# On fixed point approach to equilibrium problem

Le Dung Muu  Institute of Mathematics and Applied Sciences
Thang Long University
Hanoi
Vietnam<EMAIL_ADDRESS>Xuan Thanh Le  Institute of Mathematics
Vietnam Academy of Science and Technology
Hanoi
Vietnam<EMAIL_ADDRESS>
###### Abstract

The equilibrium problem defined by the Nikaidô-Isoda-Fan inequality contains a
number of problems such as optimization, variational inequality, Kakutani
fixed point, Nash equilibria, and others as special cases. This paper presents
a picture for the relationship between the fixed points of the Moreau proximal
mapping and the solutions of the equilibrium problem that satisfies some kinds
of monotonicity and Lipschitz-type condition.

###### Key words and phrases:

Monotone equilibria, fixed point, Moreau proximal mapping.

###### 1991 Mathematics Subject Classification:

47H05, 47H10, 90C33

##  1\. Introduction

In this paper we are concerned with the equilibrium problem stated as

$\text{Find }x^{*}\in C\text{ such that }f(x^{*},y)\geq 0\text{ for all }y\in
C,$ (EP)

in which $C$ is a nonempty closed convex subset in a Hilbert space
$\mathcal{H}$ endowed with an inner product $\langle\cdot,\cdot\rangle$ and
the induced norm $\|\cdot\|$, and
$f:\mathcal{H}\times\mathcal{H}\to\mathbb{R}\cup\\{+\infty\\}$ is a bifunction
such that $f(x,y)<+\infty$ for every $x,y\in C$. The inequality in Problem
(EP) was first used in [36] for convex noncooperative game theory. The first
result on solution existence of (EP) is due to K. Fan [16], where this problem
was called a minimax inequality. The name equilibria was first used in [33].
After the appearance of the paper by Blum and Oettli [11], the problem (EP)
has attracted much attention of many authors and a lot of algorithms have been
developed for solving the problem where the bifunction $f$ have monotonic
properties. These algorithms are based upon different methods such as penalty
and gap functions [7, 8, 9, 23, 24, 25, 29, 33], regularization [3, 20, 28,
34, 35], extragradient methods [10, 19, 26, 37, 39, 40, 41, 44, 46, 47, 48],
splitting technique [2, 15, 32]. A comprehensive reference-list on algorithms
for the equilibrium problem can be found in the interesting monograph [6].

An interesting of this problem is that, despite its simple formulation, it
contains many problems such as optimization, reverse optimization, variational
inequality, minimax, saddle point, Kakutani fixed point, Nash equilibrium
problems, and some others as special cases (see the interesting monographs [6,
22] and the papers [11, 33]).

In what follows we always suppose that $\phi:C\times C\to\mathbb{R}$ such that
$\phi(x,\cdot)$ is convex for any $x\in C$, and $\varphi:C\to\mathbb{R}$ is a
convex function on $C$. For continuity (resp. lower and upper continuity) of
$\phi$ and $\varphi$ we means the continuity (reps. lower and upper
continuity) with respect to the set $C$. Then we consider Problem (EP) with
$f(x,y):=\phi(x,y)+\varphi(y)-\varphi(x)$. In this case Problem (EP) becomes a
mixed equilibrium problem of the form

$\text{Find }x^{*}\in C\text{ such that
}f(x^{*},y):=\phi(x^{*},y)+\varphi(y)-\varphi(x^{*})\geq 0\text{ for all }y\in
C.$ (MEP)

By considering this mixed form one can employ special structures of each
$\phi$ and $\varphi$ in subgradient splitting algorithms, where the bifunction
$f(x,y)$ can be expressed by the sum of two bifunctions
$f_{1}(x,y)+f_{2}(x,y)$ and the iterates are defined by taking the proximal
mappings of each $f_{1}$ and $f_{2}$ separately, see [2, 3, 15, 35, 32].

The first fixed point approach to equilibrium problem (EP) was first developed
in 1972 by K. Fan in [16]. There, by using the KKM lemma, it has been proved
that if $C$ is compact and $f(x,\cdot)$ is quasiconvex on $C$, then Problem
(EP) admits a solution under a certain continuity property of $f$. Note that
in this result of K. Fan, it does not require any monotonicity of the
bifunction $f$.

A direct proof using the Kakutani fixed point theorem for the solution
existence of Problem (EP) is based upon the mapping $K$ defined by taking, for
each $x\in C$,

$K(x):=\text{argmin}\\{f(x,y):y\in C\\}.$ (P1)

Clearly, if $f(x,x)=0$ for every $x\in C$, then $x^{*}$ is a solution to (EP)
if and only if it is a fixed point of $K$, i.e., $x^{*}\in K(x^{*})$. Thus, if
$C$ is convex, compact, $f(x,\cdot)$ is convex on $C$ and $K$ is upper
semicontinuous on $C$, then by the well known Kakutani fixed point theorem,
the mapping $K$ has a fixed point. It can be noticed that the mapping $K$ is
set-valued in general.

In order to avoid multivalues of $K$, an auxiliary principle has been used by
defining the proximal mapping

$B_{\lambda}(x):=\text{argmin}\left\\{\lambda f(x,y)+\frac{1}{2}\langle
y-x,G(y-x)\rangle\right\\},$ (P2)

where $\lambda>0$ and $G$ is a self-adjoint positive linear bounded operator
from $\mathcal{H}$ into itself. In the sequel, for simplicity of the
presentation, we always suppose that $G$ is the identity operator. It is well
known [12] that if $f(x,\cdot)$ is convex and subdifferentiable on $C$,
Problem (P2) is uniquely solvable even for the case $C$ is not compact.
Moreover, a point $x^{*}\in C$ is a solution of Problem (EP) if and only if
$x^{*}$ is a fixed point of $B_{\lambda}$ for any $\lambda>0$. So the solution
existence of (EP) can be proved by using the Brouwer fixed point theorem
whenever $C$ is compact and $B_{\lambda}$ is continuous on $C$.

These results suggest that one can apply the existing algorithms such as ones
based on the Scaft pivoting method [27] for computing a fixed point of the
mapping $B_{\lambda}$, thereby solving the equilibrium problem (EP). However,
the computational results [27, 45] show that the pivoting methods are
efficient only for problems with moderate size. Since in the fixed point
theory, iterative methods for computing a fixed point have been successfully
applied to contractive, generalized contractive, and nonexpansive mappings, a
natural question arises that under which conditions, the mapping $B_{\lambda}$
possesses certain contraction or generalized nonexpansiveness properties?

This is a survey paper, but it also contains some new results on a fixed point
approach to equilibrium problem (MEP). Namely, first we outline results on
quasicontraction and contraction of the Moreau proximal mapping when the
bifunction $f$ is strongly monotone and satisfies a certain Lipschitz-type
condition. Next, in the case $f$ is not necessarily strongly monotone, but
monotone, we present some results on approximate nonexpansiveness of the
proximal mapping for monotone equilibrium problems satisfying a certain
strongly Lipschitz-type condition. Finally, we show a result on
quasinonexpansiveness of a composed proximal mapping defined by the
equilibrium problem. This relationship allows that the equilibrium problem can
be solved by the existing methods in the fixed point theory (see e.g. [13, 17,
21, 31, 42, 43] and the references theirein).

The paper is organized as follows. The next section contains preliminaries on
the equilibrium problem under consideration and on generalized contractions in
real Hilbert spaces. In Section 3 we present some results on contraction,
quasicontraction, nonexpansiveness, and approximate nonexpansiveness of the
Moreau proximal mapping defined for the equilibrium problem. We close the
paper by some conclusions in Section 4.

##  2\. Preliminaries

The following definitions for a bifunction is commonly used in the literature,
see e.g. [6].

