This paper presents some optimization method aspects use in telecommunications networks. The use of optimization methods by machine learning means is especially important to avoid various emergencies in networks. It is advisable to use machine learning methods to obtain information about signal quality, traffic, etc. At the same time, it is possible to make various malfunctions forecasts, routing, safety control. It is determined that the Markov random field model is effective in modeling in homogeneous networks. This approach allows an exponential distribution nodes modeling in heterogeneous networks. A proximal gradient algorithm modification is presented-a method of variable metric proximal gradient. Ensuring fast convergence is achieved by diagonal step size means, which is more efficient than scalar. The article reveals an adaptive metric selection rule, i.e., a diagonal step based on the BarzilaiBorwein (BB) method. The presented algorithm combines two approaches: the standard proximal gradient method and the proximal Newton method. The establishment of clear rules for choosing the diagonal step size for convex optimization algorithms has been implemented.
Optimization methods are traditionally used
in informational technologies, more
specifically in telecommunication networks,
that is confirmed by positive results in a wide
range of different data [
It is relevant to use machine learning
methods when it is necessary to obtain
conclusions from various monitoring results,
namely, traffic quality or signal, etc. Moreover,
using certain However, there is a problem when
heterogeneous data are used. For instance,
heterogeneous network components failure is
analyzed based on a set of various network
parameters and factors affecting it. To model
such heterogeneous networks, the Markov
random field model is used [
Pairwise exponential Markov random field
belongs to multivariate exponential
distributions class. The specified method based
on joint distribution representation, that allows
compactly introduce heterogeneous variables,
which in turn will lead to a faster studying the
structure method for nodes distribution with
unknown parameters. In other words, this
approach allows to simulate an exponential
nodes distribution in heterogeneous networks.
A modification of the proximal gradient
algorithm – the variable metric proximal
gradient method also deserves attention.
Ensuring fast convergence is achieved by
means of a diagonal step size, which is more
efficient than a scalar one [
This article represents adaptive metric
selection rule, means diagonal step, that based
on the Barzilai-Borwein method (BB). Current
algorithm combines two approaches: standard
proximately gradient method and the proximal
Newton method [
Increasing efficiency and functional system stability for heterogeneous network management provides models and machine learning methods improvement. Using machine learning methods gives a possibility to avoid destabilizing factors in networks. To simulate such networks graphical methods are used, the assessment of which is quite difficult. Thus, it is relevant to improve metric proximately gradient, scientific novelty of which is that it uses a pair-exponential Markov random field model and diagonal step selection method, which allows to provide a faster convergence of the machine learning algorithm.
The purpose of the study is to establish clear diagonal step size rules for convex optimization algorithms and using the proposed approach to improve the machine learning method. To realize a goal, it is proposed to use an adaptive metrics selection rule, means diagonal step, that is called a Barzilai-Borwein step. Current algorithm combines two approaches: standard proximate gradient method and proximate Newton method.
A design, realization, and network
management are appropriate to conduct using
methods of optimization and machine learning.
The use of optimization methods in
telecommunication network provides effective
results. A relevant optimization methods
support using by means of neural network,
allow to analyze, and study of big data arrays
and predict possible extraordinary cases in
network [
A telecommunication field is already
prepared for machine learning implementation.
Network operators work with big amount of
data: information about clients, web
performance data, Internet traffic data and
social network data, etc. At the same time,
operators use various applications for network
planning and analysis to find patterns in the
data. It causes the appearance of many
machines learning programs in the
telecommunications [
Nowadays, a machine learning can be
interpreted as a withdrawal from patterns in
future networks and systems design. The use of
machine learning methods applies in network
application for faults prediction, intrusion
detection, safety control, routing, bandwidth
reconfiguration considering traffic and more
[
Every year, the number of devices connected to the network is constantly growing. However, this is due not only to the increase a number of smartphones, tablets, etc., but also to the emergence of new and the development of existing technologies, such as Machine-toMachine (M2M) and Internet of Things (IoT), where all electronic devices are able to connect to a heterogeneous network. An equally important regularity is that a video traffic percentage in network will grow, which require an improvement in the quality and clarity of the image. A new technology, deployed at different levels and network parts will need interaction to meet user requirements. Moreover, user requirements will be very various: from lowlatency and high data transfer rate video applications to IoT devices that has very low productivity requirements. With these conditions, networks just need to add intelligence and autonomy to adapt to this enormous heterogeneity. Networks with such characteristics are called Self-Organizing Networks –SON.
