# Channel Assignment in Uplink Wireless Communication using Machine Learning
Approach

Guangyu Jia, Zhaohui Yang, Hak-Keung Lam, , Jianfeng Shi, and Mohammad Shikh-
Bahaei G. Jia, Z. Yang, H. Lam, and M. Shikh-Bahaei are with the Centre for
Telecommunications Research, Department of Engineering, King’s College London,
WC2R 2LS, UK, Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>hak-
<EMAIL_ADDRESS><EMAIL_ADDRESS>Shi is with the School of
Electronic and Information Engineering, Nanjing University of Information
Science and Technology, Nanjing 210096, China. Email:
<EMAIL_ADDRESS>
###### Abstract

This letter investigates a channel assignment problem in uplink wireless
communication systems. Our goal is to maximize the sum rate of all users
subject to integer channel assignment constraints. A convex optimization based
algorithm is provided to obtain the optimal channel assignment, where the
closed-form solution is obtained in each step. Due to high computational
complexity in the convex optimization based algorithm, machine learning
approaches are employed to obtain computational efficient solutions. More
specifically, the data are generated by using convex optimization based
algorithm and the original problem is converted to a regression problem which
is addressed by the integration of convolutional neural networks (CNNs), feed-
forward neural networks (FNNs), random forest and gated recurrent unit
networks (GRUs). The results demonstrate that the machine learning method
largely reduces the computation time with slightly compromising of prediction
accuracy.

###### Index Terms:

Resource allocation, convex optimization, machine learning, deep learning.

##  I Introduction

Driven by the rapid development of advanced multimedia applications, next-
generation wireless networks must support massive connectivity. The
connectivity always results in the integer optimization problem, which was
solved by conventional algorithms mostly operating in an off-line manner with
high computation complexity and depending largely on accurate channel state
information [1]. Machine learning tools [2, 3, 4, 5, 6, 7] can exploit big
data for wireless network state estimation and find the relationship between
the decision variables and objective functions in an online manner so as to
reduce the computational complexity.

There are many research attentions of applying machine learning for solving
integer optimization problems in wireless communication. For integer user
association problem, the authors in [8] investigated the use of deep learning
to perform user-cell association for sum-rate maximization in massive multiple
input multiple output (MIMO) networks. For integer cache placement problem,
the optimization of the caching locations was transformed to a grey-scale
image-based problem and deep convolutional neural network (CNN) was adopted
accordingly [9]. Also, the linear sum assignment problems were solved through
using the deep neural networks (DNNs) [10]. Moreover, in device-to-device
(D2D) networks, a novel graph embedding based method for link scheduling was
proposed [11]. Further reinforcing the generalization ability in D2D networks,
a DNN structure was proposed in [12] with a novel loss function to achieve
better dynamic control over optimality and computational complexity. However,
the above contributions [8, 9, 10, 11, 12] all restricted that the number of
users is assumed to be the same as wireless resources, which cannot meet the
massive number of devices in future communication systems. Besides, the
machine learning approaches in [8, 9, 10, 11, 12] are all neural networks and
have the limitations of prediction performance. To further investigate machine
learning methods’ capacity and potential in wireless communication domains,
this letter employs several machine learning algorithms as base learners and
integrates them by an ensemble learning approach. By employing different
machine learning methods, we aim to improve the diversity of base models and
the final prediction performance as well as the model’s generalization
capability and robustness when exposed to contaminations.

The contributions of this letter is summarised as follows:

  * •

The channel assignment problem is formulated for the case that the number of
users is larger than the number of subchannels. A convex optimization based
algorithm is proposed to obtain the globally optimal channel assignment
despite of the integer constraints.

  * •

The convex optimization problem is converted to a regression problem and
solved by machine learning frameworks, which yield rigorously optimal and
computationally efficient solutions.

  * •

Ensemble learning is utilized to combine different machine learning models and
improve the prediction performance. Also, different combinations of
optimization algorithms are adopted and compared.

  * •

The computation time is largely reduced by the proposed machine learning
frameworks without much compromising of prediction accuracy.