######  Definition 1.

A bifunction $f:C\times C\to\mathbb{R}$ is said to be

(i) strongly monotone with modulus $\gamma>0$ (shortly $\gamma$-strongly
monotone) on $S\subseteq C$ if

$f(x,y)+f(y,x)\leq-\gamma\|x-y\|^{2}\ \forall x,y\in S;$

(ii) monotone on $S\subseteq C$ if

$f(x,y)+f(y,x)\leq 0\ \forall x,y\in S;$

(iii) strongly pseudomonotone on $S\subseteq C$ with modulus $\gamma>0$
(shortly $\gamma$\- strongly pseudomonotone) if

$f(x,y)\geq 0\Rightarrow f(y,x)\leq-\gamma\|x-y\|^{2}\ \forall x,y\in S;$

(iv) pseudomonotone on $S\subseteq C$ if

$f(x,y)\geq 0\Rightarrow f(y,x)\leq 0\ \forall x,y\in S.$

The notions on monotonicity properties of a bifunction are generalized ones
for operators. In fact, it is easy to see that when $f(x,y):=\langle
F(x),y-x\rangle+\varphi(y)-\varphi(x)$, then $f$ is $\gamma$-strongly monotone
(resp. monotone, $\gamma$-strongly pseudomonotone, pseudomonotone) if and only
if $F$ is $\gamma$-strongly monotone (resp. monotone, $\gamma$-strongly
pseudomonotone, pseudomonotone). The following Lipschitz-type conditions has
been introduced in [29] and commonly used for Problem (EP).

######  Definition 2.

A bifunction $f:C\times C\to\mathbb{R}$ is said to be of Lipschitz-type on
$S\subseteq C$ if there exists constants $L_{1},L_{2}>0$ such that

$f(u,v)+f(v,w)\geq f(u,w)-L_{1}\|u-v\|^{2}-L_{2}\|v-w\|^{2}\ \forall u,v,w\in
S.$

By taking $u=v=w$ we see that $f(u,u)\geq 0$, so if, in addition, $f$ is
pseudomonotone, then $f(u,u)=0$.

The following concepts are well-known in the fixed point theory (see e.g.
[1]).

######  Definition 3.

Let $T:\mathcal{H}\to C$.

(i) $T$ is said to be contractive on $C$ if there exists $0<\rho<1$ such that

$\|T(x)-T(y)\|\leq\rho\|x-y\|\ \forall x,y\in C.$

If $T$ satisfies this condition with $\rho=1$, then it is said to be
nonexpansive. It is said to be quasicontractive on $C$ if

$\|T(x)-T(y)\|\leq\rho\|x-y\|\ \forall x\in{\rm Fix}(T),y\in C,$

where ${\rm Fix}(T)$ stands for the set of fixed points of $T$. If this
condition holds for $\rho=1$, then $T$ is said to be quasinonexpansive.

(ii) $T$ is said to be firmly nonexpansive on $C$ if

$\|T(x)-T(y)\|^{2}\leq\|x-y\|^{2}-\|(I-T)(x)-(I-T)(y)\|^{2}\ \forall x,y\in
C.$

(iii) $T$ is said to be $\rho$-strongly converse monotone or $\rho$-cocoercive
on $C$ with $\rho>0$, if

$\langle T(x)-T(y),x-y\rangle\geq\rho\|T(x)-T(y)\|^{2}\ \forall x,y\in C.$

##  3\. Contraction and generalized nonexpansive properties of the proximal
mapping

Let $g:C\to\mathbb{R}$ be a convex function and $\lambda>0$. The proximal
mapping $P_{\lambda}$ with respect to $C,g,\lambda$ (shortly proximal mapping)
is defined as follows (see e.g. [38]):

$P_{\lambda}(x):=\text{argmin}\left\\{\lambda g(y)+\frac{1}{2}\|y-x\|^{2}:y\in
C\right\\}.$

For the bifunction $f$ where $f(x,\cdot)$ is convex and finite on $C$, the
proximal mapping $B_{\lambda}$ is defined by taking

$B_{\lambda}(x):=\text{argmin}\left\\{\lambda
f(x,y)+\frac{1}{2}\|y-x\|^{2}:y\in C\right\\}$

for each $x\in C$. Note that when either $f(x,\cdot)$ is continuous on $C$ or
$C$ has an interior point, and $f(x,x)=0$, then it is well known from [6] that
$x^{*}$ is a fixed point of $B_{\lambda}$ if and only if it is a solution to
Problem (EP).

The following theorem says that when $f$ is strongly monotone and satisfies
the Lipschitz-type on $C$, then one can choose a regularization parameter such
that the proximal mapping is quasicontractive on $C$. For applying optimality
condition for the problem defining the proximal mapping, we always assume that
either $C$ has an interior point or, for any $x\in C$, $\phi(x,\cdot)$ is
continuous with respect to $C$ at a point of $C$.

######  Theorem 1.

Suppose that $f$ is strongly monotone on $C$ with modulus $\tau$ and satisfies
the Lipschitz-type condition with constants $L_{1},L_{2}$ satisfying
$L_{1}+L_{2}>\tau$. Then, the proximal mapping $B_{\lambda}$ is
quasicontractive on $C$, namely

$\|B_{\lambda}(x)-x^{*}\|\leq\sqrt{\alpha}\|B_{\lambda}(x)-x^{*}\|\ \forall
x\in C,x^{*}\in{\rm Fix}(B_{\lambda}),$

whenever $\lambda\in(0,\frac{1}{2L_{2}})$, where $\alpha:=1-2\lambda(\tau-
L_{1})>0$.

###### Proof.

The following proof borrows some techniques from the one in [37]. For
simplicity of notation we let

$f_{x}(y):=\lambda f(x,y)+\frac{1}{2}\|y-x\|^{2}.$

Since $\lambda>0$ and $f(x,\cdot)$ is convex on $C$ by assumption, $f_{x}$ is
strongly convex with modulus 1. As defined, $B_{\lambda}(x)$ is a minimizer of
$f_{x}(\cdot)$ over the closed convex set $C$. Therefore, we have

$f_{x}(B_{\lambda}(x))+\frac{1}{2}\|B_{\lambda}(x)-x^{*}\|^{2}\leq
f_{x}(x^{*}),$

that is

$\lambda
f(x,B_{\lambda}(x))+\frac{1}{2}\|B_{\lambda}(x)-x\|^{2}+\frac{1}{2}\|B_{\lambda}(x)-x^{*}\|^{2}\leq\lambda
f(x,x^{*})+\frac{1}{2}\|x^{*}-x\|^{2},$

or equivalently

$\|B_{\lambda}(x)-x^{*}\|^{2}\leq
2\lambda\left(f(x,x^{*})-f(x,B_{\lambda}(x))\right)+\|x-x^{*}\|^{2}-\|B_{\lambda}(x)-x\|^{2}.$
(1)

Since $f$ is strongly monotone on $C$ with modulus $\tau$, it follows from (1)
that

$\displaystyle\ \|B_{\lambda}(x)-x^{*}\|^{2}$ $\displaystyle\leq$
$\displaystyle\
2\lambda(-\tau\|x-x^{*}\|^{2}-f(x^{*},x)-f(x,B_{\lambda}(x)))+\|x-x^{*}\|^{2}-\|B_{\lambda}(x)-x\|^{2}$
$\displaystyle\leq$ $\displaystyle\
(1-2\lambda\tau)\|x-x^{*}\|^{2}-2\lambda\left(f(x^{*},x)+f(x,B_{\lambda}(x))\right)-\|B_{\lambda}(x)-x\|^{2}.$
(2)

Since $f$ satisfies Lipschitz-type condition, we have

$f(x^{*},x)+f(x,B_{\lambda}(x))\geq
f(x^{*},B_{\lambda}(x))-L_{1}\|x^{*}-x\|^{2}-L_{2}\|x-B_{\lambda}(x)\|^{2}.$

Therefore, it follows from (2) that

$\displaystyle\ \|B_{\lambda}(x)-x^{*}\|^{2}$ $\displaystyle\leq$
$\displaystyle\ (1-2\lambda(\tau-L_{1}))\|x-x^{*}\|^{2}-(1-2\lambda
L_{2})\|x-B_{\lambda}(x)\|^{2}-2\lambda f(x^{*},B_{\lambda}(x))$
$\displaystyle\leq$ $\displaystyle\ (1-2\lambda(\tau-L_{1}))\|x-x^{*}\|^{2}.$

The last inequality is due to the fact that $f(x^{*},B_{\lambda}(x))\geq 0$
and the assumption that $0\leq\lambda\leq\frac{1}{2L_{2}}$. Since
$\tau<L_{1}+L_{2}$, we see that if $0<\lambda<\frac{1}{2L_{2}}$, then
$1-2\lambda(\tau-L_{1})<1$ and hence $B_{\lambda}$ is quasicontractive. ∎

Theorem 1 allows that the contraction iterative method can be used for solving
equilibrium problem (EP). A question arises here: under which condition, the
proximal mapping is contractive? The following theorem in [18] gives an answer
for this question.

