ with a dimension of $3K$ in
total. These features are used to recover beamforming solutions, and together
with MC sampling to estimate rate quantile $\hat{r}$ which satisfied the
constraint (8). Four main modules comprise the proposed network, which are
detailed in the following:

  * •

S-Net. This network is designed to learn the interference component caused by
the channel estimation errors, i.e., ${\bf s}\in\mathbb{R}^{K}$. It can be
implemented by the aforementioned BGNN in Section III-B and is denoted as
$\mathbf{\Phi}_{\mathcal{S}}(\tilde{{\bf H}};\mathbf{\Theta}_{\mathcal{S}})$
with the parameter set
$\mathbf{\Theta}_{\mathcal{S}}=\big{\\{}\mathbf{\Theta}_{\mathcal{C}_{l}}^{\mathcal{S}},\mathbf{\Theta}_{\mathcal{M}_{l}}^{\mathcal{S}},\mathbf{\Theta}_{\mathcal{D}_{l}}^{\mathcal{S}}\forall
l=1,\cdots,L\big{\\}}$. The hyperparameter $M$ in S-Net is denoted
$M_{\mathcal{S}}$. The details of the adopted BGNN will be given in Section V.

  * •

Power-Net. Another BGNN is used to form the  Power-Net, which aims to learn
the power features of the beamforming (namely, ${\bf p}\text{ and }{\bf q}$).
To distinguish it from the S-Net, we denote Power-Net as
$\mathbf{\Phi}_{\mathcal{P}}(\tilde{{\bf H}};\mathbf{\Theta}_{\mathcal{P}})$
with the parameter set
$\mathbf{\Theta}_{\mathcal{P}}=\big{\\{}\mathbf{\Theta}_{\mathcal{C}_{l}}^{\mathcal{P}},\mathbf{\Theta}_{\mathcal{M}_{l}}^{\mathcal{P}},\mathbf{\Theta}_{\mathcal{D}_{l}}^{\mathcal{P}}\forall
l=1,\cdots,L\big{\\}}$. The hyperparameter $M$ in P-Net is denoted
$M_{\mathcal{P}}$.

  * •

Beamforming Recovery (BR) Module. The purpose of this module is to find the
robust beamforming matrix $\hat{\mathbf{W}}$ based on (13) from the downlink
and virtual uplink power ${\bf p}\text{ and }{\bf q}$ obtained in Power-Net,
the interference feature ${\bf s}$ obtained in S-Net, and the estimated CSI
$\mathbf{\tilde{H}}$. Note that this module does not contain parameters to be
trained and optimized.

  * •

Data Augmentation based Quantile Estimation (DAQE) Module. This module is used
to evaluate the rate quantile $\hat{r}$. When we calculate the output
$\hat{\mathbf{W}}$ in the BR module, the perfect channel information
$\mathbf{H}$ is unknown, which fails to realize the robust design of
beamforming. To this end, we need to calculate the outage performance to
obtain the value of loss function and to evaluate the neural network’s
performance, but the challenge lies in that the outage constraint (8) does not
have a closed-form expression. A natural idea to solve this challenge is that
we may adopt MC sampling to estimate the value of rate quantile, which has
been widely used in conventional convex optimization methods as in [46, 47].
This inspired us to provide the so called data augmentation based quantile
estimation method, as described in Algorithm 1. Specifically, we generate a
potential channel error set $\\{{\bf E}_{1},\cdots,{\bf E}_{U}\\}$ with $U$
elements through sampling from a certain distribution, e.g., circularly
symmetric complex Gaussian distribution with zero mean and variance
$\sigma_{e}^{2}$. This combined with the estimated CSI $\mathbf{\tilde{H}}$
forms a set of potential actual channel $\\{{\bf H}_{1},\cdots,{\bf H}_{U}\\}$
via (4). Then we can calculate all the possible rate set $\\{R_{k,u},\forall
k\in\mathcal{K},u=1,\cdots,U\\}$ given beamforming matrix $\hat{\mathbf{W}}$
and the potentially actual channel set $\\{{\bf H}_{1},\cdots,{\bf H}_{U}\\}$
through (3). This is followed by a minimum operation among all users and the
minimum rate set $\\{R_{min,1},\cdots,R_{min,U}\\}$ is obtained. After that,
we sort them in ascending order and yield the set
$\\{R_{min,(1)},\cdots,R_{min,(U)}\\}$ , where $R_{min,(u)}$ denotes the
$u$-th smallest element in this set. Thus, we can estimate the value of rate
quantile $\hat{r}$ via a linear interpolation method. To be specific, after
mapping $\rho\in(0,1)$ to the sample indices $[0,U]$, we can find the location
of quantile between two samples with indices $\lfloor\rho U\rfloor$ and
$\lceil\rho U\rceil$ in the sorted order, and then we can obtain the rate
quantile by

$\hat{r}=R_{min,(\lfloor U\rho\rfloor)}+\beta(R_{min,(\lceil
U\rho\rceil)}-R_{min,(\lfloor U\rho\rfloor)}),$ (22)

where $\beta$ is the fractional part of the computed quantile index $\rho U$.
In our previous studies, such methods combined with FCNNs have been applied to
the beamforming problem with SINR probabilistic constraints [34], and to the
beamforming problem with SINR and harvested energy probabilistic constraints
in SWIPT systems [48].

