# Meta-learning Based Beamforming Design for MISO Downlink

Jingyuan Xia  School of Electrical Technology
National University of Defense Technology
Changsha, China 410072
Email<EMAIL_ADDRESS>Deniz Gunduz  Department of Electrical and
Electronic Engineering
Imperial College London
London, UK
Email<EMAIL_ADDRESS>
###### Abstract

Downlink beamforming is an essential technology for wireless cellular
networks; however, the design of beamforming vectors that maximize the
weighted sum rate (WSR) is an NP-hard problem and iterative algorithms are
typically applied to solve it. The weighted minimum mean square error (WMMSE)
algorithm is the most widely used one, which iteratively minimizes the WSR and
converges to a local optimal. Motivated by the recent developments in meta-
learning techniques to solve non-convex optimization problems, we propose a
meta-learning based iterative algorithm for WSR maximization in a MISO
downlink channel. A long-short-term-memory (LSTM) network based meta-learning
model is built to learn a dynamic optimization strategy to update the
variables iteratively. The learned strategy aims to optimize each variable in
a less greedy manner compared to WMMSE, which updates variables by computing
their first order stationary points at each iteration step. The proposed
algorithm outperforms WMMSE significantly in the high signal to noise
ratio(SNR) regime and shows the comparable performance when the SNR is low.

##  I Introduction

Downlink beamforming is an essential technology for multi-antenna cellular
networks. Optimization of the beamforming vectors is typically formulated as a
weighted sum rate (WSR) maximization problem under a total transmit power
constraint. However, the WSR maximization problem is non-convex and NP-hard
[1]. Therefore, it is typically solved by iterative methods which either adopt
convex approximations [2, 3, 4], or convert the original problem into an
alternative formulation with closed-form solutions [5, 6, 7]. Among them, the
iterative weighted minimum mean square error (WMMSE) algorithm is the most
widely used approach that provides superior performance in most scenarios.

The WMMSE algorithm converts the WSR maximization problem into a weighted sum
mean square error minimization problem under the total transmit power
constraint. The equivalent minimization problem is convex with respect to each
individual variable when the others are fixed. Therefore, a locally optimal
solution can be obtained by minimizing the objective function with respect to
individual variables in an iterative manner. The WMMSE algorithm has been
shown to achieve state-of-the-art performance in a wide range of scenarios,
and is often considered as the benchmark in the literature.

Our goal in this paper is to obtain a less greedy and more flexible iterative
updating strategy, such that the solution of each individual variable moves
away from the locally optimal solution and moves closer to the globally
optimal solution. To achieve such an adaptive algorithm, we incorporate
machine learning techniques. Using machine-learning-based solutions to improve
the WMMSE algorithm has received significant recent attention. The common
approaches are built on an end-to-end learning model, where the wireless
channel coefficients are given as input to a neural network, which then
outputs the estimation of the transmitter beamformer [8, 9, 10]. These
approaches convert the original model-based problem into a data-driven
learning-based problem, and the performance depends on the the network
architecture. Alternatively, more recent works in [11, 12] propose unfolding
the WMMSE algorithm in order to design a model-inspired computational
structure for the neural network, and learn specific parameters of the
original model by training on a set of channel samples. The unfolded WMMSE
algorithm follows the original mathematical model of the WSR optimization
problem, and designs a specific network structure to map the variables and the
parameters in the original problem; therefore, the optimization of the WSR
problem is converted into the optimization of the designed neural network.
However, the performance of the aforementioned approaches [8, 9, 11, 12] do
not surpass the WMMSE algorithm.

An alternative approach is to utilize meta-learning techniques to solve
optimization problems. The work in [13] uses meta-learner networks to learn
the variable updating strategy in the form of gradient descent, which
surpasses the benchmark approaches including Adam [14] and RMSProp [15] in
convergence speed. In [16, 17], meta-learning is applied to the design of
solution algorithms for different gradient-descent-based optimization
problems. Another popular model-agnostic meta-learning (MAML) algorithm [18,
19] uses meta-learning for finding a superior initialization, which can lead
to a faster convergence. A meta-learning based global scope optimization (GSO)
method is proposed in [20] for solving non-convex multi-variable optimization
problems. The GSO algorithm exhibits significantly better performance in
solving non-convex problems, such as matrix completion without rank
information or Gaussian mixture model with high dimensionality, than the
traditional alternating minimization-based methods.

