# Unfolding WMMSE using Graph Neural Networks for Efficient Power Allocation

Arindam Chowdhury, Gunjan Verma, Chirag Rao, Ananthram Swami, and Santiago
Segarra A. Chowdhury and S. Segarra are with the Dept. of ECE, Rice
University. G. Verma, C. Rao, and A. Swami are with the US Army’s CCDC Army
Research Laboratory. Research was sponsored by the Army Research Office and
was accomplished under Cooperative Agreement Number W911NF-19-2-0269. The
views and conclusions contained in this document are those of the authors and
should not be interpreted as representing the official policies, either
expressed or implied, of the Army Research Office or the U.S. Government. The
U.S. Government is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation herein.
Emails: {ac131<EMAIL_ADDRESS>{gunjan.verma.civ, chirag.r.rao.civ,
<EMAIL_ADDRESS>
###### Abstract

We study the problem of optimal power allocation in a single-hop ad hoc
wireless network. In solving this problem, we depart from classical purely
model-based approaches and propose a hybrid method that retains key modeling
elements in conjunction with data-driven components. More precisely, we put
forth a neural network architecture inspired by the algorithmic unfolding of
the iterative weighted minimum mean squared error (WMMSE) method, that we
denote by unfolded WMMSE (UWMMSE). The learnable weights within UWMMSE are
parameterized using graph neural networks (GNNs), where the time-varying
underlying graphs are given by the fading interference coefficients in the
wireless network. These GNNs are trained through a gradient descent approach
based on multiple instances of the power allocation problem. We show that the
proposed architecture is permutation equivariant, thus facilitating
generalizability across network topologies. Comprehensive numerical
experiments illustrate the performance attained by UWMMSE along with its
robustness to hyper-parameter selection and generalizability to unseen
scenarios such as different network densities and network sizes.

###### Index Terms:

WMMSE, power allocation, graph neural networks, deep learning, algorithm
unfolding.

##  I Introduction

Power and bandwidth are fundamental resources in communication, playing a key
role in determining the effective capacity of a wireless channel [1, 2]. In
modern wireless communication systems, the scope of resources has been
broadened to include beams in a multiple-input-multiple-output (MIMO) system,
time slots in a time-division multiple access system (TDMA), frequency sub-
bands in a frequency-division multiple access (FDMA) system, spreading codes
in a code-division multiple access (CDMA) system, among several others [3].
Optimal allocation of these resources under randomly varying channel
characteristics and user demands is essential for the smooth operation of
wireless systems. In particular, power allocation in a wireless ad hoc network
is crucial to mitigate multi-user interference, one of the main performance-
limiting factors. In addition, transmission power of a mobile user is in
itself a scarce resource. Indeed, careful and efficient power allocation is
vital to prolong the battery life of mobile users [4].

A communication system should ensure fairness and quality of service to its
users through efficient resource utilization. Mathematically, the power
allocation problem can be formulated as the problem of optimizing a certain
system-level utility function – such as sum-rate, min-rate, and harmonic-rate
– subject to resource budget constraints. Despite the remarkable success of
this paradigm [5], it turns out that many of the formulated optimization
problems are non-convex and NP-hard, making them difficult to solve [6, 7, 8].
Hence, a plethora of classical methods have been developed to obtain
approximate solutions to these challenging optimization problems. For the
canonical case of sum-rate maximization, the algorithms developed over the
past decades can be broadly categorized into those based on Lagrangian dual
decomposition [9, 10], interference pricing [11, 12], successive convex
approximation [13], and weighted minimum mean squared error (WMMSE)
minimization [14]. Of special interest to us due to its widespread adoption is
this last category of classical methods. Common shortcomings of this body of
work include high computational complexity as well as the need of an accurate
system model, which may not be feasible to construct in this era of constantly
varying communication standards and user demands. High computational
complexity has two particularly adverse consequences in wireless networks: 1)
it more rapidly depletes device battery (which runs counter to the entire
point of performing efficient power allocation), and 2) the power allocation
needs to be recomputed frequently in time-varying channels, undermining the
practical utility of a complex algorithm’s time-delayed output that lags
current channel conditions. This has led the research community to look for
faster, data-driven solutions for power allocation by leveraging machine
learning tools.

Deep learning based methods have outperformed conventional approaches for
wireless resource allocation in recent works [15, 16, 17, 18, 19, 20, 21].
Broadly, there are two ways of incorporating deep learning in solving resource
allocation problems. One way is to follow a _supervised_ or _imitation_
learning approach where the functional approximation capacity of deep neural
networks is leveraged to mimic established classical methods from solved
instances [22]. For example, a deep network has been employed [23] to
approximate the classical WMMSE solution [14] for the power allocation problem
in an interference channel. The main advantage of this paradigm is
computational efficiency since, once trained, deep networks can run faster
than the classical algorithms that they are imitating. By contrast, the main
drawback is the poor generalization to scenarios (channel characteristics,
number of users, user mobility) not represented in the original training
examples. While expanding the training set to be more inclusive of the
diversity of such phenomena is desirable, it may be infeasible to obtain
groundtruth for such a large and representative set as that would involve
repeated application of the computationally heavy iterative algorithm. The
second approach is _unsupervised_ where one parameterizes the power allocation
function using a neural network and directly employs the optimization
objective as a loss function, bypassing the need for solved problem instances
[15, 24, 25, 26, 27]. This loss function is minimized during training with
respect to the parameters defining the neural network. While such a procedure
is computationally simple, no prior knowledge (e.g., based on classical
optimization techniques) is leveraged to inform the algorithm’s architectural
or hyper-parameter choices.

In this paper, we advocate a third direction, which falls between the two
described extremes and aims to address their shortcomings. The proposed method
is _unsupervised_ in the sense that no solved instances of the power
allocation problem are needed for training. However, our approach still
_imitates_ classical methods by incorporating part of their structure into the
layered architecture of the neural network. The goal is to leverage the
theoretical models developed with expert knowledge and achieve near-optimal
performance with significantly reduced execution time. To accomplish this, we
follow the paradigm of algorithm unfolding [28].

Algorithm unfolding (or unrolling) was first proposed as an application of a
parametric function to approximate the iterative soft-thresholding algorithm
[29]. The idea is to unfold the iterations as a cascade of layers, where each
layer has the same update structure as the original algorithm but the
parameters are now learned from data. This approach was later successfully
extended to several iterative algorithms that solve problems like blind
deblurring [30], non-negative matrix factorization [31], semantic image
segmentation [32], clutter suppression [33], rain removal [34] and source
separation [35]. A standard practice in the above frameworks is to
parameterize the function of interest using multi-layer perceptrons (MLPs) or
convolutional neural networks (CNNs) [36]. However, MLPs and CNNs – which have
proven to be tremendously effective in processing images, texts, and audio –
are not quite suitable for problems in wireless communication. In particular,
the performance of these methods degrades dramatically when the network size
becomes large since MLPs and CNNs fail to exploit the underlying topology of
wireless networks. This has led to several approaches that tried to adjust
CNNs to the wireless setting [15, 37, 38] capitalizing on, e.g., the spatial
disposition of wireless networks [39]. We adopt an alternative direction [25,
40, 26], where graph neural networks (GNNs) [41, 42, 43, 44, 45, 46, 47] are
used to parameterize the power allocation function, thus leveraging the
natural representation of wireless networks as graphs. GNNs utilize the
structural relationships between nodes to locally process instantaneous
channel state information.

In this context, we propose an unfolded weighted minimum mean squared error
(UWMMSE) method, which is to the best of our knowledge the first GNN based
deep unfolded architecture based on the iterative WMMSE algorithm. UWMMSE
achieves state-of-the-art performance in utility maximization and
computational efficiency for power allocation in wireless networks.

Contribution. The contributions of this paper are threefold:
i) We propose an unfolded version of WMMSE for power allocation in wireless
networks – where the learnable modules are parameterized via GNNs – along with
guidelines for its distributed implementation. Moreover, the modular approach
advocated here can be followed as a roadmap for the derivation of other
unfolded versions of WMMSE.
ii) In terms of theoretical analysis, we show that the permutation
equivariance of the learning components implies permutation equivariance of
the whole architecture; see Propositions 1 and 2. Additionally, we determine a
necessary condition to be satisfied by the learning parameters to enable
effective learning; see Theorem 1.
iii) Through extensive numerical experimentation, we illustrate the
performance of the proposed method compared with state-of-the-art
alternatives, the impact on performance of the choice of key hyper-parameters
in the determination of our architecture, and the generalizability to wireless
networks of unseen sizes and densities.

Paper outline. In Section II, we introduce the network topology and channel
model that will be studied along with a formal statement of the power
allocation problem. After introducing the classical WMMSE algorithm, in
Section III we present the proposed unfolded architecture; see Fig. 1.
Theoretical analyses of permutation equivariance and convergence are presented
in Sections III-A and III-B, respectively. We close this main section with a
discussion on the distributed implementation and computational complexity of
the proposed architecture in Section III-C. Comprehensive numerical
experiments in Section IV demonstrate the behavior of UWMMSE in comparison
with established and recently proposed methods as well as for varying hyper-
parameter and wireless network settings. Conclusions and avenues for future
research in Section V close the paper.

Notation. The entries of a matrix $\mathbf{X}$ and a vector $\mathbf{x}$ are
denoted by $X_{ij}$ and $x_{i}$, respectively; to avoid confusion, the
alternative notation $[\mathbf{X}]_{ij}$ and $[\mathbf{x}]_{i}$ will be used
occasionally. Operations $(\cdot)^{\top}$ and $\mathbb{E}(\cdot)$ represent
transpose and expected value, respectively. $\mathbf{0}$ and $\mathbf{1}$
refer to the all-zeros and the all-ones vectors, respectively, where the sizes
are clear from context. $\text{diag}({\mathbf{X}})$ denotes a diagonal matrix
which stores the diagonal elements of ${\mathbf{X}}$.
${\mathcal{N}}(0,\sigma^{2})$ represents a zero-mean normal distribution with
variance $\sigma^{2}$.