##  II System Model and Problem Formulation

Consider an uplink single cell network with $M$ users and $N$ subchannels.
Denote $\mathcal{M}=\\{1,2,\cdots,M\\}$ as the set of all users and
$\mathcal{N}=\\{1,2,\cdots,N\\}$ as the set of all subchannels. Let $p_{i}$
denote the uplink transmission power for user $i$. Binary variable $x_{ij}$
reflects the association relationship between user $i$ and subchannel $j$,
i.e., $x_{ij}=1$ means that user $i$ occupies sbuchannel $j$; otherwise
$x_{ij}=0$.

Due to massive number of users and limited number of channels, i.e., $M\geq
N$, non-orthogonal multiple access (NOMA) scheme is utilized by serving
multiple users in each subchannel [13]. With successful interference
cancellation in NOMA111The transmission power of each user is assumed to be
fixed, thus the power constraints are not involved in the optimization
problem., the sum rate of all users occupying subchannel $j$ is given by [13,
14]:

$\vspace{-0.5em}r_{j}=B\log_{2}\left(1+\frac{\sum_{i=1}^{M}x_{ij}p_{i}h_{ij}}{\sigma^{2}B}\right),$
(1)

where $B$ is the bandwidth of each subchannel, $h_{ij}$ is the channel gain
between user $i$ and the base station (BS) on subchannel $j$, and $\sigma^{2}$
is the power density of Gaussian noise.

The sum rate maximization problem for assigning users with subchannels can be
formulated as:

$\displaystyle\mathop{\max}_{\bm{x}}\quad$
$\displaystyle\sum_{j=1}^{N}B\log_{2}\left(1+\frac{\sum_{i=1}^{M}x_{ij}p_{i}h_{ij}}{\sigma^{2}B}\right)$
(2a) s.t. $\displaystyle\sum_{j=1}^{N}x_{ij}=1,\quad\forall i\in\mathcal{M},$
(2b) $\displaystyle\sum_{i=1}^{M}x_{ij}=A,\quad\forall j\in\mathcal{N},$ (2c)
$\displaystyle x_{ij}\in\\{0,1\\},\quad\forall
i\in\mathcal{M},j\in\mathcal{N},$ (2d)

where $\bm{x}=[x_{11},x_{12},\cdots,x_{MN}]^{T}$ and $A$ is the allowed number
of associated users in one subchannel. Constraints (2b) ensue that each user
only occupies one subchannel.

##  III Convex Optimization Based Algorithm

Due to integer constraints (2d), it is hard to solve linear integer problem.
By temporarily relaxing the integer constraints (2d) with $x_{ij}\in[0,1]$,
problem (2) is a linear problem, also convex. For convex problem (2) with
relaxed constraints, the optimal solution can be effectively obtained by using
the dual method [15].

Let $\lambda_{i}$ denote the dual variable associated with constraint $i$ in
(2b), we obtain the dual problem of the problem in (2), which is given by:

$\vspace{-0.5em}\mathop{\min}_{\bm{\lambda}}\quad D(\bm{\lambda}),$ (3)

where

$\vspace{-0.5em}D(\bm{\lambda})=\left\\{\begin{array}[]{ll}\\!\\!\\!\mathop{\max}\limits_{\bm{x}}&\mathcal{L}_{1}(\bm{x},\bm{\lambda})\\\
\\!\\!\rm{s.\;t.}&\sum_{i=1}^{M}x_{ij}=A,\quad\forall j\in\mathcal{N},\\\
&x_{ij}\in[0,1],\quad\forall
i\in\mathcal{M},j\in\mathcal{N},\end{array}\right.$ (4)
$\displaystyle\\!\\!\\!\mathcal{L}_{1}(\bm{x},\bm{\lambda})\\!=\\!\sum_{j=1}^{N}\\!B\log_{2}\\!\left(\\!1\\!+\\!\frac{\sum_{i=1}^{M}\\!x_{ij}p_{i}h_{ij}}{\sigma^{2}B}\\!\right)\\!\\!+\\!\sum_{i=1}^{M}\\!\lambda_{i}\\!\left(\\!\sum_{j=1}^{N}\\!x_{ij}\\!-\\!1\\!\right)\\!$
$\displaystyle=\\!$
$\displaystyle\sum_{j=1}^{N}\\!B\log_{2}\left(\\!1+\\!\frac{\sum_{i=1}^{M}x_{ij}p_{i}h_{ij}}{\sigma^{2}B}\\!\right)\\!\\!+\\!\sum_{j=1}^{N}\\!\left(\\!\sum_{i=1}^{M}\lambda_{i}x_{ij}\\!-\\!\lambda_{i}\\!\right)\\!,\\!\\!\\!\\!\\!$
(5)

and $\bm{\lambda}=[\lambda_{1},\cdots,\lambda_{N}]^{T}$.