In this paper, we propose a meta-learning aided beamformer (MLBF) algorithm to
solve the WSR maximization problem. The MLBF algorithm is built upon the
iterative block-coordinate descent method, but instead of greedily optimizing
each individual variable at each step, it can adaptively optimize each
variable with respect to the geometry of the objective function. The proposed
MLBF algorithm is established by three long-short-term-memory (LSTM) neural
networks corresponding to the three complex variables to be optimized. The
general structure of the MLBF algorithm for the WSR maximization problem is
depicted in Fig.1. The notations and the detailed explanation will be
presented in Section IV. It can be seen that the MLBF algorithm builds a
bridge between the accumulated global loss function $\mathcal{L}_{F}$ and the
update rules on each variable. This allows the gradient flow to circulate
between the inner update loops for individual variable (referring to the
update steps $i$, $j$ and $k$ for each variable) and the outer iterative steps
for different variables (referring to different $t$).

Figure 1: The MLBF algorithm builds a bridge of information flow between the
updates on each variable and the global objective function (denoted by
$\mathcal{L}_{F}$). Each color represents the gradient flow for one individual
variable. We use solid arrows to denote information flow, and dashed arrows to
indicate the lack of gradient flow. It shows that the MLBF algorithm achieves
a circular information flow between the variable update steps within each sub-
problem and the global objective function throughout iterations.

The main contribution of this work is the proposed MLBF algorithm that can
learn a less greedy update strategy for the variables, and therefore, achieves
a better performance than the WMMSE algorithm. The extensive numerical results
show that the proposed MLBF algorithm outperforms the WMMSE algorithm in terms
of WSR when the SNR is high and performs equally at low SNR. We observe that
the gain from the proposed MLBF algorithm in terms of the achieved WSR
compared to the WMMSE algorithm increase with the channel SNR.

##  II Problem Formulation

In this paper, we study a multiple-input single output (MISO) downlink
channel, where a base station with $M$ transmit antennas serves $N$ single-
antenna users. Let $x_{i}\sim\mathcal{CN}(0,1)$ denote the independent data
symbol transmitted to the $i^{\text{th}}$ user, while
$\bm{h}_{i}\sim\mathcal{CN}(\bm{0},\bm{I}_{M})$ denote the channel between the
base station and the $i^{\text{th}}$ user. Then the received signal at the
$i^{\text{th}}$ user is given by

$y_{i}=\bm{h}_{i}^{H}\bm{v}_{i}x_{i}+\sum_{j=1,j\not=i}^{N}\bm{h}_{i}^{H}\bm{v}_{j}x_{j}+n_{i},$
(1)

where $\bm{v}_{i}\in\mathbb{C}^{M}$ is the transmit beamforming vector for
user $i$ and $n_{i}\in\mathcal{CN}(0,\sigma^{2})$ denotes a complex circularly
symmetric zero mean Gaussian noise with variance $\sigma^{2}$. The signal-to-
interference-plus-noise-ratio (SINR) is given by

$\text{SINR}_{i}=\frac{|\bm{h}_{i}^{H}\bm{v}_{i}|^{2}}{\sum_{j=1,j\not=i}|\bm{h}_{i}^{H}\bm{v}_{j}|^{2}+\sigma^{2}}.$
(2)

Given the channel knowledge, the WSR maximization problem of the downlink
channel with respect to a total transmit power constraint is formulated as

$\displaystyle\max_{\bm{V}}\sum_{i=1}^{N}\alpha_{i}\log_{2}(1+\text{SINR}_{i})$
(3) $\displaystyle\text{s.t.}\;\text{Tr}(\bm{V}\bm{V}^{H})\leq P,$

where $\alpha_{i}$ is the user weight specified by the system designer, $P$ is
the maximum transmit power,
$\bm{V}\triangleq[\bm{v}_{1},\ldots,\bm{v}_{N}]^{T}$, and $\text{Tr}(\cdot)$
denotes the trace operator.

