# Learning-Based Robust Resource allocation for D2D Underlaying Cellular
Network

Weihua Wu, Runzi Liu, Qinghai Yang, and Tony Q. S. Quek, _Fellow, IEEE_ *This
work was supported in part by the NSF China under Grant 61801365, 61701365 and
61971327, in part by the China Postdoctoral Science Foundation under Grant
2018M643581, in part by the National Natural Science Foundation of Shaanxi
Province under Grant 2019JQ-152, in part by Postdoctoral Foundation in Shaanxi
Province of China, and the Fundamental Research Funds for the Central
Universities.W. Wu and Q. Yang are with State Key Laboratory of ISN, School of
Telecommunications Engineering, Xidian University, No.2 South Taibai Road,
Xi’an, 710071, Shaanxi, China. (Email:whwu@xidian.edu.cn).R. Liu is with
School of Information and Control Engineering, Xi’an University of
Architecture and Technology, Xi’an 710055, China.T. Q. S. Quek is with the
Information Systems Technology and Design Pillar, Singapore University of
Technology and Design, Singapore 487372 (e-mail: tonyquek@sutd.edu.sg).

###### Abstract

In this paper, we study the resource allocation in D2D underlaying cellular
network with uncertain channel state information (CSI). For satisfying the
diversity requirements of different users, i.e. the minimum rate requirement
for cellular user and the reliability requirement for D2D user, we attempt to
maximize the cellular user’s throughput whilst ensuring a chance constraint
for D2D user. Then, a robust resource allocation framework is proposed for
solving the highly intractable chance constraint about D2D reliability
requirement, where the CSI uncertainties are represented as a deterministic
set and the reliability requirement is enforced to hold for any uncertain CSI
within it. Then, a symmetrical-geometry-based learning approach is developed
to model the uncertain CSI into polytope, ellipsoidal and box. After that, we
derive the robust counterpart of the chance constraint under these uncertainty
sets as the computation convenient convex sets. To overcome the conservatism
of the symmetrical-geometry-based uncertainty sets, we develop a support
vector clustering (SVC)-based approach to model uncertain CSI as a compact
convex uncertainty set. Based on that, the chance constraint of D2D is
converted into a linear convex set. Then, we develop a bisection search-based
power allocation algorithm for solving the resource allocation in D2D
underlaying cellular network with different robust counterparts. Finally, we
conduct the simulation to compare the proposed robust optimization approaches
with the non-robust one.

_Key Terms_ : D2D communications, resource allocation, robust optimization,
chance constraint, SVC.

##  I Introduction

The throughput requirements in cellular networks have been exponentially
increasing in these years [1]. Device-to-device (D2D) is considered as a
promising technique to satisfy this demand so that it received much attentions
over the last few years [2]. With the deployment of D2D, the cellular users
can communicate with each other directly without bypassing the base station
(BS). This paradigm entails a wide range of benefits, such as reusing gain,
proximity gain and hop gain [3, 4]. Due to these advantages, D2D technique is
widely used in the fields of Internet of vehicles, Internet of things,
industrial Internet and so on.

There exist two paradigms for the D2D coexisting with cellular users:
underlaid paradigm and dedicated paradigm [4]. In the underlaid paradigm, the
same resource used by a traditional cellular user can be simultaneously used
by a D2D pair when it is located in a sufficiently distant part of the cell.
Hence, it can provide services for more users without adding the bandwidth.
However, there exists cross-tier interference between D2D users and cellular
users, which seriously degrades the capacity of each transmission link. In the
dedicated paradigm, the BS should allocate orthogonal resources to D2D users
and cellular users. Thus, the cross-tier interference can be effectively
cancelled and the link performance is enhanced. However, due to the limited
resources, the number of users that the network can accommodate is strictly
limited. Therefore, in order to serve more users and obtain higher spectrum
efficiency, it is of great significance to implement the underlaid coexistence
between D2D users and cellular users.

In order to fully exploit the potential benefits of underlaid D2D
communications, it is necessary to provide a judicious assignment of the
spectrum resource (e.g. time slot or resource block) to D2D users and design a
prudent power control mechanism that avoids the detrimental interference to
cellular users. This is the case in [5, 6], where the resource allocation
approaches were developed to work out the solution of subchannel assignment
and power distribution. The unified resource management schemes in [7, 8]
achieved the coexistence of D2D and cellular by jointly optimizing mode
selection, resource allocation, and power control. All of the previously
mentioned works assume the perfect channel state information (CSI) is
available at the BS. From the practical prospective, collecting the D2D’s CSI
requires a lot of cooperation between D2D pairs and the cellular users, then
substantial amount of communication and latency overhead will be added before
the BS receives the value. As a result, the BS can only obtain the uncertain
value about the CSI of D2D. The uncertainties of CSI render the deterministic
optimizations in [5, 6, 7, 8] unreliable. It has been widely demonstrated in
[9][10] that even a slight perturbation on the CSI can greatly influence the
system performance, such as leading to suboptimal objective or the violation
of quality-of-service (QoS) constraint.

Motivated by urgent requirement of handling the uncertain value about the CSI
of D2D communications, stochastic optimization and robust optimization methods
have received many attentions in recent years. With the deployment of
stochastic optimization, the works in [11, 12, 13, 14] tackled the
uncertainties of CSI by optimizing the expected link capacity. Besides that,
[15] transformed the QoS requirement with uncertain CSI into a closed-form
expression under the given statistical distribution of the uncertainty.
Evidently, the stochastic optimization technique entails complete distribution
information about the uncertain CSI. This may be unrealistic in practical
wireless networks. For the robust optimization, there exist two approaches:
the Bernstein approximation approach [16, 17] and the worst-case optimization
approach [9, 18]. The former assumes that the uncertain channels are
uncorrelated and that all of the CSI values are independent with each other.
Then, it optimizes the network utility with guaranteeing the QoS constraint
with certain probability. However, we should note that the independent
distribution assumption in Bernstein approximation is too idealistic and that
it loses a lot of useful correlation information between different channels,
so that it may lead to over conservative solution. On the other hand, the
worst-case optimization takes a deterministic and set-based approach to
grantee the worst-case performance and satisfy the constraint under any
realization of CSI in a closed uncertainty region. For the case of unbounded
uncertainty, the worst case approach is still applicable by satisfying the QoS
constraint with a certain probability [19]. Due to these appealing
characteristics, this paper exploits the worst case approach to address the
resource allocation in D2D underlaying cellular network.

As for the worst case robust optimization in D2D networks, there already exist
some pioneering works [18, 20, 21]. More specifically, [18, 20] focused on
maximizing the worst-case system utility of both cellular and D2D links under
CSI impairment. The work in [21] utilized the worst-case approach to find the
tractable forms about the chance constraint and then derived the semi-closed
expression for the power allocation in D2D networks. However, all of these
works deploy the robust optimization based on the fixed-size uncertainty CSI
set. Actually, this assumption is excessively idealistic. In practical
wireless network, the size and shape parameters of the uncertainty set would
vary with the network conditions. Thus, constructing the uncertainty region to
include probable realizations of uncertain CSI is a paramount ingredient in
robust optimization.

In this paper, we consider the resource allocation in D2D underlaying cellular
network. In order to improve the spectrum efficiency, the D2D users reuse the
uplink spectrum resource of cellular users. We assume that the CSI connect to
BS can be perfectly obtained since it can be directly estimated by the BS,
while the CSI of D2D links are reported to BS with errors. Because D2D
communication is usually used to support the emergency services, such as the
emergency electronic brake lights in vehicle networks [22], we define a chance
constraint for the D2D QoS requirement. On the contrary, the minimum QoS
requirement constraint is defined for the cellular user since it is often used
for the non-safety related service in most of case. Different from the
previous robust optimization approaches, where the uncertainties are assumed
to being independently and symmetrically distributed, we consider a more
realistic scenario that the uncertainties are intertwined and asymmetric.
Then, we investigate the different robust optimization approaches for solving
the resource allocation problem in D2D underlaying cellular network. The main
contributions of this paper can be summarized as follows:

  * •

A robust resource allocation framework is developed for solving the highly
intractable chance constraint about D2D QoS requirement. Specifically, the CSI
uncertainties are modeled as a deterministic (uncertainty) set, and the chance
constraint is satisfied by enforcing the QoS requirement to hold for any
uncertain CSI within it

  * •

We develop a symmetrical-geometry-based learning approach to construct the
uncertainty set from the samples of uncertain CSI. More specifically, the
uncertainties are constructed as polytope, ellipsoidal and box, respectively.
Then, the robust counterparts of the chance constraint corresponding to the
different uncertainty sets are derived. It should be noted that all of the
robust counterparts are definitely computationally convenient for the convex
optimization tools.

  * •

To overcome the conservatism of the symmetrical-geometry-based uncertainty
sets, we develop a support vector clustering (SVC)-based approach to construct
the uncertainties as a compact convex set. We should note the SVC-based
uncertainty set is asymmetric and can tightly enclose the uncertainties
without any superfluous space. Moreover, the induced robust counterpart is the
tractable linear convex set. In this case, we also found that the robust
optimization works as a foundation for bridging modern machine learning tools
into the resource allocation in wireless network.

  * •

A bisection search-based power allocation algorithm is developed for solving
the power allocation problem under diversity robust counterparts with low
polynomial-time complexity.

The rest of this paper is organized as follows. In Section II, we give the
system model and problem formulation. The basic uncertainty set induced robust
resource allocation is illustrated in Section III. Section IV investigates the
kernel learning induced robust allocation. Afterwards, the simulation results
are illustrated in Section V. Finally, Section VI concludes the paper.

