# A Novel Bayes’ Theorem for Upper Probabilities

Michele Caprio PRECISE Center
Department of Computer and Information Science
University of Pennsylvania
USA  Yusuf Sale  Institute of Informatics
University of Munich (LMU)
Germany  Munich Center for Machine Learning
Germany  Eyke Hüllermeier  Institute of Informatics
University of Munich (LMU)
Germany  Munich Center for Machine Learning
Germany  Insup Lee  PRECISE Center
Department of Computer and Information Science
University of Pennsylvania
USA

###### Abstract

In their seminal 1990 paper, Wasserman and Kadane establish an upper bound for
the Bayes’ posterior probability of a measurable set $A$, when the prior lies
in a class of probability measures $\mathcal{P}$ and the likelihood is
precise. They also give a sufficient condition for such upper bound to hold
with equality. In this paper, we introduce a generalization of their result by
additionally addressing uncertainty related to the likelihood. We give an
upper bound for the posterior probability when both the prior and the
likelihood belong to a set of probabilities. Furthermore, we give a sufficient
condition for this upper bound to become an equality. This result is
interesting on its own, and has the potential of being applied to various
fields of engineering (e.g. model predictive control), machine learning, and
artificial intelligence.

##  1 Introduction

Bayes’ rule (BR) is arguably the best-known mechanism to update subjective
beliefs. It prescribes the agent to elicit a prior distribution that
encapsulates their initial opinion, and to come up with a likelihood that
describes the data generating process. Combining prior and likelihood via BR
produces the posterior distribution, which captures the agent’s revised
opinion in light of the collected data.

But what happens if the agent is not able to specify a single prior
distribution? This may occur if they face ambiguity [Ellsberg, 1961, Gilboa
and Marinacci, 2013]. In [Walley, 1991, Section 1.1.4] and Caprio and Gong
[2023], the authors point out that missing information and bounded rationality
may prevent the agent from assessing probabilities precisely in practice, even
if doing so is possible in principle. This may be due to the lack of
information on how likely events of interest are, lack of computational time
or ability, or because it is extremely difficult to analyze a complex body of
evidence. Similarly, the agent may face difficulties in gauging the data
generating process, so specifying a single likelihood may become a challenging
task.

The notion of ambiguity is strictly related with that of epistemic uncertainty
in machine learning (ML) and artificial intelligence (AI). Let us illustrate
this clearly by borrowing concepts from [Caprio et al., 2023, Section 3.2].
Epistemic uncertainty (EU) corresponds to reducible uncertainty caused by a
lack of knowledge about the best model. Notice how, in the precise case – that
is, when the agent specifies a single distribution – EU is absent. In many
applications, a single probability measure is only able to capture the idea of
irreducible uncertainty, since it represents a case in which the agent knows
exactly the true data generating process, and the prior probability that
perfectly encapsulates their initial beliefs. This is a well-studied property
of sets of probabilities [Hüllermeier and Waegeman, 2021, Page 458]. Due to
the increasing relevance of reliable and trustworthy ML and AI applications,
effective uncertainty representation and quantification have become vital
research areas [Kendall and Gal, 2017, Smith and Gal, 2018, Depeweg et al.,
2018, Kapoor et al., 2022, Sale et al., 2023, Wimmer et al., 2023]. Thus,
theoretic underpinnings of imprecise probability theories emerge as a valuable
methodology for improving the representation and quantification of
uncertainties. Adopting concepts like (convex) sets of probabilities and upper
and lower probabilities foster a more sophisticated and fine-tuned
articulation of uncertainty.

We remark that EU should not be confused with the concept of epistemic
probability [de Finetti, 1974, 1975, Walley, 1991]. In the subjective
probability literature, epistemic probability can be captured by a single
distribution. Its best definition can be found in [Walley, 1991, Sections
1.3.2 and 2.11.2]. There, the author specifies how epistemic probabilities
model logical or psychological degrees of partial belief of the agent. We
remark, though, how de Finetti and Walley work with finitely additive
probabilities, while in this paper we use countably additive probabilities.

The field of imprecise probabilities [Augustin, Thomas and Coolen, Frank P.A.
and De Cooman, Gert and Troffaes, Matthias C.M., 2014, Walley, 1991], and in
particular the classic paper by Wasserman and Kadane [1990], and successive
works like Cozman [2000], Epstein and Wang [1994], Giacomini and Kitagawa
[2021], Klibanoff and Hanany [2007], study the problem of an agent updating
their beliefs using BR in the presence of ambiguity. Our paper belongs to this
body of work. Our main result, Theorem 1, generalizes the theorem in
[Wasserman and Kadane, 1990, Section 2] to the case where the agent faces
ambiguity on both what prior and what likelihood to choose to model the
phenomenon of interest. We find the upper posterior, that is, the “upper
bound” to the set of posteriors, using only the upper prior and the upper
likelihood. Thanks to the conjugacy property of upper probabilities,
introduced in the next section, we derive the lower posterior, that is the
“lower bound” to the set of posteriors. We also give a necessary condition for
the bound to hold with equality. In addition, we hint at possible
applications, in particular in the field of model predictive control, a method
of process control that is used to control a process while satisfying a set of
constraints Rakovi and Levine [2018].

The paper is divided as follows. Section 2.1 introduces the concepts that are
needed to understand our result. In Section 2.2 we present the main theorem,
and conclude our work Section 3. We prove our results in Section 4.

##  2 Bayes’ Theorem for Upper Probabilities

###  2.1 Preliminaries

In this section, we introduce the background notions from the IP literature
[Augustin, Thomas and Coolen, Frank P.A. and De Cooman, Gert and Troffaes,
Matthias C.M., 2014, Troffaes and de Cooman, 2014, Walley, 1991] that are
needed to understand our main results.

Call $\Delta(\Omega,\mathcal{F})$ the space of (countably additive)
probability measures on a generic measurable space $(\Omega,\mathcal{F})$.
Pick a generic set $\mathcal{P}\subset\Delta(\Omega,\mathcal{F})$. We denote
by $\overline{P}$ the upper probability associated with $\mathcal{P}$, that
is, $\overline{P}(A)=\sup_{P\in\mathcal{P}}P(A)$, for all $A\in\mathcal{F}$.
Its conjugate is called lower probability,
$\underline{P}(A)=1-\overline{P}(A^{c})=\inf_{P\in\mathcal{P}}P(A)$, for all
$A\in\mathcal{F}$. Because of the conjugacy property, in the remainder of this
document we focus on upper probabilities only.

We say that upper probability $\overline{P}$ is concave or $2$-alternating if
$\overline{P}(A\cup B)\leq\overline{P}(A)+\overline{P}(B)-\overline{P}(A\cap
B)$, for all $A,B\in\mathcal{F}$. Upper probability $\overline{P}$ is
compatible [Gong and Meng, 2021] with the set

$\displaystyle\text{core}(\overline{P}):=\\{$ $\displaystyle
P\in\Delta(\Omega,\mathcal{F}):P(A)\leq\overline{P}(A),\forall
A\in\mathcal{F}\\}$ $\displaystyle=\\{$ $\displaystyle
P\in\Delta(\Omega,\mathcal{F}):\overline{P}(A)\geq P(A)\geq\underline{P}(A),$
$\displaystyle\forall A\in\mathcal{F}\\}$ (1)

where (1) is a characterization [Cerreia-Vioglio et al., 2015, Page 3389]. The
core is the set of all (countably additive) probability measures on $\Omega$
that are set-wise dominated by $\overline{P}$. It is convex: it is immediate
to see that if $P$ and $Q$ are dominated by $\overline{P}$, then $\gamma P$
and $(1-\gamma)Q$ are dominated by $\gamma\overline{P}$ and
$(1-\gamma)\overline{P}$, respectively, for all $\gamma\in[0,1]$. In turn,
$\gamma P+(1-\gamma)Q$ is dominated by $\overline{P}$, thus giving the desired
convex property. In addition, throughout the present work, we assume that
$\text{core}(\overline{P})$ is nonempty and weak⋆-closed.111Recall that in the
weak⋆ topology, a net $(P_{\alpha})_{\alpha\in I}$ converges to $P$ if and
only if $P_{\alpha}(A)\rightarrow P(A)$, for all $A\in\mathcal{F}$. See also
results presented in [Walley, 1991, Appendix D3] Then, as a result of [Walley,
1991, Section 3.6.1], it is also weak⋆-compact.

######  Remark 1.

In [Walley, 1991, Section 3.6.1], the author shows that the finitely additive
core is weak⋆-compact. The latter is defined as the set of all finitely
additive probabilities that are set-wise dominated by $\overline{P}$. It is a
superset of the countably additive core in (1). To see this, notice that, in
general, there might well be a probability measure that is set-wise dominated
by $\overline{P}$, but that is merely finitely additive. In fact, there might
even be no countably additive probabilities that are set-wise dominated by
$\overline{P}$. For this reason, we have to assume that the countably additive
core in (1) is nonempty. If we further require that the countably additive
core in (1) is weak⋆-closed, then, this implies that it is a weak⋆-closed
subset of a weak⋆-compact space. By [Rudin, 1976, Theorem 2.35], closed
subsets of compact sets are compact. In turn, we have that if the countably
additive core is weak⋆-closed, it is also weak⋆-compact.

We now present a class of probabilities that (i) is well-studied and used in
robust statistics Huber and Ronchetti [2009], and (ii) is the core of a
concave upper probability. Other classes with similar properties are presented
in [Wasserman and Kadane, 1990, Examples 3-7].