##  II System model and problem formulation

We consider a single-hop ad hoc interference network having $M$ distinct
single-antenna transmitter-receiver pairs. Transmitters are denoted by $i$ and
the $i^{th}$ transmitter is associated to a single receiver denoted by $r(i)$
for $i=1,\ldots,M$. Further, denoting the signal transmitted by $i$ as
$x_{i}\in\mathbb{R}$, the received signal $y_{i}\in\mathbb{R}$ at $r(i)$ is
given by

$y_{i}=h_{ii}x_{i}+\sum_{\begin{subarray}{c}j=1\,|\,j\neq
i\end{subarray}}^{M}h_{ij}x_{j}+n_{i},$ (1)

where $h_{ii}\in\mathbb{R}$ is the channel between the $i^{th}$ transceiver
pair, $h_{ij}\in\mathbb{R}$ for $i\neq j$ represents interference between
transmitter $j$ and receiver $r(i)$, and
$n_{i}\sim{\mathcal{N}}(0,\sigma^{2})$ for $\sigma>0$ represents the additive
channel noise. To accommodate the effect of fading channels, we slot time into
discrete intervals and consider time-varying channels $h_{ij}(t)$, which we
conveniently represent as a channel-state matrix
${\mathbf{H}}(t)\in{\mathbb{R}}^{M\times M}$ where
$[{\mathbf{H}}(t)]_{ij}=h_{ij}(t)$. It will be instrumental to interpret
${\mathbf{H}}(t)$ as the time-varying and weighted adjacency matrix of a
directed graph with $M$ nodes, where node $i$ represents the transmitter-
receiver pair composed of transmitter $i$ and receiver $r(i)$.

The instantaneous data rate $c_{i}$ achievable at receiver $r(i)$ is given by
Shannon’s capacity theorem as a function of the signal-to-interference-plus-
noise-ratio (SINR),

$c_{i}({\mathbf{p}}(t),{\mathbf{H}}(t))=\log_{2}\left(1+\frac{|h_{ii}(t)|^{2}p_{i}(t)}{\sigma^{2}+\sum_{j\neq
i}|h_{ij}(t)|^{2}p_{j}(t)}\right),$ (2)

where $p_{i}(t)\in\mathbb{R}$, $p_{i}(t)\geq 0$, represents the power
allocated to transmitter $i$ at time $t$ and
${\mathbf{p}}(t)=[p_{1}(t),\ldots,p_{M}(t)]^{\top}$. Our objective is to
determine the instantaneous power allocation vector ${\mathbf{p}}(t)$ that
maximizes a network utility related to the data rates $c_{i}$. Under the
common understanding that channel states vary with time, from this point
onward we omit the explicit dependence on $t$ to simplify notation.

We formally state our power allocation problem as follows

$\displaystyle\max_{{\mathbf{p}}}\,\,\sum_{i=1}^{M}\beta_{i}(c_{i}({\mathbf{p}},{\mathbf{H}}))\quad$
(3) $\displaystyle\text{s.t.}\,\,\,\,\,0\leq p_{i}\leq
p_{\max},\,\,\,\,\text{for all}\,\,i,$

where $\beta_{i}$ is a generic increasing utility function of the data rate
$c_{i}$ [cf. (2)], and $p_{\max}$ denotes the uniform maximum available power
at every transmitter. Notice that when selecting $\beta_{i}(z)=\alpha_{i}z$
for $\alpha_{i}>0$, the objective function in (3) boils down to the well-
established (weighted) sum-rate utility. Moreover, other frequently
encountered utilities are also encompassed by the formulation in (3) including
proportional fairness $\beta_{i}(z)=\log(z)$ and harmonic mean rate
$\beta_{i}(z)=-z^{-1}$. However, even for well-established utility functions,
the optimization problem in (3) has been shown to be NP-hard [6, 48]. In this
context, the present work is driven by the following question:

How can we achieve an _effective_ , _distributed_ , and _efficient_ solution
to (3), where ${\mathbf{H}}$ is drawn from an _accessible_ distribution
${\mathcal{H}}$?

In the above statement, by _effective_ we mean a solution that achieves
comparable performance to a near-optimal classical approach, while
_accessible_ distribution means that either ${\mathcal{H}}$ is known or we can
easily sample from it.

Several approximate methods have been proposed that aim to solve a tractable
surrogate of (3) and, thence, find at least a good local maximum of the
utility of interest. This classical body of work focuses on solving a single
instance of (3) for an arbitrary ${\mathbf{H}}$, which must then be repeated
to recompute the power allocation in successive time instants.

Given that in practice we are interested in solving several instances of (3)
across time, a learning-based body of work has gained traction in the past
years. In a nutshell, based on many channel state instances, the idea is to
learn a map (i.e., a function approximator) between the channel state matrix
${\mathbf{H}}$ and the corresponding (approximate) optimal power allocation
${\mathbf{p}}$. In this way, when a new channel is drawn, the power can be
efficiently computed using the learned map without the need for solving an
optimization problem. Unlike common neural-network based functional
approximators whose inputs and outputs are often of fixed dimensionality, a
distinguishing property of the problem here is that ${\mathbf{H}}$’s dimension
is not necessarily fixed a-priori and can indeed change as nodes enter and
exit the wireless network. This fact combined with the inherent topological
structure present in ${\mathbf{H}}$ foreshadows the importance of using graph
neural networks as the functional approximator, a point we further elaborate
in Section III.

Our goal is to combine the advantages of the classical and learning-based
paradigms. More specifically, we seek to address our driving question by
leveraging the _approximate_ , _distributed_ and _interpretable_ solution
provided by the classical WMMSE method while enhancing it with the
computational _efficiency_ of trained machine learning models. We pursue this
synergistic combination under the paradigm of algorithm unfolding, as
discussed in the next section.

##  III Unfolded weighted minimum mean squared error (UWMMSE)

Figure 1: Schematic diagram of the proposed unfolded weighted minimum mean
squared error (UWMMSE) algorithm. The input to the layered structure is
${\mathbf{v}}^{(0)}=\sqrt{p_{\max}}\,\mathbf{1}$ and the computed power
allocation is given by the elementwise square of the output
${\mathbf{v}}^{(K)}$. A generic intermediate layer $k$ is detailed. The five
equations in (5)-(8) are represented by the yellow blocks and the dependence
on intermediate variables is made explicit in the flow diagram. The parameters
$\boldsymbol{\theta}_{a}^{(k)}$ and $\boldsymbol{\theta}_{b}^{(k)}$ for all
layers $k$ are learned to minimize the loss $\ell(\boldsymbol{\Theta})$ in
(9), thus promoting a faster convergence than its classical WMMSE counterpart.

Algorithm unfolding (or unrolling) [49, 50, 28, 51] refers to the general
notion of selecting an iterative model-based approach to solve a problem of
interest and building a problem-specific neural network with layers inspired
by these iterations. Backed-up by its recent success on several problems, we
present here the first unfolded algorithm for power allocation in wireless
networks. Notice, however, that the action of unfolding is not uniquely
determined. Several decisions must be made with the objective of incorporating
domain knowledge as well as striking the right balance between
interpretability and performance. These decisions include selecting what
parameters become trainable and what trainable architectures are used to
represent these parameters. In this paper, we develop and evaluate a way to
unfold WMMSE using graph neural networks.

Before presenting our unfolded architecture, we introduce the basics of WMMSE
[14]. Essential for this classical algorithm is to reformulate (3) as

$\displaystyle\min_{\mathbf{w,u,v}}\sum_{i=1}^{M}(w_{i}e_{i}-\log w_{i}),$ (4)
$\displaystyle\text{st.}\quad
e_{i}=(1-u_{i}h_{ii}v_{i})^{2}+\sigma^{2}u_{i}^{2}+\sum_{i\neq
j}u_{i}^{2}h_{ij}^{2}v_{j}^{2},$ $\displaystyle\qquad\,\,\,v^{2}_{i}\leq
p_{\max},\,\,\,\,\text{for all}\,\,i,$

where $e_{i}$ computes the mean-square error of the signal at node $i$ under
the assumption that the transmitted signal $x_{i}$ is independent from the
noise $n_{i}$ [cf. (1)]. For simplicity, in (4) we have focused on the sum-
rate utility function ($\beta_{i}(z)=z$ for all $i$). However, the
developments here presented can be extended to more general utilities
$\beta_{i}$; see Remark 1.

We say that (4) is equivalent to (3) because it can be shown [14, Thm. 3] that
the optimal solution
$\\{{\mathbf{w}}^{*},{\mathbf{u}}^{*},{\mathbf{v}}^{*}\\}$ of the former and
that of the latter ${\mathbf{p}}^{*}$ are related as
$\sqrt{{\mathbf{p}}^{*}}={\mathbf{v}}^{*}$, where the square root is applied
elementwise. Hence, if we obtain a optimal solution to the (non-convex)
problem in (4), this would immediately translate into a optimal solution to
our problem of interest (3). Additionally, the structure in (4) is amenable to
a block-coordinate descent solution, since the objective is convex in each of
the three variables when the other two are fixed. Indeed, there exist closed-
form expressions for these coordinate descent steps. Our unfolding
architecture is inspired by these iteratively applied closed-form solutions,
which we augment with learnable parameters.

We propose to compute the allocated power as a function of the channel state
matrix ${\mathbf{p}}=\Phi({\mathbf{H}};\boldsymbol{\Theta})$ through a layered
architecture $\Phi$ with trainable weights $\boldsymbol{\Theta}$.