For the statement of the theorem, first we recall from [18] that a bifunction
$f:C\times C\to\mathbb{R}$ is said to be strongly Lipschitz-type on $C$ if
there exist $\alpha_{i}:C\times C\to C$, $\beta_{i}:C\to C$, $K_{i},L_{i}>0$
(with $i=1,\ldots,p$) such that

$f(x,y)+f(y,z)\geq
f(x,z)+\sum_{j=1}^{p}\langle\alpha_{i}(x,y),\beta_{i}(y-z)\rangle\ \forall
x,y,z\in C,$

where

$\displaystyle\|\beta_{i}(x)-\beta_{i}(y)\|$ $\displaystyle\leq K_{i}\|x-y\|$
$\displaystyle\forall x,y\in C,i=1,\ldots,p,$
$\displaystyle\|\alpha_{i}(x,y)\|$ $\displaystyle\leq L_{i}\|x-y\|$
$\displaystyle\forall x,y\in C,i=1,\ldots,p,$
$\displaystyle\alpha_{i}(x,y)+\alpha_{i}(y,x)$ $\displaystyle=0$
$\displaystyle\forall x,y\in C,i=1,\ldots,p.$

As also remarked in [18], the following facts are not hard to see.

(i) If $f$ is strongly Lipschitz-type on $C$, then it is Lipschitz-type on $C$
with both constants $\frac{1}{2}\sum_{i=1}^{p}K_{i}L_{i}$.

(ii) If $f(x,y)=\langle F(x),y-x\rangle+\varphi(y)-\varphi(x)$, then $f$ is
strongly Lipschitz-type if and only if $F$ is Lipschitz.

######  Theorem 2.

Let $C$ a be nonempty closed convex set, $f:C\times C\to\mathbb{R}$. Suppose
that $f(x,\cdot)$ is lower semicontinuous, convex, $\gamma$-strongly monotone,
and strongly Lipschitz-type on $C$. Then the proximal mapping $B_{\lambda}$ is
contractive on $C$ whenever $\lambda\in(0,\frac{2\gamma}{M})$ with
$M=\sum_{i=1}^{p}K_{i}L_{i}.$

In the case of mixed variational inequality, when $f(x,y):=\langle
F(x),y-x\rangle+\varphi(y)-\varphi(x)$ with $F$ being Lipschitz continuous and
strongly monotone, the proximal mapping is contractive (see. e.g. [4]). This
result also follows from the above theorem due to the fact that
$f(x,y):=\langle F(x),y-x\rangle+\varphi(y)-\varphi(x)$ is strongly Lipschitz-
type on $C$ whenever $F$ is Lipschitz on $C$. In case that $F$ is co-coercive
(strongly inverse monotone), the proximal mapping is nonexpansive on $C$ as
stated in the following theorem.

######  Theorem 3.

Suppose that $f(x,y):=\langle F(x),y-x\rangle+\varphi(y)-\varphi(x)$ with $F$
being $\delta$-co-coercive (or strongly inverse monotone) on $C$. Then,
whenever $0<\lambda\leq\frac{1}{2\delta}$, the proximal mapping $B_{\lambda}$
is nonexpansive on $C$.

###### Proof.

From the definition of $B_{\lambda}$, by using the optimization condition, it
is easy to see that

$\|B_{\lambda}(x)-B_{\lambda}(y)\|^{2}\leq\|x-y-\lambda(F(x)-F(y))\|^{2}\
\forall x,y\in C.$ (3)

Since $F$ is co-coercive on $C$ with modulus $\delta$, we have

$\langle x-y,F(x)-F(y)\rangle\geq\delta\|F(x)-F(y)\|^{2},$

which implies

$\|x-y-\lambda(F(x)-F(y))\|\leq\|x-y\|.$

Thus by (3), we have

$\|B_{\lambda}(x)-B_{\lambda}(y)\|\leq\|x-y\|.$

∎

A question now may arise that is the proximal mapping $B_{\lambda}$
nonexpansive when $f$ is monotone? The following simple example gives a
negative answer.

Let us consider the linear variational inequality

$\text{Find}\ x\in\mathbb{R}^{2}\text{ such that }f(x,y):=\langle
Ax,y-x\rangle\geq 0\text{ for all }y\in\mathbb{R}^{2},$ (VI)

where

$A=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}.$

For all $x,y\in\mathbb{R}^{2}$ we have

$\displaystyle f(x,y)+f(y,x)$ $\displaystyle=\langle A(x-y),y-x\rangle$
$\displaystyle=(x_{2}-y_{2})(y_{1}-x_{1})-(x_{1}-y_{1})(y_{2}-x_{2})$
$\displaystyle=0,$

therefore $f$ is monotone on $\mathbb{R}^{2}$. It is easy to see that
$x^{*}=(0,0)^{t}$ is a solution to the variational inequality (VI), since
$f(x^{*},y)=0$ for all $y\in\mathbb{R}^{2}$. Furthermore, $x^{*}$ is the
unique solution to (VI). Indeed, if
$\overline{x}=(\overline{x}_{1},\overline{x}_{2})^{t}$ is a solution to (VI),
then

$\langle A\overline{x},y-\overline{x}\rangle\geq 0\quad\forall
y\in\mathbb{R}^{2}.$

By taking
$y=\overline{y}=(\overline{x}_{1}-\overline{x}_{2},\overline{x}_{1}+\overline{x}_{2})^{t}$,
we have

$0\leq\langle
A\overline{x},\overline{y}-\overline{x}\rangle=\left\langle\begin{bmatrix}\overline{x}_{2}\\\
-\overline{x}_{1}\end{bmatrix},\begin{bmatrix}-\overline{x}_{2}\\\
\overline{x}_{1}\end{bmatrix}\right\rangle=-(\overline{x}_{1}^{2}+\overline{x}_{2}^{2})\leq
0,$

which implies $\overline{x}=(0,0)^{t}=x^{*}$. Now we see that

$\displaystyle\lambda\langle Ax,y-x\rangle+\frac{1}{2}\|y-x\|^{2}$
$\displaystyle=$ $\displaystyle\lambda\left\langle\begin{bmatrix}x_{2}\\\
-x_{1}\end{bmatrix},\begin{bmatrix}y_{1}-x_{1}\\\
y_{2}-x_{2}\end{bmatrix}\right\rangle+\frac{1}{2}\left((y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}\right)$
$\displaystyle=$
$\displaystyle\lambda(x_{2}y_{1}-x_{1}y_{2})+\frac{1}{2}\left((y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(y_{1}^{2}-2(x_{1}-\lambda
x_{2})y_{1}+y_{2}^{2}-2(x_{2}+\lambda x_{1})y_{2}+x_{1}^{2}+x_{2}^{2}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left((y_{1}-x_{1}+\lambda
x_{2})^{2}+(y_{2}-x_{2}-\lambda
x_{1})^{2}-\lambda^{2}(x_{1}^{2}+x_{2}^{2})\right)$

which attains its minimum at $(y_{1},y_{2})=(x_{1}-\lambda x_{2},x_{2}+\lambda
x_{1})$. We obtain the following explicit formula for the proximal mapping of
(VI):

$B_{\lambda}(x)=\text{argmin}\\{\lambda\langle
Ax,y-x\rangle+\frac{1}{2}\|y-x\|^{2}\mid
y\in\mathbb{R}^{2}\\}=\begin{bmatrix}x_{1}-\lambda x_{2}\\\ x_{2}+\lambda
x_{1}\end{bmatrix}.$

Therefore, for any $\lambda>0$ we have

$\|B_{\lambda}(x)-B_{\lambda}(x^{*})\|=\left\|\begin{bmatrix}x_{1}-\lambda
x_{2}\\\ x_{2}+\lambda x_{1}\end{bmatrix}-\begin{bmatrix}0\\\
0\end{bmatrix}\right\|=\sqrt{1+\lambda^{2}}\sqrt{x_{1}^{2}+x_{2}^{2}}>\|x-x^{*}\|,$

which proves that $B_{\lambda}$ is not nonexpansive for any $\lambda>0$.