##  II System Model

This section provides the network model and channel uncertainty model of the
D2D underlaying cellular network, followed by the problem formulation about
the power allocation under diversity QoS requirements.

Figure 1: System model.

###  II-A Network Model

We consider a D2D underlaying cellular network as shown in Fig. 1. In the
considered system model, the users, denoted as CUE, communicate with the BS
through the cellular link. Because the uplink resources of cellular links are
less intensively used and the BS has a stronger ability to manage the
interference than users, the uplink spectrum resource of CUE is reused by the
D2D users, denoted as DUE, for improving the spectrum utilization efficiency.
In order to reduce the complexity brought by the complicated inter- and intra-
cell interference, we assume that one DUE is only allowed to access the
spectrum of one CUE and that the spectrum of one CUE can only be reused by one
DUE. It has been widely proved in [11, 12, 13] that the optimal spectrum
reusing pattern can be easily obtained by using the Hungarian method [23]
based on the optimal power allocation of all possible reusing pairs between
CUEs and D2Ds. Thus, this paper does not study the spectrum allocation
anymore, but focuses on the challenging power allocation for a possible
reusing pair. Let $p_{c}$ denote the transmit power of CUE and $p_{d}$ denote
the transmit power of the transmitter in D2D pair. Then, the channel power
gain of D2D link is assumed to follow

$\displaystyle g_{d}=h_{d}\beta_{d}\varpi
D^{-\iota}\triangleq|h_{d}|^{2}\alpha_{d},$ (1)

where $\alpha_{d}=\beta_{d}\varpi D^{-\iota}$ represents the large-scale slow
fading channel gain, $h_{d}$ is the small-scale fast fading power component,
$\varpi$ is the pathloss constant, $D$ is the distance of the DUEs in pair,
$\iota$ is the decay exponent, and $\beta_{d}$ is log-normal shadow fading
random variable with a standard deviation $\xi$. Then, the channel power gain
between CUE and BS $g_{c}$, the corsstalk channel power gain between CUE and
DUE $g_{c,d}$, and the crosstalk channel power gain between DUE and BS
$g_{d,B}$ have the similarly definitions.

We assume that the CSI between BS and users i.e., $g_{c}$ and $g_{d,B}$ can be
perfectly obtained because they can be directly estimated by the BS. For the
CSI between different users, i.e., $g_{d}$ and $g_{c,d}$, the large-scale
fading is accurately known since they usually dependent on the locations which
vary on a slow scale. On the contrary, due to the Doppler effect [12], partial
CSI acquisition [24] and the delays in CSI feedback [25], there inevitably
exists uncertainty in the small-scale CSI. In this work, the additive error
[26, 27] model of the channel imperfection is adopted, i.e.,

$\displaystyle|h_{d}|^{2}=\delta^{2}|\hat{h}_{d}|^{2}\\!+\\!(1-\delta^{2})|e_{d}|^{2},$
(2)

where $\hat{h}$ is the estimated channel gain, $e$ is the estimation error and
$\delta$ $(0<\delta<1)$ is the channel estimation error coefficient. It can be
easily understood that $\delta=0$ means that no CSI is acquired at all, while
$\delta=1$ indicates the perfect channel estimation. Most of the current works
consider that both $\delta$ and $e$ are known to the transmitter and that the
estimation errors of different channels are independent with each other. This
paper considers a more practical and general scenario where all of the
distribution information about them are unknown and the estimation errors of
different channels are correlated and intertwined with each other. Moreover,
the underlying distribution of channel errors may be intrinsically complex and
variable with the evolution of wireless network.

Based on the these assumptions, the received Signal to Interference plus Noise
Ratio (SINR) of D2D pair can be given as

$\displaystyle\gamma_{d}\\!=\\!\frac{p_{d}\alpha_{d}\left(\delta^{2}|\hat{h}_{d}|^{2}\\!+\\!(1-\delta^{2})|e_{d}|^{2}\right)}{\sigma^{2}+p_{c}\alpha_{c,d}\left(\delta^{2}|\hat{h}_{c,d}|^{2}\\!+\\!(1\\!-\\!\delta^{2})|e_{c,d}|^{2}\right)},$
(3)

where $\sigma^{2}$ is the power of the additive white Gaussian noise.
Similarly, the received SINR at BS from the CUE can be computed as

$\displaystyle\gamma_{c}=\frac{p_{c}\alpha_{c}h_{c}}{\sigma^{2}+p_{d}h_{d,B}\alpha_{d,B}}.$
(4)

###  II-B Problem Formulation

In D2D underlaying cellular network, we assume that the CUE and DUE have
different QoS requirements. Because the D2D communication is usually used for
safety related service [14], it has the objective of reducing the outage
probability. On the contrary, the main objective of CUE is to enable a more
efficient and comfortable entertainment experience, the minimum QoS
requirement is necessary for them. Based on these considerations, the problem
can be formulated as

$\displaystyle\max\limits_{p_{c},p_{d}}$ $\displaystyle
B\log_{2}(1+\gamma_{c})$ (5a) s.t.
$\displaystyle\gamma_{c}\geq\gamma_{min}^{c},$ (5e)
$\displaystyle\Pr\left\\{\gamma_{d}\geq\gamma_{min}^{d}\right\\}\geq
1-\epsilon,$ $\displaystyle 0\leq p_{c}\leq P_{max}^{c},$ $\displaystyle 0\leq
p_{d}\leq P_{max}^{d},$

where $B$ denotes the spectrum bandwidth, $\gamma_{min}^{c}$ and
$\gamma_{min}^{d}$ are the minimum required SINR for CUE and DUE,
respectively, $P_{max}^{c}$ and $P_{max}^{d}$ indicate the maximum transmit
power of CUE and DUE, respectively. In (5c), $\Pr\\{\cdot\\}$ denotes the
probability of the input, $\epsilon$ is the maximum tolerable outage
probability for D2D communication. Obviously, the chance constraint induced by
uncertain CSI poses a great challenge on solving the resource allocation
problem. In the following, we focus on solving the power allocation problem
based on the uncertain CSI.

##  III Basic Uncertainty Set Induced Robust Resource Allocation

To avoid the complexity of directly solving the resource allocation in (5),
this section employs the robust optimization approach to represent the
uncertain CSI of D2D links as a high probability region (HPR) and then
enforces the QoS constraint to hold for any CSI value in it. To obtain the
HPR, we need to collect multiple samples of the uncertain CSI of D2D links and
then learn the uncertainty set. For satisfying the chance constraint of D2D,
the uncertainty set needs to cover the samples with a certain probability. If
the obtained power allocation solution is feasible under all of the CSI values
in the uncertainty set, the chance constraint must be satisfied. Motivated
from this consideration, we first formulate problem (5) as the following
approximation form

$\displaystyle\max\limits_{p_{c},p_{d}}$ $\displaystyle
B\log_{2}(1+\gamma_{c})$ (6a) s.t.
$\displaystyle\gamma_{c}\geq\gamma_{min}^{c},$
$\displaystyle\mathbf{p}^{T}\mathbf{g}\geq\gamma_{min}^{d},\mathbf{g}\in\mathcal{G},$
$\displaystyle\textrm{(5d)},\textrm{(5e)},$

where $\mathbf{p}=[p_{d}/\sigma^{2},-p_{c}\gamma_{min}^{d}/\sigma^{2}]^{T}$,
$\mathbf{g}=[g_{d},g_{c,d}]^{T}\in\mathbb{R}^{2}$ and $\mathcal{G}$ is the HPR
that need to be leaned. It is not difficult to understand that if
$\mathcal{G}$ covers the samples of uncertain CSI $\mathbf{g}$ with
$1-\epsilon$ confidence level, any feasible power allocation solution of (6)
must satisfy

$\displaystyle\Pr\left\\{\gamma_{d}\geq\gamma_{min}^{d}\right\\}\geq\Pr\left\\{\mathbf{g}\in\mathcal{G}\right\\}\geq
1-\epsilon,$ (7)

which indicates that the chance constraint is feasible for problem (5). Based
on the above discussions, learning the uncertainty set is the key component
for solving the power allocation. Because $\mathbf{g}\in\mathcal{G}$ can be
considered a constraint condition of problem (6), the shape of uncertainty set
must be considered from the tractability in the optimization problem.
Moreover, we note that almost all of the symmetric geometries exhibit
excellent convexity. Motivated from these considerations, we give several
basic selections, e.g., polytope, ellipsoidal and box, of the uncertainty set
in the following sections and then derive the robust counterparts of the
chance constraint based on the corresponding uncertainty sets.

###  III-A Polytope Model

The polytope model is parameterized as

$\displaystyle\mathcal{P}=\\{\mathbf{g}|\mid g_{d}-\bar{g}_{d}\mid+\mid
g_{c,d}-\bar{g}_{c,d}\mid\leq\Gamma\\},$ (8)

where $\bar{g}_{d},\bar{g}_{c,d}\in\mathbb{R}$ are the center positions of the
polytope and $\Gamma>0$ is the budget parameter to control the size of
polytope. For learning these parameters, we should collect multiple samples of
the uncertain CSI $\mathbf{g}$ as
$\mathcal{N}=\\{\bm{\xi}_{1},\bm{\xi}_{2},\cdots,\bm{\xi}_{N}\\}$, where
$\bm{\xi}_{i}\in\mathbb{R}^{2}$. Then, we propose a statistical learning
approach, which includes shape learning and size calibration, for determining
the shape parameters.

####  III-A1 Shape Learning

Note that $\bar{\mathbf{g}}=[\bar{g}_{d},\bar{g}_{c,d}]^{T}$ works as the
origin of the polytope uncertainty set. Thus, without loss of generality, it
can be chosen as the sample mean, i.e.