More precisely, setting ${\mathbf{v}}^{(0)}=\sqrt{p_{\max}}\,\mathbf{1}$, we
have that for layers $k=1,\ldots,K$,

$\displaystyle{\mathbf{a}}^{(k)}$
$\displaystyle=\Psi({\mathbf{H}};\boldsymbol{\theta}_{a}^{(k)}),\qquad{\mathbf{b}}^{(k)}=\Psi({\mathbf{H}};\boldsymbol{\theta}_{b}^{(k)}),$
(5) $\displaystyle u^{(k)}_{i}$
$\displaystyle=\frac{h_{ii}v^{(k-1)}_{i}}{\sigma^{2}+\sum_{j}h_{ij}^{2}{v^{(k-1)}_{j}}v^{(k-1)}_{j}},\,\,$
$\displaystyle\text{for all }i,$ (6) $\displaystyle w^{(k)}_{i}$
$\displaystyle=\frac{a_{i}^{(k)}}{1-u^{(k)}_{i}h_{ii}v^{(k-1)}_{i}}+b_{i}^{(k)},$
$\displaystyle\text{for all }i,$ (7) $\displaystyle v^{(k)}_{i}$
$\displaystyle=\alpha\left(\frac{u^{(k)}_{i}h_{ii}w^{(k)}_{i}}{\sum_{j}h_{ji}^{2}u^{(k)}_{j}u^{(k)}_{j}w^{(k)}_{j}}\right),$
$\displaystyle\text{for all }i,$ (8)

and the output power is determined as
${\mathbf{p}}=\Phi({\mathbf{H}};\boldsymbol{\Theta})=({\mathbf{v}}^{(K)})^{2}$,
where the square is applied elementwise. The non-linear function
$\alpha(z)\coloneqq[z]_{0}^{\sqrt{p_{\max}}}$ in (8) simply ensures that
$v_{i}^{(k)}\in[0,\sqrt{p_{\max}}]$ by saturating the right-hand side of (8)
at these extreme values. This guarantees that the constraint in (3) is
satisfied. The trainable parameters $\boldsymbol{\Theta}$ are given by the
concatenation of $\boldsymbol{\theta}_{a}^{(k)}$ and
$\boldsymbol{\theta}_{b}^{(k)}$ in (5) for all layers, i.e.,
$\boldsymbol{\Theta}=[\boldsymbol{\theta}_{a}^{(1)},\boldsymbol{\theta}_{b}^{(1)},\ldots,\boldsymbol{\theta}_{a}^{(K)},\boldsymbol{\theta}_{b}^{(K)}]$.
Finally, the functions $\Psi$ parametrized by $\boldsymbol{\Theta}$ in (5) are
chosen to be graph neural networks, as further discussed in Section III-A. A
schematic view of our proposed layered architecture is presented in Fig. 1.

To better understand the mechanics of the proposed layered architecture,
notice that each layer $k$ as described in (5)-(8) is characterized by five
vectors
${\mathbf{a}}^{(k)},{\mathbf{b}}^{(k)},{\mathbf{u}}^{(k)},{\mathbf{w}}^{(k)},{\mathbf{v}}^{(k)}\in{\mathbb{R}}^{M}$.
If we set ${\mathbf{a}}^{(k)}=\mathbf{1}$ and ${\mathbf{b}}^{(k)}=\mathbf{0}$,
then expressions (6)-(8) correspond to the closed-form solutions of a block-
coordinate algorithm designed to solve (4) [14]. Putting it differently, for
these values of ${\mathbf{a}}^{(k)}$ and ${\mathbf{b}}^{(k)}$, our proposed
architecture boils down to the classical WMMSE truncated to $K$ iterations. In
this setting, ${\mathbf{u}}^{(k)}$ and ${\mathbf{v}}^{(k)}$ represent receiver
and transmitter side variables, respectively, in the sense that $u_{i}^{(k)}$
depends exclusively on the channel states _into_ receiver $r(i)$ whereas
$v_{i}^{(k)}$ depends exclusively on the channel states _out of_ transmitter
$i$. Moreover, upon convergence, these represent measures of the strength of
signals at the receiver and transmitter for our single-input single-output
(SISO) system. Their equivalent variables in the MIMO case would correspond to
receiver and transmitter-side beamformers; see [14] for additional details.
Notice also that the update of ${\mathbf{w}}^{(k)}$ in (7) is local in the
sense that $w_{i}^{(k)}$ only depends on the $i$th entries of the other
vectors of interest and the channel state $h_{ii}$ between transmitter $i$ and
its intended receiver $r(i)$.

In spite of being the default algorithm for optimal power allocation, WMMSE
entails high computational and time complexity. This complexity arises because
WMMSE requires many iterations of the updates (6)-(8) for convergence. Hence,
the objective of the newly introduced variables ${\mathbf{a}}^{(k)}$ and
${\mathbf{b}}^{(k)}$ is to accelerate this convergence while maintaining a
good performance. Intuitively, if we learn a smarter update rule for
${\mathbf{w}}^{(k)}$ that accelerates its convergence, we can achieve good
performance with only a few iterations of WMMSE. In this sense, our proposed
architecture is an unfolded version of the classical algorithm. Notice that
additional learning parameters and more sophisticated functional forms could
be included in the updates (6)-(8). However, the simple learned affine
transformation proposed in (7) achieves good performance in practice (see
Section IV) while being simple to implement.

Under the natural assumption that the weights ${\mathbf{a}}^{(k)}$ and
${\mathbf{b}}^{(k)}$ should depend on the channel state ${\mathbf{H}}$, we
advocate a learning-based method where this dependence is made explicit via
the parametric functions $\Psi$ in (5). For fixed parameters
$\boldsymbol{\Theta}$, the allocated power for a channel state ${\mathbf{H}}$
is given by $\Phi({\mathbf{H}};\boldsymbol{\Theta})$ and results in a sum-rate
utility of
$\sum_{i=1}^{M}c_{i}(\Phi({\mathbf{H}};\boldsymbol{\Theta}),{\mathbf{H}})$;
[cf. (3) for $\beta_{i}(z)=z$]. Hence, we define the loss function

$\ell(\boldsymbol{\Theta})=-\mathbb{E}_{{\mathbf{H}}\sim{\mathcal{H}}}\left[\sum_{i=1}^{M}c_{i}(\Phi({\mathbf{H}};\boldsymbol{\Theta}),{\mathbf{H}})\right].$
(9)

Even if ${\mathcal{H}}$ is known, minimizing $\ell(\boldsymbol{\Theta})$ with
respect to $\boldsymbol{\Theta}$ entails solving a non-convex optimization
problem. Under the assumption that the function
$\Psi(\cdot\,;\boldsymbol{\theta})$ in (5) is differentiable with respect to
$\boldsymbol{\theta}$ and leveraging the fact that we have access to samples
of ${\mathcal{H}}$ (cf. Section II), we seek to minimize (9) through
stochastic gradient descent. It should be noted that, unlike [23] and in
accordance with some recent works [26, 25], UWMMSE is an _unsupervised method_
since it requires access to samples of the channel state matrices
${\mathbf{H}}$ but does _not_ require access to the optimal power allocations
(labels) associated with those channels.

######  Remark 1 (Different utility functions)

Our proposed UWMMSE is well suited to solve the optimal power allocation
problem for objective functions that go beyond the sum-rate utility. Adopting
the notation of a generic utility $\beta_{i}$ in (3), UWMMSE can be adapted to
this more encompassing setting by incorporating two changes. First, the loss
function in (9) must be updated to
$\ell(\boldsymbol{\Theta})=-\mathbb{E}_{{\mathbf{H}}\sim{\mathcal{H}}}\left[\sum_{i=1}^{M}\beta_{i}(c_{i}(\Phi({\mathbf{H}};\boldsymbol{\Theta}),{\mathbf{H}}))\right]$.
Second, and more noteworthy, we must adapt the unfolded architecture to
correspond to the WMMSE solution of the modified utility problem. This is
unlike state-of-the-art methods [23, 26, 25] where the architecture is
unaffected by the utility at hand, thus indicating a limited incorporation of
domain knowledge. More precisely, defining the scalar functions
$\gamma_{i}(z)=-\beta_{i}(-\log(z))$, the update in (7) must be changed to

$w_{i}^{(k)}=\gamma_{i}^{\prime}(1-u^{(k)}_{i}h_{ii}v^{(k-1)}_{i})\,a_{i}^{(k)}+b_{i}^{(k)},$
(10)

where $\gamma^{\prime}=\frac{d\gamma}{dz}$, while the remaining equations in
(5)-(8) are left unaltered. Two observations are in order. First, for the
specific case of sum-rate utility, we have that $\beta_{i}(z)=z$. This implies
that $\gamma_{i}(z)=\log(z)$ and it becomes apparent that (7) is recovered
from (10). Second, one can leverage [14, Thm. 2] to show that the optimal
power allocation is a fixed point of our modified iteration for
$a_{i}^{(k)}=1,b_{i}^{(k)}=0$ under the condition that $\gamma_{i}(z)$ is
strictly concave for $z>0$. The empirical effect of the modification in (10)
is studied in Section IV-C.

###  III-A Graph neural networks: Permutation equivariance

Thus far, we have been purposely nonspecific in the definition of the
parametric function $\Psi(\cdot\,;\boldsymbol{\theta})$ in (5) in order to
highlight the generality of the proposed unfolded algorithm. We have only
identified differentiability with respect to $\boldsymbol{\theta}$ as a
requirement to minimize (9) via stochastic gradient descent. In this section,
we further winnow the set of candidate functions $\Psi$ by requiring
permutation equivariance. As will become apparent, _graph neural networks_
(GNNs) arise as natural candidates for the function $\Psi$. GNNs are the
natural extensions of convolutional neural networks to graph domains, where
the architectures are appropriate to exploit non-Euclidean data. In a
nutshell, the classical convolutional layers are replaced by (banks of) graph
filters [52] that describe learnable local aggregation operations.