So in general the proximal mapping may not be nonexpansive even for the
variational inequality when $f(x,y)=\langle F(x),y-x\rangle$ with $F$ being
Lipschitz and monotone. However, it is well known from [14, 42] that if $f$ is
monotone, $f(x,\cdot)$ is convex, lower semicontinuous, and $f(\cdot,y)$ is
hemicontinuous, then the regularization proximal mapping $R_{\lambda}$ is
defined everywhere, single valued, and firmly nonexpansive for any
$\lambda>0$. Here, for each $x\in C$, $R_{\lambda}(x)$ is defined as the
unique solution of the strongly monotone equilibrium problem

$\text{Find }z\in C\text{ such that }f(z,y)+\frac{1}{2\lambda}\langle
y-z,z-x\rangle\geq 0\text{ for all }y\in C.$

Moreover, the solution set of (EP) coincides with the fixed point set of the
proximal mapping $R_{\lambda}$.

The main difference between the proximal mapping and the regularization
proximal mapping is that the former is defined as the unique solution of a
strongly convex program, while the latter is defined by the unique solution of
a strongly monotone equilibrium problem.

We adopt the following definition.

######  Definition 4.

For given $\epsilon>0$ and $\lambda>0$, the proximal mapping $B_{\lambda}$ is
said to be $\epsilon$-nonexpansive on $C$ if

$\|B_{\lambda}(x)-B_{\lambda}(y)\|^{2}\leq(1+\epsilon)\|x-y\|^{2}\ \forall
x,y\in C.$

The following theorem says that for monotone equilibrium problem, the proximal
mapping is $\epsilon$-nonexpansive.

######  Theorem 4.

Suppose that the bifunction $\phi$ is monotone and satisfies the strongly
Lipschitz-type condition on $C$. Then for any $\epsilon>0$, there exists
$\lambda>0$ such that the proximal mapping $B_{\lambda}$ for Problem (MEP) is
$\epsilon$-nonexpansive.

###### Proof.

As before we see that if $\phi$ is monotone, strongly Lipschitz-type, then so
is $f(x,y):=\phi(x,y)+\varphi(y)-\varphi(x)$ for any function
$\varphi:C\to\mathbb{R}$. It is well known (see. e.g. [30, 38]) that

$\langle
B_{\lambda}(x)-x,B_{\lambda}(x)-z\rangle\leq\lambda[f(x,z)-f(x,B_{\lambda}(x)]\
\forall x,z\in C.$

Applying this inequality with $z:=B_{\lambda}(y)$ we obtain

$\langle
B_{\lambda}(x)-x,B_{\lambda}(x)-B_{\lambda}(y)\rangle\leq\lambda[f(x,B_{\lambda}(y))-f(x,B_{\lambda}(x))]\
\forall x,y\in C.$

Similarly with $B_{\lambda}(y)$, we have

$\langle
B_{\lambda}(y)-y,B_{\lambda}(y)-B_{\lambda}(x)\rangle\leq\lambda[f(y,B_{\lambda}(x))-f(y,B_{\lambda}(y))]\
\forall x,y\in C.$

Adding the two obtained inequalities we get

$\displaystyle\langle
B_{\lambda}(x)-B_{\lambda}(y)+y-x,B_{\lambda}(x)-B_{\lambda}(y)\rangle$
$\displaystyle\leq\ $
$\displaystyle\lambda[f(x,B_{\lambda}(y))-f(x,B_{\lambda}(x)+f(y,B_{\lambda}(x))-f(y,B_{\lambda}(y))].$

By simple arrangements we obtain

$\displaystyle\|B_{\lambda}(x)-B_{\lambda}(y)\|^{2}$ $\displaystyle+\ $
$\displaystyle
2\lambda\Big{[}f(x,B_{\lambda}(y))-f(x,B_{\lambda}(x)+f(y,B_{\lambda}(x))-f(y,B_{\lambda}(y))\Big{]}$
$\displaystyle\leq\ $ $\displaystyle\|x-y\|^{2}$

Now using the strongly Lipschit-type condition, by the same argument as in the
proof of Theorem 3.7 in [18] we arrive at the following inequality

$\|B_{\lambda}(x)-B_{\lambda}(y)\|^{2}\leq(1+\lambda^{2}M)\|x-y\|^{2},$

where $M=\sum_{j=1}^{p}K_{i}L_{i}$, with $K_{j}$ and $L_{j}$ being the
Lipschitz constants defined in the strongly Litschitz-type. Hence, with
$0<\lambda^{2}<\frac{\epsilon}{M}$, we obtain
$\|B_{\lambda}(x)-B_{\lambda}(y)\|^{2}\leq(1+\epsilon)\|x-y\|^{2}$ for every
$x,y\in C$. ∎

######  Corollary 1.

Consider the mixed variational inequality

$\text{Find }x^{*}\in C\text{ such that }\langle
F(x^{*}),y-x^{*}\rangle+\varphi(y)-\varphi(x^{*})\geq 0\text{ for all }y\in
C.$ (MVI)

Suppose that $F$ is monotone and Lipschitz on $C$. Then the proximal mapping
$B_{\lambda}$ defined by the bifunction $\langle
F(x),y-x\rangle+\varphi(y)-\varphi(x)$ is $\epsilon$-nonexpansive for any
$\epsilon>0$.

###### Proof.

Since $F$ is monotone and Lipschitz on $C$, the bifunction $f(x,y):=\langle
F(x),y-x\rangle+\varphi(y)-\varphi(x)$ is strongly Lipschitz and monotone.
Thus the corollary follows directly from Theorem 4. However one can prove this
result simply as follows.

From the definition of $B_{\lambda}$, by using the optimality condition for
the problem defining $B_{\lambda}$ we can show

$\|B_{\lambda}(x)-B_{\lambda}(y)\|\leq\|x-y-\lambda(F(x)-F(y))\|\ \forall
x,y\in C.$ (4)

Then, from

$\|x-y-\lambda(F(x)-F(y))\|^{2}=\|x-y\|^{2}-2\lambda\langle
x-y,F(x)-F(y)\rangle+\lambda^{2}\|F(x)-F(y)\|^{2}$

by (4) and monotonicity of $F$ we can write

$\|B_{\lambda}(x)-B_{\lambda}(y)\|^{2}\leq\|x-y\|^{2}+\lambda^{2}\|F(x)-F(y)\|^{2},$

from which, by Lipschitz continuity of $F$, it follows that

$\|B_{\lambda}(x)-B_{\lambda}(y)\|^{2}\leq(1+L^{2}\lambda^{2})\|x-y\|^{2}.$

Hence the mapping $B_{\lambda}$ is $\epsilon$-nonexpansive on $C$ whenever
$\lambda^{2}L^{2}\leq\epsilon$. ∎

Now a natural question may arise: how to modify the proximal mapping for
monotone equilibrium problems such that it has a generalized nonexpansiveness
property? In order to answer this question, let us define the mapping
$T_{\lambda}$ from $C$ to itself by taking, for every $x\in C$,

$T_{\lambda}(x):=\text{argmin}\Big{\\{}\lambda
f(B_{\lambda}(x),y)+\displaystyle\frac{1}{2}\|y-x\|^{2}:y\in C\Big{\\}}$

where $\lambda$ is a fixed positive number.

######  Theorem 5.

([5]). Let $f:C\times C\to\mathbb{R}$ be a bifunction such that $f(x,\cdot)$
is subdifferentiable, pseudomonotone and Lipschitz-type on $C$. Suppose that
the following conditions are satisfied:

(A1) $f$ is jointly weakly continuous on $C\times C$ in the sense that, if
$x,y\in C$ and $\\{x_{n}\\},\\{y_{n}\\}\subset C$ converge weakly to $x$ and
$y$, respectively, then $f(x_{n},y_{n})\to f(x,y)$ as $n\to\infty$.

(A2) The solution set of Problem (EP) is nonempty.
Then the mapping $T_{\lambda}$ is quasi-nonexpansive on $C$ if
$0<\lambda<\min\left\\{\displaystyle\frac{1}{2L_{1}},\displaystyle\frac{1}{2L_{2}}\right\\}$.
In addition, it is demiclosed at zero, in the sense that for every sequence
$\\{x_{n}\\}$ contained in $C$ weakly converging to $x$ and
$\|T(x_{n})-x_{n}\|\to 0$, then $x\in Fix(T)$.

By this theorem, the algorithms for finding a fixed point of quasi-
nonexpansive mappings (see e.g. [17, 21]) can be used for solving
pseudomonotone equilibrium problems.

A disadvantage of the composite proximal mapping $T_{\lambda}$ is that for
evaluating it at a point, it requires solving two strongly convex programming
problems. An open question is that how to define a nonexpansive or
$\epsilon$-nonexpansive mapping with any $\epsilon>0$ for pseudomotone
equilibrium problems, which requires solving only one strongly convex
programs?