Algorithm 1 Data Augmentation based Quantile Estimation

1: Obtain the estimated channel information $\tilde{{\bf H}}$ and the learned
beamforming matrix $\hat{\bf W}$.

2: Generate $U$ channel error samples $\\{{\bf E}_{1},\cdots,{\bf
E}_{U}\\}\overset{iid}{\sim}\mathcal{CN}(0,\sigma_{e}^{2})$.

3: Obtain potential actual CSI set $\\{{\bf H}_{1},\cdots,{\bf H}_{U}\\}$ by
(4) and then calculate the corresponding rate set $\\{R_{k,u},\forall
k\in\mathcal{K},u=1,\cdots,U\\}$ by (3) with learned $\hat{\bf W}$.

4: Find the minimum rate set $\\{R_{min,1},\cdots,R_{min,U}\\}$ among all
users, and then sort them in ascending order and yield the set
$\\{R_{min,(1)},\cdots,R_{min,(U)}\\}$ , where $R_{min,(u)}$ denotes the
$u$-th smallest element in this set.

5: Estimate the rate quantile $\hat{r}$ by (22).

Now we could summarize the training and testing strategies to the proposed
neural network. The loss function is defined as follows:

$\text{Loss}=-\hat{r}=-\tau\big{(}\mathbf{\Psi}_{1}(\tilde{{\bf
H}};\mathbf{\Theta})\big{)},$ (23)

where
$\mathbf{\Theta}=\\{\mathbf{\Theta}_{\mathcal{S}},\mathbf{\Theta}_{\mathcal{P}}\\}$
is the overall parameter set. The function $\tau(\cdot)$ defines a mapping
relation from $\hat{\bf W}$ to $\hat{r}$ and represents the quantile
estimation process of the DAQE module.

At the $t$-th step of the training stage, we generate a mini-batch set
$\mathcal{H}$ with the cardinality of $|\mathcal{H}|$ as the input of the
neural network, which contains the estimated CSI $\tilde{{\bf H}}$. The S-Net
and Power-Net are computed in parallel to get the beamforming feature $[{\bf
s},{\bf p},{\bf q}]$ which together with the estimated CSI form the robust
beamforming matrix $\hat{\mathbf{W}}$. Then, the estimated rate quantile
$\hat{r}$ can be obtained through the DAQE module, and $-\hat{r}$ is used as
the loss function defined in (23) to train the overall neural network by the
back-propagation algorithm. The parameter set $\mathbf{\Theta}$ can be trained
by stochastic gradient descent (SGD) algorithm [49] or its variants, such as
the Adam algorithm [50], which is updated as follows:

$\mathbf{\Theta}^{(t)}=\mathbf{\Theta}^{(t-1)}+\frac{\alpha}{|\mathcal{H}|}\sum_{\tilde{\bf
H}\in\mathcal{H}}\tau\big{(}\nabla_{\mathbf{\Theta}}\mathbf{\Psi}_{1}\big{(}\tilde{\bf
H};\mathbf{\Theta}^{(t-1)}\big{)}\big{)},$ (24)

where $\alpha$ represents the learning rate of the neural network. Next, we
will briefly discuss differentiability of rate quantile $\hat{r}$. The rate
function defined in (3) is differentiable with respect to the output of neural
network $[{\bf s},{\bf p},{\bf q}]$, which is also differentiable with respect
to the parameter $\mathbf{\Theta}$. The MC-sampling based the quantile value
is a convex combination of the $\lfloor U\rho\rfloor$-th and $\lceil
U\rho\rceil$-th smallest values of sampled min-rate value, which will not
change the differentiability. Therefore, the estimated quantile is
differentiable according to the chain rule of differentiation.

When it comes to the testing stage, we can obtain the beamforming matrix via
forward propagation in the well trained neural network. The DAQE module will
not be involved at this stage, which means that we do not need to perform MC
sampling to obtain this robust beamforming matrix, reducing the latency of the
algorithm. The computational complexity of neural network inference is
analyzed as follows. Assume that the FCNN on each vertex has only one hidden
layer with number of neurons $z$. The computational complexity of a BGNN is
$O(LNKzM)$. The computational complexity of beamforming recovery by (13) is
dominated by the inverse operation, which is $O(KN^{3})$. Therefore, the total
inference complexity for problem P1 is
$O(LNKz(M_{\mathcal{S}}+M_{\mathcal{P}})+KN^{3})$.

##  IV Power Minimization using Bisection Algorithm

Using deep learning method to deal with the power minimization problem with
the outage-based constraint (i.e., P2) is more challenging, because it
encounters MC sampling of many variables and frequent Lagrange multipliers
updating, which increase the computational and latency overhead of training
stage, as in our previous work [34]. Another problem need to be addressed is
that traditional deep learning methods are trained for a fixed transmit power,
resulting in the lack of scalability of the neural network. In practice, a
fixed power is unrealistic, and should vary within a certain range in
accordance with different network topologies and power optimization schemes.
For instance, the heterogeneous network may includes macro-, micro- and pico-
cells with different transmit power budgets [51], and a BS will adaptively
adjust its maximum transmit power according to its access user number and
service requirements to reduce energy consumption. To this end, we first
extend the above-mentioned neural network framework to incorporate the feature
of universal power. Then, we propose a bisection-based approach to the power
minimization problem.