We introduce a formal definition for permutation equivariance. Consider the
set ${\mathcal{F}}$ of all functions $f:{\mathbb{R}}^{M\times
M}\to{\mathbb{R}}^{M}$ and a generic permutation matrix
$\boldsymbol{\Pi}\in\\{0,1\\}^{M\times M}$.

###### Definition 1

A function $f\in{\mathcal{F}}$ is permutation equivariant if
$f(\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top})=\boldsymbol{\Pi}f({\mathbf{H}})$
for all matrices ${\mathbf{H}}$ and all permutations $\boldsymbol{\Pi}$.

Intuitively, if we permute the labels of the nodes in our network before
computing the permutation equivariant function $f$, the individual output
values are not changed but only permuted by this same permutation. Notice that
this is specially relevant in our context since the node indexing is arbitrary
and should not play any role in the determination of the optimal power
allocation.

Naturally, the choice of $\Psi$ in (5) affects the permutation equivariance of
our proposed method, as we state next.

###### Proposition 1

If $\Psi(\cdot\,;\boldsymbol{\theta})$ in (5) is permutation equivariant then
the UWMMSE method $\Phi(\cdot\,;\boldsymbol{\Theta})$ is also permutation
equivariant.

###### Proof:

Denote by
${\mathbf{v}}^{(k)}=\Phi_{k}({\mathbf{H}},{\mathbf{v}}^{(k-1)};\boldsymbol{\theta}^{(k)})$
the output of the $k$th layer of the UWMMSE architecture in (5)-(8). Also,
denote by
$\tilde{{\mathbf{H}}}=\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top}$ and
$\tilde{{\mathbf{v}}}^{(k-1)}=\boldsymbol{\Pi}{{\mathbf{v}}}^{(k-1)}$
arbitrary permuted versions of the channel matrix and the input to the $k$th
layer. We are first going to establish that
$\Phi_{k}(\tilde{{\mathbf{H}}},\tilde{{\mathbf{v}}}^{(k-1)};\boldsymbol{\theta}^{(k)})=\boldsymbol{\Pi}{\mathbf{v}}^{(k)}$.
From the permutation equivariance of $\Psi$ we have that
$\tilde{{\mathbf{a}}}^{(k)}=\Psi(\tilde{{\mathbf{H}}};\boldsymbol{\theta}_{a}^{(k)})=\boldsymbol{\Pi}{{\mathbf{a}}}^{(k)}$
and, similarly,
$\tilde{{\mathbf{b}}}^{(k)}=\boldsymbol{\Pi}{{\mathbf{b}}}^{(k)}$. Denoting by
$\pi(i)$ the new index of node $i$ upon permutation $\boldsymbol{\Pi}$, it
follows from (6) that

$\displaystyle\tilde{u}^{(k)}_{i}$
$\displaystyle=\frac{\tilde{h}_{ii}\tilde{v}^{(k-1)}_{i}}{\sigma^{2}+\sum_{j}\tilde{h}_{ij}^{2}{\tilde{v}^{(k-1)}_{j}}\tilde{v}^{(k-1)}_{j}}$
$\displaystyle=\frac{{h}_{\pi(i)\pi(i)}{v}^{(k-1)}_{\pi(i)}}{\sigma^{2}+\sum_{j}{h}_{\pi(i)\pi(j)}^{2}{{v}^{(k-1)}_{\pi(j)}}{v}^{(k-1)}_{\pi(j)}}={u}^{(k)}_{\pi(i)},$

which, in vector form, equates to
$\tilde{{\mathbf{u}}}^{(k)}=\boldsymbol{\Pi}{{\mathbf{u}}}^{(k)}$. Similarly,
from (7) and (8) it follows that
$\tilde{{\mathbf{w}}}^{(k)}=\boldsymbol{\Pi}{{\mathbf{w}}}^{(k)}$ and
$\tilde{{\mathbf{v}}}^{(k)}=\Phi_{k}(\tilde{{\mathbf{H}}},\tilde{{\mathbf{v}}}^{(k-1)};\boldsymbol{\theta}^{(k)})=\boldsymbol{\Pi}{{\mathbf{v}}}^{(k)}$,
as wanted.

We are now going to leverage this to show that
$\Phi(\cdot\,;\boldsymbol{\Theta})$ is equivariant. Consider the case where
$K=1$. From the definition of $\Phi(\cdot\,;\boldsymbol{\Theta})$ we obtain
that

$\displaystyle\Phi(\tilde{{\mathbf{H}}};\boldsymbol{\Theta})$
$\displaystyle=\Phi_{1}(\tilde{{\mathbf{H}}},{\mathbf{v}}^{(0)};\boldsymbol{\theta}^{(1)})=\Phi_{1}(\tilde{{\mathbf{H}}},\boldsymbol{\Pi}{\mathbf{v}}^{(0)};\boldsymbol{\theta}^{(1)})$
$\displaystyle=\boldsymbol{\Pi}\Phi_{1}({{\mathbf{H}}},{\mathbf{v}}^{(0)};\boldsymbol{\theta}^{(1)})=\boldsymbol{\Pi}\Phi({{\mathbf{H}}};\boldsymbol{\Theta}),$

where the second equality follows from the fact that ${\mathbf{v}}^{(0)}$ is a
constant vector and the third equality is a special case of the identity
previously established for $k=1$. This shows that a single layer UWMMSE is
permutation equivariant. That permutation equivariance holds for $K>1$ can be
shown via a simple induction argument here omitted. ∎

Having established that permutation equivariance is a desirable property for
UWMMSE and guided by Proposition 1, we proceed to specify a particular $\Psi$.
Inspired by the simplicity of graph convolutional networks in [42], we propose
the following function

$\displaystyle\Psi({\mathbf{H}};\boldsymbol{\theta})$
$\displaystyle=\alpha_{2}\left(\mathrm{diag}({\mathbf{H}})\,{\mathbf{Z}}\,\boldsymbol{\omega}_{21}+{\mathbf{H}}\,{\mathbf{Z}}\,\boldsymbol{\omega}_{22}\right),$
(11) $\displaystyle{\mathbf{Z}}$
$\displaystyle=\alpha_{1}\left(\mathrm{diag}({\mathbf{H}})\,\mathbf{1}\,\boldsymbol{\omega}^{\top}_{11}+{\mathbf{H}}\,\mathbf{1}\,\boldsymbol{\omega}^{\top}_{12}\right),$
(12)

where
$\boldsymbol{\omega}_{11},\boldsymbol{\omega}_{12},\boldsymbol{\omega}_{21},\boldsymbol{\omega}_{22}\in{\mathbb{R}}^{F}$
for some predefined number of hidden features $F$,
$\boldsymbol{\theta}=[\boldsymbol{\omega}_{11}^{\top},\boldsymbol{\omega}_{12}^{\top},\boldsymbol{\omega}_{21}^{\top},\boldsymbol{\omega}_{22}^{\top}]^{\top}$,
$\alpha_{1}$ is the rectified linear unit (ReLU) activation function and
$\alpha_{2}$ is the sigmoid activation function. The proposed function
satisfies our desired property, as stated next.

###### Proposition 2

Function $\Psi(\cdot\,;\boldsymbol{\theta})$ in (11) is permutation
equivariant.

###### Proof:

Notice that

$\displaystyle\alpha_{1}\left(\mathrm{diag}(\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top})\,\mathbf{1}\,\boldsymbol{\omega}^{\top}_{11}+\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top}\,\mathbf{1}\,\boldsymbol{\omega}^{\top}_{12}\right)$
$\displaystyle=$
$\displaystyle\alpha_{1}\left(\boldsymbol{\Pi}\mathrm{diag}({\mathbf{H}})\,\mathbf{1}\,\boldsymbol{\omega}^{\top}_{11}+\boldsymbol{\Pi}{\mathbf{H}}\,\mathbf{1}\,\boldsymbol{\omega}^{\top}_{12}\right)$
$\displaystyle=\boldsymbol{\Pi}{\mathbf{Z}},$

where we have used the facts that
$\mathrm{diag}(\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top})=\boldsymbol{\Pi}\mathrm{diag}({\mathbf{H}})\boldsymbol{\Pi}^{\top}$
and $\boldsymbol{\Pi}^{\top}\mathbf{1}=\mathbf{1}$. Replacing this in (11) we
have that

$\displaystyle\Psi(\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top};\boldsymbol{\theta})$
$\displaystyle\,\,\,\,=\alpha_{2}\left(\mathrm{diag}(\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top})\,\boldsymbol{\Pi}{\mathbf{Z}}\,\boldsymbol{\omega}_{21}+\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top}\,\boldsymbol{\Pi}{\mathbf{Z}}\,\boldsymbol{\omega}_{22}\right)$
$\displaystyle\,\,\,\,=\alpha_{2}\left(\boldsymbol{\Pi}\mathrm{diag}({\mathbf{H}})\,{\mathbf{Z}}\,\boldsymbol{\omega}_{21}+\boldsymbol{\Pi}{\mathbf{H}}{\mathbf{Z}}\,\boldsymbol{\omega}_{22}\right)=\boldsymbol{\Pi}\Psi({\mathbf{H}};\boldsymbol{\theta}),$

as wanted. ∎

From the combination of Propositions 1 and 2 it immediately follows that the
proposed UWMMSE is permutation equivariant. Importantly, the modular
description of UWMMSE put forth in (5)-(8) allows the practitioner to choose
their preferred GNN in defining $\Psi$. Moreover, Proposition 1 guarantees
that, as long as the chosen $\Psi$ is permutation equivariant, this desirable
feature will permeate to UWMMSE. Henceforth, we adopt $\Psi$ as in (11) with
the exception of Section IV-B where the performance of different choices of
$\Psi$ is compared.

