# Joint User Scheduling and Beamforming Design for Multiuser MISO Downlink
Systems

Shiwen He, , Jun Yuan, Zhenyu An, , Wei
Huang, ,Yongming Huang, , and Yaoxue Zhang S. He and J. Yuan are with the
School of Computer Science and Engineering, Central South University, Changsha
410083, China. S. He is also with the National Mobile Communications Research
Laboratory, Southeast University, and the Purple Mountain Laboratories,
Nanjing 210096, China. (email: {shiwen.he.hn, yuanjun}@csu.edu.cn). Z. An is
with the Purple Mountain Laboratories, Nanjing 210096, China. (email:
anzhenyu@pmlabs.com.cn). W. Huang is with the School of Computer Science and
Information Engineering, Hefei University of Technology, Hefei 230601, China.
(email: huangwei@hfut.edu.cn)Y. Huang is with the National Mobile
Communications Research Laboratory, School of Informatiecience and
Engineering, Southeast University, Nanjing 210096, China. He is also with the
Purple Mountain Laboratories, Nanjing 210096, China. (email:
huangym@seu.edu.cn). Y. Zhang is with the Department of Computer Science and
Technology, Tsinghua University, Beijing 100084, China. (email:
<EMAIL_ADDRESS>
###### Abstract

In multiuser communication systems, user scheduling and beamforming (US-BF)
design are two fundamental problems that are usually studied separately in the
existing literature. In this work, we focus on the joint US-BF design with the
goal of maximizing the set cardinality of scheduled users, which is
computationally challenging due to the non-convex objective function and the
coupled constraints with discrete-continuous variables. To tackle these
difficulties, a successive convex approximation based US-BF (SCA-USBF)
optimization algorithm is firstly proposed. Then, inspired by wireless
intelligent communication, a graph neural network based joint US-BF (J-USBF)
learning algorithm is developed by combining the joint US and power allocation
network model with the BF analytical solution. The effectiveness of SCA-USBF
and J-USBF is verified by various numerical results, the latter achieves close
performance and higher computational efficiency. Furthermore, the proposed
J-USBF also enjoys the generalizability in dynamic wireless network scenarios.

###### Index Terms:

Cross-layer optimization, user scheduling, beamforming design, graph neural
networks, non-convex optimization.

##  I Introduction

With the explosive growth of Internet of Things (IoT) devices, wireless
communication networks (WCNs) are increasingly facing the challenge of
allocating finite transmit power and bandwidth for system utility maximization
[1]. Accordingly, one needs to design advanced radio resource management
schemes to serve numerous wireless access devices. Massive multiple-input
multiple-output (MIMO) and multiuser transmission are two key enablers for
supporting larger-scale connection in future WCNs [2]. Therefore, some works
have been carried on researching the beamforming design (BF) [3], power
allocation (PA) [4], and user scheduling (US) [5], etc.

Generally speaking, US and BF (US-BF) design are two fundamental problems in
multiuser WCNs, which are implemented at the media access control layer [6]
and the physical layer [7], respectively. Unfortunately, these two issues are
always coupled, which is difficult to be solved. Therefore, they are generally
investigated separately in the existing literature, such as BF design with a
given user set [8] or US optimization combined with PA (US-PA)[9]. For
example, the authors of [10] and [11] only consider the BF problem, where the
uplink-downlink duality theory is adopted for tackling the non-convex problem
of transceivers design. The authors of [12] and [13] also solve the BF problem
for RIS-empowered Terahertz communications with deep reinforcement learning
methods. To further improve the performance of WCNs, cross-layer design is
increasingly becoming popular [14]. The authors of [15] investigate the US-BF
problem by sequentially performing the semi-orthogonal user selection (SUS)
algorithm for US optimization and the zero-forcing BF (ZFBF) algorithm for BF
design. The authors of [16] propose a low complexity US-BF scheme for 5G MIMO
nonorthogonal multiple-access systems, but the non-convex problem is separated
by tracking two subproblems, namely, BF scheme and greedy min-power US scheme,
instead of jointly solving them. The authors of [17] also discuss cross layer
optimization with statistical channel information for massive MIMO scenario,
by tackling US and BF individually.

Meanwhile, the existing researches on coordinated multiuser communication are
mainly based on the conventional Shannon theory [18], which assumes that the
communication capacity has extremely low decoding error probability with
enough long blocklength transmission. However, in the ultra-reliable low
latency communication (URLLC) senarios, such as factory automation and remote
surgery, this condition with the long blocklength transmission may not be
satisfied [19]. To take the impact of finite blocklength transmission into
account, the achievable rate has been expressed as a complicated function
composed of the received signal-to-noise (SNR), the blocklength, and the
decoding error probability, which is smaller than the Shannon rate [20].
Consequently, the optimization problem in scenarios with finite blocklength
transmission is more challenging [21]. In order to solve the problem of
interest, the algorithms designed in the aforementioned references are mainly
based on the convex optimization theory [22]. However, such model-driven
optimization algorithms usually suffer from a high computational complexity,
which may restrict their practical application ability in WCNs.

Recently, deep neural networks (DNNs) have emerged as an effective tool to
solve such challenging radio resource management problems in WCNs [23].
Different from the model-driven optimization algorithms running independently
for each instance, DNNs are trained with numerous data to learn the mapping
between radio resource optimization policies and WCN environments. Hence, the
main computational cost of DNNs is shifted into the offline training stage,
and only simple mathematical operations are needed in the online optimization
stage. The work in [24] shows that DNNs could achieve competitive performance
with lower computational complexity than existing model-driven optimization
algorithms. 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. The authors
of [26] regard resource allocation problems in the field of wireless
communications as the generalized assignment problems (GAP), and propose a
novel deep unsupervised learning approach to solve GAP in a time-efficient
manner. The authors of [27] focus on solving PA problem via ensembling several
deep neural networks. This is also an unsupervised approach and achieves
competitive results compared with conventional methods. However, the core
network is specifically designed for power control problem and it could not be
extended for US. In addition, these DNN-based architectures [27, 26, 24, 25]
are mainly inherited from image processing tasks and not tailored to radio
resource management problems, especially the fact that they fail to exploit
the prior topology knowledge in WCNs. The numerical results obtained in [28]
illustrated that the performance of DNNs degrades dramatically with increasing
WCN size.

To achieve a better scalability of learning-based radio resource management, a
potential approach is to incorporate the network topology into the learning of
neural networks, namely graph neural networks (GNNs) [29]. For instance, the
authors of [30] combined DNNs with the geographic location of transceivers,
and thereby proposed a spatial convolution model for wireless link scheduling
problems with hundreds of nodes. The authors of [31] proposed a random edge
graph neural network (REGNN) for PA optimization on graphs formed by the
interference links within WCNs. The work in [32] demonstrates that GNNs are
insensitive to the permutation of data, such as channel state information
(CSI). Further, this work was extended in [33] to solve both PA and BF
problems via message passing graph neural networks (MPGNNs), which have the
ability to generalize to large-scale problems while enjoying a high
computational efficiency. However, their proposed designs in [32, 33] only
investigated the continuous optimization problems with simple constraints. The
discrete optimization problems with complicated constraints are still an
opening issue and need to be further considered. Fortunately, the application
of primal-dual learning in [34] provides an effective way to solve the
complicated constrained radio resource management problems.

Based on the above considerations, this work studies the joint US-BF
optimization problem in the multiuser MISO downlink system. Unlike the
conventional methods, the US-BF design will be simultaneously achieved via
solving a single optimization problem, instead of different problems.
Moreover, to improve the computational efficiency and utilize network
historical data information, we propose a GNN-based Joint US-BF (J-USBF)
learning algorithm. The main contributions and advantages of this work are
summarized as follows:

  * •

A joint US-BF optimization problem for multiuser MISO downlink systems is
formulated with the goal of maximizing the number of scheduled users subject
to user rate and base station (BS) power constraints. To solve this discrete-
continuous variables optimization problem, a SCA-based US-BF (SCA-USBF)
algorithm is firstly designed to pave the way for the J-USBF algorithm.

  * •

A J-USBF learning algorithm is developed by combining the joint user
scheduling and power allocation network (JEEPON) model with the BF analytical
solution. In particular, we first formulate the investigated problem as a
graph optimization problem through wireless graph representation, then design
a GNN-based JEEPON model to learn the US-PA strategy on graphs, and utilize
the BF analytical solution to achieve joint US-BF design. Meanwhile, a primal-
dual learning framework is developed to train JEEPON in an unsupervised
manner.

  * •

Finally, numerical results is conducted to validate the effectiveness of the
proposed algorithms. Compared with the SCA-USBF algorithm, the J-USBF learning
algorithm achieves close performance and higher computational efficiency, and
enjoys the generalizability in dynamic WCN scenarios.

The remainder of this paper is organized as follows. Section II introduces a
challenging radio resource management problem in the multiuser MISO downlink
system. Section III proposes the SCA-USBF for solving the investigated
problem. Section IV designs the JEEPON and provides a primal-dual learning
framework to train it in an unsupervised manner. Numerical results are
presented in Section V. Finally, conclusions are drawn in Section VI.

$\mathbf{\mathit{Notations}}$: Throughout this paper, lowercase and uppercase
letters (such as $a$ and $A$) represent scalars, while the bold counterparts
$\mathbf{a}$ and $\mathbf{A}$ represent vectors and matrices, respectively.
$\left|\cdot\right|$ indicates the absolute value of a complex scalar or the
cardinality of a set. $\left\|\cdot\right\|_{0}$, $\left\|\cdot\right\|_{1}$,
and $\left\|\cdot\right\|_{2}$ denote the $\ell_{0}$-norm, $\ell_{1}$-norm,
and $\ell_{2}$-norm, respectively. The superscripts $(\cdot)^{T}$,
$(\cdot)^{H}$, and $(\cdot)^{-1}$ denote the transpose, conjugate transpose,
and inverse of a matrix, respectively. $\mathbb{R}$, $\mathbb{R}^{+}$, and
$\mathbb{C}$ are the sets of real, non-negative real, and complex numbers,
respectively. Finally, $\mathbb{R}^{M\times 1}$ and $\mathbb{C}^{M\times 1}$
represent $M$-dimensional real and complex column vectors, respectively.