###  IV-A Neural Network with Universal Power

Figure 3: Modified framework for $\mathbf{\Psi}_{2}(\tilde{\bf
H},P;\mathbf{\Theta})$.

To achieve robust beamforming optimization with universal power budget, an
intuitive but inefficient idea is that we could train multiple DNNs for
various transmission power level. This practice obviously suffers from high
computational complexity and storage, thus hindering the implementation of
these DNNs. To address this challenge, we take the transmit power as an
additional input for the proposed neural network, which is similar as the work
for beamforming design with perfect CSI [25]. The modified neural network can
be represented as follows:

$\\{{\bf w}_{1},\cdots,{\bf w}_{K}\\}=\mathbf{\Psi}_{2}(\tilde{\bf
H},P;\mathbf{\Theta}),$ (25)

where $\mathbf{\Psi}_{2}(\tilde{\bf H},P;\mathbf{\Theta})$ has a similar
network framework as $\mathbf{\Psi}_{1}(\tilde{\bf H};\mathbf{\Theta})$,
except that the transmit power budget is added as an input. The Loss function
of $\mathbf{\Psi}_{2}(\tilde{\bf H},P;\mathbf{\Theta})$ is
$\text{Loss}=-\hat{r}=-\tau\big{(}\mathbf{\Psi}_{2}(\tilde{{\bf
H}};\mathbf{\Theta})\big{)}$. The architecture of the modified neural network
$\mathbf{\Psi}_{2}(\tilde{\bf H},P;\mathbf{\Theta})$ is shown in Fig. 3.

To be specific, inputs of the modified neural network include the estimated
CSI and the transmit power, with the total dimension of $2NK+1$. Transmit
power can be viewed as a random variable and is uniformly sampled from a
specific range of power variations, namely,
$P\sim\text{Uniform}(\underline{P},\bar{P})$, where $\underline{P}\text{ and
}\bar{P}$ denote the potential minimum and maximum transmit power level,
respectively. Then, $[{\bf s},{\bf p},{\bf q}]$ can be learned from the S-Net
and Power-Net with a total output dimension of $3K$ for the universal power.
Note that in order to satisfy the power constraint, the output of Power-Net
needs to be normalized , i.e., $\sum_{k=1}^{K}p_{k}=\sum_{k=1}^{K}q_{k}=1$,
which are then scaled to the total transmit power $P$ in the beamforming
recovery module. In this way, we can evaluate the maximum rate quantile
$\hat{r}$ that stands for the maximum achievable rate while satisfying the
outage constraint.

###  IV-B Bisection Algorithm

Since the sum rate of the MU-MISO system increases monotonically with the
increasing transmission power of the BS, we can use the bisection method to
find the minimum transmit power from the potential power region. This
motivates us to solve the power minimization problem P2 by the bisection
method with the aid of the improved neural network proposed in the previous
sub-section, as shown in Algorithm 2. Let us first set a range of possible
transmit power $[\underline{P},\bar{P}]$ and a minimum rate target $r$, and
use a trained universal power neural network to infer the maximum rate
quantile $\hat{r}$. We utilize bisection to find the minimum power until the
iterative algorithm stopping condition (e.g., $|\hat{r}-r|\leq\varepsilon$) is
satisfied, where $\varepsilon$ is an arbitrarily small positive number. It is
worth noting that in this process, the neural network does not need to be
retrained due to its power universal property, so this method only increases
the computational overhead of the bisection method, which is often considered
to be an efficient method.

Algorithm 2 The bisection method based power minimization algorithm

1: Initial upper and lower limits of total transmit power $\bar{P}$ and
$\underline{P}$, target minimum rate $r$, the outage probability requirement
$\rho$, and stopping threshold $\varepsilon$.

2: while $|\hat{r}-r|\leq\varepsilon$ do

3: $P=\frac{\underline{P}+\bar{P}}{2}$;

4: Obtain $\hat{r}$ and $\hat{\bf W}$ by the modified network
($\mathbf{\Psi}_{2}$) inference with input $P$ .

5: if $\hat{r}<r$ then

6: $P=\underline{P}$

7: else

8: $P=\bar{P}$

9: end if

10: end while

11: return ${\bf W}^{*}=\hat{\bf W}$.

Let us now summarize the training and testing strategies of the modified
neural network to solve power minimization problem P2. During the training
stage, a mini-batch set $\mathcal{H}$ with the cardinality of $|\mathcal{H}|$
containing numerous estimated CSI and transmit power is generated for learning
the parameter set, which can be updated as follows:

$\mathbf{\Theta}^{(t)}=\mathbf{\Theta}^{(t-1)}+\frac{\alpha}{|\mathcal{H}|}\sum_{\\{\tilde{\bf
H},P\\}\in\mathcal{H}}\nabla_{\mathbf{\Theta}}\tau\big{(}\mathbf{\Psi}_{2}\big{(}\tilde{\bf
H},P;\mathbf{\Theta}^{(t-1)}\big{)}\big{)}.$ (26)

Once the network $\mathbf{\Psi}_{2}$ is trained, we can use it combined with
bisection algorithm to infer the the solution of P2. Note that unlike solving
P1, the bisection method requires the use of estimated rate quantile $\hat{r}$
when solving problem P2, where MC sampling is essential and will increase the
computation complexity. The computation complexity of rate quantile estimation
and bisection algorithm are $O(U\log_{2}(U)KN)$ and
$\log_{2}\big{(}\frac{\bar{P}-\underline{P}}{\varepsilon}\big{)}$,
respectively. Therefore, the total complexities to solve problem P2 is
$O\big{(}\log_{2}\big{(}\frac{\bar{P}-\underline{P}}{\varepsilon}\big{)}\big{[}LNKz(M_{\mathcal{S}}+M_{\mathcal{P}})+KN^{3}+U\log_{2}(U)KN\big{]}\big{)}$.
However, the execution time of this method is still much shorter than that of
traditional optimization algorithms, and this will be demonstrated in Section
V.