######  Example 1 ($\varepsilon$-contaminated class).

Consider the space $\Delta(\Omega,\mathcal{F})$ of probability measures on a
generic measurable space $(\Omega,\mathcal{F})$, and assume $\Omega$ is
compact. Pick any $P\in\Delta(\Omega,\mathcal{F})$ and any
$\varepsilon\in[0,1]$. Define

$\displaystyle\mathcal{Q}^{\text{co}}:=\\{Q\in\Delta(\Omega,\mathcal{F}):Q(A)=(1-\varepsilon)P(A)+\varepsilon
R(A),$ $\displaystyle\forall A\in\mathcal{F}\text{,
}R\in\Delta(\Omega,\mathcal{F})\\}.$ (2)

Superscript “co” stands for convex and core. $\mathcal{Q}^{\text{co}}$ is the
$\varepsilon$-contaminated class induced by $P$; it was studied in [Wasserman
and Kadane, 1990, Example 3] and references therein. We have that
$\overline{Q}(A)=(1-\varepsilon)P(A)+\varepsilon$, for all
$A\in\mathcal{F}\setminus\\{\emptyset\\}$ and
$\underline{Q}(A)=(1-\varepsilon)P(A)$, for all
$A\in\mathcal{F}\setminus\\{\Omega\\}$. In addition,
$\mathcal{Q}^{\text{co}}=\text{core}(\overline{Q})$, and $\overline{Q}$ is
concave.

The $\varepsilon$-contaminated class is also instrumental for a future
application of Theorem 1 to model predictive control. We will discuss this at
length at the end of section 2.2. There, we also explain why it is important
to account for the ambiguity in the likelihood model in real-world safety-
critical scenarios.

###  2.2 A novel Bayes’ Theorem for Upper Probabilities

Let $(\Theta,\mathcal{F})$ be the measurable parameter space of interest and
$\Delta(\Theta,\mathcal{F})$ the space of (countably additive) probability
measures on $(\Theta,\mathcal{F})$. Let $\mathcal{Y}$ be the set of all
bounded, non-negative, $\mathcal{F}$-measurable functionals on $\Theta$. Call
$\mathscr{D}$ the sample space endowed with sigma-algebra $\mathcal{A}$. That
is, for any random variable $Y$ of interest and all $\theta\in\Theta$,
$Y(\theta)=y\in\mathscr{D}$. Let the agent elicit a set of probabilities
$\mathcal{L}_{\theta}:=\\{P_{\theta}\in\Delta(\mathscr{D},\mathcal{A}):\theta\in\Theta\\}$
on $\mathscr{D}$, parameterized by $\theta\in\Theta$. This captures the
ambiguity faced by the agent in determining the true data generating process
[Ellsberg, 1961, Gilboa and Marinacci, 2013]. We write $P_{\theta}\equiv
P(\cdot\mid\theta)$ for notational convenience. Assume that each
$P_{\theta}\in\mathcal{L}_{\theta}$ has density $L(\theta)=p(y\mid\theta)$
with respect to some sigma-finite dominating measure $\nu$ on
$(\mathscr{D},\mathcal{A})$; this represents the likelihood function for
$\theta$ having observed data $y\in\mathscr{D}$.

Assumption 1. Every density $L$ corresponding to an element $P_{\theta}$ of
$\mathcal{L}_{\theta}$ belongs to $\mathcal{Y}$; that is, every density is
bounded and non-negative.

Assumption 1 is needed mainly for mathematical purposes; as we shall see later
in this section, it can be relaxed. Let the agent specify a set $\mathcal{P}$
of probabilities on $(\Theta,\mathcal{F})$. It represents their (incomplete)
prior knowledge on the elements of $\mathcal{F}$; its elements may be informed
by the collected data, thus giving the analysis an (imprecise) empirical Bayes
flavor [Casella, 1985]. Then, compute
$\overline{P}(A)=\sup_{P\in\mathcal{P}}P(A)$, for all $A\in\mathcal{F}$, and
consider $\mathcal{P}^{\text{co}}:=\text{core}(\overline{P})$, assumed
nonempty and weak⋆-closed. It represents the agent’s initial beliefs. We
assume that every $P\in\mathcal{P}^{\text{co}}$ has density $p$ with respect
to some sigma-finite dominating measure $\mu$ on $(\Theta,\mathcal{F})$, that
is, $p={\text{d}P}/{\text{d}\mu}$. We require the agent’s beliefs to be
represented by the core for two main reasons. The first, mathematical, one is
to ensure that the upper probability is compatible with the belief set. The
second, philosophical, one is the following. In Bayesian statistics, the agent
selects a specific prior to encapsulate their initial beliefs. Berger [1984]
points out how such choice is oftentimes arbitrary, and posits the dogma of
ideal precision (DIP). It states that in any problem there is an ideal
probability model $P_{T}$ which is precise, but which may not be precisely
known. To overcome this shortcoming, the agent should specify a finite
collection $\\{P_{s}\\}_{s\in\mathcal{S}}$ of “plausible” prior distributions,
and compute the posterior for each $P_{s}$. Notice how this corresponds to
selecting a finite number of elements from the core of
$\overline{P}_{\mathcal{S}}$, where
$\overline{P}_{\mathcal{S}}(A)=\sup_{s\in\mathcal{S}}P_{s}(A)$, for all
$A\in\mathcal{F}$. A criticism to the DIP was brought forward by Walley. In
[Walley, 1991, Section 2.10.4.(c)], he claims how given an upper probability
$\overline{P}$, there is no cogent reason for which the agent should choose a
specific $P_{T}$ that is dominated by $\overline{P}$, or – for that matter – a
collection $\\{P_{s}\\}_{s\in\mathcal{S}}$ of “plausible” probabilities.
Because the core considers all (countably additive) probability measures that
are dominated by $\overline{P}$, it is the perfect instrument to address
Walley’s criticism [Caprio and Gong, 2023].

Let the agent compute $\overline{P}_{\theta}$, the upper probability
associated with $\mathcal{L}_{\theta}$, and consider
$\mathcal{L}^{\text{co}}_{\theta}:=\text{core}(\overline{P}_{\theta})$,
assumed nonempty and weak⋆-closed. It represents the set of plausible
likelihoods. As Grünwald and van Ommen [2017] point out, accounting for
ambiguity around the true data generating process is crucial, as Bayesian
inference may suffer from inconsistency issues if carried out using a
misspecified likelihood.

Let

$\mathscr{L}:=\left\\{{L=\frac{\text{d}P_{\theta}}{\text{d}\nu}\text{,
}P_{\theta}\in\mathcal{L}_{\theta}^{\text{co}}}\right\\}\subset\mathcal{Y}$
(3)

be the set of pdf/pmf associated with the elements of
$\mathcal{L}^{\text{co}}_{\theta}$. Let also
$\overline{L}(\theta):=\sup_{L\in\mathscr{L}}L(\theta)$ and
$\underline{L}(\theta):=\inf_{L\in\mathscr{L}}L(\theta)$, for all
$\theta\in\Theta$. Call

$\displaystyle\mathcal{P}^{\text{co}}_{y}:=\Bigg{\\{}P_{y}$
$\displaystyle\in\Delta(\Theta,\mathcal{F}):$
$\displaystyle\frac{\text{d}P_{y}}{\text{d}\mu}$ $\displaystyle=p(\theta\mid
y)=\frac{L(\theta)p(\theta)}{\int_{\Theta}L(\theta)p(\theta)\text{d}\theta}\text{,
}$ $\displaystyle p$ $\displaystyle=\frac{\text{d}P}{\text{d}\mu}\text{,
}P\in\mathcal{P}^{\text{co}}\text{,
}L=\frac{\text{d}P_{\theta}}{\text{d}\nu}\text{,
}P_{\theta}\in\mathcal{L}_{\theta}^{\text{co}}\Bigg{\\}}$

the class of posterior probabilities when the prior is in
$\mathcal{P}^{\text{co}}$ and the likelihood is in
$\mathcal{L}_{\theta}^{\text{co}}$, and let
$\overline{P}_{y}(A)=\sup_{P_{y}\in\mathcal{P}^{\text{co}}_{y}}P_{y}(A)$, for
all $A\in\mathcal{F}$. Then, the following is a generalization of Bayes’
theorem in [Wasserman and Kadane, 1990, Section 2], and is an extension of
[Caprio et al., 2023, Theorem 7]. We prove it in Section 4.

######  Theorem 1 (Bayes’ theorem for upper probabilities).

Suppose $\mathcal{P}^{\text{co}},\mathcal{L}_{\theta}^{\text{co}}$ are
nonempty and weak⋆-closed. Then for all $A\in\mathcal{F}$,

$\displaystyle\overline{P}_{y}(A)$
$\displaystyle\leq\frac{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\mathbf{c}}$
(4)
$\displaystyle\leq\frac{\int_{0}^{\infty}\overline{P}\left(\left\\{{\theta\in\Theta:\overline{L}(\theta)\mathbbm{1}_{A}(\theta)>t}\right\\}\right)\text{d}t}{\mathbf{c}^{\prime}},$
(5)

provided that the ratios are well defined. Here $\mathbbm{1}_{A}$ denotes the
indicator function for $A\in\mathcal{F}$,
$\mathbf{c}:=\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)+\inf_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)$,
and

$\displaystyle\mathbf{c}^{\prime}:=$
$\displaystyle\underbrace{\int_{0}^{\infty}\overline{P}\left(\left\\{{\theta\in\Theta:\overline{L}(\theta)\mathbbm{1}_{A}(\theta)>t}\right\\}\right)\text{d}t}_{\text{{upper
Choquet integral} of }\overline{L}\mathbbm{1}_{A}}$ $\displaystyle+$
$\displaystyle\underbrace{\int_{0}^{\infty}\underline{P}\left(\left\\{{\theta\in\Theta:\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)>t}\right\\}\right)\text{d}t}_{\text{{lower
Choquet integral} of }\underline{L}\mathbbm{1}_{A^{c}}}.$

In addition, if $\overline{P}$ is concave, then the inequalities in (4) and
(5) are equalities, for all $A\in\mathcal{F}$.