A telecommunications field is already
prepared for machine learning implementation,
because network operators are already having a
big amount of data: the acquire and store data
about clients, Internet performance data,
internet traffic and socials networks data, etc. In
addition, operators are already use such
applications, as networks planning and root
cause analysis to find patterns in the data.
Therefore, it is not surprising that many
machine learning programs are already starting
to appear in the telecommunications sector
[
One way to use machine learning in a network is to analyze and examine network performance data to identify idle cells and trigger automatic restarts. It can seriously affect a service quality, especially in crowded areas or at loaded times of the day. A manual resetting often results in the cellular tower being turned off for several hours. To avoid such inconvenience, it is appropriate to use machine learning methods.
Nowadays, machine learning perceived as paradigm shift to future networks and systems design. These methods must allow to make conclusions from data, acquired using different types of monitoring (e.g., signal quality, traffic examples, etc.) It is relevant to use machine learning is network applications for faults predictions, invasion detection, safety control, routing, bandwidth reconfiguration considering traffic and more.
To solve in network work issues, it is
proposed a use of model in a pair-exponential
Markov random field form [
An algorithm, based on multipliers methods of variable direction and the solution of closed form restoration for each of the sub-problems are proposed. Due to the use of these fast recoveries that the solution process is accelerated, and the method of variable direction multipliers becomes more scalable than other methods. The obtained methodology was tested both on artificially created and real data.
A convex optimization as a mathematical theory has been studied for a long time and has found its application in control synthesis, signal processing, working with big data and machine learning.
For the convex optimization study, the following model was chosen as a basis: min xRn
F x f x q x (1) where x Rn is solution variable, function f : Rn R convexed and differential, function q:Rn R is convexed and can be not differentiated. Function q may be used can be used to encode constraints on a variable х.
A proposed structured model of (1) convexed optimization using for a lot of machine learning tasks: regression, classification, matrices build, etc. To solve such optimization tasks, presented in the form, the proximal gradient algorithms are very often used, which allows to improve measurement process, provide practical rules for step selection, theoretical guarantees under moderate conditions, increasing the accuracy of the result, etc.
Most proximal gradient algorithm modifications subject to the same form, that is called a metric proximate gradient method.
When selection an algorithm step, that provides a fast convergence, it is established that the diagonal step size is more effective than the scalar one. Clear rules for diagonal step size for convex optimization algorithms selection should be established.
This article proposes to use an adaptive metrics selection rule, means diagonal step, that is called a Barzilai-Borwein step. Proposed algorithm combines two approaches: a standard proximate gradient method, and a proximal Newton method. Metrical proximal gradient with diagonal step BB provides low measurement steps loses, a much better Hessian approximation at each iteration, and consequently—a fast convergence of the algorithm (in comparing with proximate gradient method with scalar step). Conducted empirical research determined, that proposed method with diagonal metrics provides improved convergence in comparing with proximal algorithm method with scalar step.
The most popular diagonal step determining should be attributed the following: spectral scalar step size, variable non-scalar metric, diagonal metric.
Usually, a spectral step method is used for
gradient BB type methods. The additive rule for
spectral metric selection uses a spectral step
method. To reduce the specified limitations, a
new adaptive diagonal metric selection strategy
with convergence guarantees using string
search is proposed [
A proximate gradient step can be considered as the minimization of the function F, where differentiated part f is approximated to its second-order form for xn, relatively Mn ϵ Ch++ [19].
proxq,Mn xn M n 1 f xn arg min q x f xn f xn T x xn 1 x xn 2 x 2 M n This approximation shows that M n 2 f xn is optimal choice after proximal Newton method. However, the Hessian use usually leads to a high cost of iterations. An alternative option and Hessian approximation with use of secant state:
M ncn yn , yn f xn f xn1 . for step cn xn xn1 and gradient change
A Barzilai-Borwein method is an approach, that estimates a scalar Hessian approximation, by setting M n n 1 , that satisfies formula (2). The method is quite popular.