##  V Numerical Results

In this section, we provide numerical results to validate and evaluate the
performance of the proposed deep learning-based robust beamforming approach.
For the sake of comparison, we adopt the same parameter setting as [35].
Unless otherwise specified, we consider a MU-MISO system consisting of $N=4$
transmit antennas and $K=4$ users. Rayleigh fading is used to model the
estimated MISO channel $\tilde{{\bf H}}$, where each single link (from an
antenna to a user) follows a circularly symmetric complex Gaussian
distribution with zero mean and unit variance, namely,
$h_{nk}\sim\mathcal{CN}(0,1),~{}\forall n\in\mathcal{N},\forall
k\in\mathcal{K}$. The total bandwidth $B=10$ MHz, the transmission power
$P=30$ dBm, and the variance of channel estimation error
$\sigma_{e}^{2}=0.075$. Considering path loss and inter-cell interference, we
set the noise power spectral density (PSD) to -75 dBm/Hz. The statistical
performance requirement induced by the channel uncertainties is set as the
largest outage probability $\rho=0.05$.

TABLE I: The Architecture and Parameters of FCNNs.
$\mathbf{\Phi}_{\mathcal{C}_{l}}^{\mathcal{S}}(\cdot;\mathbf{\Theta}_{\mathcal{C}_{l}}^{\mathcal{S}})$.
---
Layers | Input dimension | Output dimension
Dense + ReLu | $M_{\mathcal{S}}+3$ | 200
Dense + Tanh | 200 | $M_{\mathcal{S}}$
$\mathbf{\Phi}_{\mathcal{C}_{l}}^{\mathcal{P}}(\cdot;\mathbf{\Theta}_{\mathcal{C}_{l}}^{\mathcal{P}})$.
Layers | Input dimension | Output dimension
Dense + ReLu | $M_{\mathcal{P}}+4$ | 200
Dense + Tanh | 200 | $M_{\mathcal{P}}$
$\mathbf{\Phi}_{\mathcal{M}_{l}}^{\mathcal{S}}(\cdot;\mathbf{\Theta}_{\mathcal{S}})$
and
$\mathbf{\Phi}_{\mathcal{M}_{l}}^{\mathcal{P}}(\cdot;\mathbf{\Theta}_{\mathcal{P}})$.
Layers | Input dimension | Output dimension
Dense + ReLu | $3M+2$ | 200
Dense + Tanh | 200 | $M$
Note: $M=M_{\mathcal{S}}$ for $\mathbf{\Phi}_{\mathcal{M}_{l}}^{\mathcal{S}}$,
and $M=M_{\mathcal{P}}$ for $\mathbf{\Phi}_{\mathcal{M}_{l}}^{\mathcal{P}}$
$\mathbf{\Phi}_{\mathcal{D}_{l}}^{\mathcal{S}}(\cdot;\mathbf{\Theta}_{\mathcal{D}_{l}}^{\mathcal{S}})$.
Layers | Input dimension | Output dimension
Dense + ReLu | $2M_{\mathcal{S}}+1$ | 200
Dense | 200 | 1
$\mathbf{\Phi}_{\mathcal{D}_{l}}^{\mathcal{P}}(\cdot;\mathbf{\Theta}_{\mathcal{D}_{l}}^{\mathcal{P}})$.
Layers | Input dimension | Output dimension
Dense + ReLu | $2M_{\mathcal{P}}+1$ | 200
Dense + Softmax | 200 | 2

All simulation results are generated by using a computer with an Intel i7-7700
CPU and an NVIDIA Titan Xp GPU. Note that we also used the neural network
design with universal power (same as the method in Section IV) for
$\mathbf{\Psi}_{1}(\hat{\bf H};\mathbf{\Theta})$ in our simulations to allow
it to adapt to different transmit powers. We use two BGNNs with $L=5$ layers
to respectively implement S-Net and Power-Net, where the FCNN is adopted to
construct the DNN of all vertices. The architecture and parameters of neural
networks are given in TABLE I. The hyperparameters $M_{\mathcal{S}}$ and
$M_{\mathcal{P}}$ are set as 3 and 5, respectively. The rectified linear unit
(ReLU) function is used to be the activation function at all hidden layers of
FCNNs. The activation function at the hidden layer of all DNNs is the
rectified linear unit (ReLU) function. The hyperbolic tangent (Tanh) and
Softmax functions are used to be the activation functions at the output layer
of FCNNs accordingly. Since we take the transmit power as an additional input,
the input dimensions of
$\mathbf{\Phi}_{\mathcal{D}_{l}}^{\mathcal{S}}(\cdot;\mathbf{\Theta}_{\mathcal{D}_{l}}^{\mathcal{S}})$
and
$\mathbf{\Phi}_{\mathcal{D}_{l}}^{\mathcal{P}}(\cdot;\mathbf{\Theta}_{\mathcal{D}_{l}}^{\mathcal{P}})$
are to $2M_{\mathcal{S}}+1$ and $2M_{\mathcal{P}}+1$, respectively. The
initialization of messages defined in Section III-B is realized by
independently sampling from the Gaussian distribution with zero mean and unit
variance.