To minimize the objective function in (4), the optimal $x_{ij}$ can be
calculated via the Karush-Kuhn-Tucker (KKT) conditions. Since both the
objective function and constraints can be decoupled into $N$ subchannels, the
subproblem of (4) in subchannel $j$ can be given by:

$\displaystyle\mathop{\max}_{\bm{x}_{j}}\quad$ $\displaystyle
B\log_{2}\left(1+\frac{\sum_{i=1}^{M}x_{ij}p_{i}h_{ij}}{\sigma^{2}B}\right)+\sum_{i=1}^{M}\lambda_{i}x_{ij}$
(6a) s.t. $\displaystyle\sum_{i=1}^{M}x_{ij}=A,$ (6b) $\displaystyle
x_{ij}\in[0,1],\quad\forall i\in\mathcal{M},$ (6c)

where $\bm{x}_{j}=[x_{1j},x_{2j},\cdots,x_{Mj}]^{T}$.

It can be proved that (6) is a convex problem, and the Lagrangian function of
problem (6) is

$\displaystyle\vspace{-0.5em}\mathcal{L}_{2}$
$\displaystyle(\bm{x_{j}},\alpha,\bm{\beta},\bm{\gamma})=B\log_{2}\left(1+\frac{\sum_{i=1}^{M}x_{ij}p_{i}h_{ij}}{\sigma^{2}B}\right)+\sum_{i=1}^{M}\lambda_{i}x_{ij}$
$\displaystyle+\alpha\left(\sum_{i=1}^{M}x_{ij}-A\right)+\sum_{i=1}^{M}\beta_{i}x_{ij}+\sum_{i=1}^{M}\gamma_{i}(1-x_{ij}).$
(7)

The KKT conditions of problem (6) are [15]

$\displaystyle\frac{p_{i}h_{ij}}{(\ln
2)\left(\sum_{i=1}^{M}x_{ij}p_{i}h_{ij}+\sigma^{2}B\right)}\\!+\\!\lambda_{i}\\!+\\!\alpha\\!+\\!\beta_{i}\\!-\\!\gamma_{i}$
$\displaystyle=0,$ (8a) $\displaystyle\sum_{i=1}^{M}x_{ij}$ $\displaystyle=A,$
(8b) $\displaystyle\beta_{i}x_{ij}$ $\displaystyle=0,$ (8c)
$\displaystyle\gamma_{i}(1-x_{ij})$ $\displaystyle=0.$ (8d)

In the left hand side of (8a), it is founded that the denominator of the first
term is the same for every user $i$. With this finding, we introduce a new
variable $\kappa$, which is defined as:

$\vspace{-0.5em}\kappa={(\ln
2)\left(\sum_{i=1}^{M}x_{ij}p_{i}h_{ij}+\sigma^{2}B\right)}.$ (9)

With the help of $\kappa$, we can directly obtain the optimal solution of
$x_{ij}^{*}$ according to the following analysis. Substituting (9) to (8a)
yields:

$\vspace{-0.5em}\frac{p_{i}h_{ij}+\kappa\lambda_{i}}{\kappa}=\gamma_{i}-\beta_{i}-\alpha.$
(10)

Based on (8c)-(8d), we can obtain that (i) $x_{ij}=1$ if and only if
$\gamma_{i}>0$ and $\beta_{i}=0$ and (ii) $x_{ij}=1$ if and only if
$\gamma_{i}=0$ and $\beta_{i}\geq 0$, which indicates that the right hand side
of (10) should be as large as possible for the case $x_{ij}=1$. Besides, there
are only $A$ users such that $x_{ij}=1$ according to (8b). Thus, the user with
$x_{ij}=1$ should has the $A$ highest values of
$\frac{p_{i}h_{ij}+\kappa\lambda_{i}}{\kappa}$, i.e.,