##  III WMMSE Algorithm

As mentioned earlier, the WSR maximization problem in (3) is NP hard and non-
convex. The iterative WMMSE algorithm is typically implemented to solve it in
the following steps: First, problem (3) is converted into the equivalent
weighted sum mean square error minimization problem:

$\displaystyle\min_{\bm{u},\bm{w},\bm{V}}\alpha_{i}\sum_{i=1}^{N}(w_{i}e_{i}-\log_{2}w_{i})$
(4) $\displaystyle\text{s.t.}\;\text{Tr}(\bm{V}\bm{V}^{H})\leq P,$

where $e_{i}$ is the mean-square error given by

(5) $\displaystyle e_{i}$
$\displaystyle=\mathbb{E}_{x_{i},\bm{h}_{i}}[|\hat{x_{i}}-x_{i}|^{2}]$
$\displaystyle=|u_{i}\bm{h}_{i}^{H}\bm{v}_{i}-1|^{2}+\left(\sum_{j\not=i,j=1}^{N}|u_{i}\bm{h}_{i}^{H}\bm{v}_{j}|^{2}\right)+\sigma^{2}|u_{i}|^{2},$

$u_{i}$ denotes the receiver gain, $w_{i}$ is a positive user weight, and
$\bm{w}=[w_{1},\ldots,w_{N}]^{T}$, $\bm{u}=[u_{1},\ldots,u_{N}]^{T}$. Although
this new formulation is also non-convex, problem (4) is convex with respect to
each individual variable when the others are fixed. Therefore, this new
formulation can be solved by block coordinate descent. Indeed, a closed-form
solution with respect to the first-order stationary point for each variable
can be obtained. Then, the WMMSE algorithm iteratively updates each variable
$i=1,\ldots,N$ by computing their first order stationary points as follows

$w_{i}=\frac{\sum_{j=1}^{N}|\bm{h}_{i}^{H}\bm{v}_{j}|^{2}+\sigma^{2}}{\sum_{j\not=i,j=1}^{N}|\bm{h}_{i}^{H}\bm{v}_{j}|^{2}+\sigma^{2}},$
(6)

$u_{i}=\frac{\bm{h}_{i}^{H}\bm{v}_{i}}{\sum_{j=1}^{N}|\bm{h}_{i}^{H}\bm{v}_{j}|^{2}+\sigma^{2}},$
(7)

$\bm{v}_{i}=\alpha_{i}u_{i}w_{i}\bm{h}_{i}(\bm{A}+\mu\bm{I})^{-1},$ (8)

where
$\bm{A}\triangleq\sum_{i=1}^{N}\alpha_{i}w_{i}|u_{i}|^{2}\bm{h}_{i}\bm{h}_{i}^{H}$,
and $\mu\geq 0$ is a Langrange multiplier. In a nutshell, the WMMSE algorithm
first initializes $\bm{w}$ so that it satisfies the power constraint, then
iteratively updates $\bm{V}$, $\bm{u}$ and $\bm{V}$ following the equations
(6)-(8) until a stop criterion is met. Parameter $\mu$ is computed by
bisection search. Further details of the WMMSE algorithm can be found in [7].

##  IV A Meta-learning based Beamforming (MLBF) Algorithm

We propose a meta-learning based WMMSE algorithm to learn an adaptive updating
strategy of each variable in problem (4). In particular, for each variable,
the update term is generated by a meta-learner neural network whose parameters
are continuously updated with the objective of minimizing the global loss
function $F(\bm{u},\bm{w},\bm{V})$. We refer to the minimization of
$F(\bm{u},\bm{w},\bm{V})$ as the overall problem while to the minimization of
$f(\bm{u})$,$f(\bm{w})$ and $f(\bm{V})$ as the three sub-problems to be
optimized sequentially in each iteration. Following (4) the overall problem
can be written as

$\min
F(\bm{u},\bm{w},\bm{V})\triangleq\alpha_{i}\sum_{i=1}^{N}(w_{i}e_{i}-\log_{2}w_{i})+\mu\text{Tr}(\bm{V}\bm{V}^{H})-\mu
P.$ (9)