The neural network $\mathbf{\Psi}_{1}(\tilde{\bf H};\mathbf{\Theta})$ for
problem P1 is trained using over 150 epochs with early stopping according to
the validation at the end of each epoch. The batch size is 100. $10^{5}$
training data and $2\times 10^{3}$ testing data are used. To realize the data
augmentation, $U=1000$ channel errors are sampled from the circularly
symmetric complex Gaussian distribution, which then form the 1000 true
channels based on (4). Then, we can obtain the rate quantile estimation
$\hat{r}$ through the DAQE module. The $P$ is uniformly sampled a special
power range, i.e., $P\sim\text{Uniform}(\underline{P},\bar{P})$. We set
$\underline{P}=0$ dBm and $\bar{P}=35$ dBm. Adam algorithm [50] with a
learning rate $\alpha=10^{-3}$ is used for the training of both neural
networks. The training of $\mathbf{\Psi}_{2}(\tilde{\bf H},P;\mathbf{\Theta})$
for problem P2 is similar with that of $\mathbf{\Psi}_{1}(\tilde{\bf
H};\mathbf{\Theta})$.

The following benchmark schemes are considered for comparison:

  * •

The traditional optimization approach: specifically, we use BTI method as
described in [12] to obtain the solution via CVX [52], which achieves the
superior robust performance among existing methods.

  * •

The traditional optimization by evaluation: note that the obtained solution of
[12] is conservative, and therefore the achievable rate performance evaluated
by the MC method is usually better than the optimized solution. We use ‘by
Evaluation’ to denote the actual achievable rate performance.

  * •

The DNN with power optimization only approach [35]: it adopts RZF beamforming
structure to reduce the output dimension of neural network. The rate quantile
is also obtained by MC sampling.

  * •

The DNN method: the neural network directly learns the $\hat{\bf W}$ rather
than the features of any beamforming structure, which is a purely data-driven
method and requires frequent updating the Lagrange multipliers.

  * •

The proposed solution with $s_{k}=0$: to evaluate the impact of the
interference features in (13) on the communication performance, we set
$s_{k},\forall k\in\mathcal{K}$ to 0 in S-Net of the proposed method.

###  V-A Results of Rate Quantile Maximization Problem P1

Figure 4: The achievable robust data rate against the transmit power.

Fig. 4 shows that the achievable robust data rate increases with the
increasing transmit power. The proposed method achieves the highest robust
data rate among all considered methods. For instance, when the transmit power
equals 30 dBm, the robust rate of the proposed method reaches about 10.9 Mbps,
around 14% higher than the traditional optimization method. The reason for
this result is that the traditional optimization methods use convex
approximations such as convex restrictions and semidefinite programming, which
loses the global optimality. Interestingly, the traditional optimization
algorithm actually computes a conservative solution which actually performs
better. We adopt the MC method to evaluate the traditional optimization method
and found that it achieves about 0.7 Mbps higher data rate than the
traditional optimization method, but still lower than the proposed method. The
proposed method with $s_{k}=0$ obtains a lower robust rate performance, which
implies that learning interference features (i.e., $s_{k},\forall\mathcal{K}$)
can indeed achieve performance improvement. The rate of the DNN method is much
lower than above four methods because it only use data driven learning and
thus is hard to improve the performance of the neural network. The DNN with
power optimization only method recovers the beamforming matrix via the learned
power features and RZF beamforming model. Since the RZF beamforming only
focuses on interference cancellation and ignores the impact of noise. In
contrast, other methods use either the optimal structure or the rate-oriented
optimization, and thus the DNN with power optimization only method obtains the
lowest robust data rate.

Figure 5: Comparison of execution time of different rate maximization
algorithms.

Fig. 5 depicts the comparison between the traditional optimization method and
our proposed method in terms of execution time under various transmit power.
Our proposed method has a fairly low execution latency of about 10 to 15 ms,
which is about 1000 times lower than the traditional optimization method. The
reasons for this phenomenon lie in multiple folds. Our proposed method only
needs to use the learned network to infer the optimal solution. In contrast,
the traditional optimization method involves many iterations to obtain the
solution, and loads of matrix operations (such as finding the relaxation
matrices for convex restriction) in each iteration. Once it obtains the rank-1
solution, the beamforming vectors for each user is obtained via matrix
factorization due to the limits of semidefinite programming. If the solution
suffers from high-rank, the Gaussian randomization procedure is required to
obtain approximate beamforming vectors, which further deteriorates the rate
performance, the algorithm’s execution time and even feasibility of the
solution. Table II shows the probability of getting high-rank solutions for
the rate quantile maximization problem, and we found that this probability
rises with increasing transmit power. When $P=30$ dBm, obtaining a high-rank
solution is a high probability event, i.e. 89.65%. This is in line with our
previous observations in Fig. 4 and Fig. 5 about the deterioration of data
rate performance and algorithm execution latency due to high-rank solutions
for traditional optimization methods. In these cases, we choose the
beamforming directions as the principal eigenvectors of the high-rank
solution, and then optimize the power only.