$\displaystyle\bar{\mathbf{g}}=\frac{1}{N}\sum_{k=1}^{N}\bm{\xi}_{k}.$ (9)

After the shape learning, we should determine the size of polytope in the
following procedure.

####  III-A2 Size Calibration

The objective of size calibration is to calibrate the uncertainty set so that
it can satisfy $\mathbf{g}\in\mathcal{P}(\mathcal{N})$ with confidence
$1-\epsilon$. The key idea is to estimate the quantile of the transformation
of the data sample. More concretely, let

$\displaystyle
t_{p}(\bm{\xi})=\mid\bm{\xi}(1)-\bar{g}_{d}\mid+\mid\bm{\xi}(2)-\bar{g}_{c,d}\mid$
(10)

be the transformation map from random space $\mathbb{R}^{2}$ into
$\mathbb{R}$. Then, the size of $\mathcal{P}$ can be set as the estimated
$(1-\epsilon)$-quantile of the underlying distribution of $t_{p}(\bm{\xi})$
based on the sample data set $\mathcal{N}$. For achieving this objective, we
should introduce the definition of $(1-\epsilon)$-quantile $q_{1-\epsilon}$ as

$\displaystyle\Pr\left\\{t_{p}(\bm{\xi})\leq
q_{1-\epsilon}\right\\}=1-\epsilon.$ (11)

Then, by computing the function values of $t_{p}(\bm{\xi})$ on all of the
samples in $\mathcal{N}$, we can obtain the observations
$t_{p}(\bm{\xi}_{(1)}),\cdots,t_{p}(\bm{\xi}_{(N)})$. By sorting the
observations $t_{p}^{(1)}\leq\cdots\leq t_{p}^{(N)}$ in ascending order, the
$k_{p}^{*}=\lceil(1-\epsilon)N\rceil$-th value can be considered as the upper
bound of $(1-\epsilon)$-quantile of $t_{p}(\bm{\xi})$. As a result, the size
of uncertainty set $\mathcal{P}$ can be set as

$\displaystyle\Gamma=t_{p}(\bm{\xi}_{(k_{p}^{*})}).$ (12)

Using the presented two procedures, the polytope model for covering the
uncertain CSI with confidence level $1-\epsilon$ can be obtained. Moreover, we
should note that the polytope uncertainty set can be also represented as
$\mathcal{P}=\\{\mathbf{g}|\mathbf{M}_{p}(\mathbf{g}-\bar{\mathbf{g}})\leq\bm{\Gamma}\\}$,
where $\bm{\Gamma}=\Gamma\times\mathbf{1}_{4\times 1}$,
$\mathbf{M}_{p}\in\mathbb{R}^{4\times 2}$ is the weight matrix and each row of
$\mathbf{M}_{p}$ indicates a possible realization of the combination of $1$
and $-1$.

Based on the obtained polytope model, the robust counterpart of D2D QoS
constraint can be computed as

$\displaystyle\min$ $\displaystyle\mathbf{p}^{T}\mathbf{g}$ (13) s.t.
$\displaystyle\mathbf{M}_{p}(\mathbf{g}-\bar{\mathbf{g}})\leq\bm{\Gamma}.$

Since (13) is feasible, the optimal objective $\Delta^{*}$ can be obtained via
its Lagrange dual as

$\displaystyle\Delta^{*}=\max$
$\displaystyle-(\bm{\Gamma}+\mathbf{M}_{p}\bar{\mathbf{g}})^{T}\mathbf{x}$
(14) s.t.
$\displaystyle-\mathbf{M}_{p}^{T}\mathbf{x}\leq\mathbf{p},\mathbf{x}\geq 0,$

where $\mathbf{x}\in\mathbb{R}^{4}$ is the dual variable. If the feasible
solution, denoted as $\tilde{\mathbf{x}}$, of (14) satisfy
$-(\bm{\Gamma}+\mathbf{M}_{p}\bar{\mathbf{g}})^{T}\tilde{\mathbf{x}}\geq\gamma_{min}^{d}$,
then
$\Delta^{*}\geq-(\bm{\Gamma}+\mathbf{M}_{p}\bar{\mathbf{g}})^{T}\tilde{\mathbf{x}}\geq\gamma_{min}^{d}$
holds for all uncertainty of $\mathbf{g}$. In this case, the D2D QoS
constraint in (6c) can be replaced by the following constraint

$\displaystyle\left\\{\begin{array}[]{l}-(\bm{\Gamma}+\mathbf{M}_{p}\bar{\mathbf{g}})^{T}\mathbf{x}\geq\gamma^{d}_{min},\\\
-\mathbf{M}_{p}^{T}\mathbf{x}\leq\mathbf{p},\mathbf{x}\geq
0.\end{array}\right.$ (17)

It shows that the intractable chance constraint is transformed into a
combination of linear constraint. Definitely, it would provide computational
advantages for the power allocation problem.

###  III-B Ellipsoidal Uncertainty Set

The ellipsoidal uncertainty set can be parameterized as

$\displaystyle\mathcal{E}=\\{\mathbf{g}:(\mathbf{g}-\bar{\mathbf{g}})^{T}(\mathbf{g}-\bar{\mathbf{g}})\leq\Lambda\\},$
(18)

where $\bar{\mathbf{g}}$ is the center of $\mathcal{E}$ and $\Lambda>0$ is the
size of $\mathcal{E}$. In the ellipsoidal uncertainty set, the center is same
to the one in polytope. The size of $\mathcal{E}$ can be estimated by using
the same quantile method as shown in Subsection A. Therefore, we use the
following function

$\displaystyle
t_{e}(\bm{\xi})=(\bm{\xi}-\bar{\mathbf{g}})^{T}(\bm{\xi}-\bar{\mathbf{g}})$
(19)

to represent the map from random space $\mathbb{R}^{2}$ into $\mathbb{R}$. By
computing the function values of $t_{e}(\bm{\xi})$ on each sample of
$\mathcal{N}$ and sorting the observations of $t_{e}(\bm{\xi})$ in ascending
order, the $k_{e}^{*}=\lceil(1-\delta)N\rceil$-th value of the ranked
observations can be set as the size of uncertainty set $\mathcal{E}$, i.e.,

$\displaystyle\Lambda=t_{e}(\bm{\xi}_{(k_{e}^{*})}).$ (20)

According to the obtained budget parameters, the ellipsoidal uncertainty set
can be also represented as

$\displaystyle\mathcal{E}=\\{\mathbf{g}:\mathbf{g}=\bar{\mathbf{g}}+\zeta\mathbf{u},\mathbf{u}^{T}\mathbf{u}\leq
1\\},$ (21)

where $\mathbf{u}\in\mathbb{R}^{2}$ and $\zeta$ can be computed as
$\zeta=\sqrt{\Lambda}$.

Based on the ellipsoid uncertainty set, the robust counterpart of D2D QoS
constraint can be computed as

$\displaystyle\min$ $\displaystyle\mathbf{p}^{T}\mathbf{g}$ (22) s.t.
$\displaystyle\mathbf{g}=\bar{\mathbf{g}}+\zeta\mathbf{u},\mathbf{u}^{T}\mathbf{u}\leq
1.$

Since

$\displaystyle\inf_{\parallel\mathbf{u}\parallel\leq
1}\mathbf{p}^{T}(\bar{\mathbf{g}}+\zeta\mathbf{u})=\mathbf{p}^{T}\bar{\mathbf{g}}-\zeta\parallel\mathbf{p}^{T}\parallel,$
(23)

where the derivation is based on the Schwartz inequality, the D2D QoS
constraint in (6c) can be replaced by

$\displaystyle\mathbf{p}^{T}\bar{\mathbf{g}}-\zeta\|\mathbf{p}^{T}\|\geq\gamma^{d}_{min},$
(24)

which is a second-order cone. Thus, it is effectively compatible with the
convex optimization tools.

###  III-C Box Uncertainty Set

The box uncertainty set is parameterized as

$\displaystyle\mathcal{B}=\\{\mid\mathbf{g}(i)-\bar{\mathbf{g}}(i)\mid\leq\Psi,i=1,2\\},$
(25)

where $\bar{\mathbf{g}}$ is the center of box and $\Psi$ determines the size
of $\mathcal{B}$. In order to obtain $\Psi$, we should define

$\displaystyle
t_{b}(\bm{\xi}_{i})=\max_{i=1,2}\mid\bm{\xi}_{i}-\bar{\mathbf{g}}_{i}\mid$
(26)

to map the channel samples into real number. Then, the parameter $\Psi$ can be
chosen as the $k_{b}^{*}=\lceil(1-\epsilon)N\rceil$-th value of the ranked
observations $t_{b}^{(1)}\leq\cdots\leq t_{b}^{(N)}$ in ascending order, i.e.,
$\Psi=t_{b}(\bm{\xi}_{(k_{b}^{*})})$.

Then, based on the obtained parameter, the box uncertainty set can be re-
represented as

$\displaystyle\mathcal{B}=\\{\mathbf{g}|\mathbf{M}_{b}(\mathbf{g}-\bar{\mathbf{g}})\leq\bm{\Psi}\\},$
(27)

where $\bm{\Psi}=\Psi\times\mathbf{1}_{4\times 1}$ and each row of
$\mathbf{M}_{b}$ indicates a possible realization of the combination $0$ and
$\pm 1$.