$\vspace{-0.5em}x_{ij}^{*}(\kappa)=\left\\{\begin{array}[]{ll}\\!\\!1,&\text{if}\;i\in\mathcal{M}_{A}\\\
\\!\\!0,&\text{otherwise},\end{array}\right.$ (11)

where set $\mathcal{M}_{A}$ means the index of $A$ users with the top $A$
highest values in $p_{i}h_{ij}+\kappa\lambda_{i}$. Equation (11) indicates
that the optimal $x_{ij}^{*}$ is only a function of $\kappa$. To calculate the
value of $\kappa$, we combine (9) and (11), which yields:

$\vspace{-0.5em}\kappa={(\ln
2)\left(\sum_{i=1}^{N}x_{ij}^{*}(\kappa)p_{i}h_{ij}+\sigma^{2}B\right)}.$ (12)

With the increase of $\kappa$, the user with lower value in $p_{i}h_{ij}$ can
have higher value in $p_{i}h_{ij}+\kappa\lambda_{i}$, i.e., the value of
$\sum_{i=1}^{N}x_{ij}^{*}(\kappa)p_{i}h_{ij}$ decreases, which indicates that
the right hand side of (12) decreases with $\kappa$. Since the left hand side
of (12) increases with $\kappa$ and the right hand side of (12) decreases with
$\kappa$, the unique value of $\kappa$ can be obtained by the bisection
method.

Figure 1: Block diagram of the implementation procedure in this letter.

Given $\bm{x}$, the value of $\bm{\lambda}$ can be determined by the gradient
method [16]. By iteratively optimizing primal variable $\bm{x}$ and dual
variable $\bm{\lambda}$, the optimal subcahnnel allocation is obtained. The
dual method used to obtain the optimal subcahnnel allocation is given in
Algorithm 1. Notice that the optimal $x_{ij}$ is either 0 or 1 according to
(11), even though we relax $x_{ij}$ as continuous constraint in (2).
Therefore, we can obtain optimal solution to problem (2).

Algorithm 1 Dual Method for Subchannel Association

1: Initialize dual variable $\bm{\lambda}$.

2: repeat

3: Update subchannel allocation $\bm{x}$ according to (11).

4: Update dual variable $\bm{\lambda}$ by the gradient method.

5: until the objective value (2a) converges

The major complexity in Algorithm 1 lies in updating subchannel allocation
$\bm{x}$. Based on (11), the complexity of calculating $\bm{x}$ is
$\mathcal{O}\left(NM\log_{2}(1/\epsilon)\right)$, where
$\mathcal{O}\left(\log_{2}(1/\epsilon)\right)$ is the complexity of solving
(12) by using the bisection method with accuracy $\epsilon$. As a result, the
total complexity of Algorithm 1 is
$\mathcal{O}\left(LNM\log_{2}(1/\epsilon)\right)$, where $L$ is the number of
iterations in Algorithm 1.

##  IV Deep Learning Based Approaches

In this letter, the machine learning approaches are also involved to obtain
computational efficient solutions for the optimization problem. The overall
implementation procedures are shown in Fig. 1. The key of machine learning
methods is to learn the relationship between the constant input parameters
$R=\\{r_{ij}=p_{i}h_{ij}\\},i\in\mathcal{M},j\in\mathcal{N}$ and the
subchannel arrangements $X=\\{x_{ij}\\},i\in\mathcal{M},j\in\mathcal{N}$. To
make the problem solvable, we convert the target one-hot matrix
$X=\\{x_{ij}\\},i\in\mathcal{M},j\in\mathcal{N}$ to vector form
$\bar{X}=\\{\bar{x}_{i}\in\mathcal{N}\\},i\in\mathcal{M}$ so that the problem
can be considered as a regression problem. Each column in $X$, i.e.,
$\\{x_{i1},x_{i2},...,x_{iN}\\}^{T},i\in\mathcal{M}$ is transformed to an
integer $\bar{x}_{i}\in\mathcal{N},i\in\mathcal{M}$ which denotes the index of
non-zero value (i.e., 1) in the each column, meaning the index of user in the
$j$-th subchannel, i.e., $x_{i\bar{x}_{i}}=1$.

After obtaining the input and target, the optimization problem is converted to
a regression problem which can be modelled by machine learning frameworks. The
adopted methods for regression consist of CNNs, feed-forward neural networks
(FNNs), random forest and gated recurrent unit networks (GRUs). In addition,
ensemble learning method is utilized to combine different models and further
improve the regression performance. After regression, the output of machine
learning models are permutated according to the output values, thus, the
integer outputs which represent the indices of users are obtained in the end.