This result is particularly appealing. Under Assumption 1, if the prior upper
probability (PUP) is concave and the prior and likelihood sets
$\mathcal{P}^{\text{co}},\mathcal{L}_{\theta}^{\text{co}}$ are nonempty and
weak⋆-closed, then the agent can perform a (generalized) Bayesian update of
the PUP by carrying out only one operation. This is the case even when the
agent faces ambiguity around the true data generating process so that a set of
likelihoods is needed. The posterior lower probability is obtained immediately
via the conjugacy property $\underline{P}_{y}(A)=1-\overline{P}_{y}(A^{c})$.

######  Corollary 1.1.

Retain the assumptions in Theorem 1. If $\mathcal{L}^{\text{co}}_{\theta}$ is
a singleton, we retrieve Bayes’ theorem in [Wasserman and Kadane, 1990,
Section 2].

Corollary 1.1 tell us that when there is no ambiguity around the likelihood,
Theorem 1 recovers Wasserman and Kadane’s classical Bayes’ theorem. Given the
straightforward nature of the proof, we omit it here. We also have the
following lemma, that is proved in Section 4.

######  Lemma 2 (Preserved concavity).

Suppose $\mathcal{P}^{\text{co}},\mathcal{L}_{\theta}^{\text{co}}$ are
nonempty and weak⋆-closed. Then, if $\overline{P}$ is concave, we have that
$\overline{P}_{y}$ is concave as well.

This lemma is important because it tells us that the generalized Bayesian
update of Theorem 1 preserves concavity, and so it can be applied to
successive iterations. If at time $t$ the PUP is concave, then the PUP at time
$t+1$ – that is, the posterior upper probability at time $t$ – will be concave
too. Necessary and sufficient conditions for a generic upper probability to be
concave are given in [Marinacci and Montrucchio, 2004, Section 5].

In the future, we plan to forego Assumption 1 and use the techniques developed
in Troffaes and de Cooman [2014] to generalize our results to the case in
which the elements of $\mathscr{L}$ are unbounded and not necessarily non-
negative. We also intend to extend our results to the case in which the
elements of $\mathcal{Y}$ are $\mathbb{R}^{d}$-valued, for some
$d\in\mathbb{N}$. We suspect this is a less demanding endeavor since we do not
use specific properties of $\mathbb{R}$ in our proofs.

As mentioned earlier, a natural application of our results is model predictive
control (MPC). MPC is a method that is used to control a process while
satisfying a set of constraints [Rakovi and Levine, 2018]. Typically, when the
process is stable (or at least stable for the past $k$ time steps, for some
$k\geq 0$) and if the scholar decides to take the Bayesian approach, they
proceed as follows. They specify a Normal likelihood (the distribution of the
control inputs) centered at the objective function of the process, and a
Normal prior on the parameter of such function. In this framework, if
ambiguity enters the picture, then our results become relevant.

The importance of addressing prior ambiguity was discussed at length in
section 2.2. The reasons why accounting for likelihood ambiguity is important
are as follows. First, as pointed out earlier, we may run into inconsistency
issues if we perform Bayesian analysis using a misspecified likelihood
[Grünwald and van Ommen, 2017]. Second, a (precise) Normal likelihood is a
good choice only if stability of the process is ensured. MPC procedures are
used in the process industries in chemical plants, oil refineries, power
system balancing models, and power electronics. These are all safety-critical
applications where accounting for possible sudden unexpected instabilities is
of paramount importance.

The scholar may specify a class of $\epsilon$-contaminated truncated Normal
priors and a class of $\eta$-contaminated truncated Normal likelihoods, and
use Theorem 1 to compute the upper posterior. Notice that the Normals need to
be truncated in light of Assumption 1. This requirement is not too stringent,
and – as pointed out earlier in this section – our future work will allow us
do without it.

##  3 Conclusion

In this paper, we present a new Bayes’ theorem for upper probabilities that
extends the one in [Wasserman and Kadane, 1990, Section 2], and [Caprio et
al., 2023, Theorem 7]. In the future, we plan to generalize Theorem 1 by
letting go of Assumption 1, and to apply it to an MPC problem and to other
fields of engineering, and ML and AI. For example, we intend to use it to
overcome the computational bottleneck of step 2 of the algorithm that computes
the posterior set in an imprecise Bayesian neural network procedure [Caprio et
al., 2023]. There, an element-wise application of Bayes’ rule for all the
extreme elements of the prior and likelihood sets is performed. As we can see,
this is a combinatorial task that can potentially be greatly simplified in
light of Theorem 1, conveying a computationally cheaper algorithm.

##  4 Proofs

###### Proof of Theorem 1.

Assume that $\mathcal{P}^{\text{co}},\mathcal{L}_{\theta}^{\text{co}}$ are
nonempty and weak⋆-closed. Pick any $A\in\mathcal{F}$. Recall that we can
rewrite the usual Bayes’ updating rule as

$\displaystyle{P}_{y}(A)$
$\displaystyle=\frac{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)+\int_{\Theta}{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)}$
$\displaystyle=\frac{1}{1+\frac{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)}{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}},$

which is maximized when

$\frac{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)}{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}$

is minimized. But

$\frac{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)}{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}\geq\frac{\inf_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)}{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)},$

which proves the inequality in (4). The inequality in (5) is true because

$\displaystyle\inf_{P\in\mathcal{P}^{\text{co}}}$
$\displaystyle\int_{\Theta}\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)$
$\displaystyle\geq$
$\displaystyle\int_{0}^{\infty}\underline{P}\left(\left\\{{\theta\in\Theta:\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)>t}\right\\}\right)\text{d}t$

and

$\displaystyle\sup_{P\in\mathcal{P}^{\text{co}}}$
$\displaystyle\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\overline{P}\left(\left\\{{\theta\in\Theta:\overline{L}(\theta)\mathbbm{1}_{A}(\theta)>t}\right\\}\right)\text{d}t.$

Assume now that $\overline{P}$ is concave. By [Wasserman and Kadane, 1990,
Lemma 1], we have that there exists $\check{P}\in\mathcal{P}^{\text{co}}$ such
that

$\displaystyle\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}L(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)=\int_{\Theta}L(\theta)\mathbbm{1}_{A}(\theta)\check{P}(\text{d}\theta),$
(6)

for all $L\in\mathscr{L}$. In addition, by [Wasserman and Kadane, 1990, Lemma
4], we have that for all $Y\in\mathcal{Y}$ and all $\epsilon>0$, there exists
a non-negative, upper semi-continuous function $h\leq Y$ such that

$\displaystyle\begin{split}\left[\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}Y(\theta)P(\text{d}\theta)\right]-\epsilon&<\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}h(\theta)P(\text{d}\theta)\\\
&\leq\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}Y(\theta)P(\text{d}\theta).\end{split}$
(7)

Let now $Y=\overline{L}\mathbbm{1}_{A}$. Notice that since
$\mathcal{L}^{\text{co}}_{\theta}$ is weak⋆-compact (as a result of [Walley,
1991, Section 3.6.1]), by (3) so is $\mathscr{L}$. This implies that
$\underline{L},\overline{L}\in\mathscr{L}$, since a compact set always
contains its boundary, so $\underline{L},\overline{L}\in\mathcal{Y}$ as well,
and in turn
$\underline{L}\mathbbm{1}_{A^{c}},\overline{L}\mathbbm{1}_{A}\in\mathcal{Y}$.
Fix then any $L\in\mathscr{L}$ and put $h=L\mathbbm{1}_{A}$. It is immediate
to see that $h$ is non-negative and upper semi-continuous. Then, by (7), we
have that for all $\epsilon>0$

$\displaystyle\begin{split}&\left[\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)\right]-\epsilon<\\\
&\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)\leq\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta).\end{split}$
(8)

Combining (6) and(8), we obtain

$\displaystyle\begin{split}&\left[\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)\right]-\epsilon\\\
&<\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)\check{P}(\text{d}\theta)\leq\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta),\end{split}$
(9)

for all $L\in\mathscr{L}$.

Pick now any $\epsilon>0$ and put

$\displaystyle k$
$\displaystyle:=\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)$
$\displaystyle+\inf_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)>0.$

Choose any $L\in\mathscr{L}$ and $\delta\in(0,\epsilon k)$. By (9) we have
that

$\left[\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)\right]-\delta<\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)\check{P}(\text{d}\theta)$
(10)

and that

$\left[\inf_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)\right]+\delta>\int_{\Theta}{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)\check{P}(\text{d}\theta).$
(11)

The inequality in (10) comes from the fact that the first inequality in (9)
holds for all $\epsilon>0$, and – given how $k$ is defined – we have that
$\delta>0$. The inequality in (11) is obtained by re-deriving (6), (7), (8),
and (9) for the infimum of set $\mathcal{P}^{\text{co}}$ rather than the
supremum. In that case, we simply substitute $\sup$ with $\inf$,
$\overline{L}$ with $\underline{L}$, $\mathbbm{1}_{A}$ with
$\mathbbm{1}_{A^{c}}$, “$-\epsilon$” with “$+\epsilon$”, and we reverse the
inequalities.