Given this uncertainty set, the robust counterpart can be computed as

min $\displaystyle\mathbf{p}^{T}\mathbf{g}$ (28) s.t.
$\displaystyle\mathbf{M}_{b}(\mathbf{g}-\bar{\mathbf{g}})\leq\bm{\Psi}.$

The optimal objective $\Theta^{*}$ of problem (26) can be obtained via its
Lagrange dual as

$\displaystyle\Theta^{*}=\textrm{max}$
$\displaystyle-(\bm{\Psi}+\mathbf{M}_{b}\bar{\mathbf{g}})^{T}\mathbf{y}$ (29)
s.t. $\displaystyle-\mathbf{M}_{b}^{T}\mathbf{y}\leq\mathbf{p},\mathbf{y}\geq
0,$

where $\mathbf{y}\in\mathbb{R}^{4}$ is the dual variable. Similar to the
computation of robust counterpart based on polytope uncertainty set, we obtain
$\Theta^{*}\geq-(\bm{\Psi}+\mathbf{M}_{b}\bar{\mathbf{g}})^{T}\tilde{\mathbf{y}}\geq\gamma_{min}^{d}$
for all uncertainties of $\mathbf{g}$. Finally, the chance constraint of D2D
can be replaced by the following equations

$\displaystyle\left\\{\begin{array}[]{l}-(\bm{\Psi}+\mathbf{M}_{b}\bar{\mathbf{g}})^{T}\mathbf{y}\geq\gamma^{d}_{min},\\\
-\mathbf{M}_{b}^{T}\mathbf{y}\leq\mathbf{p},\mathbf{y}\geq
0.\end{array}\right.$ (32)

It shows that the robust counterpart of chance constraint supported on box
uncertainty set leads to a combination of linear constraint. It is definitely
computationally convenient for the convex optimization tools.

###  III-D Power Allocation

By modeling the uncertain CSI, the intractable chance QoS constraint of D2D
can be transformed into the corresponding tractable one. Then, power
allocation problem (6) can be reformulated as the following optimization

$\displaystyle\max\limits_{p_{c},p_{d}}$ $\displaystyle
B\log_{2}\left(1+\frac{p_{c}g_{c}}{\sigma^{2}+p_{d}g_{d,B}}\right)$ (33) s.t.
(15)or(22)or(28) $\displaystyle\textrm{(6b)},\textrm{(6d)},\textrm{(6e)}.$

Although any combination of the constraints in (29) can constitutes a convex
set, problem (29) is still nonconvex due to the fractional form of $p_{c}$ and
$p_{d}$ in objective function. For solving this nonconvex problem, we should
first fix either $p_{c}$ or $p_{d}$ to find the corresponding optimal
$p_{d}^{*}$ or $p_{c}^{*}$, respectively. Then, repeat the above process until
both $p_{c}$ and $p_{d}$ converge the optimal solution of (29).

Inspired by the above discussion, we can first consider a fixed $p_{d}$ and
formulate a subproblem to maximize the throughput of CUE, i.e.

$\displaystyle\max\limits_{p_{c}}$ $\displaystyle
B\log_{2}\left(1+\frac{p_{c}g_{c}}{\sigma^{2}+p_{d}g_{d,B}}\right)$ (34) s.t.
(15)or(22)or(28) $\displaystyle p_{c}\geq 0,\textrm{(6b)},$

which is a concave optimization problem. Thus, it can be effectively solved by
the widely used convex optimization tools. Then, before developing the
algorithm for searching $p_{d}$, we should give the following observations.

For the power allocation in D2D underlaying cellular network, if the CUE
transmit power is larger, more transmit power need to be allocated for
satisfying the QoS constraint of D2D. Therefore, $p_{d}(p_{c})$ can be
considered as an increasing function of $p_{c}$. Based on this observation,
the optimal transmit power of DUE can be searched by using the bisection
search algorithm. Moreover, the following theorem provides a condition to
terminate the searching process.

######  Lemma 1.

The optimal power allocation solution for (29) must satisfy either
$p_{c}^{*}=P_{max}^{c}$ or $p_{d}^{*}=P_{max}^{d}$.

###### Proof.

The similar proof can be found in [13]. ∎

Based on Lemma 1, the bisection search method can be developed in Algorithm 1
for searching the optimal power allocation of (29). Then, the standard
operating procedure for the symmetrical-geometry-based robust resource
allocation can be concluded as follows:

  * •

Collect a set $\mathcal{N}$ of samples of the imperfect CSI.

  * •

Learn the shape parameter $\bar{\mathbf{g}}$ based on the sample set
$\mathcal{N}$.

  * •

Set the size parameter based on quantize estimation method.

  * •

Construct the robust counterpart based on the obtained uncertainty set, and
use it to replace the chance constraint in (5c) to derive the convex
constraint set.

  * •

Solve the induced power allocation problem by using Algorithm 1.

Algorithm 1 Bisection Search-Based Power Allocation

Set termination threshold $0<\zeta<1$;

Set $p_{k,min}^{d}=0$ and $p_{k,max}^{d}=P_{max}^{d}$;

while $p_{k}^{d}<P_{max}^{d}-\zeta$ do

set $p_{k}^{d}=(p_{k,min}^{d}+p_{k,max}^{d})/2$; Solve (30) to obtain
$p_{m}^{c}$;

if $p_{m}^{c}>P_{max}^{c}+\zeta$ then

$p_{k,max}^{d}=p_{k}^{d}$

else if $p_{m}^{c}<P_{max}^{c}-\zeta$ then

$p_{k,min}^{d}=p_{k}^{d}$

else if $P_{max}^{c}-\zeta<p_{m}^{c}<P_{max}^{c}+\zeta$ then

break

end if

end while

Output the optimal transmit powers $p_{m}^{c,*}$ and $p_{k}^{d,*}$.

##  IV Kernel Learning Induced Robust Resource Allocation

In above section, all of the uncertainty sets are symmetric and radial. The
resulting robust counterparts have the succinct formulations and are
computational convenience for the resource allocation problem. However, we
should note that the classical uncertainty sets have some limitations in
practical applications. First, we should determine the moment values as well
as the support information carefully. As illustrated in the channel training
process, it is a nontrivial task to obtain the exact information of these
parameters. Second, the CSI uncertainties on different dimensions in D2D
underlaying cellular network maybe intertwined and asymmetric, as shown in
Fig. 2(a). No matter what the shape of the symmetric-geometry-based
uncertainty set is, it cannot accurately cover the distributions of the
uncertain CSI. As a result, the symmetric-geometry-based uncertainty sets lead
to large block superfluous coverage so that they will result in over-
conservative decisions. Inspired by the recently popular machine learning (ML)
methods, constructing the uncertainty set can be considered as the
unsupervised leaning problem. The support vector clustering (SVC) technique in
ML is always considered as an effective learning method to provide powerful
representations of sample distribution [28, 29]. Thus, in this section, we
propose to resort the SVC technique to estimate the uncertainty set from CSI
samples.

Figure 2: Uncertainty set construction based on the imperfect CSI samples. (a)
Symmetric-geometry-based uncertainty sets, i.e., polytope, ellipsoidal and
box. (b) Nonlinear mapping map of the data samples into a high-dimensional
features space.

###  IV-A SVC-Based Robust Optimization

It has been widely proven that the SVC is an efficient pattern recognition
approach for analyzing the complicated high-dimensional data. The SVC focuses
on the description of channel uncertainties by means of an enclosing sphere
with minimal volume. To achieve this objective, we should first collect a set
of $\mathcal{N}=\\{\bm{\xi}_{1},\bm{\xi}_{2},\cdots,\bm{\xi}_{N}\\}$ samples
of the imperfect CSI and then use a nonlinear mapping
$\phi(\bm{\xi}_{i}):\mathbb{R}^{2}\mapsto\mathbb{R}^{K}$ to map the CSI
samples into a high-dimensional features space $\mathcal{F}$, i.e., $K>2$.
After the nonlinear transformation, the CSI samples with correlation and
asymmetry can be gathered into cluster in the high-dimensional features space,
as shown in Fig. 2(b). The objective of SVC is to seek the smallest sphere to
enclose the CSI samples in high-dimensional features space, which can be
formulated as the following optimization problem.

$\displaystyle\min\limits_{R,\bm{\rho},\\{\psi_{i}\\}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!R^{2}+C\sum_{i=1}^{N}\psi_{i}$ (35a)
s.t.
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}\leq
R^{2}+\psi_{i},i=1,\cdots,N,$ (35c)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\psi_{i}\geq 0,i=1,\cdots,N,$

where $R$ denotes the radius of the sphere, $\bm{\rho}$ is the center,
$\psi_{i}$ is a slack variable and $C$ is the parameter to control the
tradeoff between the two goals.