##  II System Model and Problem Formulation

In this work, we consider a multiuser MISO downlink system with taking the
reliable and delivery latency into account, where a BS with $N$ antennas
serves $K$ single-antenna users111Since the complexity of discussed problem,
the single-cell scenario is considered in this paper. Research on more complex
scenario with multi-cells will be discussed in future work, where inter-cell
interference should be considered.. For simplicity, let
$\mathcal{K}=\\{1,2,\cdots,K\\}$ and
$\mathcal{S}=\\{1,2,\cdots,K^{\ast}\\}\subseteq\mathcal{K}$ be the set of
candidate users and scheduled users, respectively, where $K^{\ast}\leq{K}$.
The channel between user $k$ and the BS is denoted as
$\mathbf{h}_{k}\in\mathbb{C}^{N\times 1}$. Let $p_{k}\geq{0}$ and
$\mathbf{w}_{k}\in\mathbb{C}^{N\times 1}$ represent the transmit power and
unit-norm BF vector used by the BS for user $k$, respectively. Thus, the
received signal at user $k$ is given by

$y_{k}=\sum\limits_{l\in\mathcal{S}}\sqrt{p_{l}}\mathbf{h}_{k}^{H}\mathbf{w}_{l}s_{l}+n_{k},$
(1)

where $s_{l}$ is the normalized data symbol intended for the $l$-th user, and
$n_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ denotes the additive Gaussian white
noise at user $k$ with zero mean and variance $\sigma_{k}^{2}$. For notational
convenience, we define
$\overline{\mathbf{h}}_{k}=\frac{\mathbf{h}_{k}}{\sigma_{k}}$ and the downlink
signal-to-interference-plus-noise ratio (SINR) of user $k$ as

$\overrightarrow{\gamma}_{k}=\frac{p_{k}\left|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{k}\right|^{2}}{\sum\limits_{l\neq
k,l\in\mathcal{S}}p_{l}\left|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{l}\right|^{2}+1}.$
(2)

To satisfy the extreme requirements of delay, finite blocklength transmission
regime is adopted in this paper. The results in [20] show that the achievable
rate is not only a function of the received SNR (or SINR), but also the
decoding error probability $\epsilon$ and the transmission finite blocklength
$n$. Accordingly, the achievable rate of user $k$ with finite blocklength
transmission is given by222The proposed algorithms is also suitable for
solving similar optimization problems, where the user rate is based on Shannon
capacity formula.

$R(\overrightarrow{\gamma}_{k})=C(\overrightarrow{\gamma}_{k})-\vartheta\sqrt{V(\overrightarrow{\gamma}_{k})},$
(3)

where $C(\overrightarrow{\gamma}_{k})=\ln(1+\overrightarrow{\gamma}_{k})$
denotes the Shannon capacity, $\vartheta=\frac{Q^{-1}(\epsilon)}{\sqrt{n}}$,
$Q^{-1}(\cdot)$ is the inverse of Gaussian Q-function
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\mathrm{exp}(-\frac{t^{2}}{2})dt$,
and $V(\overrightarrow{\gamma}_{k})$ denotes the channel dispersion, which is
defined as

$V(\overrightarrow{\gamma}_{k})=1-\frac{1}{(1+\overrightarrow{\gamma}_{k})^{2}}.$
(4)

The target of this work is to maximize the number of users belonging to the
scheduled user set $\mathcal{S}\subseteq\mathcal{K}$ subject to the
constraints of per-user minimum rate requirement and BS maximum power budget.
Specifically, one needs to carefully select the scheduled user set
$\mathcal{S}$, and design BF vectors with reasonable transmit power333For the
ultra-dense or large-scale connective URLLC scenario, it may be a better
choice to schedule as many users as possible while satisfying reliability and
latency requirements. Accordingly, we aim to maximize the set cardinality of
scheduled users in this work.. To this end, the joint US-BF optimization
problem is formulated as follows444In our experiment, we obtain perfect CSI
via link level simulation. However, it is indeed hard to estimate CSI in the
real communication systems [35]. Although there are pilot-based and blind
channel estimation methods, the perfect CSI cannot be obtained due to the
estimation error, which may lead to performance deterioration. Statistical
CSI, including RSRP (Reference Signal Receiving Power), RSRQ (Reference Signal
Receiving Quality), RSSI (Received Signal Strength Indicator), et al., might
be helpful under this condition. We would like to further investigate the
joint US-BF problem in the future work.

$\displaystyle\max_{\\{p_{k},\mathbf{w}_{k}\\}}|\mathcal{S}|,$ (5a)
$\displaystyle\mathrm{s.t.}~{}$ $\displaystyle r_{k}\leq
R(\overrightarrow{\gamma}_{k}),~{}\left\|\mathbf{w}_{k}\right\|_{2}=1,\forall{k}\in\mathcal{S},$
(5b)
$\displaystyle\sum\limits_{k\in\mathcal{S}}p_{k}\leq{P},{\color[rgb]{1,0,0}~{}p_{k}\geq{0},\forall{k}\in\mathcal{S},}$
(5c)

where $\left|\mathcal{S}\right|$ is the cardinality of set $\mathcal{S}$,
$r_{k}$ is the per-user minimum rate requirement, and $P$ denotes the power
budget of the BS. Problem (5) is a mixed-integer continuous-variable
programming problem that involves a discrete objective function and two
continuous-variable constraints about power and unit-norm BF vectors. It is
difficult to obtain the global optimal solution of problem (5), even the near-
optimal solution. Although the greedy heuristic search based US-BF (G-USBF)
algorithm in Appendix A could be considered as a possible effective solution,
it brings extremely high computational complexity especially for large-scale
WCNs. In the sequel, the SCA-based US-BF optimization algorithm and the GNN-
based learning algorithm are successively proposed to solve the problem (5).

##  III Design of The SCA-USBF Algorithm

In this section, we pay our attention on designing an effective optimization
algorithm for problem (5) from the perspective of successive convex
approximation (SCA) optimization theory. Since problem (5) is non-convex, the
first thing is to transform it into a tractable form via some basic
mathematical transformations. One idea is to apply the uplink-downlink duality
theory [36] to equivalently transform the downlink problem (5) into a virtual
uplink dual problem (6) 555Similar to formula (1), the virtual uplink input-
output relationship could be expressed as
$\mathbf{y}=\sum\limits_{k\in{\mathcal{S}}}\sqrt{q_{k}}\overline{\mathbf{h}}_{k}s_{k}+\mathbf{n}$,
where $\mathbf{y}\in\mathbb{R}^{N\times 1}$ is the virtual uplink received
signal at BS, $s_{k}$ is the virtual uplink normalized data symbol intended
for the $k$-th user, and $\mathbf{n}\in\mathbb{R}^{N\times 1}$ is the additive
Gaussian white noise with $\mathcal{CN}(0,\mathbf{I})$. For the virtual uplink
communication systems, $\mathbf{w}_{k}$ is used as the received vector for the
$k$-th user. Thus, the virtual uplink received SINR of the $k$-th user can be
calculated via the received signal $\mathbf{w}_{k}^{H}\mathbf{y}$., i.e.,

$\displaystyle\max_{\left\\{q_{k},\mathbf{w}_{k}\right\\}}|\mathcal{S}|,$ (6a)
$\displaystyle\mathrm{s.t.}~{}$ $\displaystyle
r_{k}\leq{R}(\overleftarrow{\gamma}_{k}),\left\|\mathbf{w}_{k}\right\|_{2}=1,\forall{k}\in\mathcal{S},$
(6b)
$\displaystyle\sum\limits_{k\in\mathcal{S}}q_{k}\leq{P},{\color[rgb]{1,0,0}~{}q_{k}\geq{0},\forall{k}\in\mathcal{S},}$
(6c)

where $q_{k}$ is the virtual uplink transmit power of user $k$, and
$\overleftarrow{\gamma}_{k}$ represents the corresponding virtual uplink
received SINR, i.e.,

$\overleftarrow{\gamma}_{k}=\frac{q_{k}\left|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{k}\right|^{2}}{\sum\limits_{l\neq{k},l\in\mathcal{S}}q_{l}\left|\overline{\mathbf{h}}_{l}^{H}\mathbf{w}_{k}\right|^{2}+1}.$
(7)

Note that the definition (7) focuses on calculating SINRs for the scheduled
user set $\mathcal{S}$, with its implicit information is that the SINRs of the
unscheduled users are all zero values in theory. For convenience, we further
propose a new SINR definition ${\overleftarrow{\gamma}}_{k}^{(\mathcal{K})}$
which is directly calculated based on the candidate user set $\mathcal{K}$,
i.e.,

$\overleftarrow{\gamma}_{k}^{(\mathcal{K})}=\frac{q_{k}\left|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{k}\right|^{2}}{\sum\limits_{l\neq{k},l\in\mathcal{K}}q_{l}\left|\overline{\mathbf{h}}_{l}^{H}\mathbf{w}_{k}\right|^{2}+1}.$
(8)

To clearly indicate whether a user is scheduled or not, we introduce
$\kappa_{k}$ as a binary variable indicator of the user state, with
$\kappa_{k}=1$ if user $k$ is scheduled and $\kappa_{k}=0$ otherwise,
$k\in\mathcal{K}$. Therefore, $\kappa_{k}=1$ also means that the minimum rate
constraint is met for the $k$-th user, i.e., formulas
$r_{k}\leq{R}(\overleftarrow{\gamma}_{k})$ and $q_{k}\geq{0}$ will hold.
However, $\kappa_{k}=0$ does not mean that formulas
$R\left(\overleftarrow{\gamma}_{k}\right)=0$ and $q_{k}=0$ are always true.
For instance, for a candidate user set $\mathcal{K}$ and scheduled user set
$\mathcal{S},\mathcal{S}\subset\mathcal{K}$. Transmission power of the BS is
not always precisely exhausted for scheduled user set $\mathcal{S}$. For user
$k^{\prime}$ from the rest user set $\mathcal{K}\setminus\mathcal{S}$, if the
residual power could not meet the minimum transmission power requirement, then
we have
$\kappa_{k^{\prime}}=0,0<R({\overleftarrow{\gamma}_{k^{\prime}}})<r_{k^{\prime}},||\mathbf{w}_{k^{\prime}}||_{2}=1$.
In such circumstance,
$\kappa_{k^{\prime}}r_{k^{\prime}}\leq{R}(\overleftarrow{\gamma}_{k^{\prime}})$
holds, but $\kappa_{k^{\prime}}=0$, i.e.,
$\kappa_{k^{\prime}}\notin\mathcal{S}$. Meanwhile, for user $k\in\mathcal{S}$,
$\overleftarrow{\gamma}_{k}>\overleftarrow{\gamma}_{k}^{\mathcal{(K)}}>0$ and
$\kappa_{k}=1$ hold. For user $k\notin\mathcal{S}$, if
$k\in\mathcal{K},k\neq{k^{\prime}}$, then
$\overleftarrow{\gamma}_{k}=\overleftarrow{\gamma}_{k}^{\mathcal{(K)}}=0$ and
$\kappa_{k}=0$ hold. Let
$\bm{\kappa}=[\kappa_{1},\kappa_{2},\cdots,\kappa_{k},\cdots,\kappa_{K}]^{T}$,
problem (6) is approximately written as

$\displaystyle\max_{\left\\{\kappa_{k},q_{k},\mathbf{w}_{k}\right\\}}\left\|\bm{\kappa}\right\|_{0},$
(9a) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle\kappa_{k}\in\\{0,1\\},\forall{k}\in\mathcal{K},$ (9b)
$\displaystyle\kappa_{k}r_{k}\leq{R}(\overleftarrow{\gamma}_{k}^{\mathcal{(K)}}),\left\|\mathbf{w}_{k}\right\|_{2}=1,\forall{k}\in\mathcal{K},$
(9c)
$\displaystyle\sum\limits_{k\in\mathcal{K}}q_{k}\leq{P},~{}q_{k}\geq{0},\forall
k\in\mathcal{K}.$ (9d)

As discussed above, for user $k\in\mathcal{S}$,
$\overleftarrow{\gamma}_{k}>\overleftarrow{\gamma}_{k}^{\mathcal{(K)}}>0$
holds, and for user $k\notin\mathcal{S}$, $\kappa_{k}=0$ holds. Therefore,
(9c) is a more strict constraint than (6b), and the solution to problem (6) is
the upper bound of problem (9).