Three LSTM networks $m_{\bm{V}}$, $m_{\bm{w}}$ and $m_{\bm{u}}$ are
established for updating $\bm{V}$, $\bm{w}$, and $\bm{u}$, respectively. We
define the variable update process within each sub-problem as inner loops and
the iterative steps over different sub-problems as outer loops. We
correspondingly denote by $\bm{\theta}_{\bm{V}}$, $\bm{\theta}_{\bm{u}}$ and
$\bm{\theta}_{\bm{w}}$ the parameters of the three LSTM networks, and by
$\bm{C}_{\bm{V}}$, $\bm{C}_{\bm{u}}$, and $\bm{C}_{\bm{w}}$ their cell states.
The inputs of the LSTM networks are the gradients of the sub-problems and the
previous states of the variables which are represented by the corresponding
$\nabla f()$ and $\bm{C}$, respectively. The outputs of the LSTM networks
include updating steps for variables, along with the update of the cell state.
The cell states contain the previous variables’ information, and can be
adaptively adjusted by the inner control gates inside the corresponding LSTM
networks. Further details can be found in [20]. Denoting the inner loop update
steps by $k=1,\ldots,K$, $i=1,\ldots,I$, and $j=1,\ldots,J$ as superscripts
and the outer loop steps by $t=1,\ldots,T$ as subscripts for each sub-problem,
where $K$, $I$ and $J$ denote the maximum number of steps of the inner loops
for each variable, and $T$ denotes the maximum number of steps of the outer
loop, respectively. The meta-learner networks $m_{\bm{V}}$, $m_{\bm{w}}$ and
$m_{\bm{u}}$ carry out variable updates in the inner loops in the following
form:

$\displaystyle\bm{V}^{(k)}=\bm{V}^{(k-1)}+m_{\bm{V}}(\nabla
f(\bm{V}^{(k-1)}),\bm{C}_{\bm{V}^{(k-1)}},\bm{\theta}_{\bm{V}}),$ (10)
$\displaystyle\bm{u}^{(i)}=\bm{u}^{(i-1)}+m_{\bm{u}}(\nabla
f(\bm{u}^{(i-1)}),\bm{C}_{\bm{u}^{(i-1)}},\bm{\theta}_{\bm{u}}),$
$\displaystyle\bm{w}^{(j)}=\bm{w}^{(j-1)}+m_{\bm{w}}(\nabla
f(\bm{w}^{(j-1)}),\bm{C}_{\bm{w}^{(j-1)}},\bm{\theta}_{\bm{w}}).$

Within the inner loops, the parameters of these three networks are fixed, and
are used to generate the updates for the variables as the input of the
variable update function. In the outer loops, we update the network parameters
through backpropagation with respect to the accumulated global loss, given by

$\mathcal{L}_{F}^{s}=\frac{1}{t_{u}}\sum_{t_{s}=(s-1)t_{u}+1}^{st_{u}}\omega_{t_{s}}F(\bm{W}_{t_{s}},\bm{X}_{t_{s}}),$
(11)

where $t_{u}$ is the update interval; that is, for every $t_{u}$ outer loop
iterations, the accumulated global loss $\mathcal{L}_{F}^{s}$ is used to
update $\bm{\theta}_{\bm{V}}$, $\bm{\theta}_{\bm{u}}$ and
$\bm{\theta}_{\bm{w}}$, $s=1,2,\ldots,S$, and
$\omega_{t_{s}}\in\mathbb{R}_{\geq 0}$ denotes the weights associated with
each outer loop step. $T=St_{u}$ denotes the maximum number of outer loop
steps. Every $t_{u}$ outer loop iterations, the parameters of the LSTM
networks are updated by the Adam [14] optimizer using the accumulated global
loss $\mathcal{L}_{F}^{s}$:

$\displaystyle\bm{\theta}_{\bm{V}}^{s+1}=\bm{\theta}_{\bm{V}}^{s}+\alpha_{\bm{V}}\cdot\mathrm{Adam}(\bm{\theta}_{\bm{V}}^{s},\nabla_{\bm{\theta}_{\bm{V}}^{s}}\mathcal{L}_{F}^{s}),$
(12)
$\displaystyle\bm{\theta}_{\bm{u}}^{s+1}=\bm{\theta}_{\bm{u}}^{s}+\alpha_{\bm{u}}\cdot\mathrm{Adam}(\bm{\theta}_{\bm{u}}^{s},\nabla_{\bm{\theta}_{\bm{u}}^{s}}\mathcal{L}_{F}^{s}),$
$\displaystyle\bm{\theta}_{\bm{w}}^{s+1}=\bm{\theta}_{\bm{w}}^{s}+\alpha_{\bm{w}}\cdot\mathrm{Adam}(\bm{\theta}_{\bm{w}}^{s},\nabla_{\bm{\theta}_{\bm{w}}^{s}}\mathcal{L}_{F}^{s}),$

where $\alpha_{\bm{V}}$, $\alpha_{\bm{u}}$ and $\alpha_{\bm{w}}$ denote the
learning rate for the LSTM networks $m_{\bm{V}}$, $m_{\bm{u}}$ and
$m_{\bm{w}}$, respectively.