We introduce the Lagrangian multipliers $\bm{\lambda}$ and $\bm{\eta}$ for
constructing the following Lagrangian function

$\displaystyle
L(R,\bm{\rho},\bm{\psi},\bm{\lambda},\bm{\eta})=\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle R^{2}+C\sum_{i=1}^{N}\psi_{i}-\sum_{i=1}^{N}\eta_{i}\psi_{i}$
$\displaystyle+\sum_{i=1}^{N}\lambda_{i}(\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}-R^{2}-\psi_{i}).$

The derivatives of $L$ with respect to $R$, $\bm{\rho}$ and $\psi_{i}$ can be
computed as

$\displaystyle\left\\{\begin{array}[]{l}\frac{\partial L}{\partial
R}=0\rightarrow\sum_{i=1}^{N}\lambda_{i}=1,\\\ \frac{\partial
L}{\partial\bm{\rho}}=\mathbf{0}\rightarrow\bm{\rho}=\sum_{i=1}^{N}\lambda_{i}\bm{\phi}(\bm{\xi}_{i}),\\\
\frac{\partial
L}{\partial\psi_{i}}\rightarrow\lambda_{i}+\eta_{i}=C.\end{array}\right.$ (40)

Eq. (33) shows that the center of sphere is a linear combination of the
mapping of all uncertain channel samples. Moreover, according to the Karuch-
Kuhn-Tucker (KKT) conditions, one can obtain

$\displaystyle\eta_{i}\psi_{i}=0,\
\lambda_{i}(\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}-R^{2}-\psi_{i})=0.$ (41)

Using these relations, we may eliminate the variables $R$, $\bm{\rho}$ and
$\psi_{i}$, and turn the Lagrangian into a disciplined quadratic programming
(QP) as the dual problem

$\displaystyle\min\limits_{\bm{\lambda}}\sum_{i=1}^{N}\sum_{j=1}^{N}\lambda_{i}\lambda_{j}K(\bm{\xi}_{i},\bm{\xi}_{j})-\sum_{i=1}^{N}\lambda_{i}K(\bm{\xi}_{i},\bm{\xi}_{i})$
(42a) $\displaystyle\ \textrm{s.t.}\quad 0\leq\lambda_{i}\leq C,i=1,\cdots N,$
(42b) $\displaystyle\quad\quad\ \sum_{i=1}^{N}\lambda_{i}=1,$ (42c)

where $K(\cdot,\cdot)$ is the kernel trick and it can be computed as
$K(\bm{\xi}_{i},\bm{\xi}_{j})=\bm{\phi}(\bm{\xi}_{i})^{T}\bm{\phi}(\bm{\xi}_{j})$.
We should note that the specific expression of kernel function should not
affect the convexity of the dual problem. In the following discussions, we
will illustrate how to design the kernel in detail.

Based on the KKT conditions and complementary slackness, we can obtain the
following desirable geometric interpretations,

$\displaystyle\left\\{\begin{array}[]{l}\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}<R^{2}\rightarrow\lambda_{i}=0,\eta_{i}=C,\\\
\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}=R^{2}\rightarrow
0<\lambda_{i}<C,0<\eta_{i}<C,\\\
\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}>R^{2}\rightarrow\lambda_{i}=C,\eta_{i}=0.\end{array}\right.$
(46)

It is not difficult to observe that the CSI samples located in the interior of
sphere has no contribution on the construction of the center. We denote the
set of these samples as $\mathcal{I}$. Thus, the uncertain CSI samples with
positive $\lambda_{i}$ can be referred to as support vectors. Among the
supports, there are some CSI samples $\bm{\xi}_{i}$ with $0<\lambda_{i}<C$
located exactly on the boundary of the sphere. We call these CSI samples as
boundary support vectors. The rest of them with $\lambda_{i}=C$ are regarded
as outliers. Based on these observations, we give the following definitions

$\displaystyle\mathcal{F}=\\{i\mid\lambda_{i}>0,\forall i\\}\ \textrm{and}\
\mathcal{B}_{v}=\\{i\mid 0<\lambda_{i}<C,\forall i\\}$ (47)

to denote the index sets of all support vectors and boundary support vectors,
respectively. Then, based on (33) and (37), we obtain

$\displaystyle
1=\sum_{i\in\mathcal{I}}\lambda_{i}+\sum_{i\in\mathcal{F}-\mathcal{B}_{v}}\lambda_{i}+\sum_{i\in\mathcal{B}_{v}}\lambda_{i}>n_{out}C,$
(48)

where $n_{out}$ denote the number of outlier samples. Then, $1/(NC)$ can be
considered as the upper bound on the fraction of outliers. For satisfying the
chance constraint of D2D communication, the objective of (31) is to at least
encapsulate $(1-\epsilon)\times 100\%$ percentage of the $N$ CSI samples. At
this case, the fraction of outliers should be smaller than $\epsilon\times
100\%$. Therefore, it is naturally to set $C=1/(\epsilon N)$ to control the
sphere covering the CSI samples with $(1-\epsilon)\times 100\%$ confidence.

Then, the radius of the sphere can be determined as the distance from the
center $\bm{\rho}$ to any boundary support vector $\bm{\xi}_{l}$ in
$\mathcal{B}_{v}$, i.e.,

$\displaystyle R^{2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=\|\phi(\bm{\xi}_{l})-\bm{\rho}\|^{2}=\|\phi(\bm{\xi}_{l})-\sum_{i=1}^{N}\lambda_{i}\bm{\phi}(\bm{\xi}_{i})\|^{2}$
$\displaystyle\\!\\!\\!\\!=K(\bm{\xi}_{l},\bm{\xi}_{l})\\!-\\!2\sum_{i=1}^{N}\lambda_{i}K(\bm{\xi}_{l},\bm{\xi}_{i})\\!+\\!\sum_{i=1}^{N}\sum_{j=1}^{N}\lambda_{i}\lambda_{j}K(\bm{\xi}_{i},\bm{\xi}_{j}).$

Thus, the feasible set of uncertain CSI can be defined as the sphere with
radius $R^{2}$, i.e.

$\displaystyle\mathcal{S}_{\epsilon}(\mathcal{N})=$
$\displaystyle\\!\\!\\!\\!\\!\\!\left\\{\mathbf{g}\mid
K(\mathbf{g},\mathbf{g})-2\sum_{i=1}^{N}\lambda_{i}K(\mathbf{g},\bm{\xi}_{i})\right.$
$\displaystyle\quad\quad\quad\left.+\sum_{i=1}^{N}\sum_{j=1}^{N}\lambda_{i}\lambda_{j}K(\bm{\xi}_{i},\bm{\xi}_{j})\leq
R^{2}\right\\}.$

Indeed, the feasible set $\mathcal{S}_{\epsilon}(\mathcal{N})$ can be
recognized a data-driven uncertainty set. The kernel function in
$\mathcal{S}_{\epsilon}(\mathcal{N})$ plays a key role for ensuring the
application of $\mathcal{S}_{\epsilon}(\mathcal{N})$ in robust optimization.
The commonly used kernel function, such as the Gaussian kernel:
$K(\bm{\xi}_{i},\bm{\xi}_{j})=\exp(-q\|\bm{\xi}_{i}-\bm{\xi}_{j}\|^{2})$,
sigmoid kernel:
$K(\bm{\xi}_{i},\bm{\xi}_{j})=\tanh(a\cdot\bm{\xi}_{i}^{T}\cdot\bm{\xi}_{j}+r)$
and polynomial kernel:
$K(\bm{\xi}_{i},\bm{\xi}_{j})=(\bm{\xi}_{i}^{T}\cdot\bm{\xi}_{j}+1)^{d}$
contains some nonlinear terms [30], which inevitably complicate its
application in power allocation. Hence, when designing the kernel function, it
is imperative to consider that the designed kernel function could provide a
convex expression of $\mathcal{S}_{\epsilon}(\mathcal{N})$ and preserve the
convexity of the dual problem in (35). Based on these motivations, the
weighted generalized intersection kernel (WGIK) [28] is employed for the
robust optimization

$\displaystyle
K(\bm{\xi}_{i},\bm{\xi}_{j})=\sum_{k=1}^{2}\Xi_{k}-\|\mathbf{Q}(\bm{\xi}_{i}-\bm{\xi}_{j})\|_{1},$
(51)

where $\mathbf{Q}\in\mathbb{R}^{2\times 2}$ is a weighted matrix, $\Xi_{k}$
represents the interval width. In these parameters, $\mathbf{Q}$ can be
constructed as $\mathbf{Q}=\bm{\Sigma}^{-\frac{1}{2}}$, where $\bm{\Sigma}$ is
the covariance information from the uncertain CSI samples and it is computed
as

$\displaystyle\bm{\Sigma}\\!=\\!\frac{1}{N\\!-\\!1}\left[\sum_{i=1}^{N\\!-\\!1}\bm{\xi}_{i}\bm{\xi}_{i}^{T}\\!-\\!\frac{1}{N\\!-\\!1}\left(\sum_{i=1}^{N}\bm{\xi}_{i}\right)\left(\sum_{i=1}^{N}\bm{\xi}_{i}\right)^{T}\right].$
(52)

Since the dual problem in (35) is convex only when the kernel matrix
$\mathbf{K}=\\{K(\bm{\xi}_{i},\bm{\xi}_{j})\\}\succ\mathbf{0}$, $\Xi_{k}$ is
chosen with a simple criterion as

$\displaystyle\Pi_{k}>\max_{1\leq i\leq
N}\mathbf{q}_{k}^{T}\bm{\xi}_{i}-\min_{1\leq i\leq
N}\mathbf{q}_{k}^{T}\bm{\xi}_{i}.$ (53)

where $\mathbf{q}_{k}$ is the column vector of $\mathbf{Q}$. Based on the WGIK
model, we notice that $K(\bm{\xi}_{i},\bm{\xi}_{i})$ is a constant for any $i$
which satisfies $1\leq i\leq N$.