The most common steps in the BB method are the following: BnB1 cn22 / cn yn ; BnB2 cn , yn / yn22 where БnБ2 БnБ1 is always performed.
For this purpose, several modifications and protective measures have been adopted to the initial stage of BB. One of such numerical measures for cn and yn a hybrid choice between these two steps is suggested: BnB BB cn , yn BnB1 1 BnB2 ,if BnB1 BnB2
BnB2 , in other cases where hyperparameter R usually equals to 2. If in equation (4) is negative, then previous step is chosen = −1.
Such modification and activities designed to eliminate instability in the primary BB with step (4) for the poorly conditioned f. However, even with such modification, a scalar BB still addicted to inconsistences. It should be notes, that ( 1)−1 and ( 2)−1 can be (2) (3) (4) considered, and Hessian approximation in Euclidean space. However, in poorly conditioned conditions these scalar approximations may be far from the true nonEuclidean geometry of Hesse. In other case, e.g., after such approximations, as step direction projection cn and gradient change yn they may be close to orthogonality. This leads to degenerate scenarios with 1 → ∞ or 2 → 0. For such cases, a scalar estimation may be significantly different from the secant condition (2) and Hesse's geometry.
The proposed diagonal step of the BB is considered in the paper.
To display Hessian geometry f we enter the diagonal metric M n , that at each iteration n is calculated as follows: min mRn
Mcn yn 2 M M n1 2 2 F (5) 1 1 BnB1 J M BnB2 J
M diag m where hyperparameter 0 manages the compromise between the satisfaction of the state (2) and consistency with previous metrics M n1 . If Hessian changes so fast, it is necessary to select a low enough value μ. In this case this parameter numerical protection will be used. If Hessian won’t change much during iterations, there is a necessity to choose big value μ. Eventually, diagonal elements are restricted, means guaranteed with BB step (3).
One of proposed form (5) peculiarity is, that it has a simple closed solution form. For Mn diag mn , where mn m1n, ...., mhn Rn task (5) solving is given as follows: 1 БnБ1 , cin 2 min 1 cin yin min1 БnБ2 , cik 2 cin yin min1 1 БnБ1
1 БnБ2 (6) cin yin min1 , cin 2 in other cases.
n n where сi and yi are і element cn and yn.
A metric proximation gradient algorithm with diagonal metrics has next steps (Fig. 1). Set parameters Set the initial
points Set the initial
metric Calculate
Let'sinitialize Yes
No Yes
No Yes
No
Calculate
A metrics proximation gradient with diagonal step BB significantly goes beyond standard proximal gradient (∼ 20% of measurements) as poorly conditioned.
Proposed diagonal metric provides better estimate for poorly conditioned local Hessian in
Calculate Calculate Calculate
Calculate
Return metric
Yes Let'sget the results
End No
Stop criterion satisfied? comparing with standard scalar BarzilaiBorwein step BB, which leads to faster algorithm convergence. A metric proximal gradient method is developed, scientific novelty of which is in, that it use a pair-exponential Markov random field model and selection diagonal step method, that allows to provide faster machine learning algorithm convergency. A proposed block implementation in heterogeneous network management system allows early react to overloaded network with help of short- and medium-term forecasts build and strengthen an intelligent network management block.
A proposed metric proximate gradient method implementation in heterogenous telecommunication network will provide efficient decentralized management of heterogenous network management and reduce amount of service information in the network. It will allow to avoid network overload when extraordinary cases are appeared. However, the question of network overload on the equipment during its management, when many users present, remains open.
Markovskogo Sluchaynogo Polya, Actual Problems of Economic, vol. 1, no. 223, 2020, pp. 180–191. [18] V. Zhebka, et al., Optimizatsiya Rabotyi Algoritma Gradientnogo Bustinga s Pomoschyu Perekrestnoy Proverki, Actual Problems of Economic, vol. 12, no. 222, 2019, pp. 189–197. [19] V. Zhebka, Modeliuvannia Markivskoho Vypadkovoho Polia z Metoiu Yoho Podalshoi Optymizatsii ta Zastosuvannia, Zviazok, vol. 5, 2020, pp. 35–40.