Fig. 6 compares the cumulative distribution function (CDF) of achievable
robust rate between the proposed solution and the traditional optimization
approach by evaluation. Since the traditional optimization approach may suffer
from the high-rank solutions, for comparative fairness, we choose a channel
realization that the traditional approach gives a rank-1 solution. Then, to
evaluate the robust rate, we generate 2000 channel error samples to estimate
the robust rate at outage probability of 5%. The result shows that given the
5% outage constraint, the achievable rate of the proposed solution reaches 10
Mbps, about 2 Mbps higher than that of the traditional optimization approach
by evaluation. From another point of view, if a 10 Mbps achievable robust rate
is desired, the outage probability of traditional method is as high as about
40%, where such solutions cannot be applied in a practical system. In
contrary, the proposed solution can satisfy the outage requirement 5%. It is
worth emphasizing that the comparison environment we set up is very favorable
to traditional optimization methods, which avoid the high-rank challenge, as
well as using MC estimation instead of the original algorithmic results
employed in the comparison of most works. Despite this, our proposed method
still achieves better performance.

Figure 6: CDF of the achievable robust data rate for the rate maximization solution. TABLE II: The probability of high-rank solution for the rate maximization problem. Power | 5 | 10 | 15 | 30
---|---|---|---|---
Probability | 0.2520 | 0.6375 | 0.8060 | 0.8965
Figure 7: The achievable robust data rate against the transmit power with
large scale fading.

In the above simulations, the path loss and inter-cell interference are
combined to PSD (i.e., -75 dBm), where a similar approach is used in [12] and
[35]. Fig. 7 shows the achievable robust data rate with the increasing
transmit power under large-scale fading. Specifically, the path loss is
modeled as $128.1+37.6\log_{10}(d)$ where $d$ is the distance in kilometer
between the BS and a user. Users are uniformly located in an area between 100
and 500 meters from the BS. The small-scale fading follows the Rayleigh
distribution with zero mean and unit variance. Simulation results again show
that our proposed method is able to achieve better robust rate performance.

Figure 8: The scalability and generalizability of the proposed method.

Fig. 8 depicts the scalability and generalizability of the proposed method for
a 4-user system, where the number of transmit antennas is varied from 6 to 9.
$M_{\mathcal{S}}=3$ and $M_{\mathcal{P}}=8$ are chosen as hyperparameters for
the neural network. The robust rates of all methods are normalized by the rate
of the traditional optimization by evaluation method for the same system
parameters, and the corresponding ratios are expressed as percentages. Our
proposed method consistently outperforms traditional optimization (and by
evaluation) methods in terms of robust rate performance under different sizes
of communication systems, demonstrating excellent scalability. In order to
describe the generalizability of the proposed method, we first train the
neural network for the system size of $N=6,K=4$, and use the trained network
for a larger number of transmit antennas at the BS, where systems with
$N\in[7,9]$ were not trained by the neural network. The curve marked with
“generalized from proposed solution” describes the generalizability
performance of the above simulation. It can be seen that the robust
beamforming rate of the generalized from proposed solution outperforms
traditional optimization (and by evaluation) methods at $N=7\text{ and }8$. At
$N=9$, it achieves only slightly worse performance than the traditional
optimization by evaluation method, but still outperforms the traditional
optimization method.

###  V-B Results of Power Minimization Problem P2

Fig. 9 shows the transmit power versus required robust rate requirements. With
the increasing rate requirements, larger transmit power is needed. The
proposed method saves more power compared with the traditional optimization
method. For instance, when rate target is equal to 10 Mbps, the power
consumption of the proposed method is around 11 dBm, 2.2 dBm lower than the
traditional optimization method. In many cases, there is no feasible solution
for given rate and outage constraints, so the power shown is only averaged
over those feasible solutions.

Figure 9: The minimum transmit power against the data rate requirements.

Fig. 10 depicts the probability of obtaining feasible solutions which satisfy
the outage constraint in P2. The feasibility of both methods decreases with
the increasing target rate, because higher rate targets will cause more
outage. The robust beamforming solutions of our proposed approach have much
higher feasibility probabilities than the traditional optimization approach.
For instance, when the rate target is equal to 10 Mbps, the proposed approach
can get robust solutions with about 60% feasibility, 30% higher than the
traditional optimization method. This is because the solution obtained by the
traditional optimization method using convex restriction and semidefinite
relaxation is conservative and thus this method restricts the solution space
and reduces the chance of obtaining feasible solutions. In addition, when
high-rank solutions are obtained by semidefinite relaxation, it is in general
difficult to find a feasible beamforming solution that satisfies the outage
probability constraints. In contrast, our method utilizes neural networks to
optimize the robust beamforming, which can explore a much larger solution
space and produce less conservative solutions, thus making it easier to
ensuring feasibility.

Figure 10: Feasibility probability of the solutions versus rate target.

Fig. 11 shows the comparison of execution time between the proposed method and
the traditional optimization method for the power minimization problem under
different rate targets. The proposed solution spends much more time on the
power minimization problems than the rate maximization problem because it uses
bisection search and in each iteration, and it requires MC method to evaluate
the rate quantile value. However, it still spends much less time than the
traditional optimization method. For instance, when the rate target is equal
to 10 Mbps, the execution time of the proposed method is about 500 ms less
than that of the traditional optimization method. Besides, the execution time
of the proposed method decreases with the increasing target rates, because the
bisection method can identify the infeasible solution early to avoid
unnecessary iterations. Table III depicts the probability of high-rank
solutions of the traditional optimization method for the power minimization
problems.