As we have seen from the definition of the proximal mapping that when applying
the iterative fixed point methods for solving mixed equilibrium problem (MEP),
at an iterative point $x^{k}\in C$, we have to solve a strongly convex program
of the form

$\min\left\\{f(x^{k},y):=\phi(x^{k},y)+\varphi(y)-\varphi(x^{k})+\frac{1}{2\lambda}\|y-x^{k}\|^{2}:y\in
C\right\\}.$ $None$

This problem can be solved by efficient algorithms of convex programming (see
[12]).

##  4\. Conclusions

The mixed equilibrium problem (MEP) and the regularized Moreau proximal
mapping $B_{\lambda}$ defined for it are equivalent in the following senses:

(i) The solution set of (MEP) coincides with the fixed point set of
$B_{\lambda}$ for any $\lambda>0$.

In addition:

(ii) If (MEP) is strongly monotone and satisfies the Lipschitz-type condition,
then one can choose $\lambda$ such that $B_{\lambda}$ is quasicontractive.

(iii) If (MEP) is strongly monotone and satisfies the strongly Lipschitz-type
condition, then one can choose $\lambda$ such that $B_{\lambda}$ is
contractive.

(iv) If (MEP) is monotone and satisfies the strongly Lipschitz-type, then one
can choose $\lambda$ such that $B_{\lambda}$ is $\epsilon$-nonexpansive for
any $\epsilon>0$.

(v) If (MEP) is pseudomonotne and satisfies the Lipschitz-type, then the
composite proximal mapping is quasinonexpansive, and its fixed point-set
coincides the solution-set of Problem (MEP).

Applications to mixed variational inequality problems with Lipschitz cost
operator have been presented.

The following question seems to be interesting: How to extend these results
for the equilibrium problem when the bifunction involved is quasiconvex with
respect to its second variable?

## Acknowledgements

This research is funded by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under grant number 101.01-2020.06.

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# Traffic Flow Estimation using LTE Radio Frequency Counters and Machine
Learning

Forough Yaghoubi  Ericsson ABStockholmSweden<EMAIL_ADDRESS>,
Armin Catovic  Schibsted Media GroupStockholmSweden
<EMAIL_ADDRESS>,  Arthur Gusmao  Ericsson ABStockholmSweden
<EMAIL_ADDRESS>,  Jan Pieczkowski  Ericsson ABStockholmSweden
<EMAIL_ADDRESS>and  Peter Boros  Ericsson ABStockholmSweden
<EMAIL_ADDRESS>
###### Abstract.

As the demand for vehicles continues to outpace construction of new roads, it
becomes imperative we implement strategies that improve utilization of
existing transport infrastructure. Traffic sensors form a crucial part of many
such strategies, giving us valuable insights into road utilization. However,
due to cost and lead time associated with installation and maintenance of
traffic sensors, municipalities and traffic authorities look toward cheaper
and more scalable alternatives. Due to their ubiquitous nature and wide global
deployment, cellular networks offer one such alternative. In this paper we
present a novel method for traffic flow estimation using standardized LTE/4G
radio frequency performance measurement counters. The problem is cast as a
supervised regression task using both classical and deep learning methods. We
further apply transfer learning to compensate that many locations lack traffic
sensor data that could be used for training. We show that our approach
benefits from applying transfer learning to generalize the solution not only
in time but also in space (i.e., various parts of the city). The results are
very promising and, unlike competing solutions, our approach utilizes
aggregate LTE radio frequency counter data that is inherently privacy-
preserving, readily available, and scales globally without any additional
network impact.

Intelligent Transportation Systems, Traffic Flow, LTE, Radio Frequency,
Machine Learning, Transfer Learning

††copyright: none††doi: ††isbn: ††conference: ; 2021††journalyear: ;††price:

##  1\. Introduction

The increasing number of vehicles in the public roadway network, relative to
the limited construction of new roads, has caused recurring congestion in the
U.S. and throughout the industrialized world (Lawrence A. Klein, 2006). In the
U.S. alone, the total cost of lost productivity caused by traffic congestion
was estimated at $87 billion in 2018 (Fleming, [n.d.]). While one solution is
to build new and expand existing roads, this is costly and takes time. A
complementary approach is to implement strategies that improve the utilization
of existing transport infrastructure. These strategies are found in
Intelligent Transportation Systems (ITS) roadway and transit programs that
have among their goals reducing travel time, easing delay and congestion,
improving safety, and reducing pollutant emissions (Lawrence A. Klein, 2006).

Traffic flow sensor technology forms a key component of ITS. Traffic sensors
can be categorized as in-roadway (e.g. inductive loop sensors and
magnetometers), or over-roadway (e.g. traffic cameras, radar, infrared and
laser sensors). More recently there has been a surge of ad-hoc over-roadway
sensor technology, including road-side cellular network masts, Bluetooth and
Wi-Fi sensors, as well as telemetry collected from connected vehicles,
smartphones and GPS devices. Cellular network masts are particularly
appealing, combining ubiquity of cellular network technology (e.g. LTE or more
specifically E-UTRA), with strict high availability requirements.

While there have been previous approaches in utilizing cellular networks for
understanding traffic flow, they’ve either been intrusive due to using user
data, or not practical from a network operations perspective. In this paper we
present a novel method for traffic flow estimation that leverages standard
LTE/E-UTRA performance management (PM) counters, as defined by the 3rd
Generation Partnership Project (3GPP) (3GPP, 2019). Namely, we utilize two
radio frequency (RF) measurements - path loss distribution, and timing advance
distribution counters, aggregated over 15 minute intervals. These counters are
inherently privacy-preserving, and are continuously collected by nearly all
LTE networks around the world, independent of network vendor. Thus our
solution is non-invasive, and highly practical as it can be scaled across vast
geographic regions with no live network impact.

Our contributions in this paper are threefold:

  1. (1)

We present a novel method for estimating traffic flow using classical and deep
learning regression models trained on E-UTRA RF counters (features) and
vehicle counts from actual traffic sensors (targets).

  2. (2)

We evaluate the performance of our models by applying the learned model to
different time samples, referred to as temporal generalization in this paper.

  3. (3)

We evaluate the performance of our models by applying the learned model to
different road segments lacking ground truth data, referred to as spatial
generalization in this paper; it is shown that due to difference in traffic
distribution, the performance of our models is sub-optimal, hence we improve
the accuracy using two transfer learning approaches.

The rest of the paper is organized as follows: in section 2 we summarize the
related works in the area of traffic flow estimation; our overall solution
including feature selection/transformation and learning algorithms is
explained in section 3; section 4 introduces two different transfer learning
approaches; in section 5 we evaluate the performance of our models in terms of
temporal and spatial generalization; ethical aspects are considered in section
6; finally the key takeaways are summarized in section 7.

##  2\. Related Works

Traffic flow estimation has and continues to be a popular research topic. In
(Zewei et al., 2015; George et al., 2013; Ma et al., 2013; Haferkamp et al.,
2017; Nam et al., 2020) traffic flow estimation approaches are presented using
data gathered from different sources such as cameras, acoustic sensors,
magnetometers and spatially separated magnetic sensors. These solutions are
not efficient due to coverage limitations and effort required in terms of
installation and maintenance. To cope with these problems (Hansapalangkul et
al., 2007; Pattara-Atikom and Peachavanish, 2007; Hongsakham et al., 2008;
Caceres et al., 2012; Xing et al., 2019; Ji and Hong, 2019; Wang et al., 2020)
propose the use of mobile subscriber data in traffic flow estimation. In
(Hansapalangkul et al., 2007; Pattara-Atikom and Peachavanish, 2007;
Hongsakham et al., 2008) the cell dwelling time and global positioning system
(GPS) coordinates of a mobile subscriber are used to estimate the traffic
congestion. These methods however have the disadvantage of high power
consumption on a mobile device due to constant use of GPS, and are inherently
intrusive. Another type of cellular data is considered in (Caceres et al.,
2012) where authors propose a traffic flow estimation algorithm based on the
number of subscribers in cars making a voice call. With today’s heavy usage of
streaming and social media services however, voice calls are hardly
representative of the traffic density, which limits the accuracy of such an
approach. The authors in (Xing et al., 2019) use the travel trajectory of
different mobile subscribers to detect in-vehicle users and henceforth compute
the number of vehicles on a specific road. Tracking individual mobile users
however is highly contentious and in most countries any user-identifiable or
user-sensitive information limits the real-time usage of such data. In more
recent work (Ji and Hong, 2019), the authors propose a method to predict the
traffic speed and direction using wireless communication access logs including
S1 application protocol (S1AP) collected from multiple radio base stations
(RBS) located within a predetermined distance from the road. Due to high-
intensity nature of S1AP signalling, tracing on the S1 interface in every
single RBS leads to increase in network load, which is undesirable as it could
lead to network overload with potentially catastrophic consequences.
Furthermore, S1AP exposes potentially sensitive subscriber information
allowing for the user to be fingerprinted or tracked. Another approach as
presented in (Wang et al., 2020), describes a ”data fusion” approach, i.e.
combining taxi GPS data with vehicle counts from license plate recognition
(LPR) devices. The scalability of such approach is constrained due to limited
availability of LPR and taxi GPS data.

