AU2020103326A4 - A robust diffusion kernel risk-sensitive loss (d-KRSL) algorithm for asynchronous networks - Google Patents

Abstract

: With the developments in wireless sensor networks, embedded systems, and low-power communications, the Internet of Things (IoT) have become a reality. However, in the Internet of Things (IoT), asynchronous networks with varying topology are quite common. Besides, Gaussian noise and impulsive noise also widely exist in asynchronous networks. As we all know, distributed strategies become more and more attractive in IoT, due to their robustness against imperfections, low complexity, and low-power demands. However, existing works on distributed estimation problems in networks primarily consider fixed topologies and Gaussian noise. Thus, these algorithms are not suitable for distributed parameter estimation in asynchronous networks. To overcome this issue, this patent proposes a distributed diffusion kernel risk-sensitive loss (d-KRSL) algorithm that is a new cost function based on the information-theoretic learning (ITL), which can achieve a good performance in asynchronous networks with varying topology, and maintains the robustness to both Gaussian and impulsive noise. The mean and mean square performances of the proposed algorithm are analyzed theoretically and verified by numerical simulation results. The results show that the algorithm has good performance in the field of Internet of things with variable network structure. Figi. Atmo.her. .................. phen-en, so r 0 trt ceAdapaon #nre Aftack&Souroe 'nis 14(i Information exchan e Combination -4 2 i Fig2. 1.2 -------------- --------- ------------- 1.2 ............. ---------................... P 0 --.--- 7. . . . x-coordi n ale

Description

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Editorial Note 2020103326 There is 8 pages of Description only.

1. Background and Purpose

Recent developments in wireless sensor networks, embedded systems, and low-power

communications have made the Internet of Things (IoT) a reality. With a huge and increasing number of smart devices connected to the Internet, IoT networks are increasingly popular in various scenarios. Facilitated with IoT technology, our world is becoming smarter and smarter. Moreover,

distributed strategies become more and more attractive in IoT, due to their robustness against imperfections, low complexity, and low-power demands. As an important issue in the field of distributed learning, distributed parameter estimation over asynchronous networks plays an

essential role in many applications. The related works aim to estimate some essential parameters from noisy observation measurements through cooperation between nodes. The distributed methods include incremental strategies, consensus strategies, and diffusion strategies. Among these

distributed schemes, in the incremental strategy, a cyclic path is defined over the nodes, and data are processed in a cyclic manner through the network until optimization is achieved. However, determining a cyclic path that runs across all nodes, which is generally a challenging (NP-hard) task

to perform. In the consensus strategy, vanishing step sizes are used to ensure that nodes can reach consensus and converge to the same optimizer in a steady state. In the diffusion strategy, information is processed locally and simultaneously at all nodes. The processed data are diffused through a real

time sharing mechanism that ripples through the network continuously. The diffusion strategies are particularly attractive because they are robust, flexible, and fully distributed compared with incremental and consensus strategies, so we adopt diffusion strategies in this algorithm.

Some prior works concerned nodes estimating the parameter vector collaboratively over synchronous networks with fixed topology, and the background noise was assumed to be Gaussian distributed. In this scenario, numerous distributed algorithms have been developed for parameter

estimation, such as diffusion RLS, diffusion least mean square (LMS), diffusion sign-error LMS (DSE-LMS), and diffusion sparse LMS. To improve the transient performance, a preconditioned graph diffusion LMS algorithm was introduced. A general regularization framework for inference

over multitask networks was proposed, which considered the problem of distributed sequential detection using wireless sensor networks in the presence of imperfect communication channels between the sensors and the fusion center. The mentioned algorithms can achieve the smallest mean

square error (MSE) by optimizing the cost function. Some people considered the random link failure in diffusion networks. To address the issue of distributed estimation in networks, they used to incorporate equalization coefficients into the diffusion combination step and updated the

combination weights dynamically. In some works, time-varying wireless channels were discussed, a novel centralized least mean squares (CLMSs) algorithm was proposed, with a refined version of the transmitted data and benefits from a link failure alarm strategy to discard severely distorted data.