By substituting (39) into (40), the uncertainty set can be further computed as
follows

$\displaystyle\mathcal{S}_{\epsilon}(\mathcal{N})\\!=\\!\left\\{\\!\mathbf{g}\\!\mid\sum_{i=1}^{N}\lambda_{i}K(\mathbf{g},\bm{\xi}_{i})\\!\geq\\!\sum_{i=1}^{N}\lambda_{i}K(\bm{\xi}_{l},\bm{\xi}_{i}),l\\!\in\\!\mathcal{B}_{v}\right\\}.$
(54)

Recalled the discussion in Eq. (36), when sample $i$ located in the interior
of the sphere, there exists $\lambda_{i}=0$. It indicates that only the CSI
samples in the exterior of the sphere can contribute to the construction of
uncertainty set. Thus, (44) can be further simplified as

$\displaystyle\mathcal{S}_{\epsilon}(\mathcal{N})\\!=\\!\left\\{\\!\mathbf{g}\\!\mid\sum_{i\in\mathcal{F}}\lambda_{i}K(\mathbf{g},\bm{\xi}_{i})\\!\geq\\!\sum_{i\in\mathcal{F}}\lambda_{i}K(\bm{\xi}_{l},\bm{\xi}_{i}),l\\!\in\\!\mathcal{B}_{v}\right\\}.$
(55)

Then, by substituting Eq. (41) into Eq. (45), the explicit expression of data-
driven uncertainty set is represented as

$\displaystyle\mathcal{S}_{\epsilon}(\mathcal{N})\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\\{\\!\mathbf{g}\\!\mid\\!\sum_{i\in\mathcal{F}}\\!\lambda_{i}\|\\!\mathbf{Q}(\mathbf{g}\\!-\\!\bm{\xi}_{i})\|_{1}\\!\\!\leq\\!\\!\sum_{i\in\mathcal{F}}\\!\lambda_{i}\|\\!\mathbf{Q}(\bm{\xi}_{l}\\!\\!-\\!\bm{\xi}_{i})\|_{1},l\\!\in\\!\mathcal{B}_{v}\right\\}.$

It is not difficult to observe that the right hand side of (46) is a constant.
Then, let
$\varrho=\sum_{i\in\mathcal{F}}\lambda_{i}\|\mathbf{Q}(\bm{\xi}_{l}-\bm{\xi}_{i})\|_{1},l\in\mathcal{B}_{v}$
and introduce auxiliary variable
$\mathbf{V}=[\mathbf{v}_{1},\mathbf{v}_{1},\cdots,\mathbf{v}_{N}]$, where
$\mathbf{v}_{i}\in\mathbb{R}^{2}$, the uncertainty set
$\mathcal{S}_{v}(\mathcal{N})$ can be further rewritten as

$\displaystyle\mathcal{S}_{\epsilon}(\mathcal{N})=\left\\{\mathbf{g}\left|\begin{array}[]{c}\exists\mathbf{v}_{i},i\in\mathcal{F}\
\textrm{s.t.}\\\
\sum_{i\in\mathcal{F}}(\lambda_{i}\cdot\mathbf{v}_{i}^{T}\mathbf{1}_{2\times
1})\leq\varrho\\\
-\mathbf{v}_{i}\leq\mathbf{Q}(\mathbf{g}-\bm{\xi}_{i})\leq\mathbf{v}_{i},i\in\mathcal{F}\end{array}\right.\right\\}.$
(60)

Figure 3: Transformation of uncertainty set from high-dimensional feature
space to two-dimensional CSI space. (a) Sphere in high-dimensional feature
space. (b) Polytope in the two-dimensional CSI space.

Eq. (47) shows that the sphere in high-dimensional feature space is
transformed into a polytope in two-dimensional CSI space. Hence, it will
provide computational advantages for the power allocation problem. This is
shown in Fig. 3. The obtained polytope is not symmetric and radial, moreover
it can cover the uncertain CSI without surplus. Besides, the constraints
$-\mathbf{v}_{i}\leq\mathbf{Q}(\mathbf{g}-\bm{\xi}_{i})\leq\mathbf{v}_{i},i\in\mathcal{F}$
constitute the edges of a polytope. The number of edges depends on the number
of support vectors in $\mathcal{F}$.

Then, under the polytope uncertain set, the chance QoS constraint of D2D can
be rewritten as the following linear programming (LP)

$\displaystyle\min_{\mathbf{g},\\{\mathbf{v}_{i}\\}}$
$\displaystyle\mathbf{p}^{T}\mathbf{g}$ (61a) s.t.
$\displaystyle\sum_{i\in\mathcal{F}}(\lambda_{i}\cdot\mathbf{v}_{i}^{T}\mathbf{1}_{2\times
1})\leq\varrho,$ (61c)
$\displaystyle-\mathbf{v}_{i}\leq\mathbf{Q}(\mathbf{g}-\bm{\xi}_{i})\leq\mathbf{v}_{i},i\in\mathcal{F}.$

By introducing the Lagrange multipliers $\kappa$, $\bm{\varphi}_{i}$ and
$\bm{\omega}_{i}$, the dual problem of (48) can be given as

$\displaystyle\max\limits_{\kappa,\\{\bm{\varphi}_{i}\\},\\{\bm{\omega}_{i}\\}}$
$\displaystyle\sum_{i\in\mathcal{F}}(\bm{\omega}_{i}-\bm{\varphi}_{i})^{T}\mathbf{Q}\bm{\xi}_{i}-\varrho\kappa$
(62a) s.t. $\displaystyle\sum_{i\in
F}(\bm{\omega}_{i}-\bm{\varphi}_{i})^{T}\mathbf{Q}-\mathbf{p}=\mathbf{0},$
(62d)
$\displaystyle\bm{\omega}_{i}+\bm{\varphi}_{i}-\lambda_{i}\cdot\kappa\cdot\mathbf{1}=\mathbf{0},\forall
i\in\mathcal{F},$ $\displaystyle\kappa\geq
0,\bm{\varphi}_{i},\bm{\omega}_{i}\in\mathbb{R}^{2}_{+}.$

In this case, the intractable chance constraint in (5c) can be replaced by the
following linear constraint

$\displaystyle\left\\{\begin{array}[]{l}\sum_{i\in\mathcal{F}}(\bm{\omega}_{i}-\bm{\varphi}_{i})^{T}\mathbf{Q}\bm{\xi}_{i}-\varrho\kappa\geq\gamma_{min}^{d},\\\
\sum_{i\in\mathcal{F}}(\bm{\omega}_{i}-\bm{\varphi}_{i})^{T}\mathbf{Q}-\mathbf{p}=\mathbf{0},\\\
\bm{\omega}_{i}+\bm{\varphi}_{i}-\lambda_{i}\cdot\kappa\cdot\mathbf{1}=\mathbf{0},\forall
i\in\mathcal{F},\\\ \kappa\geq
0,\bm{\varphi}_{i},\bm{\omega}_{i}\in\mathbb{R}^{2}_{+}.\end{array}\right.$
(67)

Compared with the chance constraint, the deterministic constraint is tractable
but the introduced auxiliary variables will increase the complexity of the
problem. In words, the complexity increases with the number of support
vectors. Recalled the statements below Eq. (38), $\epsilon$ is the parameter
to control the size of the sphere in high-dimensional features space. Namely,
the obtained sphere will at least encapsulate $(1-\epsilon)\times 100\%$
percentage of the training samples. Thus, there are roughly $\epsilon N$
samples located in the exterior of the sphere, i.e.,
$\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}\geq R^{2}$, which is referred to as
support supports. Therefore, the number of auxiliary variables, i.e.
$\bm{\upsilon}_{i}$ and $\bm{\omega}_{i}$, is at least $\lceil
2N\epsilon\rceil$. This indicates that the parameter $\epsilon$ is also can be
used to manipulate the complexity of the induced robust optimization.

###  IV-B Power Allocation

After clustering the uncertain CSI, the intractable chance QoS constraint of
D2D can be transformed into a combination of linear constraints. Then, the
power allocation problem can be reformulated as

$\displaystyle\max\limits_{p_{c},p_{d}}$ $\displaystyle
B\log_{2}\left(1+\frac{p_{c}g_{c}}{\sigma^{2}+p_{d}g_{d,B}}\right)$ (68) s.t.
$\displaystyle\textrm{(50)},\textrm{(5b)},\textrm{(5d)},\textrm{(5e)}.$

Evidently, the problem in (51) is similar to the one in (29). Thus, the same
bisection search method as illustrated in Algorithm 1 can be deployed for
searching the optimal solution. To sum up, we propose the following standard
operating procedure for the SVC-based power allocation.

  * •

Collect a set $\mathcal{N}$ of samples of the imperfect CSI;

  * •

Calculate the covariance matrix $\bm{\Sigma}$ based on the CSI sample set
$\mathcal{N}$ and obtain the weight matrix
$\mathbf{Q}=\bm{\Sigma}^{\frac{1}{2}}$;

  * •

Compute the interval width $\Xi_{k}$ according to the proposed criterion in
(43);

  * •

Obtain the kernel trick based on the WGIK in (41);

  * •

Solve the disciplined QP in (35) for determining the solution $\bm{\lambda}$
and the indices of support vectors;

  * •

Use the support vectors $\\{\bm{\xi}_{i},i\in\mathcal{F}\\}$ and the
corresponding Lagrange multipliers $\\{\lambda_{i},i\in\mathcal{F}\\}$ to
construct the robust counterpart in (50), and use it to replace the chance
constraint in (6c) to derive the convex constraint set.

  * •

Solve the induced power allocation problem by using Algorithm 1.

##  V Simulation Results

In this section, we conduct the simulation to verify the performance of the
proposed approaches. In the simulation, we consider a single cell simulation
model as the one in Fig. 1. The distances between different nodes are set as
$D_{c}=42$m, $D_{c,d}=50.16$m, $D_{d,B}=85$m and $D_{d}=44$m, respectively.
The carrier bandwidth is set as $B=10$MHz. The large-scale channel gain in the
cellular network is composed of shadow fading and path loss, where the
shadowing standard deviation is set as 8dB and the path loss is modeled as the
macrocell propagation model $128.1+37.6\log_{10}d(\textrm{km})$. On the
contrary, the path loss for D2D communication is set as the WINNER+B1 [31]
model and the shadowing standard deviation is 4dB. Besides, all of the small-
scale channel gains are modeled as Rayleigh fading and the noise power is set
as $-134$dBm. In the simulation, we consider two types of uncertainties under
different distributions: (i) exponential distribution truncated on a polytope;
(ii) bivariate Gaussian. The sample number for channel training is $N=1000$.
The resource allocation solutions are tested in a test set with 10000 samples
of the uncertain CSI. The other parameters are given separately for each
experiment. We compare our proposed approaches with the Non-Robust policy as a
baseline, where the power allocation problem is solved based on the average
channel gain $\bar{\mathbf{g}}$. Besides that, we also develop a Quantile-SVC
approach in the following as a baseline to illustrate the effectiveness of the
proposed SVC-based robust optimization.