Therefore in this paper, we propose a new solution to traffic flow estimation
problem based on aggregate LTE/E-UTRA radio frequency counter data that is
inherently privacy preserving, readily available, and does not impose any
extra load on the network.

##  3\. Method

In this paper we describe two different approaches to using E-UTRA RF counters
for traffic flow estimation. One approach involves using uplink path loss
distribution as a feature vector in our model. Here we reason that different
number of vehicles, i.e. obstacles on the road, can be represented by
different path loss distributions. By optimizing for the number of vehicles
the model should be able to discriminate between vehicles and all other users
in the vicinity.

In the second approach we use radio propagation delay, or more specifically
timing advance (TA), as our feature vector. LTE radio base stations (eNBs)
estimate the propagation delay on every random access (RA) initiated by a
user. These propagation delays are aggregated from all RAs and represented as
a distribution over discretized distances, where every bin represents a
certain distance range from the eNB. By selecting only the bins corresponding
to the known distances between the eNB and the relevant road segments, we can
directly capture the road users, i.e. vehicles.

Figure 1. Traffic flow estimation system presented in this paper.

Path loss and timing advance features are described in detail in section 3.1.
In both cases we use supervised regression techniques to train and evaluate
our models. Fig. 1 shows the high level view of our system. In this paper we
work in two domains: the source domain consists of training and validation
data - the feature and target variables; the second domain, referred to as
target domain, is where we perform inference using only the feature variables
- however we use ground truth data for evaluation purposes.

Our solution depends on the following assumptions:

  * •

The eNB, or more specifically the sector antenna, is located within line-of-
sight (LOS) of the relevant road segment,

  * •

The distance between the road segment and the sector antenna is known,

  * •

The relevant road segment consists of predominantly vehicular traffic,

  * •

Traffic sensors used to supply ground truth data completely capture the
traffic flow along the relevant road segment.

In the following subsections we describe our feature and target variables, and
learning algorithms.

###  3.1. Feature and Target Variables

Figure 2. An example of a laser-based traffic sensor used in this paper;
photograph by Holger Ellgaard, distributed under a CC BY-SA 3.0 license.

####  3.1.1. Traffic Sensor Data

Target variables, i.e. ground truth data, consist of total vehicle counts
aggregated over 15-minute intervals. The data is collected from a number of
laser-based traffic sensors around inner Stockholm. Fig. 2 shows an example of
such a sensor. Each sensor scans one lane of the road. For each road segment
we sum the values from each lane to obtain total vehicle counts per 15-minute
interval. We remove any samples where one of the sensor’s values are missing
(e.g. due to a malfunction).

####  3.1.2. Path Loss Features

Path loss (PL) is the attenuation of electromagnetic wave caused by free-space
losses, absorption (e.g. by atmospheric particles), and scattering off various
obstacles and surfaces. Radio propagation models attempt to account for this
attenuation, and are a pivotal component in cellular network planning. Hata-
Okamura (Hata, 1980) models are one such family of radio propagation models
used to approximate cellular network coverage in different environments. In
LTE, eNBs estimate PL values for all the scheduled users on every transmit
time interval (TTI), which is typically 1ms. These estimates, represented as
decibel (dB) values are then placed into discretized PL bins; in our case we
have 21 bins, where each bin covers a range of 5dB, starting from $<$50dB and
going up to $>$140dB. These estimates are done per-frequency band; in our case
we have three bands, 800MHz, 1800MHz (two separate antennas working in this
band) and 2600MHz, so we concatenate PL bins for all bands, resulting in total
of 4 x 21 = 48 PL features. We don’t apply any filtering or transformation to
PL features, and we treat all PL bins equally. Even though PL estimates are
done every 1ms, the actual data available to us is aggregated in 15-minute
intervals.

The motivation behind our use of PL features is that different traffic
conditions will result in different radio wave scattering characteristics,
leading to different path loss distributions. A condition where there are no
vehicles on the road will be represented by path loss distribution $PL_{a}$,
which would be representative of radio wave losses due to predominantly indoor
users and pedestrians. On the other hand a condition where the traffic flow is
greater than zero would result in path loss distribution $PL_{b}$ where
$PL_{b}\neq PL_{a}$, since radio wave scattering off vehicle surfaces would
yield a different path loss ”signature”. Our trained algorithms should be able
to discriminate between such conditions.

Figure 3. Spatial granularity of an eNB.

####  3.1.3. Timing Advance Features

Timing advance (TA) is estimated for every user connection request, or more
specifically on every random access. TA estimation is dependent on successful
completion of an RRC Connection Request procedure, and the 11-bit TA command.
The name is a slight misnomer, since TA features are actually represented as
discretized distance bins/ranges, representing the distance between the user
and the sector antenna. In our case we have 35 bins, starting from $<$80m up
to 100km; typically only the first few bins are incremented, as users are
normally within 500m of the antenna (otherwise they will be handed over to
another sector, or another eNB). Fig. 3 shows the spatial granularity of an
eNB and how the TA features may be represented.

Unlike PL features, we do actually apply a distance selection filter to TA
features. Our aim is to consider only road users (vehicles), which means
selecting TA bins/features that represent the known distances between the
relevant road segment and the sector antenna. Lets assume that TA value ranges
are indicated by $M$ bins where bin $ta_{i}$ corresponds to a distance
interval shown by $[d_{i};d_{i+1})$ given a known distance $d$ between the
road segment and the sector antenna, we choose the TA bin index $ta_{i}$ where
$d_{i}\leq d\leq d_{i+1}$.

Just like traffic sensor data and PL features, TA features are also aggregated
in 15-minute intervals.

####  3.1.4. Cyclic Time Features

As traffic exhibits strong seasonality, it is beneficial to give our models
temporal information. To encode this information, a common method is to
transform the date-time representation into cyclic time features using a
$\sin$ and $\cos$ transformation as follows:

$\displaystyle x_{\sin}=$ $\displaystyle\sin(\frac{2\pi x}{\max(x)})$
$\displaystyle x_{\cos}=$ $\displaystyle\cos(\frac{2\pi x}{\max(x)})$

where $x$ can be hour, day and month. By using the above equation, we convert
time-of-day, day-of-week and week-of-month to the corresponding cyclic time
features. As the time granularity for our data is in minutes, we set the
$\max(x)$ to $24*60$, $7*24*60$, and $4*7*24*60$ respectively.

####  3.1.5. Road-dependent Features

As traffic flow depends on the road characteristics, we also apply different
road-dependent features. These features are easily extracted from e.g.
OpenStreetMap services. In this paper we use the following road-dependent
features: number of lanes, maximum speed limit, and road category, i.e.
highway, large city road and small city road.

###  3.2. Learning Algorithms

In this paper we compare two supervised learning approaches for traffic flow
estimation. In the first approach we evaluate a number of different classical
regression algorithms. In the second approach we take into account the history
of time samples using gradient based Long Short-term Memory (LSTM).

####  3.2.1. Classical Regression Models

Classical regression assumes independence between time samples. Since we’re
working with fairly coarse 15-min aggregate intervals, it is reasonable to
assume this independence. Regression then amounts to estimating a function
$f(\bm{x};\bm{\theta})$, which transforms a feature vector $\bm{x}$ to a
target variable $y$. Function parameters
$\bm{\theta}=(\theta_{0},\theta_{1},...,\theta_{k})$ are found by minimizing
expected loss, typically a mean squared error (MSE) of the form
$MSE(\bm{\theta})=\frac{1}{N}\sum_{n=1}^{N}(y_{n}-f(\bm{x_{n}};\bm{\theta}))^{2}$,
where $N$ corresponds to total number of 15-min aggregate samples. We evaluate
a number of different regression algorithms including Support Vector Regressor
(SVR), Kernel Ridge (KR), Decision Tree (DT), and Random Forest (RF). Each
algorithm also requires setting its internal parameters, or hyperparameters.
Since the total number of hyperparameters is small, we use grid search method
to exhaustively search through the hyperparameter space and pick the
combination of parameters that yield the best performance. We apply a time-
dependent train/test split, e.g. by selecting the first 6 weeks for training,
and the following 2 weeks for testing; compared to a random assignment of
train/test data, our approach is more in line with how the algorithm would be
used in practice, and is more representative of the generalization capability
in the real-world setting.