To strictly satisfy the total power constraint when neural networks are used
to update the variables, we define
$\mathcal{D}\triangleq\\{\bm{V}|\text{Tr}(\bm{V}\bm{V}^{H})\leq P\\}$ as the
set of beamforming matrices that satisfy the power constraint. Then, we apply
the following projection of the beamforming matrix $\bm{V}$ to set
$\mathcal{D}$ in each outer loop step when the inner loop updating for
$\bm{V}$ is completed.

$\Omega_{\mathcal{D}}\\{\bm{V}\\}=\left\\{\begin{array}[]{l}\bm{V},\;\;\;\;\;\;\;\;\;\;\;\text{if}\;\bm{V}\in\mathcal{D},\\\
\frac{\bm{V}}{||\bm{V}||_{F}}\sqrt{P},\text{otherwise}.\end{array}\right.$
(13)

In conclusion, the MLBF algorithm builds a connection between the variable
update functions and the global loss function. Specifically, at the outer
loops, the three networks update their parameters with the objective of
maximizing the WSR (equivalently minimizing the global loss function in (9)),
which then determine the generation of the variable update terms with the
inner loops. In this way, the knowledge of the global loss is reflected on the
update rules of each variable within the inner loops; that is,
$\bm{\theta}_{\bm{V}}$, $\bm{\theta}_{\bm{u}}$ and $\bm{\theta}_{\bm{w}}$
convey the global loss knowledge from the outer loops for the variable update
functions. This allows the individual variable update functions to be less-
greedy, and to return updates that are more aligned with the global loss
function instead of the first-order stationary points of the corresponding
sub-problems. Therefore, the learned updating strategy may not necessarily
minimize the loss function of each sub-problem, but the overall problem could
be better accommodated compared to the WMMSE algorithm. The general structure
of the proposed MLBF algorithm for the WSR maximization problem is presented
in Algorithm 1.

We highlight that, here the proposed MLBF algorithm is applied directly as a
solution algorithm. For each WSR problem to be optimized, the LSTM networks’
parameters of the implemented MLBF algorithm start from scratch and are
trained during the iterative steps, such that the finally obtained variables
are assigned as the solutions. We note that the computational complexity of
the MLBF algorithm is higher than the WMMSE algorithm, nevertheless, it is
directly used as a solution algorithm for WSR problem. In this way, the
proposed MLBF approach is less time-consuming and resource-dependent to be
applied than the previous deep-learning based approaches [11, 9, 10, 12], as
the training process is essentially embedded in the solution process while the
traditional deep-learning based approaches have to train the parameters of the
networks with a set of training samples for a certain period of time (which is
typically quite long) before they can be used to solve a particular instance
of the problem.

1 Given: global loss function $F(\bm{u},\bm{w},\bm{V})$, $\bm{H}$.

2Initialized: $\bm{V}_{0}$, $\bm{u}_{0}$, and $\bm{w}_{0}$.

3for  _$t\leftarrow 1$ ,2, $\ldots$, T_ do

4 $i,j,k,s=1$

5 while  _$i\leq I$_ do

6 $\Delta\bm{u}=m_{\bm{u}}(\nabla
f(\bm{u}^{(i-1)}),\bm{C}_{\bm{u}}^{(i-1)},\bm{\theta}_{\bm{u}}^{s})$

7 $\bm{u}^{(i)}\leftarrow\bm{u}^{(i-1)}+\Delta\bm{u}$ $i=i+1$

8 end while

9 $\bm{u}_{t}=\bm{u}^{(I)}$

10 generate $f(\bm{w})\text{with}\;\bm{u}_{t},\bm{V}_{t-1}$

11 while  _$j\leq J$_ do

12 $\Delta\bm{w}=m_{\bm{w}}(\nabla
f(\bm{w}^{(j-1)}),\bm{C}_{\bm{w}}^{(j-1)},\bm{\theta}_{\bm{w}}^{s})$