######  Remark 2 (Incorporating node features)

GNNs are well suited to leverage the structure of a given graph to process
values defined on its nodes, often referred to as node features or graph
signals [42, 53]. However, in our proposed implementation [cf. (5) and (11)],
there are no node features involved. In the presence of node features, our
proposed framework can be easily enriched to encompass this information. More
precisely, collecting $F^{\prime}$ node features in the matrix
${\mathbf{Q}}\in{\mathbb{R}}^{N\times F^{\prime}}$ where $q_{if}$ stores the
values of feature $f$ at node $i$, we may consider functions of the type
$\Psi({\mathbf{H}},{\mathbf{Q}};\boldsymbol{\theta})$ in (5). In particular,
we still propose the GNN in (11) replacing $\mathbf{1}$ in (12) by
${\mathbf{Q}}$ and resizing the parameter matrices accordingly. The theory
discussed in this section still holds, where the modified notion of
permutation equivariance boils down to
$\Phi(\boldsymbol{\Pi}{\mathbf{H}}\boldsymbol{\Pi}^{\top},\boldsymbol{\Pi}{\mathbf{Q}};\boldsymbol{\Theta})=\boldsymbol{\Pi}\Phi({\mathbf{H}},{\mathbf{Q}};\boldsymbol{\Theta})$.
Examples of node features in the context of power allocation include relative
node weights in the determination of a weighted sum-rate utility, queue
lengths, traffic rates, and topological features of the underlying graph; see
Section IV-C for an implementation of this enriched setup.

###  III-B UWMMSE convergence: A necessary condition

To further study the proposed method, we analyze its behavior as the number of
layers $K$ grows. Intuitively, since the optimal power allocation is a fixed
point of the classical WMMSE iteration [14], one would expect deeper layers of
the UWMMSE to more closely resemble its classical counterpart. To formalize
this intuition, let us introduce the following notation. For any given channel
state ${\mathbf{H}}$, let us define the optimal power allocation, i.e., the
solution to (3), as ${\mathbf{p}}^{*}$. Accordingly, we define
$v_{i}^{*}=\sqrt{p_{i}^{*}}$, $u_{i}^{*}$ as given by (6) when
${\mathbf{v}}^{(k-1)}$ is replaced by ${\mathbf{v}}^{*}$, and
$w_{i}^{*}=(u_{i}^{*}h_{ii}v_{i}^{*})^{-1}$. Also, for simplicity we consider
the case where ${\mathbf{a}}^{(k)}=\mathbf{1}$ for all $k$ and study the
convergence of ${\mathbf{b}}^{(k)}$. With this notation in place, the
following result can be shown.

###### Theorem 1

Consider a UWMMSE architecture (5)-(8) of infinite depth with
${\mathbf{a}}^{(k)}=\mathbf{1}$ for all $k$. If
${\mathbf{v}}^{(k)}\to\sqrt{{\mathbf{p}}^{*}}$ uniformly as $k\to\infty$ then,
for all $i$ such that $0<p_{i}^{*}<p_{\max}$, it must hold that

$\sum_{j\neq
i}h_{ji}^{2}(u_{j}^{*})^{2}(w_{i}^{*}b_{j}^{(k)}-w_{j}^{*}b_{i}^{(k)})\to
0\,\,\text{as}\,\,k\to\infty.$ (13)

###### Proof:

Since ${\mathbf{v}}^{(k)}\to\sqrt{{\mathbf{p}}^{*}}={\mathbf{v}}^{*}$
uniformly, for all $\eta>0$ there must exist a layer index $K_{1}$ such that
for all $k>K_{1}$, $v_{i}^{(k-1)}=v_{i}^{*}+\epsilon_{i}$ and
$v_{i}^{(k)}=v_{i}^{*}+\epsilon^{\prime}_{i}$ where $|\epsilon_{i}|<\eta$ and
$|\epsilon^{\prime}_{i}|<\eta$ for all $i$. Resorting to the notation used in
the proof of Proposition 1, we have that
${\mathbf{v}}^{(k)}=\Phi_{k}({\mathbf{H}},{\mathbf{v}}^{(k-1)};{\mathbf{b}}^{(k)})$,
where we have made explicit the dependence on ${\mathbf{b}}^{(k)}$. From the
uniform convergence, it follows that

${\mathbf{v}}^{*}+\boldsymbol{\epsilon}^{\prime}={\mathbf{v}}^{(k)}=\Phi_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)}).$
(14)

Notice that if we find an expression
$g_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)})$
such that
$\Phi_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)})={\mathbf{v}}^{*}+g_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)})$,
we might replace this in (14) to obtain

$\boldsymbol{\epsilon}^{\prime}=g_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)}).$
(15)

Since (15) must hold for arbitrary small but positive $\eta$, this implies
that
$g_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)})\to
0$ as $k\to\infty$. In the remainder of the proof, we find an explicit form
for
$g_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)})$
and show that this implies (13).

We begin by replacing ${\mathbf{v}}^{(k-1)}$ by
${\mathbf{v}}^{*}+\boldsymbol{\epsilon}$ in (6) to obtain

$\displaystyle u^{(k)}_{i}$
$\displaystyle=\frac{h_{ii}(v_{i}^{*}+\epsilon_{i})}{\sigma^{2}+\sum_{j}h_{ij}^{2}(v_{j}^{*}+\epsilon_{j})^{2}},$
(16)
$\displaystyle=\frac{h_{ii}v_{i}^{*}+h_{ii}\epsilon_{i}}{\sigma^{2}+\sum_{j}h_{ij}^{2}[(v_{j}^{*})^{2}+2v_{j}^{*}\epsilon_{j}+\epsilon^{2}_{j}]}$
$\displaystyle=\frac{h_{ii}v_{i}^{*}}{\sigma^{2}\\!+\\!\sum_{j}h_{ij}^{2}(v_{j}^{*})^{2}}\\!+\\!\frac{h_{ii}\epsilon_{i}}{\sigma^{2}+\sum_{j}h_{ij}^{2}[(v_{j}^{*})^{2}+2v_{j}^{*}\epsilon_{j}+\epsilon^{2}_{j}]}$
$\displaystyle-\frac{h_{ii}v_{i}^{*}}{\sigma^{2}+\sum_{j}h_{ij}^{2}(v_{j}^{*})^{2}}\,\,\frac{\sum_{j}h_{ij}^{2}[2v_{j}^{*}\epsilon_{j}+\epsilon^{2}_{j}]}{\sigma^{2}+\sum_{j}h_{ij}^{2}[(v_{j}^{*})^{2}+2v_{j}^{*}\epsilon_{j}+\epsilon^{2}_{j}]}$
$\displaystyle=u_{i}^{*}+\epsilon_{u_{i}},$

where the last equality follows from the definition of $u_{i}^{*}$ and
$\epsilon_{u_{i}}$ contains all the terms depending on $\boldsymbol{\epsilon}$
so that $|\epsilon_{u_{i}}|\to 0$ as $\eta\to 0$. We then move to (7) and
replace $u^{(k)}_{i}$, $v^{(k-1)}_{i}$, and $a^{(k)}_{i}$ by
$u_{i}^{*}+\epsilon_{u_{i}}$, $v_{i}^{*}+\epsilon_{i}$ and $1$, respectively.
Following a procedure similar to that in (16) we get that

$\displaystyle w^{(k)}_{i}=w_{i}^{*}+\epsilon_{w_{i}}+b^{(k)}_{i},$ (17)

where $|\epsilon_{w_{i}}|\to 0$ as $\eta\to 0$. We repeat the procedure for
(8), where we notice that, for large enough $k$, the non-linearity $\alpha$ is
moot since $v_{i}^{(k)}\to\sqrt{p_{i}^{*}}$ and $0<p_{i}^{*}<p_{\max}$. With
this in mind, it follows from (8) that

$\displaystyle v^{(k)}_{i}$
$\displaystyle=\frac{u^{*}_{i}h_{ii}w^{*}_{i}}{\sum_{j}h_{ji}^{2}(u^{*}_{j})^{2}w^{*}_{j}}+\epsilon_{v_{i}}$
(18)
$\displaystyle+\frac{h_{ii}u^{*}_{i}}{\sum_{j}\\!h_{ji}^{2}(u^{*}_{j})^{2}w^{*}_{j}}\,\frac{\sum_{j\neq
i}h_{ji}^{2}(u_{j}^{*})^{2}(w_{j}^{*}b_{i}^{(k)}-w_{i}^{*}b_{j}^{(k)})}{\sum_{j}h_{ji}^{2}[(u_{j}^{*}+\epsilon_{u_{j}})^{2}(w^{*}_{j}+\epsilon_{w_{j}}\\!+b_{j}^{(k)})]}$
(19)

where $|\epsilon_{v_{i}}|\to 0$ as $\eta\to 0$. From the fact that
$({\mathbf{u}}^{*},{\mathbf{w}}^{*},{\mathbf{v}}^{*})$ is a fixed point of the
classical WMMSE iteration [14], it follows that the first term in the right-
hand side of (18) is equal to $v_{i}^{*}$. This implies that the function
$g_{k}({\mathbf{H}},{\mathbf{v}}^{*}+\boldsymbol{\epsilon};{\mathbf{b}}^{(k)})$
that we were looking for is given, elementwise, by the second and third terms
in the right-hand side of (18). Since $\epsilon_{v_{i}}\to 0$ as $\eta\to 0$
(equivalently, as $k\to\infty$), it must be that the last term in (18) goes to
$0$ as $k$ goes to infinity, from where our result follows. ∎

Theorem 1 asserts that if the UWMMSE is uniformly learning the true power
allocation ${\mathbf{p}}^{*}$ as it goes deeper, then the learned weights
${\mathbf{b}}^{(k)}$ must asymptotically satisfy (13). Intuitively, we
expected ${\mathbf{b}}^{(k)}\to\mathbf{0}$ so that the UWMMSE layer boils down
to the classical WMMSE iteration. Indeed, ${\mathbf{b}}^{(k)}\to\mathbf{0}$
satisfies the necessary condition in (13). Notice, however, that other limits
are permitted by this necessary condition, e.g., $b_{i}^{(k)}\to w_{i}^{*}$
for all $i$ would also satisfy (13). Nonetheless, this would require the
function $\Psi$ in (5) to be sufficiently expressive to return
${\mathbf{w}}^{*}$ (an explicit function of the true optimal power allocation)
from ${\mathbf{H}}$. This is not the case in general for the simple function
proposed in (11)-(12) so that, in practice, we expect (13) to be satisfied by
${\mathbf{b}}^{(k)}\to\mathbf{0}$. As is customary in the analysis of deep
learning algorithms, the infinite depth architecture is amenable to
theoretical analysis but the derived conclusions are also valid in the finite
regime. Indeed, as further discussed in Section IV-D, $b_{i}^{(k)}$ yields
values close to zero for most nodes $i$ when $k$ approaches $K$.