####  3.2.2. LSTM

LSTM is a specific kind of recurrent neural network (RNN) that has the ability
to capture long-term time dependencies and bridge time intervals in excess of
1000 steps even in case of noisy, in-compressible input sequences (Du et al.,
2017). Similar to other types of RNNs, LSTM has a chain structure with
modified repeating modules. In each module, instead of having a single neural
network layer, there are four layers that interact with each other. More
detailed information about LSTM architecture can be found in (Smagulova and
James, 2019).

The architecture of our LSTM based traffic flow estimator consists of two LSTM
layers, followed by a dropout regularization layer, and then finally the two
fully-connected (FC) layers. The two LSTM, as well as the two FC layers, use
the rectified linear unit (ReLU) activation function, while the output layer
activates with the linear function.

##  4\. Transfer Learning Approaches

The learning approaches mentioned above optimize the model for temporal
generalization where we use all available locations in our training set, but
withhold a contiguous period of time (e.g. two weeks) for test purposes.
However, we would like our models to generalize well across all possible
locations, even never-before-seen locations, which may potentially have
completely different traffic patterns/distributions. We refer to this problem
as spatial generalization. To cope with this problem we use transfer learning
(TL) approaches. TL focuses on transferring the knowledge between different
domains and can be a promising solution to overcome the spatial generalization
problem. Recently, there has been lot of work focusing on transfer learning
and proposing efficient solutions (Zhuang et al., 2019; Pan and Yang, 2009;
Tan et al., 2018). These studies categorize TL into three subcategories based
on different situations involving source and target domain data and the tasks,
including inductive, transductive, and unsupervised transfer learning. Our
work can be fitted into transductive transfer learning where the source label
data are available while no label data for target domain is provided. Here the
assumption is that the task between target and source domain is the same, but
the domain marginal or conditional distributions are different. Among the
proposed transductive TL algorithms, we evaluate two approaches - one based on
instant weighting and the second one based on deep domain adaptation. We
explain each of the algorithms in detail in the following sections.

###  4.1. Instant Weighting

The data-based TL approaches, such as instant weighting, focus on transferring
the knowledge by adjustment of the source data. Assuming that the source and
target domain only differ in marginal distribution, a simple idea for
transformation is to assign weights to source domain data equal to the ratio
of source and target domain marginal distribution. Therefore the general loss
function of the learning algorithm is given by:

(1)
$\min_{\theta}\frac{1}{N_{s}}\sum_{1}^{N_{s}}\alpha_{i}J(\theta(x_{i}^{s}),y_{i}^{s})+\lambda\gamma(\theta)$

where $J$ represents the loss of source data and $\alpha_{i}$ is the weighting
parameter and is equal to:

(2) $\alpha_{i}=\frac{P^{T}(x)}{P^{S}(x)}.$

In the literature, there exist many ways to compute $\alpha_{i}$; in (Huang et
al., 2006) the authors used Kernel Mean Matching (KMM) to estimate the ratio
by matching the means of target and source domain data in the reproducing-
kernel Hilbert space where the problem of finding weights can be written as
follows:

(3) $\displaystyle\min_{\alpha}\frac{1}{2}\alpha^{T}K\alpha-\kappa\alpha$
$\displaystyle s.t\sum_{i}^{N_{s}}\alpha_{i}-N_{s}\leq N_{s}\epsilon$
$\displaystyle\alpha\in[0,B]$

where $N_{s}$ shows the number of sample in source domain data and $K$ is
kernel matrix and is defined as:

(4) $K=\begin{bmatrix}k_{ss}&k_{st}\\\ k_{ts}&k_{tt}\end{bmatrix},$

while $k_{ss}=k(x_{s},x_{s})$ and
$\kappa_{i}=\frac{N_{s}}{N_{T}}\sum_{i}^{N_{T}}k(x_{i},x_{Tj})$.

###  4.2. Domain Adaptation

Deep learning algorithms have received lot of attention from researchers
having successfully outperformed many traditional machine learning methods in
tasks such as computer vision and natural language processing (NLP). Therefore
in the TL area many researchers also utilize deep learning techniques.

In this paper, we use discrepancy-based domain adaptation, where a deep neural
network is used to learn the domain-independent feature representations. In
deep neural networks, the early layers tends to learn more generic
transferable features, while domain-dependent features are extracted in the
terminal layers. Therefore, to decrease the gap between the distribution in
the last layers, we add multiple adaptation layers with discrepancy loss as
regularizer.

The deep learning model used for feature extraction is the LSTM model
explained in previous section. The pretrained LSTM model will be used to
extract the features for both source and target domains. After that the
primary goal is to reduce the difference between target and source domain
distribution. The term maximum mean discrepancy (MMD) is widely used in TL
literature as a metric to compute the distance between two distribution (Wang
et al., 2017; Gretton et al., 2012). Fig. 4 shows the architecture of our
domain adaptation network based on LSTM.

Figure 4. The domain adaptation network based on LSTM.

Let $f$ denote the function for feature representation of our pretrained
model, then the distance between the feature distribution of source and target
domain is given by:

(5) $d(p,q)=\sup_{f\in F}{E_{p}\\{f(x)\\}-E_{q}\\{f(y)\\}}$

where $\sup$ defines the supremum, $E$ denotes the expectation and $x$ and $y$
are independently and identically distributed (i.i.d) samples from $p$ and
$q$, respectively. The above equation can be easily computed using the kernel
trick where it can be expressed by expectation of kernel functions. Therefore,
the square of equation (5) can be reformulated as follows:

(6)
$d^{2}_{k}(p,q)=E_{x_{p}^{s}x_{p}^{s}k(x_{p}^{s},x_{p}^{s})}+E_{x_{q}^{t}x_{q}^{t}k(x_{q}^{t},x_{q}^{t})}-2E_{x_{p}^{s}x_{q}^{t}k(x_{p}^{s},x_{q}^{t})},$

where $x_{p}^{s}$ and $x_{q}^{t}$ are the samples from source and target
domain respectively, and $k$ is the kernel defined as $\exp(\frac{-\left\lVert
x_{i}-x_{j}\right\rVert^{2}}{\gamma})$.

To adapt the pretrained model for the target data samples, the objective
function of our TL algorithm is given by (Long et al., 2015):

(7)
$\min_{\theta}\frac{1}{N_{s}}\sum_{1}^{N_{s}}J(\theta(x_{i}^{s}),y_{i}^{s})+\lambda\sum_{l=l_{1}}^{l_{2}}d_{k}^{2}(D^{s}_{l},D^{t}_{l}),$

where $J$ is the loss for source domain in LSTM network, $l_{1}$ and $l_{2}$
indicate the layer indices between which the regularization is effective, and
$D^{s}_{l}$ and $D^{t}_{l}$ are $l$ layer representation of the source and
target samples, respectively. The parameter $\lambda$ is a trade off term so
that the objective function can benefit both from TL and deep learning.

##  5\. Results

We use approximately 8 weeks worth of data, where every data sample
corresponds to a 15-min interval, so we have $\scriptstyle\sim$ 96 * 7 * 8 =
5376 data samples. The data are collected from six different locations around
inner Stockholm; each location corresponds to a road segment with a traffic
sensor and a nearby LTE eNB. We evaluate models using PL and TA features
independently and across a range of regression algorithms. When evaluating
temporal generalization we use all locations during training and split the
data into 80/20 train/test sets, which corresponds to approximately 6 weeks of
contiguous training data, and 2 weeks of test data. When evaluating spatial
generalization we use all time samples for training but we randomly assign
road segments into source and target domains. For evaluation purposes we use
coefficient of determination $R^{2}$ defined as follows:

(8) $R^{2}=1-\frac{SS_{res}}{SS_{tot}}$

where

$\displaystyle SS_{tot}=\sum_{i}\left(y_{i}-\bar{y}\right)^{2}$ (9)
$\displaystyle SS_{res}=\sum_{i}\left(y_{i}-\hat{y}_{i}\right)^{2}$

$SS_{tot}$ represents total sum of squares, and $SS_{res}$ represents residual
sum of squares, while $\bar{y}$ and $\hat{y_{i}}$ are the observed data mean
and the predicted traffic flow respectively. A model that always predicts
observed data mean will have $R^{2}=0$; models with observations worse than
the observed data mean will have negative values; the most optimal value is
$R^{2}=1$, so we want our models to be as close to 1 as possible.