The goal of problem (9) is to maximize the number of scheduled users under the
given constraints. Further, constraints (9b) and (9c) can be equivalently
transformed into continuous constraint type and SINR form [21], respectively.
Let $\widetilde{\gamma}_{k}>0$ be the minimum SINR associated with achieving
the minimum achievable rate $r_{k}$ for the $k$-th user. Thus, problem (9) can
be equivalently transformed as

$\displaystyle\max_{\\{\kappa_{k},q_{k},\mathbf{w}_{k}\\}}\left\|\bm{\kappa}\right\|_{0},$
(10a) $\displaystyle\mathrm{s.t.}~{}$ $\displaystyle
0\leq\kappa_{k}\leq{1},\forall{k}\in\mathcal{K},$ (10b)
$\displaystyle\sum\limits_{k\in\mathcal{K}}\left(\kappa_{k}-\kappa_{k}^{2}\right)\leq{0},$
(10c)
$\displaystyle\kappa_{k}\widetilde{\gamma}_{k}\leq\overleftarrow{\gamma}_{k}^{(\mathcal{K})},\left\|\mathbf{w}_{k}\right\|_{2}=1,\forall{k}\in\mathcal{K},$
(10d)
$\displaystyle\sum\limits_{k\in\mathcal{K}}q_{k}\leq{P},~{}q_{k}\geq{0},\forall{k}\in\mathcal{K}.$
(10e)

Constraints (10b) and (10c) assure that the value of $\kappa_{k}$ equals to
either one or zero, i.e., $\kappa_{k}\in\\{0,1\\}$, $\forall k\in\mathcal{K}$.
According to [37, Proposition 2], the strong Lagrangian duality holds for
problem (10). Introducing similar mathematical tricks on handling constraint
(10c), problem (10) is reformulated as follows

$\displaystyle\min_{\\{\kappa_{k},q_{k},\mathbf{w}_{k}\\}}-\sum\limits_{k\in\mathcal{K}}\kappa_{k}+g\left(\bm{\kappa}\right)-h\left(\bm{\kappa}\right),$
(11a) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\eqref{Eq.(10b)},~{}\eqref{Eq.(10d)},~{}\eqref{Eq.(10e)},$
(11b)

where $\lambda$ is a proper non-negative constant, and
$g\left(\bm{\kappa}\right)$ and $h\left(\bm{\kappa}\right)$ are defined
respectively as

$\displaystyle g\left(\bm{\kappa}\right)$
$\displaystyle\triangleq\lambda\sum\limits_{k\in\mathcal{K}}\kappa_{k}+\lambda\left(\sum\limits_{k\in\mathcal{K}}\kappa_{k}\right)^{2},$
(12a) $\displaystyle h\left(\bm{\kappa}\right)$
$\displaystyle\triangleq\lambda\sum\limits_{k\in\mathcal{K}}\kappa_{k}^{2}+\lambda\left(\sum\limits_{k\in\mathcal{K}}\kappa_{k}\right)^{2}.$
(12b)

Note that the optimal receiver BF vector $\mathbf{w}_{k}^{(\ast)}$ for
maximizing the uplink SINR $\overleftarrow{\gamma}_{k}^{(\mathcal{K})}$ of the
$k$-th user is the minimum mean square error (MMSE) filter with fixed
$\\{q_{k}\\}$, i.e.,

$\mathbf{w}_{k}^{(\ast)}=\frac{\left(\mathbf{I}_{N}+\sum\limits_{k\in\mathcal{K}}q_{k}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\right)^{-1}\overline{\mathbf{h}}_{k}}{\left\|\left(\mathbf{I}_{N}+\sum\limits_{k\in\mathcal{K}}q_{k}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\right)^{-1}\overline{\mathbf{h}}_{k}\right\|_{2}},$
(13)

where $\mathbf{I}_{N}$ denotes $N$-by-$N$ identity matrix. For fixed
$\\{\mathbf{w}_{k}\\}$, problem (11) is rewritten as

$\displaystyle\min_{\\{\kappa_{k},q_{k}\\}}-\sum\limits_{k\in\mathcal{K}}\kappa_{k}+g\left(\bm{\kappa}\right)-h\left(\bm{\kappa}\right),$
(14a) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle\widetilde{\gamma}_{k}\kappa_{k}-q_{k}\left|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{k}\right|^{2}+\varphi_{k}(\bm{\kappa},\mathbf{q})-\phi_{k}(\bm{\kappa},\mathbf{q})\leq{0},\forall{k}\in\mathcal{K},$
(14b) $\displaystyle~{}\eqref{Eq.(10b)},~{}\eqref{Eq.(10e)},$ (14c)

where $\varphi_{k}\left(\bm{\kappa},\mathbf{q}\right)$ and
$\phi_{k}\left(\bm{\kappa},\mathbf{q}\right)$ are defined as

$\displaystyle\varphi_{k}(\bm{\kappa},\mathbf{q})$
$\displaystyle\triangleq\frac{1}{2}\left(\widetilde{\gamma}_{k}\kappa_{k}+\sum\limits_{l\in\mathcal{K},l\neq
k}q_{l}\left|\overline{\mathbf{h}}_{l}^{H}\mathbf{w}_{k}\right|^{2}\right)^{2},$
(15a) $\displaystyle\phi_{k}(\bm{\kappa},\mathbf{q})$
$\displaystyle\triangleq\frac{1}{2}\widetilde{\gamma}_{k}^{2}\kappa_{k}^{2}+\frac{1}{2}\left(\sum\limits_{l\in\mathcal{K},l\neq
k}q_{l}\left|\overline{\mathbf{h}}_{l}^{H}\mathbf{w}_{k}\right|^{2}\right)^{2}.$
(15b)

Problem (14) belongs to the class of difference of convex programming problem,
since the objective function (14a) and constraint (14b) are the difference of
two convex functions. In the sequel, we resort to the classic SCA-based
methods [38]. Using the convexity of functions $h\left(\bm{\kappa}\right)$ and
$\phi\left(\bm{\kappa},\mathbf{q}\right)$, we have

$\begin{split}&h(\bm{\kappa})\geq\psi\left(\bm{\kappa}\right)\triangleq
h(\bm{\kappa}^{(\tau)})+\sum\limits_{k\in\mathcal{K}}h^{\prime}(\bm{\kappa}^{(\tau)})(\kappa_{k}-\kappa_{k}^{(\tau)}),\\\
&\phi_{k}(\bm{\kappa},\mathbf{q})\geq\varrho_{k}(\bm{\kappa},\mathbf{q})\triangleq\phi_{k}(\bm{\kappa}^{(\tau)},\mathbf{q}^{(\tau)})\\\
&+\widetilde{\gamma}_{k}^{2}\kappa_{k}^{(\tau)}(\kappa_{k}-\kappa_{k}^{(\tau)})+\sum\limits_{l\in\mathcal{K},l\neq{k}}\rho_{k,l}(\mathbf{q}^{(\tau)})(q_{l}-q_{l}^{(\tau)}),\end{split}$
(16)

where $h^{\prime}\left(\kappa_{k}\right)\triangleq
2\lambda\left(\kappa_{k}+\sum\limits_{l\in\mathcal{K}}\kappa_{l}\right)$,
$\rho_{k,m}\left(\mathbf{q}\right)\triangleq\left|\overline{\mathbf{h}}_{m}^{H}\mathbf{w}_{k}\right|^{2}\sum\limits_{n\in\mathcal{K},n\neq
k}q_{n}\left|\overline{\mathbf{h}}_{n}^{H}\mathbf{w}_{k}\right|^{2}$, and
superscript $\tau$ is the $\tau$-th iteration of the SCA-USBF algorithm
presented shortly. From the aforementioned discussions, the convex
approximation problem solved at the $(\tau+1)$-th iteration of the proposed
algorithm is given by

$\displaystyle\min_{\\{\kappa_{k},q_{k}\\}}-\sum\limits_{k\in\mathcal{K}}\kappa_{k}+g(\bm{\kappa})-\psi(\bm{\kappa}),$
(17a) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle\widetilde{\gamma}_{k}\kappa_{k}-q_{k}\left|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{k}\right|^{2}+\varphi_{k}(\bm{\kappa},\mathbf{q})-\varrho_{k}(\bm{\kappa},\mathbf{q})\leq{0},\forall{k}\in\mathcal{K},$
(17b) $\displaystyle~{}\eqref{Eq.(10b)},~{}\eqref{Eq.(10e)}.$ (17c)

Based on the above mathematical transformation, the SCA-USBF is summarized in
Algorithm 1. In the description of Algorithm 1, $\delta$ denotes the maximum
permissible error, and $\upsilon^{(\tau)}$ and $\zeta^{(t)}$ denote the
objective value of problem (11) at the $\tau$-th iteration and problem (17) at
the $\tau$-th iteration and the $t$-th iteration, respectively. Note that SCA-
USBF is also suitable for problem scenarios based on the Shannon capacity
formula, as we just need to replace the $R(\overleftarrow{\gamma}_{k})$ with
$C(\overleftarrow{\gamma}_{k})$ in problems (5), (6), and (9), and replace the
minimum SINR $\widetilde{\gamma}_{k}$ in problem (10b) for achieving minimum
achievable rate $r_{k}$ with $\widetilde{\gamma}^{\prime}_{k}=2^{r_{k}}-1$.
The convergence of SCA-USBF can be guaranteed by the monotonic boundary
theory. To speed up the convergence of SCA-USBF, we can first filter out the
users which meets constraints (5b) and (5c) by adopting a single user
communication with the maximum ratio transmission (MRT) and full power
transmission, thus, at least one user could be scheduled in such circumstance.