###  IV-A Data Information

####  IV-A1 Data generation

We deploy $M$ users uniformly in a square area of size $500$ m $\times$ $500$
m with the BS located at its center. In each independent run, the channel
gains of users on the subcahnnels are determined, i.e., $R$ is obtained. The
output $X$ is accordingly calculated by using Algorithm 1. The total number of
generated samples is 1 million.

####  IV-A2 Data splitting

The generated data are splitted to training, validation and test datasets. 70%
of the data are used for training while 10% for test. 20% of data are held out
for validation which plays an important role in model ensemble stage, i.e.,
the stacking procedure.

###  IV-B Machine Learning Methods

Different machine learning methods are employed to solve the regression of the
input $R$ and the target $\bar{X}$. Ensemble learning is then utilized to make
different models complement each other and improve the regression performance.
Different base learners are used as they can render diversity of the whole
structure, which is beneficial for models’ prediction accuracy, robustness and
generalization capability when exposed to contaminations. The models are
selected mainly through referencing published literatures of the successful
models on similar datasets and trial and error of existed models. After
comparing the performance of existed machine learning models, FNNs, CNNs,
random forest and GRUs are employed as the base learners. The configurations
of adopted models are shown in Table I, which are determined mainly through
trial and error and taking into consideration of the input size.

TABLE I: Machine Learning Models’ Specifications.

FNNs |  CNNs |  GRU |  Random Forest
---|---|---|---
Structure: basic FNN structure is shown in Fig. 2. Three hidden layers used in this letter are of the size ([10, 10, 10], [10, 10, 10, 10], [20, 20, 20]); Optimizer: LM∗; Epoch: 300; Loss function: MSE∗∗. |  Structure: basic CNN structure is shown in Fig. 3. Layers’ depth and filter size vary according to the input size; Optimizer: Adam + SGD with momentum; Batchsize: 500; Epoch: 40–60; Loss function: MSE. |  Structure: Two GRU layers with 50 units each, followed by dropout layer and 4 Dense layers; Optimizer: Adam; Batchsize: 200; Epoch: 40; Loss function: MSE. |  The number of trees in forest: 500; Criterion: Gini Impurity; The minimum number of samples required to split an internal node and to be at a leaf node: 2, 1, respectively.

  * *

LM: Levenberg-Marquardt method.

  * **

MSE: mean squared error.

####  IV-B1 Deep Learning Models

FNNs are widely used in function approximation, pattern recognition and data
classification. Due to its superiority in data fitting problems, we adopted
FNNs as one of models to solve the regression problem. The general structure
of FNNs used in this letter is shown in Fig. 2.

Figure 2: The structure of FNNs.

CNNs have advantages in processing the high-dimensional data, such as images
and time-series data due to (a) its convolutional setting in which the hidden
units are not fully connected to the input but instead divided into locally
connected segments; (b) pooling methods which reduce the dimensionality of the
feature space and achieve invariance to small local distortions [17]. One CNN
architecture used in this letter is shown in Fig. 3.

Figure 3: One CNN structure used in this letter.

GRU architectures have superiority in capturing dependences of different time
scales. GRUs are similar to Long Short-Term Memories (LSTMs) which have gating
units to adjust the information flow inside the unit but do not have a
seperate memory cells [18] and thus are faster to be trained than LSTMs but on
a par with LSTMs’ performance. When applying GRUs, the input data
$R=\\{r_{ij}\\},i,j\in\mathcal{N}$ are reshaped to vectors so as to form the
sequence inputs for GRU networks. As the correlation of channel model is based
on the fact that the channel gain of each user is correlated in time domain
[19], the GRU model is selected accordingly as one of the base learners.

Figure 4: Stacking process.

Stacking is a widely used ensemble method to boost the prediction performance
[20] of base learners. In this letter, we applied stacking method to a holdout
dataset, i.e., the validation dataset, the predictions of base learners fed by
validation data are then stacked to form a new dataset for training the top
model. The stacking structure used in this letter is shown in Fig. 4. The
complexity of base learners and top models is no more than
$\mathcal{O}(N^{2})$ [10]. The details of base models and top models in each
case will be illustrated in the following section.