Recall that
$\mathbf{c}:=\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)+\inf_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\underline{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)$,
and define

$\mathbf{d}:=\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)\check{P}(\text{d}\theta)+\int_{\Theta}{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)\check{P}(\text{d}\theta).$

Then we have,

$\displaystyle\check{P}_{y}(A)$
$\displaystyle=\frac{\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)\check{P}(\text{d}\theta)}{\mathbf{d}}$
$\displaystyle\geq\frac{\left[\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)\right]-\delta}{\mathbf{c}+\delta-\delta}$
$\displaystyle=\frac{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\mathbf{c}}-\frac{\delta}{k}$
$\displaystyle>\frac{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\mathbf{c}}-\epsilon.$

Since this holds for all $\epsilon>0$, we have that

$\sup_{P_{y}\in\mathcal{P}_{y}^{\text{co}}}P_{y}(A)=\frac{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\mathbf{c}},$

concluding the proof of inequality (4) being an equality when $\overline{P}$
is concave. Inequality (5) being an equality when $\overline{P}$ is concave
comes immediately from [Wasserman and Kadane, 1990, Lemma 4], and the fact
that
$\overline{L}\mathbbm{1}_{A},\underline{L}\mathbbm{1}_{A^{c}}\in\mathcal{Y}$,
as pointed out above. ∎

###### Proof of Lemma 2.

In their works Walley [1981], Wasserman and Kadane [1990], the authors show
that concave upper probabilities are closed with respect to the generalized
Bayes’ rule. In particular, this means that, if we let
$\mathbf{b}:=\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)+\inf_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}{L}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)$,
for any fixed $A\in\mathcal{F}$, if $\overline{P}$ is concave, then for all
$L\in\mathscr{L}$

$\overline{P}_{y}(A)=\frac{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\mathbf{b}}$
(12)

is concave. But since $\mathcal{L}^{\text{co}}_{\theta}$ is weak⋆-compact (as
a consequence of [Walley, 1991, Section 3.6.1]), by (3) so is $\mathscr{L}$.
This implies that $\underline{L},\overline{L}\in\mathscr{L}$, since a compact
set always contains its boundary. Call then
$L^{\prime}=\overline{L}\mathbbm{1}_{A}+\underline{L}\mathbbm{1}_{A^{c}}$. It
is immediate to see that $L^{\prime}\in\mathscr{L}$. Then, by (12) we have
that if we call
$\mathbf{b}^{\prime}:=\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}{L^{\prime}}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)+\inf_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}{L^{\prime}}(\theta)\mathbbm{1}_{A^{c}}(\theta)P(\text{d}\theta)$,
it follows that

$\displaystyle\overline{P}_{y}(A)$
$\displaystyle=\frac{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}{L^{\prime}}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\mathbf{b}^{\prime}}$
$\displaystyle=\frac{\sup_{P\in\mathcal{P}^{\text{co}}}\int_{\Theta}\overline{L}(\theta)\mathbbm{1}_{A}(\theta)P(\text{d}\theta)}{\mathbf{c}}$

is concave, concluding the proof. ∎

Michele Caprio and Yusuf Sale contributed equally to this paper.

###### Acknowledgements.

Michele Caprio would like to acknowledge partial funding by the Army Research
Office (ARO MURI W911NF2010080). Yusuf Sale is supported by the DAAD programme
Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the
Federal Ministry of Education and Research.

## References

  * Augustin, Thomas and Coolen, Frank P.A. and De Cooman, Gert and Troffaes, Matthias C.M. [2014] Augustin, Thomas and Coolen, Frank P.A. and De Cooman, Gert and Troffaes, Matthias C.M., editor.  _Introduction to imprecise probabilities_.  Wiley Series in Probability and Statistics. John Wiley and Sons, 2014\.
  * Berger [1984] James O. Berger.  The robust Bayesian viewpoint.  In Joseph B. Kadane, editor, _Robustness of Bayesian Analyses_. Amsterdam : North-Holland, 1984.
  * Caprio and Gong [2023] Michele Caprio and Ruobin Gong.  Dynamic imprecise probability kinematics.  _Proceedings of Machine Learning Research_ , 215:72–83, 2023.
  * Caprio et al. [2023] Michele Caprio, Souradeep Dutta, Radoslav Ivanov, Kuk Jang, Vivian Lin, Oleg Sokolsky, and Insup Lee.  Imprecise Bayesian neural networks.  _Available atarXiv:2302.09656_, 2023.
  * Casella [1985] George Casella.  An introduction to empirical Bayes data analysis.  _The American Statistician_ , 39(2):83–87, 1985\.
  * Cerreia-Vioglio et al. [2015] Simone Cerreia-Vioglio, Fabio Maccheroni, and Massimo Marinacci.  Ergodic theorems for lower probabilities.  _Proceedings of the American Mathematical Society_ , 144:3381–3396, 2015.
  * Cozman [2000] Fabio Gagliardi Cozman.  Credal networks.  _Artificial Intelligence_ , 120:199–233, 2000.
  * de Finetti [1974] Bruno de Finetti.  _Theory of Probability_ , volume 1.  New York : Wiley, 1974.
  * de Finetti [1975] Bruno de Finetti.  _Theory of Probability_ , volume 2.  New York : Wiley, 1975.
  * Depeweg et al. [2018] Stefan Depeweg, Jose-Miguel Hernandez-Lobato, Finale Doshi-Velez, and Steffen Udluft.  Decomposition of uncertainty in bayesian deep learning for efficient and risk-sensitive learning.  In _International Conference on Machine Learning_ , pages 1184–1193. PMLR, 2018.
  * Ellsberg [1961] Daniel Ellsberg.  Risk, ambiguity, and the Savage axioms.  _The Quarterly Journal of Economics_ , 75(4):643–669, 1961.
  * Epstein and Wang [1994] Larry G Epstein and Tan Wang.  Intertemporal Asset Pricing Under Knightian Uncertainty.  _Econometrica_ , 62(2):283–322, 1994.
  * Giacomini and Kitagawa [2021] Raffaella Giacomini and Toru Kitagawa.  Robust Bayesian inference for set-identified models.  _Econometrica_ , 89(4):1519–1556, 2021.
  * Gilboa and Marinacci [2013] Itzhak Gilboa and Massimo Marinacci.  Ambiguity and the Bayesian paradigm.  In Daron Acemoglu, Manuel Arellano, and Eddie Dekel, editors, _Advances in Economics and Econometrics, Tenth World Congress_ , volume 1. Cambridge : Cambridge University Press, 2013.
  * Gong and Meng [2021] Ruobin Gong and Xiao-Li Meng.  Judicious judgment meets unsettling updating: dilation, sure loss, and Simpson’s paradox.  _Statistical Science_ , 36(2):169–190, 2021.
  * Grünwald and van Ommen [2017] Peter Grünwald and Thijs van Ommen.  Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It.  _Bayesian Analysis_ , 12(4):1069 – 1103, 2017\.
  * Huber and Ronchetti [2009] Peter J. Huber and Elvezio M. Ronchetti.  _Robust statistics_.  Wiley Series in Probability and Statistics. Hoboken, New Jersey : Wiley, 2nd edition, 2009.
  * Hüllermeier and Waegeman [2021] Eyke Hüllermeier and Willem Waegeman.  Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods.  _Machine Learning_ , 3(110):457–506, 2021.
  * Kapoor et al. [2022] Sanyam Kapoor, Wesley J Maddox, Pavel Izmailov, and Andrew Gordon Wilson.  On uncertainty, tempering, and data augmentation in bayesian classification.  _arXiv preprint arXiv:2203.16481_ , 2022.
  * Kendall and Gal [2017] Alex Kendall and Yarin Gal.  What uncertainties do we need in bayesian deep learning for computer vision?  _Advances in neural information processing systems_ , 30, 2017.
  * Klibanoff and Hanany [2007] Peter Klibanoff and Eran Hanany.  Updating preferences with multiple priors.  _Theoretical Economics_ , 2(3):261–298, 2007\.
  * Marinacci and Montrucchio [2004] Massimo Marinacci and Luigi Montrucchio.  Introduction to the mathematics of ambiguity.  In Itzhak Gilboa, editor, _Uncertainty in economic theory: a collection of essays in honor of David Schmeidler’s 65th birthday_. London : Routledge, 2004.
  * Rakovi and Levine [2018] Saa V. Rakovi and William S. Levine.  _Handbook of Model Predictive Control_.  Birkhäuser Basel, 1st edition, 2018.
  * Rudin [1976] Walter Rudin.  _Principles of Mathematical Analysis_.  New york : McGraw-Hill, 3rd edition, 1976.
  * Sale et al. [2023] Yusuf Sale, Michele Caprio, and Eyke Höllermeier.  Is the volume of a credal set a good measure for epistemic uncertainty?  In _Uncertainty in Artificial Intelligence_ , pages 1795–1804. PMLR, 2023.
  * Smith and Gal [2018] Lewis Smith and Yarin Gal.  Understanding measures of uncertainty for adversarial example detection.  _arXiv preprint arXiv:1803.08533_ , 2018.
  * Troffaes and de Cooman [2014] Matthias C.M. Troffaes and Gert de Cooman.  _Lower Previsions_.  Chichester, United Kingdom : John Wiley and Sons, 2014.
  * Walley [1981] Peter Walley.  Coherent lower (and upper) probabilities.  Technical report, University of Warwick, Coventry, 1981.
  * Walley [1991] Peter Walley.  _Statistical Reasoning with Imprecise Probabilities_ , volume 42 of _Monographs on Statistics and Applied Probability_.  London : Chapman and Hall, 1991.
  * Wasserman and Kadane [1990] Larry A. Wasserman and Joseph B. Kadane.  Bayes’ theorem for Choquet capacities.  _The Annals of Statistics_ , 18(3):1328–1339, 1990.
  * Wimmer et al. [2023] Lisa Wimmer, Yusuf Sale, Paul Hofman, Bernd Bischl, and Eyke Hüllermeier.  Quantifying aleatoric and epistemic uncertainty in machine learning: Are conditional entropy and mutual information appropriate measures?  In _Uncertainty in Artificial Intelligence_ , pages 2282–2292. PMLR, 2023.