Algorithm 1 The SCA-USBF Algorithm for Problem (10)

1: Let $t=0$, $\tau=0$, $\lambda=10^{-2}$ and $\delta=10^{-5}$. Initialize the
BF vectors $\\{\mathbf{w}_{k}^{(0)}\\}$ and downlink power vectors
$\\{p_{k}^{(0)}\\}$, such that constraint (5b) and (5c) are satisfied.

2: Initialize $\zeta^{(0)}$ and $\upsilon^{(0)}$, calculate the downlink SINR
$\overrightarrow{\gamma}_{k}$ via $\\{p_{k}^{(0)},\mathbf{w}_{k}^{(0)}\\}$ and
Eq. (2), and obtain the uplink power vector
$\mathbf{q}=[q_{1},\cdots,q_{K^{\ast}}]^{T}$ with
$\mathbf{q}=\bm{\Psi}^{-1}\mathbf{I}_{K^{\ast}},$ (18) where
$\mathbf{I}_{K^{\ast}}$ is the all-one vector with $K^{\ast}$ dimensions, and
matrix $\bm{\Psi}$ is given by  $[\bm{\Psi}]_{k,l}=\left\\{\begin{aligned}
\frac{|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{k}|^{2}}{\overrightarrow{\gamma}_{k}},k=l,\\\
-|\overline{\mathbf{h}}_{l}^{H}\mathbf{w}_{k}|^{2},k\neq{l}.\end{aligned}\right.$
(19)

3: Let $t\leftarrow{t+1}$. Solve problem (17) to obtain
$\\{\kappa_{k}^{(t)},q_{k}^{(t)}\\}$ and $\zeta^{(t)}$.

4: If $|\frac{\zeta^{(t)}-\zeta^{(t-1)}}{\zeta^{(t-1)}}|\leq\delta$, go to
Step 5. Otherwise, go to Step 3.

5: Let $\tau\leftarrow\tau+1$, update $\\{\mathbf{w}_{k}^{(\tau)}\\}$ with
$\\{q_{k}^{(t)}\\}$ and Eq. (13), and obtain the objective value
$\upsilon^{(\tau)}$. If
$|\frac{\upsilon^{(\tau)}-\upsilon^{(\tau-1)}}{\upsilon^{(\tau-1)}}|\leq\delta$,
stop iteration and go to Step 6. Otherwise, go to Step 3.

6: Calculate the uplink SINR $\overleftarrow{\gamma}_{k}$ via
$\\{q_{k}^{(t)},\mathbf{w}_{k}^{(\tau)}\\}$ and Eq. (7), and obtain the
downlink power vector $\mathbf{p}=[p_{1},\cdots,p_{K^{\ast}}]^{T}$ with
$\mathbf{p}=\mathbf{D}^{-1}\mathbf{I}_{K^{\ast}},$ (20) where matrix
$\mathbf{D}$ is given by  $[\mathbf{D}]_{k,l}=\left\\{\begin{aligned}
\frac{|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{k}|^{2}}{\overleftarrow{\gamma}_{k}},k=l,\\\
-|\overline{\mathbf{h}}_{k}^{H}\mathbf{w}_{l}|^{2},k\neq{l}.\end{aligned}\right.$
(21)

7: Calculate the objective function value, then output the US, PA and BF
vectors $\\{\kappa_{k},p_{k},\mathbf{w}_{k}\\}$.

##  IV Design of The J-USBF Algorithm

In this section, the transformation of problem (5) to problem (10) is
inherited, where the BF vector has an analytical solution. We focus on
proposing the J-USBF learning algorithm to output the joint US-BF strategy.
Specifically, we first introduce the graph representation method of single-
cell WCNs, then design the JEEPON model to learn the US-PA strategy, and
combine the BF analytical solution to achieve the J-USBF learning algorithm,
which is summarized as Algorithm 2. In the sequel, we focus on studying the
design of JEEPON and the corresponding training framework.

Algorithm 2 The J-USBF Learning Algorithm

0: $\mathcal{D}=\\{\mathbf{h}_{k}\\}$: Testing sample with $K$ users;
$\bm{\Theta}$: The trainable parameters of JEEPON.

0: The optimization strategy
$\\{\kappa_{k}^{(\ast)},q_{k}^{(\ast)},\mathbf{w}_{k}^{(\ast)}\\}$ of sample
$\mathcal{D}$.

1: Construct graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ for sample
$\mathcal{D}$ via the graph representation module.

2: Input graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ to JEEPON and obtain the
US-PA strategy $\\{\kappa_{k}^{(\ast)},q_{k}^{(\ast)}\\}$.

3: Calculate the BF vectors $\\{\mathbf{w}_{k}^{(\ast)}\\}$ via Eq. (13), and
output the strategy
$\\{\kappa_{k}^{(\ast)},q_{k}^{(\ast)},\mathbf{w}_{k}^{(\ast)}\\}$.

4: Calculate the uplink SINR $\overleftarrow{\gamma}_{k}$ via
$\\{q_{k}^{(\ast)},\mathbf{w}_{k}^{(\ast)}\\}$ and Eq. (7), and obtain the
downlink power vector $\\{p_{k}^{(\ast)}\\}$ via Eq. (21).

###  IV-A Problem Transformation and Loss Function Definition

Different from the proposed SCA-USBF that alternately updates the BF vectors,
in the sequel, it is regarded as intermediate variables about the virtual
uplink power vectors. Taking (13) into (8), the uplink received SINR of user
$k$ is rewritten as

$\widehat{\gamma}_{k}=\frac{q_{k}|\overline{\mathbf{h}}_{k}^{H}\bm{\Lambda}^{-1}\overline{\mathbf{h}}_{k}|^{2}}{\sum\limits_{l\neq{k},l\in\mathcal{K}}q_{l}|\overline{\mathbf{h}}_{l}^{H}\bm{\Lambda}^{-1}\overline{\mathbf{h}}_{k}|^{2}+|\bm{\Lambda}^{-1}\overline{\mathbf{h}}_{k}|_{2}^{2}},$
(22)

where
$\bm{\Lambda}=\mathbf{I}_{N}+\sum\limits_{k\in\mathcal{K}}q_{k}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}$.
Thus, problem (10) is formulated as follows

$\displaystyle\max_{\\{\kappa_{k},q_{k}\\}}\sum\limits_{k\in\mathcal{K}}\kappa_{k},$
(23a) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle{0}\leq\kappa_{k}\leq{1},\forall{k}\in\mathcal{K},$ (23b)
$\displaystyle\sum\limits_{k\in\mathcal{K}}(\kappa_{k}-\kappa_{k}^{2})\leq{0},$
(23c)
$\displaystyle\kappa_{k}\widetilde{\gamma}_{k}\leq\widehat{\gamma}_{k},\forall{k}\in\mathcal{K},$
(23d)
$\displaystyle\sum\limits_{k\in\mathcal{K}}q_{k}\leq{P},~{}q_{k}\geq{0},\forall{k}\in\mathcal{K}.$
(23e)

To facilitate the design of JEEPON, incorporating partially the constraints
into the objective function, the violation-based Lagrangian relaxation method
[39] is adopted to formulate problem (23) as an unconstrained optimization
problem. Observe the constraints constraints (23b) and (23e) only contain
single optimization variables that can be satisfied by subsequent projection-
based methods. For constraints (23c) and (23d), we introduce the non-negative
Lagrangian multipliers $\\{\mu,\nu\in\mathbb{R}^{+}\\}$ to capture how much
the constraints are violated. Thus, the partial Lagrangian relaxation function
of problem (23) is given by

$\displaystyle\mathcal{L}(\bm{\kappa},\mathbf{q},\mu,\nu)=-\sum\limits_{k\in\mathcal{K}}\kappa_{k}+\mu\sum\limits_{k\in\mathcal{K}}\chi_{c}^{\geq}(\kappa_{k}-\kappa_{k}^{2})+\nu\sum\limits_{k\in\mathcal{K}}\chi_{c}^{\geq}(\kappa_{k}\widetilde{\gamma}_{k}-\widehat{\gamma}_{k}),$
(24)

where $\chi_{c}^{\geq}(x)=\max\\{x,0\\}$ is the violation degree function.
Further, the Lagrangian dual problem of (23) is formulated as

$\displaystyle\max_{\\{\mu,\nu\\}}\,\min_{\\{\kappa_{k},q_{k}\\}}\mathcal{L}(\bm{\kappa},\mathbf{q},\mu,\nu).$
(25)

To update the trainable parameters of JEEPON, a primal-dual learning framework
(PDLF) is proposed to train it in an unsupervised manner, and the loss
function is defined as $\mathrm{Loss}=\mathcal{L}/K$. In the sequel, we focus
on describing the architecture of JEEPON and PDLF.

###  IV-B Graph Representation and Model Design

WCNs can be naturally divided into undirected/directed graphs depending on the
topology structures, and homogeneous/heterogeneous graphs depending on types
of the communication links and user equipments (UEs) [29]. For notational
convenience, a graph with node set $\mathcal{V}$ and edge set $\mathcal{E}$ is
defined as $\mathcal{G}(\mathcal{V},\mathcal{E})$, where the node
$v\in\mathcal{V}$ and edge $e_{u,v}\in\mathcal{E}$ (between node $u$ and node
$v$) feature vectors are represented as $\mathbf{x}_{v}$ and
$\mathbf{e}_{u,v}$, respectively. In the graph representation of single-cell
cellular networks, we can consider the UEs as nodes and the interfering links
between different UEs as edges, as shown in Fig. 1. In general, the node and
edge features of graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ mainly include
CSI and other environmental information, such as user weights and Gaussian
noise. In order to reduce the dimensionality of node and edge feature vectors,
we consider using the orthogonality (modulo value) of CSI to represent channel
gains and interferences. Therefore, the features of node $v$ and edge
$e_{u,v}$ are defined as
$\mathbf{x}_{v}=|\overline{\mathbf{h}}_{v}^{H}\mathbf{h}_{v}|$ and
$\mathbf{e}_{u,v}=|\overline{\mathbf{h}}_{u}^{H}\mathbf{h}_{v}|$,
respectively.

Figure 1: A wireless channel graph with four UEs.