Figure 11: Comparison of execution time of different power minimization
algorithms. Figure 12: CDF of the achievable robust data rate for the power
minimization solution.

In Fig. 12, again we choose a channel for which the traditional optimization
gives a rank-1 solution and compare the CDF and the power consumption. The
rate requirement is 10 Mbps. The results are interesting: first, verifies that
the traditional is conservative which means it achieves higher than 10 Mbps at
outage of 5% while our proposed solution can achieve exactly 10 Mbps, avoiding
waste of resources; second, our solution also uses nearly 6 dBm less power,
which confirms its efficiency in using limited resources to achieve better and
more robust solutions.

TABLE III: The probability of high-rank solution for the power minimization problem. Rate (Mbps) | 5.86 | 7.06 | 10 | 11.76
---|---|---|---|---
Probability | 0.1149 | 0.1577 | 0.1573 | 0.1479

##  VI Conclusions

This paper proposed a data and model-driven learning approach to address
robust beamforming optimization for the MU-MISO system, considering Gaussian
channel uncertainties. The paper investigated two optimization problems with
the outage probability constraint: the rate quantile maximization problem and
the power minimum problem. For the rate quantile maximization problem, we
proposed a model-driven learning approach combined with data augmentation to
handle the challenging outage constraint. The modified robust beamforming
structure is employed to reduce the neural network’s output dimension from
$2NK$ to $3K$. For the power minimum problem, we introduced a bisection-based
learning method by incorporating the transmit power budget into the input of
the proposed neural network, achieving the feature of universal transmit
power. Numerical results demonstrated that the proposed approach outperforms
state-of-the-art counterparts in terms of rate improvement, power reduction,
and execution time.

[Mathematical Interpretation of (13)] The overall neural network framework
illustrated in Fig. 2 of the manuscript solves the robust beamforming design
based on outage constraints in the following two steps.

  * •

Neural networks (including S-Net and Power-Net) together with beamforming
recovery model obtain optimal beamforming for generalized problems, where the
outage probability constraint and the minimum rate are not involved here.
Denote $\Gamma_{k}$ as the SINR at user $k$, and the generalized problems
corresponding to P1 and P2 are respectively given as follows:

$\displaystyle\textbf{E1:}\max_{\mathbf{w}_{k},\forall\mathcal{K}}f(\Gamma_{1},\cdots,\Gamma_{K})\
\ \ \ \mbox{s.t.}\ \ \ \sum_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}\leq P,$ (27)
$\displaystyle\textbf{E2:}\min_{\mathbf{w}_{k},\forall\mathcal{K}}\sum_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}\
\ \ \ \mbox{s.t.}\ \ \ R_{k}\geq r,$ (28)

where $f(\cdot)$ stands for an arbitrary utility function that is strictly
increasing in each user’s SINR as defined in [23].

  * •

The DAQE module estimates the quantile of the minimum rate among users
$\hat{r}$ which satisfies the outage constraint. By using $-\hat{r}$ as the
loss function, the neural network gradually learns to output beamforming that
can satisfy the outage probability constraints during training until
convergence.

We then explore expressions for the optimal solution structure for E1 and E2
taking into account the channel error. According to [53, 23], the optimal
beamforming solution for E2 is also the optimal solution for E1, so we use E2
to deduce the robust optimal beamforming structure. The constraints of E2 can
be recast as:

$\displaystyle\Gamma_{k}=\frac{|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{k}|^{2}}{\sum_{j\neq
k}|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{j}|^{2}+\sigma^{2}_{k}}\geq
2^{r/B}-1\triangleq\gamma$ (29) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{1}{\gamma\sigma_{k}^{2}}|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{k}|^{2}\geq\sum_{j\neq
k}\frac{1}{\sigma_{k}^{2}}|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{j}|^{2}+1.$
(30)

Obviously, the strong duality and Karush-Kuhn-Tucker (KKT) conditions are held
for problem E2 [54], so we consider the Lagrangian function as

$\displaystyle\mathcal{L}({\bf w}_{1},\cdots,{\bf w}_{K},q_{1},\cdots,q_{k})$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{K}\|{\bf
w}_{k}\|^{2}+\sum_{k=1}^{K}q_{k}\bigg{(}\sum_{j\neq
k}\frac{1}{\sigma_{k}^{2}}|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{j}|^{2}+1$ (31)
$\displaystyle-\frac{1}{\gamma\sigma_{k}^{2}}|\mathbf{h}_{k}^{\dagger}\mathbf{w}_{k}|^{2}\bigg{)},$

where $q_{k},\forall\mathcal{K}$ is the Lagrange multiplier. Based on KKT
conditions, the optimal solution will satisfy
$\frac{\partial\mathcal{L}}{\partial{\bf w}_{k}}=0$. Then we have