#  Artificial Intelligence Powered Mobile Networks: From Cognition to Decision

Guiyang Luo, Quan Yuan, Jinglin Li, Shangguang Wang, and Fangchun Yang This
work was supported the National Natural Science Foundation of China under
Grant 6210070488, Grant 61902035 and Grant 61876023. (Corresponding author:
Yuan Quan.)G. Luo, Q. Yuan, J. Li, S. Wang, and F. Yang are with the State Key
Laboratory of Networking and Switching Technology, Beijing University of Posts
and Telecommunications, Beijing, 100876 China (e-mail: {luoguiyang, yuanquan,
jlli, sgwang, fcyang}@bupt.edu.cn).

###### Abstract

Mobile networks (MN) are anticipated to provide unprecedented opportunities to
enable a new world of connected experiences and radically shift the way people
interact with everything. MN are becoming more and more complex, driven by
ever-increasingly complicated configuration issues and blossoming new service
requirements. This complexity poses significant challenges in deployment,
management, operation, optimization, and maintenance, since they require a
complete understanding and cognition of MN. Artificial intelligence (AI),
which deals with the simulation of intelligent behavior in computers, has
demonstrated enormous success in many application domains, suggesting its
potential in cognizing the state of MN and making intelligent decisions. In
this paper, we first propose an AI-powered mobile network architecture and
discuss challenges in terms of cognition complexity, decisions with high-
dimensional action space, and self-adaption to system dynamics. Then,
potential solutions that are associated with AI are discussed. Finally, we
propose a deep learning approach that directly maps the state of MN to
perceived QoS, integrating cognition with the decision. Our proposed approach
helps operators in making more intelligent decisions to guarantee QoS.
Meanwhile, the effectiveness and advantages of our proposed approach are
demonstrated on a real-world dataset, involving $31261$ users over $77$
stations within $5$ days.

###### Index Terms:

Artificial intelligence, Machine learning, Mobile network, Network managements

##  I Introduction

The next-generation mobile network 5G has already started hitting the market
and will continue to expand worldwide, by providing everything (e.g., massive
internet of things, camera drones, robotics, etc.) from superfast bandwidth
speeds, to ultra-low latency, and to ultra-high reliability [1]. 5G will
efficiently support three generic services, i.e., enhanced mobile broadband
(eMBB), massive machine-type communications (mMTC), and ultra-reliable low-
latency communications (uRLLC) [2]. The ever-increasing traffic demand, the
rapid growth of the number of network applications and services, and the
blossoming new service requirements, are the driving forces for the complexity
of mobile systems. This complexity resides not only in the sheer number of
components and interactions, but also in the operations on multiple levels and
time-scales [3]. The daunting complexity of mobile systems appears as a major
hurdle for large-scale modeling efforts, which try to cognize user equipment,
radio environment, and mobile network elements, and then apply it for network
planning, service deployment, management functions, and resource scheduling.

Currently, cognition is built on rules derived from system analysis and
simulation with prior domain knowledge and experiences. For example, during
LTE radio network planning, a general propagation model that doesn’t
incorporate the actual geographical information (terrain model) is applied to
estimate coverage. As for capacity, link budget calculation, and theoretically
assumed traffic are adopted to estimate cell size and capacity. After the cell
has been deployed, it would operate and behave according to preprogrammed
rules (protocols). In a nutshell, the mobile network is treated as a white-
box, which requires an idealized abstraction and simplification of the
underlying network. However, this method faces increasing challenges due to
today’s dynamic and diverse traffic, and the complexity of network
architectures and resource structures. Besides, protocols cannot make
intelligent decisions, and only react when a problem occurs. Consequently,
conventional mobile networks are reactive rather than proactive [4], e.g.,
transmission control protocol (TCP) just slows down the transfer speed when
packets are lost and tries to slowly increase the speed until the next packet
is lost. If it had knowledge of network status and condition before the
transfer, it could optimize transfer speed and avoid packet loss.

Artificial intelligence (AI), with its emphasis on understanding how the human
mind organizes and processes massive amounts of information, has made
considerable achievements in the areas of computer vision [5, 6], speech
recognition, natural language processing, and audio recognition [7].
Specifically, AI has produced results superior to human experts in strategic
game systems (e.g., Deep Blue, AlphaGo, and AlphaStar, etc.), which
demonstrates the potential ability in terms of cognition and decision.
Therefore, powered with mobile network big data (e.g., signaling data, user
access data, connection data, etc.) and a plethora of computational power
(mobile edge computing, and cloud computing), AI has the potential to cognize
complex network environment, and arm it with the ability “see through the
future”. Then, proactive strategies and intelligent decisions can be taken to
manage network resources, considering both user experiences and network
performance. AI will revolutionize the current mobile network from the
following aspects: 1) comprehensive cognition of user equipment, radio
environment, and mobile network elements. With AI, it is possible to
understand the user preferences and user traffic patterns, to construct a
radio map [8], and to cognize the state of mobile network elements. 2)
intelligent decisions with orchestration between individual welfare and social
welfare. AI can analyze and mine historical data to discover the reasons
behind the past successes and failures, and then, exploit the extracted
knowledge and experiences to properly allocate resources, guaranteeing end
users with a high quality of experience (QoE). 3) self-adaption to system
renovation and dynamics. By learning from past behaviors, and similar entities
in the network, AI will equip a mobile network with the ability of self-
learning and adaptive to the dynamics.

In this paper, we focus on the potential opportunity offered by harnessing AI
into MN. Specifically, we first propose an AI-powered mobile network
architecture, which has incorporated AI to the tasks of perception, cognition,
and decision, under the emerging paradigm of mobile edge computing. Then, we
investigate challenges and discuss potential solutions, from the aspects of
cognition complexity, decisions with high dimensional action space, and self-
adaption to system dynamics. Finally, we exploit deep learning to directly map
the state and condition of mobile networks to the perceived QoS, thus helping
network operators making efficient and intelligent admission control
strategies for QoS provision. Our experiments on the real-world data collected
from a China telecom operator demonstrate the effectiveness and advantages of
the proposed method, which involves $31261$ users over $77$ stations within
$5$ days.

The remainder of this paper is organized as follows: Section II introduces the
architecture of AI-powered mobile network. Section III focuses on the
challenges and corresponding solutions are proposed in Section IV. Finally,
Section V presents a case study that applies deep learning approach to map the
state of MN to the perceived QoS.

##  II The Architecture of AI Powered Mobile Network

Figure 1: The proposed architecture for AI-powered mobile network, which
consists of perception, cognition, and decision, forming a closed-form control
system. These three parts are located in the mobile edge computing center,
which can reduce latency, lessen the burden of the backbone network, and
provide requisites (big data and computing resource) to releasing the power of
AI. Consequently, the mobile network is equipped with the ability of real-time
analytics, better customer experience, and lower operational expenditures.

Computing power and big data are essential ingredients in releasing the power
of AI. Mobile edge computing, which is a key technology toward 5G, distributes
computing, and storage resources anywhere along the continuum from a remote
cloud down to things [9]. Meanwhile, a significant amount of data has been
generated and accumulated every second at the mobile network system. With
careful orchestration of these two ingredients, we propose the architecture of
an AI-powered mobile network, which is shown in Fig. 1. Here, the Packet Core
Network can either be the Evolved Packet System (EPS) in 4G and Next-
Generation Core for 5G, which manages all operations needed to transfer voice
and data to and from external networks. As for the base station (BS), it could
be either the eNodeB in 4G and the next-generation NodeB (gNB) in 5G. The
proposed architecture exploits the resources (computing, storage, and big
data) available to mobile network operators (MNO), to ameliorate the planning,
optimization, operation, and management of MN. The proposed architecture
consists of three parts, i.e., perception, cognition, and decision, which form
a closed-loop system.

###  II-A Perception

Massive data that represent the condition of MN can be collected by mobile
network operators, which we divide into three categories, i.e., mobile
terminal data (MTD), radio access network data (RAND), and packet core network
data (PCND). MTD are collected by user equipment, consisting of service
characteristics, device information, and wireless network parameters, cell ID,
signal strength, download/upload rate, etc. The subscriber profiles and
geographic information data are also included in MTD, which are beneficial for
understanding and exploiting the features of mobile data from a social network
perspective [10]. These data depict the state and condition of the underlying
user, and radio propagation environment. RAND capture signaling events
concerning any radio resource control (RRC) operation. This allows recording
fine-grained state changes of each device, and thus detects device network
attachment and detachment operations, start and conclusion of sessions,
related to any call, texting, or data transfer activity. Moreover, they allow
collecting performance indicators on data transmission, such as the uplink and
downlink throughput experienced by the user equipment. The signaling data can
indicate the aggregate-level status and condition of the mobile system, e.g.,
RRC connection success rate, EPS radio access bearer connection success rate.
Besides, RAND also include the settings and configuration data, i.e.,
location, coverage and capacity, frequency band, maximum transmission power,
antenna bearing, which are necessary parameters to initialize a mobile network
so that it can operate as designed. PCND collect information include the IP
session start and end time, device and user identifiers, traffic volume, type
of service (i.e., web, email, streaming audio/video). PCND can also be used to
infer user preference and user attributes, by monitoring the session length
(number of pages a user clicks through) and abandonment rate (if a user leaves
the website after visiting the landing page) [11]. Besides, the feedback
channel quality is a good perception of the wireless environment.