Following the completion of the WCN graph representation, we focus on the
design of JEEPON to output the US-PA strategy, where the optimization vectors
are carefully defined in the representation vector of nodes. Specifically,
JEEPON applies a message passing mechanism based graph convolutional network
(GCN) [40] to iteratively update the representation vector of node
$v\in\mathcal{V}$ by aggregating features from its neighbor nodes and edges.
The GCN consists of two steps, first generating and collecting messages from
the first-order neighborhood nodes and edges of node $v$, and then updating
the representation vector of node $v$ with the aggregated messages. After
$\ell$ times of graph convolutions, the representation vector of node $v$
captures the messages within its $\ell$-hop neighborhood nodes and edges. To
be specific, the update rule of the $\ell$-th GCN layer at node $v$ is
formulated as

$\displaystyle\mathbf{m}_{u,v}^{(\ell)}$
$\displaystyle=\mathbf{M}_{\theta}^{(\ell)}\left(\bm{\beta}_{u}^{(\ell-1)},\mathbf{x}_{u},\mathbf{e}_{u,v}\right),u\in\mathcal{N}_{v},$
(26) $\displaystyle\mathbf{g}_{v}^{(\ell)}$
$\displaystyle=\mathbf{G}\left(F_{\mathrm{max}}\left(\\{\mathbf{m}_{u,v}^{(\ell)}\\}\right),F_{\mathrm{mean}}\left(\\{\mathbf{m}_{u,v}^{(\ell)}\\}\right)\right),v\in\mathcal{V},$
$\displaystyle\bm{\beta}_{v}^{(\ell)}$
$\displaystyle=\mathbf{U}_{\theta}^{(\ell)}\left(\bm{\beta}_{v}^{(\ell-1)},\mathbf{x}_{v},\mathbf{g}_{v}^{(\ell)}\right),v\in\mathcal{V},$

where $\mathcal{N}_{v}$ is the first-order neighborhood set of node $v$,
$\bm{\beta}_{v}^{(\ell)}\triangleq[\kappa_{v},q_{v}]\in\mathbb{R}^{2}$
represents the pairwise optimization vector of node $v$ at the $\ell$-th GCN
layer, and $\bm{\beta}_{v}^{(0)}$ is initialized with an all-zero vector.
Therefore, when the update of the $\ell$-th graph convolution operation is
completed, the representation vector of node $v$ could be formulated as
$[\bm{\beta}_{v}^{(\ell)},\mathbf{x}_{v}]$. Fig. 2 illustrates the state
update process of node $v$ at the $\ell$-th GCN layer. Here,
$\mathrm{M}_{\theta}^{(\ell)}(\cdot)$ is a message construction function
defined on each edge to generate edge message
$\mathbf{m}_{u,v}^{(\ell)}\in\mathbb{R}^{m}$ by combining incoming node and
edge features, where $m$ is the dimension size. $\mathbf{G}(\cdot)$ is a
message aggregation function that uses the concatenation of the max function
$F_{\mathrm{max}}(\cdot)$ and the mean function $F_{\mathrm{mean}}(\cdot)$ to
gather the relevant edge messages
$\\{\mathbf{m}_{u,v}^{(\ell)}|u\in\mathcal{N}_{v}\\}$ and output the
aggregated message $\mathbf{g}_{v}^{(\ell)}\in\mathbb{R}^{2m}$.
$\mathbf{U}_{\theta}^{(\ell)}(\cdot)$ is a state update function defined on
each node, which is used to update the node representation through the
aggregated message $\mathbf{g}_{v}^{(\ell)}$, node feature $\mathbf{x}_{v}$
and optimization vector $\bm{\beta}_{v}^{(\ell-1)}$. In JEEPON, function
$\mathrm{M}_{\theta}^{(\ell)}(\cdot)$ and function
$\mathbf{U}_{\theta}^{(\ell)}(\cdot)$ are parameterized by different neural
network modules.

Figure 2: The state update process of node $v$ at the $\ell$-th GCN layer.

Through the combination of several GCN layers, JEEPON can gather multi-hop
node and edge features. An illustration of JEEPON is given in Fig. 3, which
consists of $N_{\mathrm{L}}$ GCN layers and one projection activation (PAC)
layer. Each GCN layer includes an input layer, an output layer, and two
different MLPs which are composed of linear (LN) layers, batch normalization
(BN) layers and activation (AC) layers. In the final part of JEEPON, we
utilize the PAC layer to put
$\\{\kappa_{k}^{(N_{\mathrm{L}})},q_{k}^{(N_{\mathrm{L}})}\\}$ into the
feasible region $\bm{\Omega}$, i.e.,

$\displaystyle\bm{\Omega}\triangleq\\{\bm{\kappa},\mathbf{q}:{0}\leq\kappa_{k}\leq{1},\sum\limits_{k\in\mathcal{K}}q_{k}\leq{P},q_{k}\geq{0},\forall{k}\in\mathcal{K}\\}.$
(27)

To this end, the projection functions of the PAC layer are defined as

$\displaystyle\kappa_{k}^{(\ast)}$
$\displaystyle=F_{\mathrm{relu}}(\kappa_{k},1),~{}q_{k}^{{}^{\prime}}=F_{\mathrm{relu}}(q_{k},P),\forall{k}\in\mathcal{K},$
(28) $\displaystyle q_{k}^{(\ast)}$
$\displaystyle=\frac{P}{\mathrm{max}\\{P,\sum\limits_{k\in\mathcal{K}}q_{k}^{{}^{\prime}}\\}}q_{k}^{{}^{\prime}},\forall{k}\in\mathcal{K},$

where function
$F_{\mathrm{relu}}(\mathbf{z},\mathbf{b})=\max\\{\min\\{\mathbf{z},\mathbf{0}\\},\mathbf{b}\\}$,
and $\mathbf{b}$ is the upper bound of the input. Furthermore, due to the
matrix inversion operation in the uplink SINR equation (22), it leads to a
high computational overhead. To speed up the computation, the following Lemma
1 is applied to replace the direct matrix inversion by $K$ matrix iterations.
Specifically, it reduces the computational complexity to $\mathcal{O}(KN^{2})$
instead of matrix inversion complexity $\mathcal{O}(KN^{2}+N^{3})$, where
$\mathcal{O}(\cdot)$ is the big-$\mathcal{O}$ notation for describing the
computational complexity.

Figure 3: The architecture of JEEPON.

######  Lemma 1.

According to the Sherman-Morrison formula [41], for an invertible square
matrix $\mathbf{A}\in\mathbb{C}^{N\times{N}}$, if there exists two column
vectors $\mathbf{u},\mathbf{v}\in\mathbb{C}^{N\times 1}$,
$1+\mathbf{v}^{H}\mathbf{A}^{-1}\mathbf{u}\neq 0$, then the inverse of
$\mathbf{A}$ is given by

$(\mathbf{A}+\mathbf{u}\mathbf{v}^{H})^{-1}=\mathbf{A}^{-1}-\frac{\mathbf{A}^{-1}\mathbf{u}\mathbf{v}^{H}\mathbf{A}^{-1}}{1+\mathbf{v}^{H}\mathbf{A}^{-1}\mathbf{u}}.$
(29)

Based on this formula, let
$\mathbf{T}_{n}=\bm{\Lambda}^{-1},n\in\\{0,1,\cdots,K\\}$, then
$\mathbf{T}_{n}$ can be converted into an iterative matrix product form, which
is formulated as follows

$\mathbf{T}_{n}=\left\\{\begin{aligned} \mathbf{I}_{N}&,n=0,\\\
\mathbf{T}_{n-1}-\frac{\mathbf{T}_{n-1}q_{n}\overline{\mathbf{h}}_{n}\overline{\mathbf{h}}_{n}^{H}\mathbf{T}_{n-1}}{1+q_{n}\overline{\mathbf{h}}_{n}^{H}\mathbf{T}_{n-1}\overline{\mathbf{h}}_{n}}&,n>0.\end{aligned}\right.$
(30)

###  IV-C Primal-Dual Learning Framework

With regard to the aforementioned aspects, PDLF is developed for training the
JEEPON model to solve the Lagrangian dual problem (25), which is composed of
two parts, the primal update part and the dual update part, as shown in Fig.
4. The primal update part takes the user’s historical channel data sample
$\mathcal{D}\triangleq\\{\mathbf{h}_{k}\\}$ as input, and outputs the related
US-PA strategy
$\bm{\Phi}(\mathcal{D},\bm{\Theta})\triangleq\\{\kappa_{k},q_{k}\\}$, where
$\bm{\Theta}$ is the trainable parameters of JEEPON. Specifically, it includes
a graph representation module for WCN topology construction, a JEEPON model
for US-PA optimization, and a loss function module for updating $\bm{\Theta}$.
In the dual part, the Lagrangian multipliers $\\{\mu,\nu\\}$ are updated by
the sub-gradient optimization method. PDLF runs two parts alternately, the
former minimizes function $\mathcal{L}$ with fixed $\\{\mu,\nu\\}$ by updating
$\bm{\Theta}$ to obtain a suitable $\\{\kappa_{k},q_{k},\mathbf{w}_{k}\\}$,
and the latter maximizes function $\mathcal{L}$ with fixed
$\\{\kappa_{k},q_{k},\mathbf{w}_{k}\\}$ by updating $\\{\mu,\nu\\}$.
Therefore, the update rule of the Lagrangian multipliers $\\{\mu,\nu\\}$ at
the $\tau$-th epoch is

$\displaystyle\mu^{(\tau+1)}$
$\displaystyle=\mu^{(\tau)}+\varepsilon_{\mu}\sum\limits_{k\in\mathcal{K}}\chi_{c}^{\geq}\left(\kappa_{k}^{(\tau)}-(\kappa_{k}^{(\tau)})^{2}\right),$
(31) $\displaystyle\nu^{(\tau+1)}$
$\displaystyle=\nu^{(\tau)}+\varepsilon_{\nu}\sum\limits_{k\in\mathcal{K}}\chi_{c}^{\geq}\left(\kappa_{k}^{(\tau)}\widetilde{\gamma}_{k}-\widehat{\gamma}_{k}\right),$

where $\varepsilon_{\mu}$ and $\varepsilon_{\nu}$ is the update step-size of
$\mu$ and $\nu$, respectively. In addition, the Lagrangian multipliers are
updated every epoch based on the violation degree of the training datasets.
For the inner optimization of problem (25), JEEPON is proposed to transform it
into a statistical learning problem, which aims to obtain appropriate
optimization vectors $\\{\kappa_{k},q_{k}\\}$ by updating the trainable
parameters of JEEPON.

Figure 4: The architecture of the PDLF. Algorithm 3 PDLF for Training JEEPON.

0: $N_{\mathrm{e}}$: Number of epochs; $\bm{\Theta}$: The trainable parameters
of JEEPON; $\varepsilon_{\mu},\varepsilon_{\nu}$: Step size of Lagrangian
multipliers;
$\mathcal{D}\triangleq\\{\mathcal{D}_{i}\\}_{i=1}^{N_{\mathrm{ta}}}$: Training
dataset with $N_{\mathrm{ta}}$ samples.

0: The trained JEEPON model.

1: Initialize the trainable parameters $\bm{\Theta}$ and the Lagrangian
multipliers $\\{\mu^{(0)},\nu^{(0)}\\}$.