Finally, it should be noticed that Theorem 1 is independent of the specific
function $\Psi$ chosen in (5). That is, the above result is a characteristic
of the architecture in (5)-(8), irrespective of the functional form chosen to
parameterize ${\mathbf{b}}^{(k)}$.

###  III-C Distributed implementation, scalability, and complexity analysis

It is possible to deploy the trained UWMMSE in a distributed fashion as the
allocated power at a given node $i$ can be computed without explicit knowledge
of the states of channels that do not include transmitter $i$ or receiver
$r(i)$. However, similar to WMMSE [14], we need to make some key assumptions
regarding the information available to each transceiver pair. Primarily, each
transmitter $i$ should have estimates of the local channel state $h_{ji}$ for
all receivers $r(j)$. Additionally, there should be a feedback mechanism
through which receiver $r(i)$ can transmit information to all transmitters in
the network. Under these assumptions, each receiver $r(i)$ can compute its
individual $u_{i}^{(k)}$ and $w_{i}^{(k)}$ and pass them on to the
transmitters to compute $v_{i}^{(k)}$ for all $k$ unrolling layers in all
nodes. Each node is also assumed to have access to the full set of trainable
weights of both GCN-based learning modules in each unrolling layer $k$. It is
important to note here that our proposed method cannot be trained in a
distributed fashion as the used formulation of GCN [42] requires access to the
entire CSI matrix and thus necessitates centralized training. But once
trained, it is sufficient to have the indexed row and column available for
computing the output of any given node, allowing distributed implementation of
the trained model. Such a setup would still incur additional communication
overheads and the rate of power allocation would be limited by the speed of
the feedback link between a given transceiver pair.

Figure 2: Performance comparison of the proposed UWMMSE with WMMSE [14], a
truncated version thereof, an MLP [23], IGCNet[26], and REGNN [25]. (a)
Histogram of the achieved utilities for $6400$ randomly drawn channel state
matrices in the low-noise regime ($\sigma=$2.6\text{\times}{10}^{-5}$$).
UWMMSE achieves performance comparable to WMMSE but requiring the same number
of iterations than the underperforming truncated version. (b) Box plots
associated with the histograms in (a), where the comparisons with MLP, IGCNet,
and REGNN have been added. (c) Counterpart of (b) but for the high-noise
regime ($\sigma=1.0$).

The computational complexity per WMMSE iteration has been shown to be
$\mathcal{O}(M^{2})$ for an $M$-user single-antenna interference network. This
complexity translates directly to each unrolled layer in UWMMSE as we follow
the same update rules as WMMSE. In addition, each GCN in a given unrolling
layer has a complexity of $\mathcal{O}(M^{2}F)$ where $F$ is the hidden layer
dimension [42]. Therefore, the complexity of UWMMSE scales with the size of
the network as $\mathcal{O}(M^{2})$ per unrolled layer, which is the same as
the per-iteration complexity of WMMSE. As a result, we achieve a significant
gain in terms of processing time by limiting the number of unrolling layers in
UWMMSE. This will be validated empirically in Section IV-A.

##  IV Numerical experiments

We present extensive numerical experiments to illustrate the performance of
UWMMSE in different settings and to validate our understanding of the
architecture.111Code to replicate the numerical experiments here presented can
be found at https://github.com/archo48/unrolled-wmmse.git. In Section IV-A, we
compare the performance of UWMMSE with competing approaches whereas in Section
IV-B we analyze how this performance varies for different choices of key
parameters in the proposed architecture. In Section IV-C, we demonstrate how
the considered framework can be enriched by incorporating node features and
varying the utility function. Our theoretical result in Theorem 1 is
empirically validated in Section IV-D. Finally, in Section IV-E we analyze the
generalizability of the proposed approach to changes in the density and size
of the wireless network under consideration.

For all our experiments, we consider a Rayleigh fading channel model. To that
end, we construct a random geometric graph in two dimensions having $M$
transceiver pairs. First, each transmitter $i$ is dropped uniformly at random
at locations $\mathbf{t}_{i}\in[-M,M]^{2}$. Then, its paired receiver $r(i)$
is dropped uniformly at random at location
$\mathbf{r}_{i}\in[{\mathbf{t}}_{i}-\frac{M}{4},{\mathbf{t}}_{i}+\frac{M}{4}]^{2}$.
Under fading conditions, the channel between a transmitter $i$ and any
receiver $r(j)$ is composed of two components $H_{ij}=H_{ij}^{P}H_{ij}^{F}$,
where the path gain is given by
$H_{ij}^{P}=\lVert\mathbf{t}_{i}-\mathbf{r}_{j}\rVert^{-2.2}$ and the fading
coefficient is randomly drawn $H_{ij}^{F}\sim\text{Rayleigh}(1)$. We fix
$H_{ij}^{P}$ and sample $H_{ij}^{F}$ at each power allocation instance. In all
but the experiments presented in Section IV-E, we assume that the underlying
topology of the network is fixed. This can be interpreted as a set of static
nodes communicating with each other over a period of time. At each discrete-
time instant, fading conditions change and the instantaneous channel
information is used by the model to generate the optimal power allocation. In
all but the experiment presented in Section IV-C, we set the node features to
unity (cf. Remark 2).

Our proposed solution model, UWMMSE, is composed of $4$ unrolled WMMSE layers
with each layer $k$ having two $2$-layered GCNs (11)-(12) as its learning
modules for the affine transform parameters ${\mathbf{a}}^{(k)}$ and
${\mathbf{b}}^{(k)}$. Hidden layer dimension of all GCNs is set to $4$. ADAM
[54] optimizer is employed for training across $10000$ steps with each
training step being performed on a batch of $64$ randomly sampled channel
fading states. Learning rate is set to $1\text{\times}{10}^{-3}$ and training
is performed for a maximum of $20$ epochs, with early stopping. For testing,
we randomly sample $6400$ channel fading states. Unless otherwise specified,
the network size $M$ is fixed at $20$.

###  IV-A Performance comparison

We begin by comparing the performance attained by UWMMSE with that of
established baselines and state-of-the-art methods. We perform these
comparisons in two different regimes that are defined on the basis of additive
channel noise variance. In the high-noise regime, the standard deviation
${\sigma}$ of the noise [cf. (2)] is set as ${1}$ whereas in the low-noise
regime $\sigma=$2.6\text{\times}{10}^{-5}$$.

We choose the following methods for comparison, all of which address the
problem of optimal power allocation in wireless networks with minor variations
in noise regimes and channel models, achieving state-of-the-art performance on
standard benchmarks.

Figure 3: Impact of variation in depth, width and GNN architecture on UWMMSE
performance. (a) Box plots of sum-rate achieved by UWMMSE with number of
unrolled layers varying from $2$ to $7$. (b) Box plots of sum-rate achieved by
UWMMSE with the dimension of GCN hidden features varying from $2$ to $15$. (c)
Histogram of sum-rate achieved by UWMMSE with a GCN-based learning module
versus that with an REGNN-based learning module.

  1. 1.

WMMSE [14] forms the baseline for our experiments as it is the most commonly
followed classical approach to address this problem. We set a maximum of $100$
iterations for WMMSE per sample.

  2. 2.

Truncated WMMSE (Tr-WMMSE) provides a performance lower bound to UWMMSE in
terms of the sum-capacity that WMMSE achieves with as many iterations as the
unrolled layers without the learning component. We fix the number of
iterations of Tr-WMMSE to $4$ to match UWMMSE unrollings.

  3. 3.

MLP [23] is employed in a supervised setting to learn a functional mapping
between channel state information and corresponding power allocation, using
WMMSE output as the training signal.

  4. 4.

REGNN [25] addresses the specific problem of binary power allocation in
Rayleigh interference channels. It tackles this problem in the high-noise
regime where it achieves state-of-the-art performance.

  5. 5.

IGCNet [26] addresses the power allocation problem in Gaussian interference
channels, specifically, in the high-noise regime, where it achieves state-of-
the-art performance.