(a) Symmetrical-geometry-based uncertainty set under truncated exponential.

(b) Symmetrical-geometry-based uncertainty set under gaussian.

(c) Quantile-SVC approach under truncated exponential.

(d) SVC under truncated exponential.

(e) Quantile-SVC approach under gaussian.

(f) SVC under under gaussian.

Figure 4: Learning results of the uncertainty sets, where the solid line and
the dotted line correspond to $\epsilon=0.01$ and $\epsilon=0.05$ respectively

###  V-A Quantile-SVC Approach

The Quantile-SVC approach adopts the hard margin to seek the smallest sphere
to enclose the data in high-dimensional features space, which can be
formulated as the following optimization problem

$\displaystyle\min\limits_{R,\bm{\rho}}$ $\displaystyle R^{2}$ (69) s.t.
$\displaystyle\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}\leq R^{2},i=1,\cdots,N.$

Using the similar method in Section IV, the dual problem is computed as

$\displaystyle\min\limits_{\bm{\lambda}}$
$\displaystyle\sum_{i=1}^{N}\sum_{j=1}^{N}\lambda_{i}\lambda_{j}K(\bm{\xi}_{i},\bm{\xi}_{j})-\sum_{i=1}^{N}\lambda_{i}K(\bm{\xi}_{i},\bm{\xi}_{i})$
(70) s.t. $\displaystyle\sum_{i=1}^{N}\lambda_{i}=1,\lambda_{i}\geq 0.$

According to the KKT conditions and complementary slackness, when the $i$-th
sample in the interior of the sphere
$\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}<R^{2}$, we must have $\lambda_{i}=0$. On
the contrary, when the $i$-th sample located on the boundary of the sphere
$\|\phi(\bm{\xi}_{i})-\bm{\rho}\|^{2}=R^{2}$, there must exist
$\lambda_{i}>0$. Note that only the sample $\bm{\xi}_{i}$ with positive
$\lambda_{i}$ can contribute the construction of the center. We refer to this
samples as support vectors and the index set can be expressed as

$\displaystyle\mathcal{S}_{v}=\\{i\mid\lambda_{i}>0,\forall i\\}.$ (71)

Besides, the center of sphere is also computed as a linear combination of the
mapping of all support vectors, i.e.,
$\bm{\rho}=\sum_{i\in\mathcal{S}_{v}}\lambda_{i}\bm{\phi}(\bm{\xi}_{i})$.
Then, we should calibrate the size of the sphere so that it can cover the
uncertain CSI samples in high-dimensional features space with confidence
$1-\epsilon$. For using the quantile estimation method, we define

$\displaystyle t_{svc}(\bm{\xi}_{l})=\|\phi(\bm{\xi}_{l})-\bm{\rho}\|$ (72)

to map the channel sample into real number. Then, the size parameter $R^{2}$
will be chosen as $n_{\epsilon}=\lceil(1-\epsilon)N\rceil$-th value of the
ranked observations $t_{svc}^{(1)}\leq\cdots\leq t_{scv}^{(N)}$ in ascending
order, i.e., $R^{2}=t_{svc}^{(n_{\epsilon})}(\bm{\xi}_{l_{\epsilon}})$. After
that, the feasible set of uncertain CSI can be defined as the sphere with
radius $R^{2}$, i.e.

$\displaystyle\mathcal{S}_{\epsilon}(\mathcal{N})=\\{\mathbf{g}\mid\phi(\mathbf{g})-\bm{\rho}\leq
R^{2}\\}.$ (73)

Then, using the similar process as shown in Section IV, the robust counterpart
of chance constraint based on Quantile-SVC approach can be computed as the
following linear constraint

$\displaystyle\left\\{\begin{array}[]{l}\sum_{i\in\mathcal{S}_{v}}(\bm{\omega}_{i}-\bm{\varphi}_{i})^{T}\mathbf{Q}\bm{\xi}_{i}-\varrho\kappa\geq\gamma_{min}^{d},\\\
\sum_{i\in\mathcal{S}_{v}}(\bm{\omega}_{i}-\bm{\varphi}_{i})^{T}\mathbf{Q}-\mathbf{p}=\mathbf{0},\\\
\bm{\omega}_{i}+\bm{\varphi}_{i}-\lambda_{i}\cdot\kappa\cdot\mathbf{1}=\mathbf{0},\forall
i\in\mathcal{S}_{v},\\\ \kappa\geq
0,\bm{\varphi}_{i},\bm{\omega}_{i}\in\mathbb{R}^{2}_{+},\end{array}\right.$
(78)

where
$\varrho=\sum_{i\in\mathcal{S}_{v}}\lambda_{i}\|\mathbf{Q}(\bm{\xi}_{l_{\epsilon}}-\bm{\xi}_{i})\|_{1}$.
Similar to the robust counterpart in Eq. (47), the number of edges of the
obtained polytope in (57) also depends on the number of support vectors in
$\mathcal{S}_{v}$. However, it can be known from the computation of the dual
problem in (57) that the number of support vectors has nothing to do with the
outage probability requirement $\epsilon$. That means that the number of edges
of the obtained polytope in (57) is the same for all possible values of
$\epsilon$. From the quantile estimation in (55), we knows that the change of
$\epsilon$ leads to the case that different CSI sample
$\bm{\xi}_{l_{\epsilon}}$ will be selected to compute the size of the sphere
in high-dimensional features space. As follows, the parameter
$\varrho=\sum_{i\in\mathcal{S}_{v}}\lambda_{i}\|\mathbf{Q}(\bm{\xi}_{l_{\epsilon}}-\bm{\xi}_{i})\|_{1}$,
which can be considered the size parameter of the polytope, will change so
that the obtained polytope can cover the uncertain CSI with confidence
$1-\epsilon$.

###  V-B Experiment Results

In the first experiment, we model the uncertain CSI using the proposed
learning approaches with different outage probabilities, and the graphical
results are shown in Fig. 4. We observe from Figs. 4(a) and (b) that for
covering the uncertain CSI with the specific confidence level, the
sysmmetrical-geometry-based uncertainty set will result in large superfluous
coverage. This potential leads to over-conservative power allocation
solutions. However, it is shown in Figs. 4(c)-(f) that both Quantile-SVC and
SVC yield the convex and asymmetric uncertainty sets which can compactly cover
the uncertain CSI without too much superfluous coverage. This verifies the
effectiveness of the ML technique in clustering the uncertain CSI samples.
From these figures, we also observe that when the outage probability
$\epsilon$ is set as 0.01, regardless of the distribution of the uncertain
CSI, the polytope shapes learned by Quantile-SVC and SVC are almost the same.
However, with the increase of outage probability, the polytope learned by
Quantile-SVC approach will shrink proportionally without changing its shape.
This verifies our analysis about the robust counterpart in (57). For the SVC
approach, when the outage probability increases, the size of the enclosing
envelop decreases, while the number of edges of the uncertainty set increases.
As a result, the rim of the polytope tends to be more and more smooth. Thus,
more samples are considered as outliers residing outside the uncertainty set
and the induced polytope becomes less conservative.

In Figs. 5 and 6, we evaluate the convergence of the bisection search-based
power allocation algorithm. The figures show that the proposed algorithm
converges within only at most 17 iterations. This verifies the low complexity
of the proposed algorithm. We also observe from the figures that no matter
under which robust optimization approach, the transmit power of CUE can reach
the maximum value. This verifies the solution in Lemma 1. Besides that, Fig. 6
shows that the DUE transmit powers under Box, Ellipsoidal and Polytope
approaches are much larger than the ones under SVC, Quantile-SVC and Non-
Robust. That is because all of the sysmmetrical-geometry-based uncertainty
sets will result in superfluous coverage. For satisfying the chance D2D QoS
constraint, the DUE with the deployment of these approaches must consume more
transmit power than others. Benefiting from the advanced uncertainty set
learning method, the DUE with SVC and Quantile-SVC can satisfy the QoS
constraint with smaller transmit powers. Although the Non-Robust approach
achieves the smallest transmit power, the chance D2D QoS requirement is not
necessarily guaranteed.

Figure 5: Convergence of the bisection search-based algorithm, assuming
$P_{max}^{c}=P_{max}^{d}=20$dBm, QoS requirements $\gamma_{min}^{c}=5$,
$\gamma_{min}^{d}=0.1$ and outage probability $\epsilon=0.05$. Figure 6:
Convergence of the bisection search-based algorithm, assuming
$P_{max}^{c}=P_{max}^{d}=20$dBm, QoS requirements $\gamma_{min}^{c}=5$,
$\gamma_{min}^{d}=0.1$ and outage probability $\epsilon=0.05$. Figure 7: DUE
SINR cumulative distributions under Gaussian uncertainties, assuming
$P_{max}^{c}=P_{max}^{d}=20$dBm, QoS requirements $\gamma_{min}^{c}=5$,
$\gamma_{min}^{d}=0.1$ and outage probability $\epsilon=0.05$. Figure 8: DUE
SINR cumulative distributions under exponential uncertainties, assuming
$P_{max}^{c}=P_{max}^{d}=20$dBm, QoS requirements $\gamma_{min}^{c}=5$,
$\gamma_{min}^{d}=0.1$ and outage probability $\epsilon=0.05$.