The set of classical regression algorithms used for training are Support
Vector Regression (SVR), Kernel Ridge (KR), Decision Trees (DT) and Random
Forest (RF). We also train a deep learning model with two LSTM layers followed
by a dropout layer and two fully-connected layers activated with the ReLU
function. The hyperparameters providing the best $R^{2}$ score on the test set
for our models are found using grid search and presented in Table 1. The
corresponding results for both temporal and spatial generalization performance
are shown in Table 2.

Table 1. Hyperparameters used in this paper. Models | Parameters
---|---
| Kernel = rbf
SVR |  $C$ = 10
|  $\gamma$ = 0.001
| Kernel = rbf
KR |  $\alpha$ = 1
|  $\gamma$ = 0.01
DT | Maximum depth = 10
RF | Maximum depth = 30
| Learning rate = 0.0009
LSTM | Hidden size = 100
| Epochs = 300
| Dropout rate = 0.2
| Window = 5
Table 2. $R^{2}$ score for temporal and spatial generalization performance using TA and PL features. Higher scores are better. Models | Temporal Generalization | Spatial Generalization
---|---|---
TA | PL | TA | PL
SVR | 0.754 | 0.786 | 0.12 | -0.62
KR | 0.862 | 0.888 | -0.79 | -0.63
DT | 0.938 | 0.946 | -3.22 | -0.37
RF | 0.946 | 0.959 | -0.96 | 0.017
LSTM | 0.845 | 0.901 | 0.087 | -1.67

The results in Table 2 indicate that all regression algorithms perform
reasonably well in terms of temporal generalization, using either TA or PL
features. The Random Forest (RF) model outperforms all the others, including
the LSTM model, with an average $R^{2}$ score of 0.95. These results validate
our initial assumption that due to a fairly coarse 15-min aggregate interval,
it is safe to assume independence between time steps, hence deep learning
based LSTM does not add any additional value. A more visual representation of
the RF algorithm performance is shown in Fig. 5 where we compare traffic flow
estimates from our model against the actual values across three different
locations. The algorithm does not always capture the peaks - our hypothesis is
that more training samples with varied traffic flow distributions are needed
for the model to generalize even better.

(a) (b) (c)

Figure 5. Traffic flow estimates in terms of number of vehicles per 15-min
interval using our Random Forest (RF) model compared to the ground truth, for
5(a) Road 1, 5(b) Road 2 and 5(c) Road 3.

Despite good temporal generalization performance, the average $R^{2}$ score
for spatial generalization is very low for all regression models. This poor
performance is due to inherent difference between the source and target domain
distributions. In order to improve spatial generalization we use two types of
transfer learning (TL) algorithms, namely instant weighting and deep domain
adaptation.

In the first approach we implement the instant weighting for classical
regression. For each test location, we compute the weights solving the
quadratic optimization problem, and then retrain the model using these
weights. Since the RF model yields the highest $R^{2}$ score on temporal
generalization we apply instant weighting to RF only.

Table 3 presents the $R^{2}$ scores of RF model for both TA and PL features
with and without applying the instant weighting. The results indicate that
instant weighting can only improve the performance when TA features are used.
Since the TA features represent the road users more explicitly we expect there
to be some minimum similarity between all domain distributions. On the other
hand PL represents all users, including indoor users, and therefore PL
features are highly sensitive to physical layout of the environment, i.e.
number of buildings, thickness of walls, heights of buildings etc.

Table 3. $R^{2}$ score for spatial generalization using the Random Forest (RF) model with and without transfer learning (TL). Higher scores are better. Test Road | PL Features | TA Features
---|---|---
No TL | TL | No TL | TL
1 | 0.02 | -0.47 | -0.96 | 0.71
2 | 0.02 | -0.75 | -0.96 | 0.72
3 | 0.02 | -0.86 | -0.96 | 0.42
Mean | 0.02 | -0.69 | -0.96 | 0.62
Table 4. $R^{2}$ score for spatial generalization using the LSTM model with and without transfer learning (TL). Higher scores are better. Test Road | PL Features | TA Features
---|---|---
No TL | TL | No TL | TL
1 | -0.20 | 0.24 | 0.24 | 0.66
2 | -1.73 | -1.02 | -0.62 | 0.61
3 | -3.09 | -0.52 | -0.13 | 0.61
Mean | -1.67 | -0.43 | -0.17 | 0.63

In the second approach, we implement the deep domain adaptation algorithm as
shown in Fig. 4. We freeze the two LSTM layers and the two fully-connected
layers using the pre-trained weights, while we train the final two fully-
connected layers using the MMD regularizer. As there is no target domain label
data available only the source output is considered in the loss function.

Table 4 shows the performance of spatial generalization using the LSTM and
deep domain adaptation. The LSTM model performs reasonably well using TA
features, with average $R^{2}$ score very similar to what we saw using RF and
instant weighting.

##  6\. Ethical Considerations

One of the main motivations for the work presented in this paper concerns user
privacy and integrity. Traffic cameras and automated license plate recognition
devices allow for unprecedented levels of identification and tracking. This is
all the more true for user data obtained from cellular networks and mobile
devices. Our approach as presented in this paper uses data that is inherently
privacy-preserving - we use readily available radio frequency counters that
are aggregated on cell level and per definition do not contain any information
about individual users, nor could this information be reconstructed. It is
therefore impossible to identify or track any individual user based on this
data. With that in mind we can state that the work presented in this paper
does not raise any ethical issues.

##  7\. Conclusion

Traffic flow estimation has traditionally involved forecasting methods based
on observations from dedicated traffic sensors. Firstly these methods don’t
scale well since we require large number of sensors. Secondly we would need a
separate forecasting model for every road, since roads don’t exhibit
homogeneous behaviour. Finally our traffic estimation performance would be
susceptible to drastic changes in driver behaviour or road conditions, such as
traffic accidents and road works. To overcome these limitations alternative
approaches have been proposed, including using various forms of cellular
network data to estimate traffic flow. However existing approaches are either
user invasive, or can potentially result in adverse operational impacts to
cellular networks.

In this paper we propose a traffic flow estimation method using inherently
anonymous and widely available LTE/E-UTRA radio frequency counters, namely
path loss and timing advance counters, effectively turning LTE eNBs into
traffic sensors. We cast traffic flow estimation as a supervised regression
problem, where path loss and timing advance counters are used as primary
features, and vehicle counts from actual traffic sensors as target or ground
truth variables. We demonstrated excellent performance using both Random
Forest and LSTM regression models. Since we have limited amount of ground
truth data, i.e. we only had access to six different locations, we also
evaluated the performance of two different transfer learning approaches,
namely instant weighting, and deep domain adaptation. With transfer learning
we demonstrated reasonable performance using either Random Forests or LSTMs,
but using only timing advance features. Our hypothesis is that with more data
and more locations the performance will improve further still.

While our models are not perfect estimators, they are still extremely useful -
they capture the shape of the traffic very well, and for most purposes provide
a good-enough estimate of the traffic flow. The output of these models can be
used for anomaly detection, for example for detecting traffic congestion or
accidents. All this can be achieved without having to install any additional
sensors - we simply re-use LTE radio base stations that are permanently fixed
in their locations with near 100% uptime.

## Acknowledgments

We would like to thank the following people for their support throughout the
project, and for facilitating the network and traffic sensor data without
which none of this would be possible: Elin Allison, Madeleine Körling and
Jyrki Lehtinen from Telia Company AB; Anders Broberg and Tobias Johansson from
City of Stockholm; Annika Engström from KTH Royal Institute of Technology and
Digital Demo Stockholm; Chris Deakin and Chris Holmes from WM5G Limited; Mo
Elhabiby and Mike Grogan from Vodafone UK. We would also like to extend our
gratitude to Leif Jonsson, Jesper Derehag, Carolyn Cartwright and Simone
Ferlin, for reviewing our paper and providing valuable feedback.

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