The comparisons are shown in Fig 2. Owing to the randomness of channel state
information, there can be significant variations in the utility function value
across test samples for any given power allocation algorithm. Each individual
channel realization can be more or less challenging based on how the
interference strengths compare with the channel strengths for the specific
fading conditions. To illustrate this, we show a histogram of utility values
over the entire test set in the low-noise regime; see Fig 2. The empirical
distribution of utilities across different channel instances shows that UWMMSE
matches the performance of WMMSE for most realizations, while even surpassing
it in a few cases. To be more precise, UWMMSE, WMMSE, and Tr-WMMSE
respectively achieve average sum-rates of $83.21$, $82.94$, and $76.49$ as
shown in Table I with corresponding standard deviations of $6.10$, $5.76$, and
$6.28$. The performance gain in WMMSE compared to its truncated version was
expected due to additional iterations that add to the complexity of the
solution. UWMMSE bridges this gap with a learning module that performs more
intelligent update steps to converge to the optimal within as many iterations
as the underperforming Tr-WMMSE.

Having compared UWMMSE with the classical method, we now expand the analysis
to include other learning-based approaches in the low-noise regime; see Fig 2.
MLP, which learns a functional mapping of the WMMSE output, beats Tr-WMMSE
with an average sum-rate of $78.17$, still falling short of WMMSE by a
significant margin. This shows that supervised methods are limited by the
quality of their training signals and often do not generalize to out-of-sample
data. On the other hand, both REGNN and IGCNet – originally designed and
tested in high-noise regimes – prove to be inadequate to match the performance
of WMMSE in the more challenging low-noise setting. For instance, REGNN was
designed for binary power allocation – either $0$ or $p_{\max}$ – but finer
precision is needed for state-of-the-art performance in the low-noise regime.
Indeed, in this regime the interference component dominates in the sum-
capacity formulation [cf. (2)] and, therefore, minor variations in power
allocation result in significant performance alteration.

By contrast, we now shift focus to the high-noise regime, where noise
dominates over the interference component in the SINR formulation. Hence, the
sum-capacity achieved is less dependent on the precise power allocation at
each node. In this setting, all the algorithms tend to follow a binary
approach in which each node would either transmit with full power or not
transmit at all. Due to the high noise in this setting, low performance is
generally achieved as clearly manifested by the low sum-rate values obtained
by all the algorithms in Fig 2. Also, given the high noise, it is always
optimal to transmit with almost full power at most nodes. Consequently, there
is little performance variation across methods.

Figure 4: Experimental validation of Remarks 1 and 2. (a) Boxplot comparing the performance of UWMMSE, WMMSE, Tr-WMMSE, and a version of UWMMSE trained with less data (UWMMSE-Ld), in terms of the sum of squared rates, without the proposed modification in (10) for the update of ${\mathbf{w}}^{(k)}$. (b) Counterpart of (a) when the proposed modification in (10) is used. (c) Histogram of UWMMSE performance with all-one node features ${\mathbf{Q}}=\mathbf{1}$ versus that with customized node features.  TABLE I: Performance and time comparisons of all methods Algorithm | Training | Test | Test
---|---|---|---
| time (min) | sum-rate | time (msec)
WMMSE [14] | - | 82.94 | 16
Tr-WMMSE | - | 76.49 | 1.0
MLP [23] | 0.5 | 78.17 | 3.2
REGNN [25] | 15 | 57.92 | 2.5
IGCNet [26] | 5 | 55.30 | 3.0
UWMMSE | 15 | 83.21 | 2.0

Achieving a sum-rate that is close to optimal is not enough as the time
required for power allocation has to be synchronized with fading conditions
that are fast evolving. It is therefore essential to consider the time taken
by each algorithm to offer a power allocation output given a channel state
input. To that end, we provide a computation time comparison222All
computations were performed on an Nvidia Quadro T2000 GPU. among the various
algorithms in Table I. UWMMSE, which takes close to $2$ ms per sample, is
significantly faster as compared to WMMSE which takes around $16$ ms per
sample. All the other learning-based methods [26, 23, 25] have a processing
time similar to that of UWMMSE, however, none of them achieve the same
performance in that time duration, which is the main advantage of our method
over existing algorithms. Unlike WMMSE and similar to any other learning-based
method, UWMMSE has the additional requirement of training, which takes close
to $15$ mins, but in most applications this is not a problem since we envision
the computational effort to be done prior to implementation. MLP [23] has
significantly lower training time, but requires a full iteration of WMMSE over
its training data to get the ground-truth power allocation which adds to the
time complexity.

###  IV-B Impact of depth, width, and GNN functional form

Our next objective is to analyze the dependence of our proposed method on the
various design parameters in the architecture. To that end, we study the
variation of UWMMSE performance specifically with respect to two architectural
components, namely the number of unrolled layers (depth) and the dimension of
hidden features (width). We evaluate UWMMSE performance separately for each of
these hyper-parameters taking a fixed set of values,
$depth\in\\{2,3,4,5,6,7\\}$ and $width\in\\{2,5,10,15\\}$, in an experimental
setup that is otherwise identical to that in Section IV-A. As shown in Fig 3,
mean sum-rate improves from $2$-layered to $5$-layered architectures and then
decays as more layers are added to the model. This could be attributed to the
need of more data to successfully train deeper architectures. On the other
hand, in Fig 3 we observe overlapping distributions of sum-rate for all the
configurations with minor improvement on using a hidden feature dimension of
$10$. The observed robustness in terms of depth and width of the architecture
is of practical importance, providing the practitioner with a wide design
space without compromising performance.

We further evaluate the performance of our proposed unrolling scheme by
employing an alternative learning module, i.e., selecting a different function
$\Psi$ in (5). Our goal is to characterize how the achieved performance
depends on the specific selection of the GNN. To that end, we compare our
canonical choice (11)-(12) with an REGNN [25], primarily due to its simple and
lightweight architecture. REGNN also interprets the CSI as the adjacency
matrix of a graph and constructs a layered architecture by intertwining linear
graph filters with point-wise non-linearities. To be more precise, if we
denote by ${\mathbf{z}}_{l}$ the output of an intermediate layer in an REGNN,
we have that

${\mathbf{z}}_{l}=\alpha\left(\sum_{k=0}^{K}\nu_{lk}{\mathbf{H}}^{k}{\mathbf{z}}_{l-1}\right),$
(20)

where $\alpha$ is a ReLU activation function and the coefficients $\nu_{lk}$
are trained. The REGNN is defined by recursive application of (20) and we test
it here as an alternative to GCNs for the computation of ${\mathbf{a}}^{(k)}$
and ${\mathbf{b}}^{(k)}$ in (5).

As shown in Fig. 3, the empirical distribution of sum-rate achieved by the
REGNN-based UWMMSE model has significant overlap with that of the GCN-based
baseline. These neural networks can therefore be used interchangeably to
achieve very similar performances revealing that the proposed unfolded
architecture is not restricted by a specific choice of GNN. In this way, the
practitioner can choose the GNN of preference given the application-specific
computation and memory constraints. It is important to notice that once the
REGNN is incorporated into our unfolded architecture, the performance markedly
increases to its standalone implementation [cf. Figs. 2 and 3] unveiling the
importance of including domain-specific knowledge in the neural architecture.

Figure 5: Histograms of learned variables ${\mathbf{a}}^{(k)}$ and
${\mathbf{b}}^{(k)}$ in layers $k\in\\{1,3,5\\}$ of a fully trained
$5$-layered UWMMSE model. (a) In layer-$1$, the majority of
${\mathbf{a}}^{(1)}$ and ${\mathbf{b}}^{(1)}$ values are concentrated at the
lower end while almost one-third of the ${\mathbf{b}}$ values are distributed
at the higher end of the range. (b) Layer-$3$ sees a transition of
${\mathbf{b}}^{(3)}$ values towards the lower end of the range. (c) Layer-$5$
presents a clear separation in which the majority of ${\mathbf{a}}^{(5)}$
values are concentrated near $1$ while almost all ${\mathbf{b}}^{(5)}$ values
move to $0$, marking the convergence of UWMMSE to the original WMMSE updates.

###  IV-C Incorporating node features and different utility functions

We discuss in Remark 1 the extension of our unrolled scheme to several other
utility functions $\beta(z)$ provided that they satisfy the condition that
$\gamma(z)$ is strictly concave for $z>0$, where $\gamma(z)=-\beta(-\log(z))$.
To validate this, we select the sum of squares of data rate as our utility
function $\beta(z)=z^{2}$. In this case, $\gamma(z)=-(\log(z))^{2}$ is
strictly concave for $z>0$, as required. Under the same experimental setup as
in Section IV-A, our model now learns to maximize the sum of squared rates as
its utility function. From comparing Figs. 4 and 4, one can observe a clear
improvement in performance of WMMSE and its truncated version when the
modification proposed in (10) is employed, as expected. More interestingly,
UWMMSE can attain high performance even when the modified update is not used
[cf. Fig. 4]. This points towards the fact that the learnable parameters
${\mathbf{a}}^{(k)},{\mathbf{b}}^{(k)}$ can rely on data to enforce the right
update trajectory for ${\mathbf{w}}^{(k)}$ even in the presence of a model
mismatch. Indeed, the improvement of UWMMSE with the modified update compared
to the version without it is minimal (means of $1141.38$ and $1131.86$,
respectively). This difference is slightly magnified in the presence of less
training data, where the ability of the learning parameters to make up for the
model mismatch is restricted. To illustrate this, we employed a variant of
UWMMSE which was trained on half of the available data (UWMMSE-Ld). As
expected, UWMMSE-Ld underperforms with respect to UWMMSE in both settings.
When comparing the rates for UWMMSE-Ld with and without the modified update,
we observe that the modified update yields a utility of 1124.58 whereas
without the update the utility is of 1112.42. It should be noticed that there
is still a major improvement with respect to the classical WMMSE, underscoring
the importance of the learning module even with limited training data.