Figs. 7-9 illustrate the DUE SINR cumulative distributions for different
robust optimization approaches under the test set. It is not difficult to
understand that for all of the curves, the point $(0.1,F(0.1))$ can be
considered as the outage probability of D2D communication. Then, we observe
from the figures that no matter what the distribution of the channel error is,
the outage probability of Non-robust approach is almost larger than 0.5. Such
terrible performance is very dangerous for many application scenarios. For
example, the reliability requirement for D2D-enabled vehicular communications
at the vehicle platooning use case can reach up to 99% [22]. On the contrary,
by substituting the chance constraint as the proposed robust counterparts, all
of the robust optimization approaches achieve satisfactory outage performance.
However, we should note that the achieved SINR of Box, Polytope and
Ellipsoidal is almost larger than 0.3 for the Gaussian uncertainties and 0.4
for the exponential uncertainties. This performance is too conservative, and
will lead to the waste of wireless resources. Thus, by using the ML method on
learning the uncertainty set, the conservatism can be overcame very well. We
also observe from the figures that the achieved SINR under Quantile-SVC is
larger than the one under SVC. This shows that Quantile-SVC is more
conservative than SVC. This phenomenon can be explained in two aspects. First,
at different outage probabilities, the polytope specified by the robust
counterpart of Quantile-SVC can only shrink proportionally. However, the
envelope of the polytope specified by SVC tends to be more and more smooth
when the outage probability increases. Thus, SVC shows higher flexibility than
Quantile-SVC, so it has better ability for overcoming the conservatism.
Second, we note that the size of uncertainty set in Quantile-SVC is estimated
by using the quantile estimation method in high-dimensional features space.
Recalled the definition of $\phi(\bm{\xi})$, the uncertain CSI from two-
dimensional space is mapped into to three-dimensional space by the nonlinear
mapping. Therefore, the order about the radius of sphere in high-dimensional
space may not be applicable in the two-dimensional CSI space. This lead to the
result that the polytope in two-dimensional CSI space cannot cover the
uncertain CSI samples with confidence level as $1-\epsilon$, so that it is
more conservative than SVC.

Figure 9: CUE throughput versus tolerable outage probability, assuming
$P_{max}^{c}=P_{max}^{d}=20$dBm and QoS requirements $\gamma_{min}^{c}=5$,
$\gamma_{min}^{d}=0.1$. Figure 10: DUE SINR versus tolerable outage
probability, assuming $P_{max}^{c}=P_{max}^{d}=20$dBm and QoS requirements
$\gamma_{min}^{c}=5$, $\gamma_{min}^{d}=0.1$.

In Figs. 9 and 10, we illustrate the CUE throughput and the DUE SINR versus
the tolerable outage probability. Because all of the robust optimization
approaches show the similar performance under different uncertainty
distribution, thus this and the following experiment only exhibit the results
under truncated exponential distribution. In addition, both the CUE throughput
and the DUE SINR under Non-Robust approach are much worse than the ones under
other optimization approaches and it is difficult to exhibit them in the same
figure. Thus, the performance of Non-Robust approach will not be shown any
more in this and the following experiments. In Figs. 9 and 10, we observe that
when the tolerable outage probability is small, the symmetrical-geometry-based
robust optimization approaches obtain the smaller CUE throughput than others.
This is because these approaches are very conservative. They greedily consume
a lot of power to guarantee the chance constraint of D2D. As a result, the
obtained CUE throughput is smaller than others. On the contrary, the ML-based
robust optimization approaches overcome the conservatism very well so that
they achieve higher CUE throughput whilst ensuring the chance QoS constraint
of D2D. The figures also show that with the increase of tolerable outage
probability, the DUE SINR under all of the robust optimization approaches
decreases while the CUE throughput increases. This is because the size of
uncertainty set decreases with the increase of tolerable outage probability,
so the DUE no longer needs to consume much power for guaranteeing the chance
QoS requirement. As a result, the obtained CUE throughput increases.

Figure 11: CUE throughput versus DUE QoS requirement, assuming
$P_{max}^{c}=P_{max}^{d}=20$dBm, CUE QoS requirement $\gamma_{min}^{c}=5$ and
outage probability $\epsilon=0.05$. Figure 12: DUE SINR versus DUE QoS
requirement, assuming $P_{max}^{c}=P_{max}^{d}=20$dBm, CUE QoS requirement
$\gamma_{min}^{c}=5$ and outage probability $\epsilon=0.05$.

Figs. 11 and 12 illustrate the CUE throughput and the DUE SINR versus the DUE
QoS requirement $\gamma_{min}^{d}$. We observe from these figures that, with
the increase of DUE QoS requirement, the achieved DUE SINR increases while the
CUE throughput decreases. This is because the DUE needs to consume more
transmit power to support the increased DUE SINR. As a result, its
interference to the CUE increases so that the CUE throughput decreases. The
figures also show that at under any DUE QoS requirement, the ML-based robust
optimization approach can overcome the conservatism of the symmetrical-
geometry-based approaches so that they achieve the larger CUE throughput.
Moreover, due the effectiveness of SVC on the representations of sample
distribution, it always shows better performance on the DUE SINR and achieves
larger CUE throughput than Quantile-SVC. We also observe from the figures that
when the DUE QoS requirement is larger than 1.4, the proposed robust
optimization approaches will collapse one after another. This is because the
total power resources of the network are limited, so they cannot support such
a large D2D QoS requirement. Because the symmetrical-geometry-based approaches
consume too much power for protecting the chance constraint, so that they
collapse earlier than the others. On the contrary, both SVC and Quantile-SVC
can support a larger DUE QoS requirement. This illustrates the effectiveness
of the ML-based robust optimization approaches.

Figure 13: CUE throughput versus DUE transmit power, assuming
$P_{max}^{c}=20$dBm, CUE QoS requirement $\gamma_{min}^{c}=5$, DUE QoS
requirement $\gamma_{min}^{c}=1$ and outage probability $\epsilon=0.05$.
Figure 14: DUE SINR versus DUE transmit power, assuming $P_{max}^{c}=20$dBm,
CUE QoS requirement $\gamma_{min}^{c}=5$, DUE QoS requirement
$\gamma_{min}^{c}=1$ and outage probability $\epsilon=0.05$.

In Figs. 13 and 14, we illustrate the CUE throughput and the DUE SINR versus
the DUE transmit power. In these figures, we observe that when the DUE
transmit power is small, all of the robust optimization approaches cannot find
the feasible solution for the power allocation problem. With the increase of
DUE transmit power, the ML-based robust optimization algorithms revive first,
followed by the symmetrical-geometry-based approaches. This is because the ML-
based robust optimization approaches can overcome the conservatism of the
symmetrical-geometry-based approaches, so they need the smaller transmit power
than others for finding the feasible solution to satisfy the D2D QoS
requirement. Moreover, we also observe from the figures that although the
Quantile-SVC approach can revive together with SVC, its CUE throughput
performance is worsen than SVC. Figs. 15 and 16 plot the CUE throughput and
DUE SINR versus the channel estimation error coefficient. As the statements of
channel model in Section II, with the decrease of channel estimation error
coefficient, the channel uncertainty gets more and more larger. Thus, the DUE
need to allocate more transmit power to get the larger SINR for satisfying the
chance QoS constraint. As a result, as shown in Fig. 16, the achieved DUE SINR
of all approaches increases. Undoubtedly, the increased DUE transmit power
will produce serious interference to CUE. Thus, we observe from Fig. 15 that
the CUE throughput decreases with the decrease of channel estimation error
coefficient. Besides, Figs. 15 and 16 illustrate the similar phenomenon as
Figs. 11 and 12. With the increase of channel uncertainty, the robust
optimization approaches will collapse one after another. However, benefiting
from the effectiveness of the ML-based robust optimization on representation
of the uncertainty distribution, both SVC and Quantile-SVC approaches can deal
with larger channel uncertainty than the symmetrical-geometry-based robust
optimization approaches. This verifies the effectiveness of the proposed ML-
based robust optimization approaches again from another perspective.

Figure 15: CUE throughput versus channel estimation error coefficient,
assuming $P_{max}^{c}=P_{max}^{d}=20$dBm, CUE QoS requirement
$\gamma_{min}^{c}=5$, DUE QoS requirement $\gamma_{min}^{c}=1.5$ and outage
probability $\epsilon=0.05$. Figure 16: DUE SINR versus channel estimation
error coefficient, assuming $P_{max}^{c}=P_{max}^{d}=20$dBm, CUE QoS
requirement $\gamma_{min}^{c}=5$, DUE QoS requirement $\gamma_{min}^{c}=1.5$
and outage probability $\epsilon=0.05$.

##  VI Conclusions

In this paper, we studied the resource allocation in D2D underlaying cellular
network. The problem was formulated as maximizing the CUE throughput under the
channel uncertainties whilst guaranteeing a minimum SINR requirement
constraint for D2D. We proposed a robust resource allocation framework for
solving the highly intractable chance constraint about D2D service
requirement. Then, we modeled the uncertain CSI into polytope, ellipsoidal and
box and derived the robust counterparts of the chance constraint under these
uncertainty sets. To overcome their conservatism, we developed a support
vector clustering (SVC)-based approach to model uncertain CSI as a compact
convex uncertainty set. Finally, we developed a bisection search-based power
allocation algorithm for solving the resource allocation in D2D underlaying
cellular network with different robust counterparts.

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