###  II-B Cognition

The massive data offer tremendous opportunities in learning and understanding
user equipment, radio propagation environment, and mobile network elements
(i.e., EPS and E-UTRAN in LTE). User preferences, user mobility, and user
traffic behaviors are among the most important features that affect the
performance of the cellular network. User preferences are associated with the
QoE perceived by the user. The QoE is of significant importance, since the
ultimate goal of the cellular network is to improve the overall perceived QoE.
User mobility, and user traffic behavior influence the traffic load of each
base station, thus affecting the overall performance. This user-related
knowledge can be learned and mined from RAND and MTD [11]. Radio propagation
environment account for the propagation mechanisms of electromagnetic waves on
a wireless link, which is affected by environment dynamics, obstacles
distribution, noise, weather, etc. Such environment information can be
leveraged to not only improve the performance of existing wireless services
(e.g., via proactive resource provisioning) but also enable new applications,
such as wireless spectrum surveillance. Almost every radio link activity is
associated with a signaling exchange inside the mobile network system. The
signaling data equip us with real-time awareness of the traffic load, resource
utilization, and network state. By associating these data to mobile users, the
relationship between user traffic and mobile network state can be properly
learned.

###  II-C Decision

With the extracted knowledge of the user equipment, radio environment, and
mobile network elements, more optimized network planning and intelligent
scheduling strategies can be achieved. However, the strategies involve actions
within different layers, granularity, and even mixed strategies. For example,
based on the analysis of historical traffic load, we could plan the network by
adjusting the capacity and coverage of a cell site (i.e., increase coverage,
or deploy more small base stations). Besides, we could also optimize the
antenna direction and down tilts, mobility (handover and cell re-selection)
parameters for each cell, based on learned traffic patterns. Except for long-
term network planning, there exist many scheduling strategies that deal with
real-time situations. When there is a slight traffic load, we could choose to
reduce energy consumption by dynamically switching off base stations. With the
traffic load increased, various load balancing scheme can be applied, e.g.,
offloading a part of traffic to a neighboring cell, or other networks (WiFi,
3G, etc.). Continuously increased traffic load would lead to traffic
congestion, and this can be detected via learning the relationship between
traffic load and the state of the LTE system. To handle this situation, we
could reject a part of low-priority data requests in advance to prevent the
congestion from affecting all the users within its coverage.

##  III Challenges

###  III-A Complexity and correlation in cognition

The mobile network system is affected by the performance of individual nodes
and protocols through which information travels, due to the use of packet
switching. Consequently, joint cognition of user equipment, radio environment,
and mobile network elements are essential to make intelligent and optimized
decisions. However, the cognition is not only complicated but also
significantly correlated. For example, cognition of user includes both per-
user level (mobile user viewpoint) and aggregate level traffic (mobile
operator viewpoint) [12]. Per-user level traffic is related to user
preference, mobility, and varying traffic demands. The aggregation of these
data requests exhibits traffic patterns, e.g., spatio-temporal dynamics,
anomalous behaviors induced by particular social events. The radio environment
serves as a bridge for exchanging data between the user equipment and mobile
network elements, which is characterized by a radio map (a continuous function
of location, frequency, and time) [13]. It contains rich and useful
information regarding the spectral activities and propagation channels in
wireless networks. Constructing the map is challenging since it requires real-
time measurements and constantly updated. Mobile network elements manage and
allocate the resources through sophisticated software and protocols,
considering the traffic demands of all users and the radio environment.
Therefore, the user equipment, radio environment, and mobile network elements
are heavily correlated.

The complexity and correlations pose tremendous challenges in smoothing and
optimizing network performance. Take the perceived QoE of a certain user as an
example. It is related to the user, radio environment, and mobile network
elements. Besides, it is also related to both user characteristics (different
perceived QoE even with the same QoS) and user traffic (different services
require different QoS). The radio environment affects the transmission rate of
the connection. The mobile network elements, which forwards the packets to the
Internet, would also influence the QoS of the connections. Furthermore, the
situation beyond the mobile network can also deteriorate the QoS. Therefore,
comprehensive cognition of the mobile network system is challenging, since it
not only complex but also heavily correlated.

###  III-B High dimensional state/action spaces in decision

The future mobile network must empower itself with more advanced technologies
to support a tremendous number of applications, meet the significantly
contrasting QoS requirements (bandwidth, latency, and reliability), and
improve the spectrum efficiency. Therefore, operators necessarily consider
decisions over multiple dimensions and granularities. At first, networks are
becoming increasingly heterogeneous, and user devices can be served by more
than one technology, such as 2G, 3G, 4G, Wi-Fi, etc. Within each technical
domain, operators have options to combine multiple layers of cells (i.e., base
station, small cell, femtocell), and various radio beams can be organized to
better serve expected data requests. Furthermore, in each connection, there
exist multiple technologies (carrier aggregation, beamforming, device to
device communication, non-orthogonal multiple access, etc.), offering
tremendous opportunities to exploit resources in time, spectrum, space, and
power domain. To make full use of these resources and better improve the
perceived QoE, the user characteristics and radio environment have to be taken
into account. For example, millimeter wave, which is a key technology enabler
for 5G and can be used for high-speed wireless broadband communications,
travels by line of sight, since its short wavelengths (wavelengths from 10
millimeters to 1 millimeter) can be blocked by physical objects like buildings
and trees. Additionally, base stations can be turned on or off dynamically to
manage load or interference, thus reducing capital expenditures and
operational expenditures. Furthermore, the decisions among neighboring cells
are dependent, e.g., interference to users in neighboring cells, offloading
from overload cell to underload cell. Therefore, it is significantly
challenging to make intelligent and optimized decisions due to the high
dimensional state/action space.

###  III-C Self-adaption to the evolving network dynamics

Driven by the prosperity and flourishing of information technology, and by the
proliferation of connected devices, the network is always evolving. The
evolving dynamics can be induced by users, the radio environment, mobile
network elements, events, etc. Born in the early 1980s, the first generation
of wireless cellular technology 1G supports voice calls only, applying analog
technology (poor battery life and voice quality, little security, and prone to
dropped calls). The introduction of 3G networks in 1998 ushers in more data-
demanding ways such as for video calling and mobile internet access. The
imminent arrival of 5G promises significantly faster data rates, higher
connection density, much lower latency, and energy savings, among other
improvements. The evolution of the mobile network has significantly enriched
the scope and horizon for mobile users, giving birth to a plethora of
applications and services with distinct QoS requirements. Consequently, user
traffic and content requirements variation is constantly evolving [14], driven
by both the evolution of technology and user characteristics. The radio
environment can be influenced by the construction, the weather, and moving
objects. Besides, national festivities (e.g., Thanksgiving, Christmas, New
Year’s Eve, Easter, etc.), events (e.g., the Olympic games, sports, solo vocal
concert) would significantly affect the traffic content and user patterns,
thus imposing influence on network performance. These dynamics could fail the
deployed strategies or rules. To handle these dynamics, self-adoptive and
agile strategies and decisions shall be taken to better meet communication
requirements.

##  IV Solutions

AI makes it possible for machines to accomplish specific tasks by processing
large amounts of data and recognizing patterns in the data. Thanks to
increased data volumes, advanced algorithms, and improvements in computing
power and storage, AI has achieved considerable improvements in the area of
computer vision, natural language processing, bioinformatics, etc. AI can be
loosely interpreted to mean incorporating human intelligence to machines,
which consists of everything from Good Old-Fashioned AI (GOFAI) all the way to
futuristic technologies such as deep learning. GOFAI, which is also known as
symbolic AI, can identify the influences and reasons of a certain event in the
mobile network, e.g., determine the influence when a base station is broken
down. Machine learning, as a subset of AI techniques that do not require human
intervention to make certain changes, was coined by Arthur Samuel in 1959.
Deep learning (also known as deep structured learning or hierarchical
learning), being a subset of machine learning, is inspired by the information
processing patterns found in the human brain and consists of thousands or even
millions of simple processing nodes that are densely interconnected. Deep
learning, usually with more (millions of) parameters, can represent much more
complicated functions, thus standing out as a powerful approach for image
classification, natural language processing, etc. Besides, deep learning
methods obviate feature engineering by translating the data into compact
intermediate representations, and derive layered structures that remove
redundancy in representation.

Machine learning empowers machines with the ability to learn without being
explicitly programmed to perform specific tasks. With the availability of
massive volume data, machine learning is widely hugged, since it can learn
from data, identify patterns, and make decisions with minimal human
intervention. Machine learning can be divided into three classes, i.e.,
supervised learning (SL), unsupervised learning (UL), and reinforcement
learning (RL), respectively. In a SL model, the algorithm learns on a labeled
dataset, providing an answer key that the algorithm can use to evaluate its
accuracy on the training data. An unsupervised model, in contrast, provides
unlabeled data that the algorithm tries to make sense of by extracting
features and patterns on its own. As for the RL model, the agent relies both
on learning from past feedback and exploration of new tactics that may present
a larger payoff, to make its decisions [1]. Machine learning can also be
divided into offline learning (OFL) and online learning (ONL), based on
whether to change their approximation of the target function during the
training. The whole training data of an OFL model must be available at the
time of model training. Only when training is completed can the model be used
for predicting. ONL is a very special kind of learning approach, which
processes data sequentially. They produce a model and put it in operation
without having the complete training dataset available at the beginning. The
model is continuously updated during operation as more training data arrives,
which is shown in Fig. 2.