2: for epoch $\tau\leftarrow 1,2,\cdots,N_{\mathrm{e}}$ do

3: Initialize dual gradient variables
$\\{\nabla_{\mu}^{(0)},\nabla_{\nu}^{(0)}\\}$.

4: for each sample $\mathcal{D}_{i}:i\leftarrow 1,2,\cdots,N_{\mathrm{ta}}$ do

5: Construct the graph $\mathcal{G}_{i}(\mathcal{V},\mathcal{E})$ for sample
$\mathcal{D}_{i}$.

6: Obtain the US-PA strategy $\\{\kappa_{k}^{(i)},q_{k}^{(i)}\\}$ via JEEPON,
and then update $\bm{\Theta}$ via the loss function module.

7: Update dual gradient variables:

8:
$\nabla_{\mu}^{(i)}\leftarrow\nabla_{\mu}^{(i-1)}+\sum\limits_{k\in\mathcal{D}_{i}}\chi_{c}^{\geq}(\kappa_{k}^{(i)}-(\kappa_{k}^{(i)})^{2})$,
$\nabla_{\nu}^{(i)}\leftarrow\nabla_{\nu}^{(i-1)}+\sum\limits_{k\in\mathcal{D}_{i}}\chi_{c}^{\geq}(\kappa_{k}^{(i)}\widetilde{\gamma}_{k}-\widehat{\gamma}_{k})$.

9: end for

10: Obtain the Lagrangian multipliers $\\{\mu^{(\tau)},\nu^{(\tau)}\\}$ via
Eq. (31).

11: end for

PDLF is designed for training JEEPON. Unlike the penalty-based supervised
training method in [25], the proposed PDLF alternately updates $\bm{\Theta}$
and $\\{\mu,\nu\\}$ in an unsupervised manner, as summarized in Algorithm 3.
Specifically, we generate a training dataset
$\mathcal{D}\triangleq\\{\mathcal{D}_{i}\\}_{i=1}^{N_{\mathrm{ta}}}$ with
$N_{\mathrm{ta}}$ samples, and each sample with the same size. The training
stage of PDLF lasts for $N_{\mathrm{e}}$ epochs in total. In the primal update
part, PDLF first constructs the graph representation for sample
$\mathcal{D}_{i}$ (Setp 5), and takes it as the input of JEEPON. Then, JEEPON
outputs the US-PA strategy
$\bm{\Phi}(\mathcal{D}_{i},\bm{\Theta})\triangleq\\{\kappa_{k}^{(i)},q_{k}^{(i)}\\}$
of sample $\mathcal{D}_{i}$ (Step 6), and adopt the loss function module to
update $\bm{\Theta}$ (Step 7). The sub-gradient values of $\\{\mu,\nu\\}$ are
also stored to avoid repeated traversal of the training dataset (Steps 8-10).
Therefore, in the dual update part, $\\{\mu,\nu\\}$ are updated by the
recorded dual gradient variables
$\\{\nabla_{\mu}^{(N_{\mathrm{ta}})},\nabla_{\nu}^{(N_{\mathrm{ta}})}\\}$ and
equation (31) (Step 13).

##  V Numerical Results

In this section, numerical results are presented for the joint US-BF
optimization problem in the multiuser MISO downlink system. We first introduce
the experimental environment and system parameters. Next, the convergence of
SCA-USBF and J-USBF is evaluated. Then, the performance of G-USBF, SCA-USBF,
and J-USBF is discussed in different system scenarios, as well as the
generalizability of J-USBF. In addition, the performance of J-USBF and the
convolutional neural network based US-BF (CNN-USBF) algorithm (see Appendix B)
are also compared. Finally, the computational complexity of the algorithms is
presented and discussed, which clearly validates the speed advantage of
J-USBF.

###  V-A Experimental Setup

In the experiment666Offline training for J-USBF is necessary and important,
where numerous data is required. However, real data is quite difficult to
obtain although some researchers are committed to solving this problem [42,
43, 44], so we could only apply simulated data instead., the $K$ single-
antenna users are randomly distributed in the range of $(d_{l},d_{r})$ from
the BS, $d_{l},d_{r}\in(d_{\mathrm{min}},d_{\mathrm{max}})$, where
$d_{\mathrm{min}}=50\mathrm{m}$ is the reference distance and
$d_{\mathrm{max}}=200\mathrm{m}$ denotes the cell radius, as shown in Fig. 5.
The channel of user $k$ is modeled as
$\mathbf{h}_{k}=\sqrt{\rho_{k}}\widetilde{\mathbf{h}}_{k}\in\mathbb{C}^{N\times
1}$ where
$\widetilde{\mathbf{h}}_{k}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})$ is the
small-scale fading, $\varrho=3$ is the path-loss exponent, and
$\rho_{k}=1/(1+(d_{k}/d_{\mathrm{min}})^{\varrho})$ denotes the long-term
path-loss between user $k$ and the BS with $d_{k}$ representing the distance
in meters (m). For simplicity, we assume that all users have the same additive
noise variance, i.e., $\sigma_{k}^{2}=\sigma^{2},\forall{k}\in\mathcal{K}$,
thus, the signal-to-noise ratio (SNR) is defined as
$\mathrm{SNR}=10\log_{10}(P/\sigma^{2})$ in dB.

Figure 5: User distribution of the multiuser MISO downlink system.

In the neural network module, J-USBF is implemented by $N_{\mathrm{L}}=2$ GCN
layers via _Pytorch_ , and the functions $\mathrm{M}_{\theta}(\cdot)$ and
$\mathbf{U}_{\theta}(\cdot)$ in each GCN layer are parameterized by MLPs with
sizes $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, respectively. Training and test
stages for J-USBF are sequentially implemented. The learing rate of J-USBF and
Lagrangian multipliers are set to $\eta=5\times 10^{-5}$ and
$\varepsilon_{\mu},\varepsilon_{\nu}=10^{-5}$, respectively. For each
configuration, we respectively prepare $N_{\mathrm{ta}}=5000$ training samples
and $N_{\mathrm{te}}=500$ testing samples, where the validation split is set
to $0.2$ and the training samples are randomly shuffled at each epoch. The
entire training stage lasts for $N_{\mathrm{e}}=200$ epochs. According to the
conclusion in [21, Corollary 1], the user minimum achievable SINR
$\widetilde{\gamma}$ will be set by the system parameters
$\\{D,n,\epsilon\\}$. Note that unless mentioned otherwise, the experiments
adopt the default system parameters in Table I.

TABLE I: Default system parameters. Parameters | Values
---|---
Range of SNR | $10~{}\mathrm{dB}$
Blocklength $n$ | $128$
Decoding error probability $\epsilon$ | $10^{-6}$
Transmission data bits $D$ | $256~{}\mathrm{bits}$
BS antenna number $N$ | $32$
Number of candidate users $K$ | $30$
Maximum permissible error $\delta$ | $10^{-5}$
Sizes of MLPs in $\mathrm{M}_{\theta}(\cdot)$ | $\mathcal{H}_{1}=\\{4,256,128,64,32,16,m\\}$
Sizes of MLPs in $\mathbf{U}_{\theta}(\cdot)$ | $\mathcal{H}_{2}=\\{3+2m,256,128,64,32,16,3\\}$

###  V-B Convergence Analysis of SCA-USBF and J-USBF

The convergence of SCA-USBF and J-USBF is evaluated in this section, where
part of the system parameters are set to $K=50$ and
$(d_{l},d_{r})=(60\mathrm{m},100\mathrm{m})$. Fig. 6(a) illustrates the
objective value curve of SCA-USBF for different random channel realizations,
indicating that SCA-USBF can reach the convergence state through iterations.
Fig. 6(b) illustrates the objective value curve of J-USBF during the training
stage, where the objective value of the training samples varies in the range
(light blue line), and the average objective value curve (blue line) converges
as the number of iterations increases to $3.5\times 10^{5}$. During the
testing stage, the constraint violation ratios of J-USBF for different testing
samples are shown in Table II. It is observed that the percentage of illegal
results is $2.268\%$, which is much lower than the results satisfying the
constraints. Note that the value of $\kappa_{k},\forall{k}\in\mathcal{K}$ will
be set to $1$ if $0<\kappa_{k}<1$ is obtained, and all the scheduled users are
filtered again with the per-user minimum SINR requirement.

(a) SCA-USBF

(b) J-USBF

Figure 6: The objective value curves of SCA-USBF and J-USBF. TABLE II: Different constraint situations of J-USBF. Different constraint situations | Percentage of total samples
---|---
$\kappa_{k}=0,q_{k}\geq{0},\forall{k}\in\mathcal{K}$ | $75.436\%$
$0<\kappa_{k}<1,q_{k}\geq{0},\forall{k}\in\mathcal{K}$ | $2.264\%$
$\kappa_{k}=1,\widetilde{\gamma}_{k}>\widehat{\gamma}_{k},\forall{k}\in\mathcal{K}$ | $0.004\%$
$\kappa_{k}=1,\widetilde{\gamma}_{k}\leq\widehat{\gamma}_{k},\forall{k}\in\mathcal{K}$ | $22.296\%$

###  V-C Performance and Generalizability Evaluation

In this subsection, the performance of J-USBF, SCA-USBF and G-USBF with
different system parameters are evaluated and compared. For intuitive
comparison, the obtained results of SCA-USBF and J-USBF are normalized by the
results of G-USBF, defined as
$R_{1}=\frac{N_{\mathrm{S}}}{N_{\mathrm{G}}}\times 100\%$ and
$R_{2}=\frac{N_{\mathrm{J}}}{N_{\mathrm{G}}}\times 100\%$, where
$N_{\mathrm{S}}$, $N_{\mathrm{J}}$ and $N_{\mathrm{G}}$ are the average number
of scheduled users obtained through SCA-USBF, J-USBF and G-USBF, respectively.
In addition, we also define the result percentage of CNN-USBF and J-USBF as
$R_{3}=\frac{N_{\mathrm{C}}}{N_{\mathrm{J}}}\times 100\%$, where
$N_{\mathrm{C}}$ is the number of scheduled users obtained through CNN-USBF.

####  V-C1 Performance with Various $K$ and $(d_{l},d_{r})$

This experiment investigates the influences of $K$ and $(d_{l},d_{r})$ and
compares the performance of J-USBF with G-USBF and SCA-USBF, as well as with
CNN-USBF in large-scale user scenarios. Table III shows that when $K$ is
small, the performance of J-USBF is closer to that of G-USBF, because
sufficient system resources are conducive to model optimization. J-USBF
remains stable when $K$ changes from 20 to 50, and there exist only $2.56\%$
performance degradation at most. Besides, the performance gain of J-USBF
improves with the distance interval changes from $20\mathrm{m}$ to
$40\mathrm{m}$, since the smaller distance interval leads to the lack of
diversity for each user, which brings more difficulties to the learning of
J-USBF. In Fig. 7, we show the average performance of these three algorithms
with different $(d_{l},d_{r})$. It suggests that J-USBF could achieve a more
stable and closer performance compared with G-USBF as $(d_{l},d_{r})$
increases. Owing to the fact that the number of scheduled users is reduced
with the increase of $(d_{l},d_{r})$, and the obtained results are more
homogeneous, which is beneficial to the learning of J-USBF.