Remark 2 analyzes the possibility of enriching UWMMSE with node attributes
that may encode essential information pertaining to a specific node such as
priority or queue length. To validate the effectiveness of such a formulation,
we perform a simple experiment in which ${\mathbf{Q}}=\mathbf{1}$ (cf. Remark
2) is augmented with two custom features for each node, namely, physical
distance between a transmitter and its corresponding receiver
($\lVert{\mathbf{t}}_{i}-{\mathbf{r}}_{i}\rVert$) and the shortest distance
between each transmitter and all other receivers ($\min_{j\neq
i}\lVert{\mathbf{t}}_{i}-{\mathbf{r}}_{j}\rVert$). Intuitively, a
transmitter’s average power allocation depends on the distance to its intended
receiver as well as the distance to other receivers, since the latter
determines the interference effect. Hence, we expect the learning method to
take advantage of having explicit access to these features, as illustrated in
Fig. 4. Indeed, the mean sum-rate achieved using the features is of $84.73$
compared to the baseline utility of $83.21$. This result further emphasizes
the benefits of infusing domain knowledge in learning-based solutions, where
the specific choice of features relies on our understanding of the problem at
hand.

###  IV-D Empirical validation of Theorem 1

To develop a deeper understanding of the working dynamics of our proposed
approach, we observe the values that the learned variables take in each
unrolled layer. In this process, we also develop an empirical validation of
Theorem 1. To that end, we retrieve ${\mathbf{a}}^{(k)}$ and
${\mathbf{b}}^{(k)}$ values for layers $k\in\\{1,3,5\\}$ of a fully trained
$5$-layered UWMMSE model and plot the corresponding histograms in Fig 5. Each
histogram is built based on $12800$ values, corresponding to $6400$ test
vectors ${\mathbf{a}}^{(k)}$ or ${\mathbf{b}}^{(k)}$, each of size $M=20$. As
discussed previously, owing to a sigmoid non-linearity on the neural network
output, all values are limited to the range $[0,1]$.

This setup illustrates a transition of the learned variables across a sequence
of layers. As shown, the values of ${\mathbf{a}}^{(k)}$ are mainly
concentrated at the lower end of the range in the first and third layers with
a major shift in the final layer where it peaks at values close to $1$. On the
other hand, ${\mathbf{b}}^{(k)}$ values decrease across layers with the peak
at the lower end increasing in height as one goes to deeper layers with a
majority of ${\mathbf{b}}^{(5)}$ values converging near $0$ on the fifth
layer. Notice that this validates our result in Theorem 1 and the succeeding
discussion. It becomes apparent that the proposed architecture leverages the
trainable values of ${\mathbf{a}}^{(k)}$ and ${\mathbf{b}}^{(k)}$ to expedite
convergence in the first few layers and then resembles the WMMSE update rule
by learning ${\mathbf{a}}^{(k)}=\mathbf{1}$ and
${\mathbf{b}}^{(k)}={\mathbf{0}}$ in the final layers.

###  IV-E Generalization to variations in network density and size

Figure 6: Generalization performance of UWMMSE and a robust version thereof
(Ro-UWMMSE) to variations in spatial density and network size compared against
WMMSE and Tr-WMMSE baselines. (a) Mean sum-rate achieved by UWMMSE, Ro-UWMMSE
and baselines as density factor $d$ varies in $[0.5,1.0]$, generating sparser
networks. (b) Mean sum-rate achieved by UWMMSE, Ro-UWMMSE and baselines as
density factor $d$ varies in $[1.0,5.0]$, generating denser networks. (c) Mean
sum-rate achieved by UWMMSE, Ro-UWMMSE and baselines as network size $M$
varies in $[10,30]$.

So far we have considered a setup in which the underlying topology of a
wireless network remains constant across scheduling cycles, while the fading
coefficients are sampled randomly at each step. However, a more realistic
scenario to consider would be one in which the nodes are in motion and thus
change their relative positions at each step. We simulate this dynamic
environment by varying the spatial density of the network. We consider two
cases. One in which nodes move closer to each other resulting in an increase
in spatial density and the other in which nodes move further apart resulting
in a sparser topology. To that end, we define a density factor $d$ that is
used to control this setup. More precisely, for each value of $d$, a
transmitter is dropped at location
$\mathbf{t}_{i}^{d}={{\mathbf{t}}_{i}}/{d}$. Its paired receiver $r(i)$,
however, is still dropped uniformly at random at location
$\mathbf{r}^{d}_{i}\in[{\mathbf{t}}_{i}^{d}-\frac{M}{4},{\mathbf{t}}_{i}^{d}+\frac{M}{4}]^{2}$.
Effectively, the spatial density is varied according to relative positions of
the transmitters while maintaining a degree of stochasticity in the receiver
positions. This is done to create a more realistic and thereby more
challenging setting in which the relative distance between a moved transmitter
and its receiver is not simply a scaled version of their original distance.
Further, to simulate both decrease and increase in density, $d$ is varied
linearly from $0.5$ to $5.0$. We separately plot the variations in the ranges
$0.5\leq d\leq 1.0$ for sparser topologies and $1.0\leq d\leq 5.0$ for denser
topologies in Fig 6.

In this experimental setup, we compare UWMMSE trained on a fixed network
topology against WMMSE and Tr-WMMSE as baselines. Our goal is to analyze the
robustness of our proposed approach against variations in the network
topology. To illustrate this, we introduce a robust version of UWMMSE (Ro-
UWMMSE), that is trained on multiple network topologies with varying spatial
density. At each training step, we generate a batch of independent network
topologies as previously described with a randomly sampled $d\in[0.5,5.0]$.
For each individual topology, we then sample Rayleigh fading coefficients to
simulate the channel state information. Unlike the baseline UWMMSE that relies
on a constant network topology over multiple training cycles, Ro-UWMMSE is
regularized in training to allow generalization to multiple network densities.
In the sparser setting, as evident from Fig. 6, there is a constant but
moderate gap in performance between WMMSE and UWMMSE, while its robust version
follows WMMSE performance closely at almost all points. This is a slightly
simpler scenario as the relative distance between transmitters increase, thus,
limiting the effect of interference. The effectiveness of Ro-UWMMSE is even
more pronounced in the contrasting scenario where spatial density increases
progressively, bringing the transmitters closer together and, thus, enhancing
the effect of interference. In this more challenging setting, UWMMSE trained
on $d=1.0$ performs poorly on higher $d$ values, lagging even behind the
truncated version of WMMSE; see Fig. 6. However, Ro-UWMMSE is still able to
follow the WMMSE performance closely, illustrating the generalization capacity
of our unrolling scheme under regularized training.

In addition to variation in spatial positions, wireless networks also undergo
changes in size as nodes are either added or removed from the network. To
study the performance of UWMMSE in such a setting, we simulate a channel model
that undergoes random insertions and deletions of nodes. To that end, in each
scheduling cycle, a set of transmitters and their corresponding receivers are
either removed from the network or a new set of transceivers are added to the
network. It is important to note that the new transmitters and receivers are
still added in the interval ${\mathbf{t}}_{i}=[-N,N]^{2}$ for all $i>N$, where
$N$ is the original network size. In this way, this experiment does not
enforce any expansion or contraction of the overall area of the multi-sized
topologies. Corresponding receivers are dropped
${\mathbf{r}}_{i}\in[{\mathbf{t}}_{i}-\frac{N}{4},{\mathbf{t}}_{i}+\frac{N}{4}]^{2}$.
Rayleigh fading coefficients are sampled independently for each individual
topology to construct the CSI matrix. In this experiment, we train the UWMMSE
model on a network of fixed size $N=20$. We then evaluate its performance on
networks of size $M\in[10,30]$ by either removing nodes from the original
network ($M<N$) or adding new unseen nodes to it ($M>N$). Similar to the
previous experiment, we compare our method against WMMSE and Tr-WMMSE as
baselines. Further, to make UWMMSE robust against these variations, we train
Ro-UWMMSE on networks of multiple sizes by randomly generating a batch of
networks of size $M\in[10,30]$ at each training step.

As shown in Fig. 6, there is a steady drop in UWMMSE performance compared to
the WMMSE baseline for both settings $M<N$ and $M>N$, while the best
performance is achieved at $M=N$. This is not entirely unexpected as in either
direction, the underlying topology of the network undergoes significant
changes. While new nodes that are added to the graph for $M>N$ form hitherto
unseen interference patterns, removing a set of transceivers $M<N$ and re-
drawing the positions of the receivers for the remaining transmitters also
disrupts the existing interference patterns. However, Ro-UWMMSE is still able
to maintain a performance that is close to WMMSE and, thus, illustrates the
generalization capacity of our method to variations in network size under
regularized training.

##  V Conclusions

We proposed UWMMSE, a novel neural network approach to solve the problem of
power allocation in wireless networks. The layered architecture of the
presented method was derived from the algorithmic unfolding of the classical
WMMSE, thus, UWMMSE naturally incorporates domain-specific elements augmented
by trainable components. These components are parameterized by GNNs to account
for and leverage the inherent graph representation of communication networks.

Revisiting the motivating question in Section II, we have demonstrated that
UWMMSE yields solutions that are: i)  _Effective_ , since the performance
attained is comparable with well-established benchmarks; ii)  _Distributed_ ,
since the layered architecture based on GNNs is amenable to a decentralized
implementation; and iii)  _Efficient_ , since, once trained, a forward pass on
the neural architecture requires less time and computation than model-based
methods.

A natural extension of our proposed method would be to obtain efficient power
allocation in case of complex-valued signals and channels. Additionally, this
work opens the door to multiple exciting avenues for future research. Firstly,
alternative ways of incorporating learning components to the classical WMMSE
iterations can be studied, putting special emphasis in understanding –
theoretically and empirically – the trade-off between computation and
performance for varying number of trainable weights. Secondly, we can pursue
the application of unfolded solutions as the one here developed for other
resource allocation problems in wireless communications. Finally, for cases
where security is a concern upon deployment, a relevant direction is the
development of architectures that are well-suited for power allocation with
missing, noisy, or even adversarial channel information.

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