$\displaystyle{\bf w}_{k}+\sum_{j\neq k}\frac{q_{j}}{\sigma_{j}^{2}}{\bf
h}_{j}{\bf h}_{j}^{\dagger}{\bf w}_{k}-\frac{q_{k}}{\gamma\sigma_{k}^{2}}{\bf
h}_{k}{\bf h}_{k}^{\dagger}{\bf w}_{k}=\mathbf{0}$
$\displaystyle\overset{(a)}{\Leftrightarrow}$ $\displaystyle{\bf
w}_{k}^{\dagger}+\sum_{j\neq k}\frac{q_{j}}{\sigma_{j}^{2}}{\bf
w}_{k}^{\dagger}{\bf h}_{j}^{\dagger}{\bf
h}_{j}-\frac{q_{k}}{\gamma\sigma_{k}^{2}}{\bf w}_{k}^{\dagger}{\bf
h}_{k}^{\dagger}{\bf h}_{k}=\mathbf{0}^{T}$ (32)

where (a) holds because the conjugate transpose is taken simultaneously for
both sides of (32). Consider ${\bf h}_{j}=\tilde{\bf h}_{j}+{\bf e}_{j}$, we
have

$\displaystyle{\bf h}_{j}^{\dagger}{\bf h}_{j}=\tilde{\bf
h}_{j}^{\dagger}\tilde{\bf h}_{j}+\underbrace{2Re\big{(}\tilde{\bf
h}_{j}^{\dagger}{\bf e}_{j}\big{)}+{\bf e}_{j}^{\dagger}{\bf
e}_{j}}_{\text{denote as }\beta_{j}\in\mathcal{R}}.$ (33)

Substituting (33) into (32), we obtain

$\displaystyle{\bf w}_{k}^{\dagger}+\sum_{j\neq
k}\frac{q_{j}}{\sigma_{j}^{2}}{\bf w}_{k}^{\dagger}(\tilde{\bf
h}_{j}^{\dagger}\tilde{\bf
h}_{j}+\beta_{j})-\frac{q_{k}}{\gamma\sigma_{k}^{2}}{\bf
w}_{k}^{\dagger}(\tilde{\bf h}_{k}^{\dagger}\tilde{\bf h}_{k}+\beta_{k})$ (34)
$\displaystyle=\mathbf{0}^{T}$ $\displaystyle\overset{(a)}{\Leftrightarrow}$
$\displaystyle{\bf w}_{k}^{\dagger}\bigg{(}{\bf
I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf
h}_{j}^{\dagger}\tilde{\bf h}_{j}+$ (35)
$\displaystyle\underbrace{\bigg{(}\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\beta_{j}-\frac{q_{k}}{\sigma_{k}^{2}}\bigg{(}1+\frac{1}{\gamma}\bigg{)}\beta_{k}\bigg{)}}_{\text{denote
as }s_{k}\in R}{\bf I}\bigg{)}$
$\displaystyle=\frac{q_{k}}{\sigma_{k}^{2}}\bigg{(}1+\frac{1}{\gamma}\bigg{)}{\bf
w}_{k}^{\dagger}\tilde{\bf h}_{k}^{\dagger}\tilde{\bf h}_{k}$
$\displaystyle\overset{(b)}{\Leftrightarrow}$ $\displaystyle{\bf
w}_{k}=\bigg{(}\big{(}1+s_{k}\big{)}{\bf
I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf h}_{j}\tilde{\bf
h}_{j}^{\dagger}\bigg{)}^{-1}\tilde{\bf h}_{k}$ (36)
$\displaystyle\times\underbrace{\frac{q_{k}}{\sigma_{k}^{2}}\bigg{(}1+\frac{1}{\gamma}\bigg{)}\tilde{\bf
h}_{k}^{\dagger}{\bf w}_{k}}_{=\text{scalar}},$

where $(a)$ is held by adding term $\frac{q_{k}}{\sigma^{2}_{k}}{\bf
w}_{k}^{\dagger}\tilde{\bf h}_{k}^{\dagger}\tilde{\bf h}_{k}$ to both side of
(34), and (b) is established by taking conjugate transpose of (35) and left
multiplying the inverse matrix of $\big{(}1+s_{k}\big{)}{\bf
I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf h}_{j}\tilde{\bf
h}_{j}^{\dagger}$. Note that
$\frac{q_{k}}{\sigma_{k}^{2}}\bigg{(}1+\frac{1}{\gamma}\bigg{)}\tilde{\bf
h}_{k}^{\dagger}{\bf w}_{k}$ is a scalar, so the optimal ${\bf w}_{k}$ must be
parallel to $\bigg{(}\big{(}1+s_{k}\big{)}{\bf
I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf h}_{j}\tilde{\bf
h}_{j}^{\dagger}\bigg{)}^{-1}\tilde{\bf h}_{k}$. In view of this, the optimal
robust beamforming vectors have the following structures:

$\displaystyle\tilde{\bf
w}_{k}=\sqrt{p_{k}}\frac{\bigg{(}(1+s_{k})\mathbf{I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf
h}_{j}\tilde{\bf h}_{j}^{\dagger}\bigg{)}^{-1}\tilde{\bf
h}_{k}}{\bigg{\|}\bigg{(}(1+s_{k})\mathbf{I}+\sum_{j=1}^{K}\frac{q_{j}}{\sigma_{j}^{2}}\tilde{\bf
h}_{j}\tilde{\bf h}_{j}^{\dagger}\bigg{)}^{-1}\tilde{\bf h}_{k}\bigg{\|}},$
(37)

where $p_{k}$ is the beamforming power for user $k$.

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