####  IV-B2 Training method

The optimization algorithm utilized for CNNs’ training in this letter is the
combination of Adam and stochastic gradient descent (SGD) with momentum. In
implementation, Adam is used first to realize fast gradient decent, then SGD
is used for further optimizing the parameters.

As shown in Fig. 5, the conversion from Adam to SGD is implemented in two
ways: i) SWATS: adopting the SWATS strategy proposed in [21] which automates
the process of switching over by determining both the switchover point and the
learning rate of SGD; ii) Customize: using Adam in the first half of training,
i.e., the first 20 epochs of the example shown in Fig. 5; then use SGD in the
second half of training, i.e., the last 20 epochs. From Fig. 5, it can be seen
that at epoch 20 (the switchover point of the Customize method), there occurs
obvious decrease of training loss from Adam to SGD. These two methods are
compared in each case and the one having the better performance is employed in
this letter.

##  V Experiment Configuration

This section will introduce the experiment procedure, platforms and the
methodology for calculating the final accuracy.

Figure 5: Training loss of two training methods. SWATS is the method proposed
in [21], Customize refers to the combination of Adam and SGD that the first
half of training adopts Adam while the second half uses SGD algorithm.

The experiments were implemented in Matlab R2019a, Tensorflow 1.13.1 and
Nvidia GPUs. Matlab is for training FNNs with Levenberg-Marquardt (LM)
algorithm, other classifiers were trained in Tensorflow and sped up by Nvidia
P100/GTX 1050 GPUs.

###  V-A Experiment Procedures

After transforming the optimization problem to a regression problem, the
machine learning methods including FNNs, CNNs, GRUs, random forest as well as
the stacking method are employed to learn the pattern of channel assignment.
Two type of cases were considered in this letter: 1) $M=N,$ meaning the same
number of users and subchannels, which is a conventional problem; 2) $M>N$,
i.e., different subchannels and users. They are investigated in this letter as
the channel resource tends to be limited, which means the number of
subchannels could be smaller than the number of users. The details of each
case are as follows.

####  V-A1 $N=M$

We consider three situations for this case, $M=N=2,M=N=4$ and $M=N=8.$

When $N=M=2$, there are two users and two subchannels in the optimization
problem. Denote the index of an arbitrary sample by $m$, each input is a
matrix with the size of $2\times 2$: $R^{m}=\\{r_{ij}^{m}\\},i,j\in{1,2}$ and
the target is a vector of the size $1\times
2:\bar{X}^{m}=\\{x_{1}^{m},x_{2}^{m}\\}$. The targets have two kinds of
values: $\\{1,2\\}$ and $\\{2,1\\}$. Three FNNs are employed as base models,
the top model is also a FNN fed by the stacked outputs of the base learners.

When $N=M=4$, for each sample $(R^{m},\bar{X}^{m})$, $R^{m}$ and $\bar{X}^{m}$
respectively have the size of $4\times 4$ and $1\times 4$. Three base models
are composed of two CNNs and a GRU. The top model is a CNN which is fed by the
input with the size $12\times 1$ obtained by stacking the outputs of three
base models.

When $N=M=8$, the samples are $(R^{m},\bar{X}^{m})$ where $R^{m}$ and
$\bar{X}^{m}$ respectively have the size of $8\times 8$ and $1\times 8$. After
comparing the performance of CNNs, GRUs, FNNs and random forest, we chose CNNs
as the base and top learners. The top model is fed by the stacked outputs of
three base models, hence the input size of the top model is $1\times 24.$

####  V-A2 $M>N$

We also consider three situations for this case: $M=4,N=2$; $M=8,N=4$ and
$M=10,N=5$. With the same procedures, the base models include CNNs, NNs,
random forest, and the top model (CNN or NN) is used for models’ stacking.

###  V-B Permutation

After regression, machine learning models usually output decimal values,
however, integer outputs are needed as we need to know the exact indices of
users. Thus, permutation (or sorting) approach is employed in the end to
process the outputs of machine learning models so as to obtain outputs in
integer form. More specifically, assume that the target of one sample is
$\\{3,1,4,2\\}$ and the real output is $\\{2.15,0.76,4.43,1.55\\}$, the real
output should be change to integers. The intuitive operations here might be
floor, ceil and round. However, these operations are not appropriate in
implementation, as rounding provides the result $\\{2,1,4,2\\}$, floor renders
$\\{2,0,4,1\\}$ and ceiling leads to $\\{3,1,5,2\\}$, all of which are
unsatisfied outcomes. Permutation sorts the four values based on their
relative size and obtains $\\{3,1,4,2\\}$ as the final prediction result,
which is more reasonable and capable to provide better performance in
practical applications.