TABLE III: Performance normalized by G-USBF with various $K$. $K$ |  $R_{1}$ and $R_{2}$ with varying $(d_{l},d_{r})$
---|---
| $(50\mathrm{m},70\mathrm{m})$ | $(60\mathrm{m},80\mathrm{m})$ | $(50\mathrm{m},90\mathrm{m})$ | $(60\mathrm{m},100\mathrm{m})$
| $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$
10 | $100\%$ | $95.68\%$ | $99.98\%$ | $94.70\%$ | $100\%$ | $93.94\%$ | $99.89\%$ | $93.41\%$
20 | $99.67\%$ | $90.04\%$ | $99.64\%$ | $91.35\%$ | $99.52\%$ | $92.02\%$ | $99.22\%$ | $92.32\%$
30 | $99.94\%$ | $89.68\%$ | $99.63\%$ | $90.33\%$ | $98.80\%$ | $90.57\%$ | $98.77\%$ | $91.25\%$
40 | $99.86\%$ | $88.91\%$ | $99.54\%$ | $89.86\%$ | $98.52\%$ | $90.27\%$ | $98.24\%$ | $91.08\%$
50 | $99.84\%$ | $88.15\%$ | $98.73\%$ | $88.79\%$ | $97.48\%$ | $89.46\%$ | $97.10\%$ | $90.15\%$
Figure 7: Average number of scheduled users with various $(d_{l},d_{r})$.

Considering large-scale user scenarios, we focus on the performance comparison
of J-USBF and CNN-USBF, whereas ignoring G-USBF and SCA-USBF due to the high
computational overhead. Table IV shows that the performance gap between CNN-
USBF and J-USBF widens as $K$ increases, especially when $K=200$ and
$(d_{l},d_{r})=(60\mathrm{m},100\mathrm{m})$, the performance of the former
can only reach $87.36\%$ of the latter. This indicates that incorporating WCN
topology information into model learning is helpful for performance
improvement and stability maintenance.

TABLE IV: Performance normalized by J-USBF with various $K$. $K$ |  $R_{3}$ with varying $(d_{l},d_{r})$
---|---
| $(50\mathrm{m},70\mathrm{m})$ | $(60\mathrm{m},80\mathrm{m})$ | $(50\mathrm{m},90\mathrm{m})$ | $(60\mathrm{m},100\mathrm{m})$
50 | $93.06\%$ | $96.26\%$ | $92.42\%$ | $93.95\%$
100 | $92.89\%$ | $90.83\%$ | $91.42\%$ | $90.80\%$
150 | $89.14\%$ | $90.53\%$ | $90.59\%$ | $89.46\%$
200 | $88.75\%$ | $88.38\%$ | $87.51\%$ | $87.36\%$

####  V-C2 Performance with Various SNR Settings

This experiment compares the performance of J-USBF, SCA-USBF and G-USBF with
different SNR settings, and the obtained results are summarized in Table V. It
is observed that J-USBF achieves competitive performance (larger than
$90.77\%$) with $\mathrm{SNR}=5$ dB, while SCA-USBF maintains over $95.73\%$
near-optimal performance compared with G-USBF. Although the performance gap of
J-USBF is enlarged as $K$ increase, the trend of degradation is rather slow.
For the configuration $\mathrm{SNR}=15$ dB and
$(d_{l},d_{r})=(50\mathrm{m},100\mathrm{m})$, J-USBF obtains only a $1.78\%$
relative performance gap with G-USBF when $K$ changes from 20 to 50. Even when
$(d_{l},d_{r})=(100\mathrm{m},150\mathrm{m})$, J-USBF still maintains a stable
performance. Moreover, Fig. 8 illustrates the gap between the SCA-USBF and
J-USBF increases while SNR changes from 0 dB to 20 dB. With the increase of
SNR, channel condition becomes better and more users might meet the
requirement of QoS. Therefore, the solution space for problem (5) enlarges and
SCA-USBF shows its advantages under this condition, because it obtains
optimal/suboptimal results. On the other hand, J-USBF is difficult to capture
the optimal value as the solution space increases in such circumstance.
However, the gap between the SCA-USBF and J-USBF decreases when SNR changes
from 20 dB to 30 dB, since much better channel condition is sufficient for
serving all the users.

TABLE V: Performance normalized by G-USBF with various SNR settings. $\mathrm{SNR}(\mathrm{dB})$ |  $K$ |  $R_{1}$ and $R_{2}$ with varying $(d_{l},d_{r})$
---|---|---
|  | $(60\mathrm{m},90\mathrm{m})$ | $(90\mathrm{m},120\mathrm{m})$ | $(50\mathrm{m},100\mathrm{m})$ | $(100\mathrm{m},150\mathrm{m})$
|  | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$
5 |  10 | $98.96\%$ | $92.13\%$ | $98.43\%$ | $94.50\%$ | $99.37\%$ | $92.56\%$ | $97.74\%$ | $93.17\%$
|  20 | $98.22\%$ | $91.08\%$ | $97.82\%$ | $92.43\%$ | $98.82\%$ | $91.15\%$ | $96.59\%$ | $92.03\%$
|  30 | $97.17\%$ | $90.77\%$ | $96.58\%$ | $91.04\%$ | $97.75\%$ | $90.83\%$ | $95.73\%$ | $91.66\%$
15 |  20 | $99.99\%$ | $90.15\%$ | $99.60\%$ | $90.69\%$ | $100\%$ | $90.48\%$ | $99.17\%$ | $91.15\%$
|  30 | $99.87\%$ | $89.20\%$ | $99.61\%$ | $89.78\%$ | $99.97\%$ | $89.52\%$ | $98.46\%$ | $90.60\%$
|  50 | $99.64\%$ | $88.34\%$ | $99.55\%$ | $89.06\%$ | $99.80\%$ | $88.70\%$ | $97.36\%$ | $89.94\%$

(a) $(d_{l},d_{r})=(60\mathrm{m},90\mathrm{m})$

(b) $(d_{l},d_{r})=(100\mathrm{m},150\mathrm{m})$

Figure 8: Performance of the algorithms with various SNR settings.

####  V-C3 Performance with Various SINR Requirements

The ultimate scheduling results of the investigated problem are significantly
affected by the per-user minimum SINR requirement, where value
$\widetilde{\gamma}=F_{\mathrm{\gamma}}(D,n,\epsilon)$ is obtained with
different system parameters $D$, $n$, and $\epsilon$, and the results are
summarized in Table VI. From the table, It is observed that the average
performance of J-USBF remains above $88.97\%$ compared with G-USBF under
different SINR requirements and user distribution distances. However, one
needs to point out that with the reduction of SINR requirements, the
performance improvement of J-USBF is lower than G-USBF, especially when
$(d_{l},d_{r})=(60\mathrm{m},80\mathrm{m})$. Therefore, J-USBF shows a slight
performance degradation compared with G-USBF when the SINR requirement is
reduced, while the performance improvement of SCA-USBF increases at the same
time.

TABLE VI: Performance normalized by G-USBF with various SINR requirements. $F_{\mathrm{\gamma}}(D,n,\epsilon)$ |  $\widetilde{\gamma}$ |  $R_{1}$ and $R_{2}$ with varying $(d_{l},d_{r})$
---|---|---
|  | $(60\mathrm{m},80\mathrm{m})$ | $(80\mathrm{m},100\mathrm{m})$ | $(60\mathrm{m},100\mathrm{m})$ | $(80\mathrm{m},120\mathrm{m})$
|  | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$ | $R_{1}$ | $R_{2}$
$(256,256,10^{-6})$ |  $1.633$ | $99.92\%$ | $88.97\%$ | $99.96\%$ | $90.36\%$ | $99.97\%$ | $89.82\%$ | $99.91\%$ | $91.03\%$
$(256,128,10^{-6})$ |  $5.054$ | $99.63\%$ | $90.33\%$ | $98.30\%$ | $91.87\%$ | $98.77\%$ | $91.25\%$ | $98.36\%$ | $92.78\%$
$(256,96,10^{-6})$ |  $9.291$ | $96.41\%$ | $90.79\%$ | $94.22\%$ | $92.94\%$ | $95.84\%$ | $91.84\%$ | $95.38\%$ | $93.62\%$
$(256,64,10^{-6})$ |  $27.97$ | $95.58\%$ | $91.05\%$ | $93.95\%$ | $93.05\%$ | $94.55\%$ | $92.76\%$ | $94.19\%$ | $94.08\%$

####  V-C4 Generalizability with Various User Distributions

Generalizability is another critical evaluation property for J-USBF, and it
focuses on investigating whether the trained network model has the ability to
perform well in unknown WCN scenarios. To test the generalizability, J-USBF is
trained from a certain scenario whose system parameters are different from the
test ones. Specifically, J-USBF is trained with
$(d_{l},d_{r})=(100\mathrm{m},120\mathrm{m})$, then the trained model is
applied to the test scenarios with different $(d_{l},d_{r})$, without any
further training777For scenarios with different number of users $K$, number of
antennas $N$ and SINR requirements $\widetilde{\gamma}$, the generalizability
of J-USBF performs poorly and needs to be further optimized.. Table VII shows
comparison results of G-USBF and J-USBF, where
$R_{4}=\frac{N_{\mathrm{J},(100,120)}}{N_{\mathrm{G}}}\times 100\%$ represents
the average performance of J-USBF normalized by G-USBF and
$N_{\mathrm{J},(100,120)}$ is the average number of scheduled users using
J-USBF. Form the table, it is observed that J-USBF performs well over the
neighboring user distribution distances when the test distance interval is
40m. Moreover, when $(d_{l},d_{r})=(80\mathrm{m},100\mathrm{m})$ and there is
no intersection with $(100\mathrm{m},120\mathrm{m})$, the performance of
J-USBF is still acceptable at $K=10$. Based on the aforementioned analysis,
our proposed J-USBF can be well generalized to scenarios with neighboring user
distribution distances.