13 $\bm{w}^{(j)}\leftarrow\bm{w}^{(j-1)}+\Delta\bm{w}$ $j=j+1$

14 end while

15

16 $\bm{w}_{t}=\bm{w}^{(J)}$

17 generate $f(\bm{V})\text{with}\;\bm{w}_{t},\bm{u}_{t}$

18 while  _$k\leq K$_ do

19 $\Delta\bm{V}=m_{\bm{V}}(\nabla
f(\bm{V}^{(k-1)}),\bm{C}_{\bm{V}}^{(k-1)},\bm{\theta}_{\bm{V}}^{s})$

20 $\tilde{\bm{V}}^{(k)}\leftarrow\bm{V}^{(k-1)}+\Delta\bm{V}$

21 $\bm{V}^{(k)}=\Omega_{\mathcal{D}}\\{\tilde{\bm{V}}^{(k)}\\}$

22 $k=k+1$

23 end while

24

25 $\bm{V}_{t}=\bm{V}^{(K)}$

26 generate $f(\bm{u})\text{with}\;\bm{V}_{t},\bm{w}_{t}$

27
$F(\bm{V}_{t},\bm{u}_{t},\bm{w}_{t})\leftarrow\bm{u}_{t},\bm{w}_{t},\bm{V}_{t}$

28 while  _$s\leq t/t_{u}$_ do

29
$\mathcal{L}_{F}^{s}=\frac{1}{t_{u}}\sum_{t_{s}=(s-1)t_{u}+1}^{st_{u}}\omega_{t_{s}}F(\bm{V}_{t_{s}},\bm{w}_{t_{s}},\bm{u}_{t_{s}})$

30
$\bm{\theta}_{\bm{V}}^{s+1}=\bm{\theta}_{\bm{V}}^{s}-\alpha_{\bm{V}}\nabla_{\bm{\theta}_{\bm{V}}^{s}}\mathcal{L}_{F}^{s}$

31
$\bm{\theta}_{\bm{u}}^{s+1}=\bm{\theta}_{\bm{u}}^{s}-\alpha_{\bm{u}}\nabla_{\bm{\theta}_{\bm{u}}^{s}}\mathcal{L}_{F}^{s}$

32
$\bm{\theta}_{\bm{w}}^{s+1}=\bm{\theta}_{\bm{w}}^{s}-\alpha_{\bm{w}}\nabla_{\bm{\theta}_{\bm{w}}^{s}}\mathcal{L}_{F}^{s}$

33 $s=s+1$

34 end while

35

36 end for

Algorithm 1 General Structure of the MLBF algorithm for WSR maximization
problem.

##  V Simulation Results

In this section, the performance of the proposed MLBF algorithm is evaluated
by extensive simulations. MLBF algorithm is implemented in Python 3.6 with
Pytorch 1.6. The WMMSE algorithm from [11] is also implemented in Python 3.6
with Tensorflow 1.1 for comparison. We consider the number of transmit
antennas $M=4$ and the number of single-antenna users $N=4$, with user weights
$\alpha_{i}=1$, $\forall i$. We initialize the MLBF and WMMSE with the same
initialization of $\bm{V}$ such that $\text{Tr}(\bm{V}\bm{V}^{H})\leq P$. Both
methods are directly evaluated on 1000 channel realizations generated
independent and identically distributed (i.i.d.) from a complex standard
Gaussian distribution.

For the MLBF algorithm, we set the maximum number of outer loops to $T=500$
when solving each problem and set the maximum number of inner loops to
$K=I=J=10$ when updating each variable within the inner loop. The update
interval $t_{u}$ is set to $5$, thus the maximum update times within each
sample in the test set is $S=100$. Our meta-learners employ two-layer LSTM
networks with 200 hidden units in each layer. The learning rates of Adam [14]
for updating the LSTM networks are set to $10^{-4}$. The weight of the outer
loop step $\omega_{t_{s}}$ is set to 1 for all $t_{s}$. We note that our MLBF
algorithm is trained during the optimization process and the final obtained
results are presented to evaluate the performance. At the beginning of solving
each problem, our networks start from scratch. In this way, the MLBF algorithm
is implemented as a solution algorithm to the optimization problem at hand,
relieved from the time-consuming training process.