###  V-C Accuracy Calculation

In order to provide comprehensive observations of models’ capacity and
predictive results, the proportion of correctly predicted users is defined as
the prediction accuracy. More specifically, assuming that the number of
samples is $K$, the accuracy is formulated as follows.

$\vspace{-.5em}\text{Accuracy}=\frac{\text{The number of correctly predicted}\
\bar{x}_{j}^{m}}{KN}.$

TABLE II: Test performance of the cases $N=M=2,4,8$.  Cases | Base model 1 | Base model 2 | Base model 3 | Top Model | Training time
---|---|---|---|---|---
N = 2 | 99.97% | 99.91% | 99.95% | 99.99% | 24.8s
N = 4 | 97.90% | 98.41% | 97.90% | 98.95% | 604.5s
N = 8 | 89.83% | 89.83% | 89.83% | 90.67% | 1218.6s
TABLE III: Test performance of the cases which have different subchannels and users: $(\text{$M,N$})\\!\\!=(4,2),(8,4),(10,5)$.  Cases | Base model 1 | Base model 2 | Base model 3 | Top Model | Training time
---|---|---|---|---|---
$M=4,N=2$ |  99.51% |  99.26% |  95.20% |  99.73% |  87.47 s
$M=8,N=4$ |  94.59% |  93.09% |  93.40% |  96.13% |  595.06 s
$M=10,N=5$ |  86.03% |  85.74% |  85.09% |  86.69% |  867.85 s
TABLE IV: Time consumed (sec) of 10000 samples using Algorithm 1 and machine learning methods for 6 cases discussed in this letter.  Methods Cases |  $M=2$, $N=2$ |  $M=4$, $N=4$ |  $M=8$, $N=8$ |  $M=4$, $N=2$ |  $M=8$, $N=4$ |  $M=10$, $N=5$
---|---|---|---|---|---|---
Algorithm 1 |  41.85s |  48.28s |  48.64s |  50.12s |  52.66s |  56.59s
ML methods |  0.04s |  0.93s |  2.85s |  0.09s |  1.26s |  2.69s

##  VI Test Results

The test results are shown in Tables II and III. From these tables, it can be
seen that although the prediction accuracy decreases as $N$ increases, the
prediction results are satisfactory and the general relationship between the
optimization matrix $R$ and the final assignments $X$ can be learned by the
integration of base and stacking models. When $N=M=2;N=M=4;M=4,N=2;M=8,N=4$,
the prediction accuracy is more than 95% which is better than the published
results in literature [10]. The test accuracy of the cases $N=4$ and $N=8$ in
literature [10] is respectively 92.76% and 77.86% (in both cases the data
generation and the training of the deep learning models are done offline in
[10]), which is lower than the performance of the proposed method about 6% and
12%.

To compare the time efficiency of Algorithms 1 and machine learning methods,
we tested the computation time of 10000 samples using the same computer
(i7-7700 CPU, Nvidia Geforce GTX 1050). The results are shown in Table IV,
from which we can see that the transformation to regression problem which is
solved by machine learning techniques largely improves the time efficiency of
computation and provides a new perspective of optimization problem solving.

##  VII Conclusion

This letter employs convex optimization based algorithm and machine learning
based methods to solve the channel assignment problem. The optimization
problem is converted to regression problems for different cases and solved by
machine learning methods: CNNs, FNNs, GRUs, random forest and stacking
approach. The results demonstrate that machine learning based methods have
superiority in improving time efficiency without compromising much prediction
accuracy. The future studies may investigate more comprehensive machine
learning techniques to achieve higher performance in more complicated
scenarios.