TABLE VII: Generalizability with various user distributions. $K$ |  $N_{\mathrm{G}}$ and $R_{4}$ with varying $(d_{l},d_{r})$
---|---
| $(100\mathrm{m},120\mathrm{m})$ | $(80\mathrm{m},100\mathrm{m})$ | $(80\mathrm{m},120\mathrm{m})$ | $(100\mathrm{m},140\mathrm{m})$
| $N_{\mathrm{G}}$ | $R_{4}$ | $N_{\mathrm{G}}$ | $R_{4}$ | $N_{\mathrm{G}}$ | $R_{4}$ | $N_{\mathrm{G}}$ | $R_{4}$
10 | $5.068$ | $94.97\%$ | $7.504$ | $86.06\%$ | $6.36$ | $91.86\%$ | $4.394$ | $92.13\%$
20 | $5.624$ | $93.92\%$ | $8.548$ | $84.63\%$ | $7.496$ | $89.58\%$ | $5.068$ | $91.46\%$
30 | $5.924$ | $92.69\%$ | $9.054$ | $83.34\%$ | $8.16$ | $88.31\%$ | $5.372$ | $88.16\%$
40 | $6.038$ | $91.24\%$ | $9.304$ | $82.75\%$ | $8.508$ | $87.92\%$ | $5.608$ | $87.73\%$
50 | $6.15$ | $90.80\%$ | $9.538$ | $83.04\%$ | $8.818$ | $85.87\%$ | $5.782$ | $86.09\%$

###  V-D Computational Complexity Analysis

In this subsection, the computational complexity of G-USBF, SCA-USBF and
J-USBF is analyzed and compared. Considering the differences in implementation
platforms and algorithm design languages, we count the floating-point
computation of the proposed algorithms. Firstly, G-USBF includes the US
optimization and BF design, whose floating-point computation is about
$\sum\limits_{\hat{k}=2}^{K}4(K-\hat{k}+1)(I_{1}(\hat{k}^{3}N+5\hat{k}^{2}N)+\hat{k}^{2})$,
where $\hat{k}$ and $I_{1}$ represent the number of scheduled users and
iterations, respectively. Secondly, SCA-USBF includes the inner and outer
optimizations, whose floating-point computation is about
$4I_{3}(I_{2}(7K^{2}N+4KN+14K^{2})+K(N^{3}+2N^{2}+2N))$, where $I_{2}$ and
$I_{3}$ represent the number of iterations for both parts. For J-USBF, since
the JEEPON model is trained offline, we mainly consider the computation of the
testing stage, including the graph representation module, the GCN module and
the SINR module. For simplicity, we assume that the GCN module is composed by
MLPs with the dimensions $\mathcal{H}\triangleq\\{h_{i}\\}$. Therefore, the
floating-point computation of J-USBF is about
$2(2K^{2}N+2KN^{2}+K\sum\limits_{\ell=1}^{L}\sum\limits_{i=1}^{|\mathcal{H}|}(2+h_{\ell,i-1})h_{\ell,i})$.
For intuitive comparison, Fig. 9 illustrates the comparison of the floating-
point computational magnitude of each algorithm for different number of users
and iterations. The computational magnitude of J-USBF is lower than that of
G-USBF and SCA-USBF, which indicates its computational efficiency advantage.

Figure 9: The floating-point computational magnitude of the algorithms.

##  VI Conclusions

In this paper, the joint US-BF optimization problem is studied for the
multiuser MISO downlink system. Specifically, with the help of uplink-downlink
duality theory and mathematical transformations, we formulate the original
problem into a convex optimization problem, and propose the G-USBF, SCA-USBF
and the J-USBF. Numerical results show that J-USBF achieves close performance
and higher computational efficiency. Additionally, the proposed J-USBF also
enjoys the generalizability in dynamic WCN scenarios. For future directions,
solving the problem of unbearable CSI acquisition burden and signaling
overhead caused by the instantaneous perfect CSI applied in this work is
interesting and meaningful. Deep learning based resource allocation algorithm
needs to be redesigned, and statistical CSI may be helpful to achieve the
goal.

##  Appendix A Design of The G-USBF Algorithm

In this appendix, the G-USBF algorithm is proposed to slove problem (5), which
is inspired by the work in [45] and the near-far effect of WCNs. The
feasibility problem in reference [21, problem (35)] forms the basis of G-USBF,
which is formulated as follows

$\displaystyle\min\limits_{\\{\mathbf{w}_{k}\\}}\sum\limits_{k\in\mathcal{S}}||\mathbf{w}_{k}||^{2},$
(32a)
$\displaystyle\mathrm{s.t.}~{}{r_{k}}\leq{R}(\overrightarrow{\gamma}_{k}),$
(32b)

where $\mathbf{w}_{k}\in\mathbb{C}^{N\times{1}}$ is the BF vector of user $k$,
and its downlink power is denoted as $p_{k}=\|\mathbf{w}_{k}\|^{2}$. Solving
problem (32) can be used to determine whether the scheduled user set
$\mathcal{S}$ is feasible, i.e., the user rate constraint and the BS power
budget need to be satisfied. G-USBF is designed with two stages, namely, the
conventional greedy search stage and the user set optimization stage, as
summarized in Algorithm 4. Here, G-USBF expands the scheduled user set
$\mathcal{S}$ from the candidate user set $\mathcal{K}$ in the first stage,
and then optimizes $\mathcal{S}$ in the second stage to achieve the goal of
scheduling more users. Since the G-USBF algorithm has close performance and
lower computational complexity compared with the exhaustive search algorithm,
therefore, it is used as the baseline.

Algorithm 4 The G-USBF Algorithm for Problem (5)

1: Input candidate user set $\mathcal{K}$ and user CSI $\\{\mathbf{h}_{k}\\}$,
and initialize scheduled user set $\mathcal{S}=\varnothing$.

2: Sort the user channels of $\mathcal{K}$ from good to bad via the MRT
method, and add the top-ranked user to $\mathcal{S}$.

3: In the greedy search stage, move one user from $\mathcal{K}$ to
$\mathcal{S}$ in sequence without repetition, and obtain temporary user sets
with $|\mathcal{K}\backslash\mathcal{S}|$ groups.

4: For each temporary user set, solve problem (32) to obtain
$\\{p_{k},\mathbf{w}_{k}\\}$, and preserve the user set
$\mathcal{S}_{1}^{(\ast)}$ with the smallest required power.

5: Let
$\mathcal{K}\leftarrow\mathcal{K}\backslash\mathcal{S}_{1}^{(\ast)},\mathcal{S}\leftarrow\mathcal{S}_{1}^{(\ast)}$
if $\mathcal{K}\neq\varnothing$ and
$\sum\limits_{k\in\mathcal{S}}p_{k}\leq{P}$ is obtained, then go to step 3.
Otherwise, go to step 6.

6: In the user set optimization stage, move one user with the largest power
consumption from $\mathcal{S}$ to $\mathcal{K}$, and obtain the user set
$\mathcal{S}_{2}$.

7: Let $\mathcal{S}\leftarrow\mathcal{S}_{2}$ and run the greedy search again
to obtain a new user set $\mathcal{S}_{2}^{(\ast)}$. If
$|\mathcal{S}_{1}^{(\ast)}|=|\mathcal{S}_{2}^{(\ast)}|$, stop iteration then
output $\mathcal{S}_{2}^{(\ast)}$ and $\\{p_{k},\mathbf{w}_{k}\\}$. Otherwise,
let $\mathcal{S}\leftarrow\mathcal{S}_{2}^{(\ast)}$ and go to step 6.

##  Appendix B Design of The CNN-USBF Algorithm

In this appendix, the CNN-USBF algorithm is proposed to slove problem (23),
which is inspired by the work in [46]. In particular, CNN-USBF takes the WCN
graph representation as input and outputs the US-PA optimization strategy and
BF vectors. To be specific, the update rule of CNN-USBF for node $v$ in graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$ is formulated as

$\displaystyle\mathrm{Input:}$
$\displaystyle~{}\mathbf{D}_{v}^{(0)}=[\mathbf{x}_{v},F_{\mathrm{max}}(\\{\mathbf{e}_{u,v}\\}),F_{\mathrm{mean}}(\\{\mathbf{e}_{u,v}\\})],u\in\mathcal{N}_{v},$
(33) $\displaystyle\mathrm{CNN\raisebox{0.0pt}{-}stage:}$
$\displaystyle~{}\mathbf{D}_{v}^{(i)}=F_{\mathrm{std}}(\mathrm{Cov1d}(\mathbf{D}_{v}^{(i-1)})),i=1,2,\cdots,N_{\mathrm{1}},$
$\displaystyle\mathrm{DNN\raisebox{0.0pt}{-}stage:}$
$\displaystyle~{}\mathbf{D}_{v}^{(i)}=F_{\mathrm{std}}(\mathrm{LNN}(\mathbf{D}_{v}^{(i-1)})),i=N_{\mathrm{1}}+1,\cdots,N_{\mathrm{1}}+N_{\mathrm{2}},$
$\displaystyle\mathrm{Output:}$
$\displaystyle~{}\mathbf{D}_{v}^{(N_{\mathrm{2}})}=[\kappa_{v}^{(\ast)},q_{v}^{(\ast)}],\mathrm{and~{}BF~{}vector}~{}\mathbf{w}_{v}^{(\ast)},v\in\mathcal{V},$

where $N_{\mathrm{1}}$ and $N_{\mathrm{2}}$ denote the layers of CNN and DNN,
respectively. $\mathbf{D}_{v}^{(0)}$ is the features of node $v$ and its
neighborhood edges, $\mathbf{D}_{v}^{(N_{\mathrm{2}})}$ is the US-PA strategy
of node $v$, and
$F_{\mathrm{std}}(\mathbf{z})=F_{\mathrm{AC}}(F_{\mathrm{BN}}(\mathbf{z}))$ is
the standardization function used to standardize the network input to
accelerate training process and reduce generalization error, which is
implemented by BN and AC layers. The neural network module of CNN-USBF is
constructed through CNN and DNN, which are implemented and trained by
_Pytorch_ and PDLF, respectively. The algorithm steps of CNN-USBF refer to
J-USBF. Note that unless mentioned otherwise, the neural network structure of
CNN-USBF refer to Table VIII and is trained separately for different WCN
scenarios.

TABLE VIII: The neural network structure of CNN-USBF. Layer | Parameters
---|---
Layer 1 (Input) |  Input of size $3K$, batch of size $K$, $N_{\mathrm{e}}$ epochs
Layer 2 (Cov1d, BN and AC) |  Input=$3$, output=$256$; Input=$256$; LReLU
Layer 3 (Cov1d, BN and AC) |  Input=$256$, output=$128$; Input=$128$; LReLU
Layer 4 (Cov1d, BN and AC) |  Input=$128$, output=$64$; Input=$64$; LReLU
Layer 5 (LNN, BN and AC) |  Input=$64$, output=$32$; Input=$32$; LReLU
Layer 6 (LNN, BN and AC) |  Input=$32$, output=$16$; Input=$16$; LReLU
Layer 7 (LNN, BN and AC) |  Input=$16$, output=$2$; Input=$2$; LReLU
Layer 8 (Output and PAC) |  Output of size $2K+KN$, Adam optimizer

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