Since complex variables are currently not supported by the deep learning
tools, we decompose the complex variables into their real and imaginary parts.
Thus we make an alternative reformulation that maps the complex computations
to the equivalent computations in reals. Let the real matrix
$\bm{V}^{{}^{\prime}}=[\bm{V}_{re},\bm{V}_{im}]$ denote the variables updated
by the meta-learner network. We employ $\bm{M}_{re}$ and $\bm{M}_{im}$ as the
masks matrices to recover the real and imaginary parts, respectively; that is,
we set $\bm{V}_{re}=\bm{V}^{{}^{\prime}}\bm{M}_{re}$ and
$\bm{V}_{im}=\bm{V}^{{}^{\prime}}\bm{M}_{im}$. Then, it is explicit that
$\bm{V}=\bm{V}_{re}+j\bm{V}_{im}$, and the computations in the complex space,
such as in (9), can be reformulated as real computations with matrices
$\bm{V}_{re}$ and $\bm{V}_{im}$.

For the WMMSE algorithm, we follow the settings in [11], including the
stopping criterion that the deviation $\epsilon$ in the WSR between two
iterations satisfies $\epsilon\leq 10^{-4}$, and employ the bisection search
method for computing $\mu$. Differently, to ensure the convergence of the
WMMSE algorithm, we increase the iterative steps $L$ of the WMMSE algorithm to
100. Furthermore, for each sample, we randomly initialize the algorithm for 10
times and choose the best performance for both of the tested algorithms as the
obtained result to mitigate the potential impact of poor initializations.

In Fig. 2, we present the obtained WSR results of the MLBF and WMMSE
algorithms averaged over 1000 channel realizations. The red solid line is the
WSR results over iterations when using the MLBF algorithm and the dash blue
line is from the WMMSE algorithm that is considered as the baseline (the WMMSE
usually converges quickly within 10 iterations, the maximum iterative steps
for the WMMSE is typically set to 6 in previous works [11, 12]). It shows that
the MLBF algorithm clearly surpasses the WMMSE performance in the high SNR
regiem of SNR=20-40dB, and achieves comparable performance when SNR=10dB. It
can be seen that, at higher SNR, the gap between the WSR obtained by the MLBF
algorithm and the WMMSE algorithm becomes larger. In Fig. 3 we further show
the impact of channel SNR on the obtained WSR by the MLBF and WMMSE
algorithms, respectively. The red solid line with triangles denotes the
averaged WSR obtained by the MLBF algorithm as a function of the channel SNR
and the blue dash line with circles refers to the results from the WMMSE
algorithm. Each point corresponds to the averaged WSR across 1000 channel
realizations. It can be seen that when SNR$\leq 15$dB, the WSR obtained by the
MLBF and WMMSE algorithms are approximately equivalent. With the increase in
SNR, the advantage of the MLBF algorithm is more significant. We infer that
the low SNR allows both the MLBF and the WMMSE algorithms converge
successfully to the optimal solution, thus both algorithms obtain similar WSR
results. On the other hand, with the increasing SNR, the non-convexity in the
WSR maximization problem is also amplified. Therefore, more local optimal
points might exist in the geometry surface of the WSR maximization problem,
and the WMMSE is surpassed by the MLBF algorithm significantly as the latter
has a more effective search algorithm along the complex geometry.

Figure 2: Red curve denotes the average WSR obtained by the MLBF algorithm,
while the blue dash line is the average WSR obtained by the WMMSE algorithm,
with L=100. Both curves are the average WSR results across 1000 channel
realizations. Figure 3: WSR obtained by the MLBF and WMMSE algorithms as a
function of the channel SNR. Each point represents the average WSR across 1000
channel realizations.

##  VI Conclusion

In this paper, we proposed a meta-learning based solution to the WSR
maximization problem in downlink MISO broadcast channels. The proposed MLBF
algorithm is built upon the iterative block-coordinate descent method, but
instead of greedily optimizing for each individual variable at each step, it
can adaptively optimize each variable with respect to the geometry of the
objective function. In this way, the proposed MLBF algorithm seeks to find a
better update of individual variables for the NP-hard and non-convex WSR
maximization problem compared to exhaustively minimizing each sub-problem.
Simulation results confirm that the MLBF algorithm outperforms WMMSE
particularly in the high SNR regiem, and achieves a comparable performance
when SNR is small. Future works will focus on extending the MLBF algorithm to
the multiple-input multiple-out (MIMO) and interference channel scenarios, and
the consideration of more general utility functions as studied in [7]. We
expect that the gains from the proposed meta-learning approach will be even
more significant in those problems due to the increased non-convexity and
complexity.

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