## References

  * [1] B. Di, L. Song, and Y. Li, “Sub-channel assignment, power allocation, and user scheduling for non-orthogonal multiple access networks,” _IEEE Trans. Wireless Commun._ , vol. 15, no. 11, pp. 7686–7698, Sep. 2016.
  * [2] M. Chen, U. Challita, W. Saad, C. Yin, and M. Debbah, “Artificial neural networks-based machine learning for wireless networks: A tutorial,” _IEEE Commun. Surveys Tut._ , vol. 21, no. 4, pp. 3039–3071, Fourthquarter 2019.
  * [3] P. Dong, H. Zhang, G. Y. Li, I. Gaspar, and N. Naderializadeh, “Deep CNN-based channel estimation for mmwave massive MIMO systems,” _IEEE J. Sel. Topics Signal Process._ , vol. 13, no. 5, pp. 989–1000, Sep. 2019.
  * [4] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, “A joint learning and communications framework for federated learning over wireless networks,” _arXiv preprint arXiv:1909.07972_ , 2019.
  * [5] Z. Yang, M. Chen, W. Saad, C. S. Hong, and M. Shikh-Bahaei, “Energy efficient federated learning over wireless communication networks,” _arXiv preprint arXiv:1911.02417_ , 2019.
  * [6] Y. Wang, M. Chen, Z. Yang, T. Luo, and W. Saad, “Deep learning for optimal deployment of uavs with visible light communications,” _arXiv preprint arXiv:1912.00752_ , 2019.
  * [7] Q. Shi, H.-K. Lam, B. Xiao, and S.-H. Tsai, “Adaptive pid controller based onq-learning algorithm,” _CAAI Transactions on Intelligence Technology_ , vol. 3, no. 4, pp. 235–244, 2018.
  * [8] A. Zappone, L. Sanguinetti, and M. Debbah, “User association and load balancing for massive MIMO through deep learning,” in _Proc. IEEE Asilomar Conference on Signals, Systems, and Computers_ , Pacific Grove, CA, USA, Oct. 2018, pp. 1262–1266.
  * [9] Y. Wang and V. Friderikos, “Caching as an image characterization problem using deep convolutional neural networks,” _arXiv preprint arXiv:1907.07263_ , 2019\.
  * [10] M. Lee, Y. Xiong, G. Yu, and G. Y. Li, “Deep neural networks for linear sum assignment problems,” _IEEE Wireless Commun. Lett._ , vol. 7, no. 6, pp. 962–965, June 2018.
  * [11] M. Lee, G. Yu, and G. Y. Li, “Graph embedding based wireless link scheduling with few training samples,” _arXiv preprint arXiv:1906.02871_ , 2019.
  * [12] ——, “Learning to branch: Accelerating resource allocation in wireless networks,” _arXiv preprint arXiv:1903.01819_ , 2019.
  * [13] N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink nonorthogonal multiple access in 5G systems,” _IEEE Commun. Lett._ , vol. 20, no. 3, pp. 458–461, Mar. 2016.
  * [14] Z. Yang, W. Xu, Y. Pan, C. Pan, and M. Chen, “Energy efficient resource allocation in machine-to-machine communications with multiple access and energy harvesting for IoT,” _IEEE Internet Things J._ , vol. 5, no. 1, pp. 229–245, Feb. 2018.
  * [15] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge University Press, 2004.
  * [16] D. P. Bertsekas, _Convex Optimization Theory_. Athena Scientific Belmont, 2009.
  * [17] M. Längkvist, L. Karlsson, and A. Loutfi, “A review of unsupervised feature learning and deep learning for time-series modeling,” _Pattern Recognition Letters_ , vol. 42, pp. 11–24, June 2014.
  * [18] J. Chung, C. Gulcehre, K. Cho, and Y. Bengio, “Empirical evaluation of gated recurrent neural networks on sequence modeling,” _arXiv preprint arXiv:1412.3555_ , 2014.
  * [19] W. Su, Z. Safar, and K. J. R. Liu, “Space-time signal design for time-correlated rayleigh fading channels,” in _Proc. IEEE Int. Conf. Commun._ , vol. 5, Anchorage, AK, USA, May 2003, pp. 3175–3179 vol.5.
  * [20] D. H. Wolpert, “Stacked generalization,” _Neural networks_ , vol. 5, no. 2, pp. 241–259, 1992.
  * [21] N. S. Keskar and R. Socher, “Improving generalization performance by switching from adam to sgd,” _arXiv preprint arXiv:1712.07628_ , 2017.