Figure 2:  Machine learning algorithms are divided into three broad
categories, i.e., SL, UL, and RL, respectively, depending on the nature of the
signal or feedback available to the learning system. They can also be
categorized into ONL and OFL based on the static dataset or sequentially
generated dataset. Each model has different requirements on the type of data,
and can be applied to different tasks.

###  IV-A Supervised learning and unsupervised learning contribute to better
cognition

To perform a task, a SL model requires a set of training examples, and each
sample in the dataset consists of an input object (typically a vector) and the
desired output value (ground truth). A SL algorithm analyzes the training data
and produces an inferred function, which can be used for mapping new examples.
SL problems can be further grouped into regression (e.g., linear regression,
logistic regression) and classification (e.g., support vector machines, neural
networks) problems. It can be adopted to cognize the traffic load of a mobile
network system, by ingeniously applying QoS changes as labels. The traffic
load of a mobile network significantly affects the QoS of connections, since
users have to share the network resources. However, the traffic load is
correlated to many values in key performance indicators (KPI) of mobile
systems, e.g., physical resource blocks utility rate, physical downlink
control channel utility rate, physical random access channel utility rate,
radio resource control success rate, etc. As for the label, we could treat the
average perceived QoS as the indicator for the condition of traffic load,
since more traffic load might lead to traffic congestion, which significantly
degrades QoS. Then, we could apply a classification algorithm over KPI to
identify the traffic load condition. Based on this traffic load, strategies
that prevent traffic congestion can be automatically adapted, which will be
elaborated in V.

The training data for UL does not possess prior labels. Consequently, the UL
agent has to depend on its capability to learn the embedded structure or
pattern in data. UL is very popular, since it does not require a labeled or
classified dataset, which is very time-consuming and labor-intensive.
Clustering (e.g., K-means, hierarchical clustering, OPTICS algorithm, etc.)
and dimensionality reduction (e.g., principal component analysis, singular
value decomposition, etc.) are two types of UL approaches. Clustering can be
applied to understand the aggregate-level traffic activity. For example, base
stations that are associated with different types of regions (e.g, work,
residential, hybrid, nightlife, and leisure) exhibit significantly different
traffic profiles. For the work areas, the activity takes place mainly during
weekdays, especially during working hours, while weekend activity is almost
nonexistent. However, for leisure areas, activity on weekends is more than
twice that of weekdays, and this activity is focused during daylight hours.
Exploiting clustering algorithms, we could divide base stations into different
clusters, which are characterized by aggregating traffic activities. We could
optimize the strategies for each cluster according to the traffic
characteristics.

Figure 3: Online learning equips each base station with self-adaption ability.

###  IV-B Deep reinforcement learning for intelligent and optimized decisions

RL is the area of machine learning that deals with sequential decision-making
[15], which can be formulated as a Markov decision process. At first, an
agent, with a given environment state $s_{t}$ from the state space
$\mathcal{S}$, attempts to take an action $a_{t}$ in the state space
$\mathcal{A}$. Then, the agent receives a reward and transits to the next
state $s_{t+1}\in\mathcal{S}$. In an episodic problem, this process continues
until the agent reaches a terminal state, and then it restarts. The agent aims
to maximize the accumulated reward of the long term return from each state. An
RL agent needs a trade-off between the exploration of uncertain policies and
exploitation of the current best policy, which is a fundamental dilemma in RL.
Deep reinforcement learning, which is a combination of RL and deep learning,
equips RL with higher generalization ability. Consequently, deep RL can model
high dimensional state/action spaces in decision. For example, a deep RL based
AlphaGo, which has defeated a Go world champion, has shown its potentials in
modeling high dimensional state and action space, since Go is a game of
profound complexity and there is an astonishing $10^{170}$ possible board
configurations. Furthermore, games like StarCraft II that are harder for
computers to play than board games like chess or Go, have also been conquered
by deep RL based models.

Deep RL can be applied to learn intelligent resource allocation and scheduling
decisions. We model the state of the users (application type, QoS
requirements), radio map (channel state information), and KPI in mobile
network as state space $\mathcal{S}$. The action space $\mathcal{A}$ consists
of resource allocation (allocated resource blocks, power) and scheduling
(access to which cells, D2D or beamforming techniques). The goal of deep RL is
to learn a policy $\pi:\mathcal{S}\rightarrow\mathcal{A}$, maximizing long
term rewards, which can be defined as the average throughput, perceived QoS,
or users’ satisfaction (QoE).

###  IV-C Online learning to build a self-adaptive system

An ONL agent makes online, real-time decisions and continuously improve
performance with the sequential arrival of data. It has a significant
difference over OFL (also known as batch learning), which is trained over the
entire dataset. ONL algorithms are widely applied for advertisement placement,
movie recommendation, and node or link prediction in evolving networks. It can
help the model to adapt to dynamics in the users, radio environment, and
mobile network system, aiming at dynamically optimizing the quality of
experience. The dynamics can be divided into slow variation (e.g., user
behavior and aggregate-level traffic, radio environment) and drastic variation
(e.g., mobile network system, events). The former one alters very slowly, such
as traffic patterns, which reveals differences during daytime and night, or
weekdays and weekends. Sequentially generated data can be applied to update
the model, and each cell is associated with a model, as shown in Fig. 3.
However, drastic variation changes sharply due to newly installed cells or
national festivities or events. For these variations, we require the
experience and knowledge from historical data or a similar area. For example,
a solo vocal concert would gather a mountain of people in a certain area
within a short duration. The traffic pattern within this area has changed
significantly, and the mobile network system has to be carefully configured.
To achieve this, we should first search historical events or data in the
remote cloud, and extract the knowledge for such events. Then, based on the
radio environment, and estimated traffic, we can reconfigure the mobile
network in this area or deploy a temporary access point to offload part of
traffic.

##  V A User Case: QoS provision from network operator point view

Network operators aim at providing better QoS connections to users while
admitting as many users as possible. However, users within a cell share the
available resources, and more users could potentially lead to traffic
congestion, which significantly influences the perceived QoS of all users.
Therefore, the network operator should first cognize the condition and state
of the cell, and then deal with (admission control) the connection requests
based on network condition, thus preventing network congestion from happening.
To achieve this goal, based on real-world data collected from a
telecommunication operator in China, we propose a deep learning based model
that integrates cognition with decision, to provision QoS. Our proposed model
directly maps the condition and state of the mobile network systems to
admission control strategies. The well-trained of such model can be applied to
help the cell in making more intelligent and optimized decisions, thus
preventing it from traffic congestion.

###  V-A Mobile Network Dataset

The real-world data are collected from a China telecom operator, containing
data from $31261$ users, over $77$ stations within $5$ days. It consists of
$4$ parts, i.e., KPI, measure reports (MR), traffic detail records (TDR), and
data monitored by user equipment (DMU). KPI records the performance values of
components in the mobile network, and is already defined in 3GPP TR 32.814.
KPI helps network operators to monitor and optimize the radio network
performance, providing better subscriber quality, and achieving a higher
utilization ratio of the network resources. KPI includes radio KPI and
transport KPI, e.g., RRC connection establishment success rate (obtained by
the number of all successful RRC establishments divided by the total number of
attempted RRC establishments and abbreviated as
$rrc\\_attconnestab\\_ue\\_rate$ in Fig. 4.), soft handover success rate
(obtained by the number of successful radio link additions divided by the
total number of radio link addition attempts and abbreviated as
$ho\\_succoutintraenb\\_rate$ in Fig. 4.). MR is a statistical data report
that contains information about channel quality, which is generated by
measuring the signal quality, e.g., average signal to interference plus noise
ratio, weak coverage range, average reference signal receiving quality, etc.
TDR records the event and the type for all data traffic, e.g., the traffic is
served by which cell, and at which time, and the traffic characteristics. DMU
records the real-time QoS (delay, jitter, packet loss, etc.) of each
connection, which is reported by user equipments.

Figure 4: Heat map of the KPI Pearson correlation coefficients.

###  V-B Correlation and Hidden Patterns

KPI parameters are dependent. For a KPI parameter $u$, the value at $t$-th
timestamp is denoted by $x^{u}_{t}$. Therefore, the values of $5$ days forms a
vector $X^{u}=[x^{u}_{0},x^{u}_{1},\cdots,x^{u}_{T}]$. For two KPI parameters
$u$ and $v$, Pearson correlation coefficient (PCC) of $X_{u}$ and $X_{v}$ is
denoted as $\rho_{u,v}=\frac{cov(X_{u},X_{v})}{\sigma_{X_{u}}\sigma_{X_{v}}}$,
where $cov(X_{u},X_{v})$ is the covariance and $\sigma_{X_{u}}$ is the
standard deviation of $X_{u}$. We have a set of KPI parameters and adopt
absolute PCC as indicator for correlations, i.e.,$|\rho_{u,v}|$. A matrix is
applied to represent the correlations of these parameters, as shown in Fig. 4.
As can be seen from this figure, the KPI parameters are heavily correlated,
e.g., in the fifth row of Fig. 4, $rrc\\_userconnmax$ is correlated with
several other parameters. Due to this correlations, we could not depend on a
single parameter to identify the state and condition of a cell, e.g, applying
uplink physical resource block average usage rate (abbreviated as
$puschprbtotmeanul\\_rate$ in Fig. 4) as the indicator for congestion
evaluation, since it is related to other KPIs. If all KPI parameters are
considered, a more accurate state and condition can be obtained, and more
accurate performance (e.g., congestion evaluation) can be achieved.

Besides, there exist some hidden features and patterns within these data. For
example, a decreasing RRC connection establishment success rate could
potentially degrade perceived QoS. We propose to discover these patterns
through deep learning technologies.

###  V-C QoS Provision Based on Deep Learning

Our objective is to design admission control strategies for QoS provision. We
achieve it by identifying the state and condition of the mobile network system
that will lead to poor performance in QoS. Then, corresponding strategies
could be applied to provision QoS, e.g., early reject low-priority data
requests. The KPI and MR data can well represent the state and condition of
the mobile network system, while the DMU and TDR data contain the information
regarding traffic connection, including the perceived QoS. The change in
average perceived QoS is treated as the learning target. In the
implementation, we treat such problem as a binary classification problem,
where KPI and MR of a cell, and characteristics of the data connections are
treated as input and the change of average perceived QoS as the output (QoS
improvement or deterioration). These input-output pairs consist of labeled
data. Among these data, $70\%$ is treated as the training dataset, $20\%$ as
the test dataset, and the rest as the validation dataset.

To capture the spatio-temporal dependences, the KPI parameters across
different timestamps and adjacent cells are treated as a three-dimensional
tensor, as shown in Fig. 5. Convolution neural networks (CNN), which take
advantage of the hierarchical patterns in data and assemble more complex
patterns using smaller and simpler patterns, are applied to extract features
and then map them to the ground truth labels. The adopted CNN model consists
of two convolutional layers, an average pooling layer, and two fully connected
layers, and is shown in Fig. 5. Furthermore, we have adopted several machine
learning algorithms, which are k-nearest neighbors (KNN), support vector
machine (SVM), decision tree (DT), and light gradient boosting machine (LGB).

The experiments results are shown in Fig. 6. For each traditional machine
learning method, we have deliberately chosen the features that could lead to
better performance, i.e., with careful feature engineering. A different number
of input features is evaluated and the performance of the best is shown in
Fig. 6. Traditional machine learning methods achieve a poor performance,
mainly due to: 1) the model is shallow, which can’t handle very complex
function approximation and learn the non-linear and complex relationship
between the state of MN and admission control decision; 2) the model is with
little trainable variables, which is prone to be over-fitting. CNN achieves
the best performance, with $78.84\%$ accuracy on the test dataset. CNN is
preferred in this case, not only for its better performance, but also for its
simplicity, i.e., without any feature engineering. This well-trained CNN model
associates the state of the mobile network system with the perceived QoS of
users, and can be applied to help the operator in deciding on how to better
allocate resources considering the state of the mobile network system.

Fig. 6(a) visualize the performance enhancements by applying the CNN model for
QoS provision, where the triangle means base stations, and the circle stands
for a user equipment. The left figure of Fig. 6(a) expresses the originally
perceived delay, while the right shows the perceived delay applying our
proposed CNN model for QoS provision. The results verify that the proposed CNN
model could decrease the perceived delay and provide better QoS to users.

Figure 5: We propose a deep learning algorithm that directly map the state and
condition of mobile network system to the perceived QoS. The proposed
algorithm can identify hidden features and patterns within MN, i.e., the state
of mobile network system that could lead to traffic congestion.

(a)

Figure 6:  We propose a deep learning algorithm CNN that directly map the
state and condition of the mobile network system to perceived QoS. The
proposed algorithm can identify hidden features and patterns within MN, i.e.,
the state of the mobile network system that could lead to traffic congestion.
(a) shows the performance of CNN as compared with several machine learning
baselines, i.e., KNN, SVM, DT, and LGB. Our proposed CNN achieves the best
performance. (b) illustrates the perceived delay without (left) and with
(right) CNN to provision QoS. The color bar maps the delay (millisecond) to
different colors, indicating our proposed CNN could potentially decrease the
perceived delay.

##  VI Conclusion and Future Work

AI has offered tremendous opportunities for network operators to plan, deploy,
manage, and maintain the network, which can not only better satisfy user
requirements but also significantly reduce capital expenditures and
operational expenditures. This paper first proposes an AI-powered
architecture. Then, challenges and potential solutions regarding the
application of AI in the mobile network are investigated. Finally, we have
proposed a deep learning approach that directly maps the state of the mobile
network to perceived QoS and demonstrated its effectiveness on real-world data
collected from a China telecom operator. To fully leverage the power of AI in
the mobile network, further research directions are discussed as follows:

Integration of terminal intelligence, edge intelligence, and cloud
intelligence — With the development of information and communications
technology, the terminal (i.e., mobile phone, wearable devices, automated
vehicles, etc.) are evolved with higher intelligence. These terminals are
aware of the usage patterns and preferences of the user, thus can make more
intelligent decisions in advance. For example, they could automatically cache
videos or news in advance when the network condition is good, taking the user
preference and network condition into account. Integration of the intelligence
from terminal, edge, and cloud could significantly improve the efficiency of
the mobile network system.

Privacy and security in utilizing the massive data — Although machine
learning, especially deep learning, has shown enriched performance results in
a variety of fields, the model can easily lead to mispredictions or
misclassifications due to malicious external influences. Besides, the massive
data in the mobile network show promising potential in the areas of social
networks, geography, and urban planning. Therefore, it is challenging to
protect privacy in traffic data, since it contains sensible information on
individual subscribers.

Advanced technologies and platforms to exploit and manage the massive data —
Users and mobile network entities are always generating data, which
distributes from the things, to the edge and the remote cloud. Due to its
enormous quantity, research challenges range from data collection, data
storage and transmission, data transformation and processing, and finally data
usage. Advanced technologies and platforms shall be carefully orchestrated to
efficiently store, harness, and disseminate these massive data.

## References

  * [1] G. Luo, Q. Yuan, H. Zhou, N. Cheng, Z. Liu, F. Yang, and X. S. Shen, “Cooperative vehicular content distribution in edge computing assisted 5g-vanet,” _China Communications_ , vol. 15, no. 7, pp. 1–17, 2018.
  * [2] B. Qian, H. Zhou, T. Ma, K. Yu, Q. Yu, and X. Shen, “Multi-operator spectrum sharing for massive iot coexisting in 5g/b5g wireless networks,” _IEEE Journal on Selected Areas in Communications_ , pp. 1–1, 2020.
  * [3] R. Shafin, L. Liu, V. Chandrasekhar, H. Chen, J. Reed, and J. C. Zhang, “Artificial intelligence-enabled cellular networks: A critical path to beyond-5g and 6g,” _IEEE Wireless Communications_ , vol. 27, no. 2, pp. 212–217, 2020.
  * [4] J. Li, G. Luo, N. Cheng, Q. Yuan, Z. Wu, S. Gao, and Z. Liu, “An end-to-end load balancer based on deep learning for vehicular network traffic control,” _IEEE Internet of Things Journal_ , vol. 6, no. 1, pp. 953–966, Feb 2019.
  * [5] H. Zhang, G. Luo, Y. Tian, K. Wang, H. He, and F.-Y. Wang, “C2FDA: Coarse-to-fine domain adaptation for traffic object detection,” _IEEE Transactions on Intelligent Transportation Systems_ , 2021.
  * [6] H. Zhang, G. Luo, J. Li, and F.-Y. Wang, “C2fda: Coarse-to-fine domain adaptation for traffic object detection,” _IEEE Transactions on Intelligent Transportation Systems_ , pp. 1–15, 2021.
  * [7] N. Cheng, F. Lyu, W. Quan, C. Zhou, H. He, W. Shi, and X. Shen, “Space/aerial-assisted computing offloading for iot applications: A learning-based approach,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 5, pp. 1117–1129, 2019.
  * [8] Y. Hu and R. Zhang, “A spatiotemporal approach for secure crowdsourced radio environment map construction,” _IEEE/ACM Transactions on Networking_ , vol. 28, no. 4, pp. 1790–1803, 2020.
  * [9] S. Deng, H. Zhao, W. Fang, J. Yin, S. Dustdar, and A. Y. Zomaya, “Edge intelligence: The confluence of edge computing and artificial intelligence,” _IEEE Internet of Things Journal_ , vol. 7, no. 8, pp. 7457–7469, 2020.
  * [10] W. Guo, “Explainable artificial intelligence for 6g: Improving trust between human and machine,” _IEEE Communications Magazine_ , vol. 58, no. 6, pp. 39–45, 2020.
  * [11] K. Keller, P. Felka, J. Fornoff, O. Hinz, and A. Rizk, “Quality of experience measurements of multipath tcp applications on ios mobile devices,” in _Proceedings of the 11th ACM Multimedia Systems Conference_ , 2020, pp. 285–290.
  * [12] J. S. Wey, Y. Luo, and T. Pfeiffer, “5g wireless transport in a pon context: An overview,” _IEEE Communications Standards Magazine_ , vol. 4, no. 1, pp. 50–56, 2020.
  * [13] S. Bi, J. Lyu, Z. Ding, and R. Zhang, “Engineering radio maps for wireless resource management,” _IEEE Wireless Communications_ , pp. 1–9, 2019.
  * [14] F. Lyu, J. Ren, N. Cheng, P. Yang, M. Li, Y. Zhang, and X. Shen, “Lead: Large-scale edge cache deployment based on spatio-temporal wifi traffic statistics,” _IEEE Transactions on Mobile Computing_ , pp. 1–1, 2020\.
  * [15] Y. Xu, H. Zhou, J. Chen, B. Qian, W. Zhuang, and X. Shen, “V2x empowered non-signalized intersection management in the ai era: Opportunities and solutions,” _IEEE Communications Standards Magazine_ , vol. 4, no. 4, pp. 18–25, 2020.