They considered the wireless sensor networks where nodes should transmit their data over fading channels to the central processing unit. Such a setting makes the network more dependent on the central unit. However, these algorithms' cost functions only considered the second-order moment

of the error. Accordingly, MSE-based methods can achieve desirable performance when the measurement noise is Gaussian. Moreover, there have been several useful studies on distributed parameter estimation with non-Gaussian measurements in synchronous networks. The research

proposed a variable weighting coefficients diffusion LMS algorithm (VWC-LMS) against impulse noise. Whereas their research assumed that the impulsive noise was successfully detected, the cost function only considered uncontaminated data. The RobustAF algorithm was developed that

effectively learns and tracks the output error distribution to improve estimation performance, which made a significant contribution to estimating parameter in the impulsive noise environment. To address the problem of modeling with non-Gaussian data, similarity measures must go beyond

second-order statistics. The C-loss functions were proposed, such as d-MCC and d-MEE. An adaptive algorithm under the maximum correntropy criterion was proposed, since correntropy was a measure of local similarity and was insensitive to outliers. The diffusion LLAD algorithm adopts

both the logarithm operation and sign operation to the error, which can elegantly and gradually adjust the conventional cost functions in its optimization based on the error variation. However, these studies do not take the varying topology into account. Besides, a majority of existing works

on distributed estimation problems are MSE-based methods, which shows good estimation performance in the Gaussian case, but the performance may degrade in an impulsive noise environment. They only considered the second-order moment of the error in the cost function. Under

the asynchronous network model, nodes in the network may stop exchanging information or updating their estimates at a random time, which results in outliers. To further promote the estimation performance in asynchronous networks with varying topology,

a novel algorithm needs to be designed. The KRSL is a new cost function based on the information theoretic learning (ITL). As compared to the MSE criterion, the ITL provides a more general framework and can achieve desirable performance when the measurements are non-Gaussian.

Various cost functions have been designed based on ITL and applied for adaptive non-Gaussian signal processing in WSNs. Among these cost functions, the KRSL performs well with the similarity measure in kernel space, which can obtain a stable convergence with a faster speed through a

gradient-based method. Moreover, the KRSL algorithm guarantees a higher estimation accuracy and is robust to large outliers. Inspired by the desirable features of KRSL, we apply the KRSL cost function in a distributed diffusion manner over asynchronous networks. The proposed algorithm in this patent is called the diffusion kernel risk-sensitive loss (d-KRSL) algorithm.

2. System Model

Consider a connected network with N nodes, every node k can only exchange information from all its neighbors N. at every time instant i. The sensor at each node k takes a scalar measurement

dk(i), and {d (i), u. (i) Iare the realization of some zero-mean random process {d ) 1uk 0) where u.(i)is an 1xMregression vector. The date {d(i), u.(i) Iare collected by all the nodes in the network. The relationship between dk (i) and u. (i) satisfies the following linear model: dk(i)= Uk(i)o>° +Vk(i)

Where wo is an M x1 unknown parameter vector to be estimated, while the background noise is v. (i) with variance v,k each node has different -, We assume the regressor u. (i) and the

measurement noise v.(i) are spatially independent and identically distributed (i.i.d.). vk(i) is

independent of u. (i) for all k and i.

The adapt-then combine (ATC) form of the diffusion strategy is considered primarily in this

patent. In this scheme, nodes in networks combine information from their immediate neighbors first

and then employ updates by the following steps.

Step 1: In order to obtain an intermediate estimate, a step-size parameters is introduced to control changes in the estimation from one iteration to the next. To imitate the behavior of asynchronous

networks, we assume that step sizes is changing with a certain possibility over time.

P p, > 0,with probabilitypk to, withprobability1-Pk Step 2: Given the topology of asynchronous networks, only the local estimator (, (i) will be transmitted between node k and all its neighbors. This step corresponds to the fourth part in Fig. 1.

Step 3: To obtain a new estimate, each node gathers its own intermediate estimate from all its neighbors.

Ic N,

To imitate the behavior of asynchronous networks, we assume that combination

coefficients between nodes are changing with a certain possibility over time. The combination

coefficients {c,(i)} are distributed as follows:

r C,k > 0,with probability qk o, withprobability 1-q, for all IE N\{k}, which are i.i.d. for different i. At each iteration, node k adjusts its own weight

Ckk by clk(i)= 1- cY i) IE Nk [I

This constraint ensures

ZIFNkCk 3. Diffusion Kernel Risk-sensitive Loss Algorithm Description

Step 1: The cost function can be defined as follows:

J KRSL () CHRSL (el(i))= CkH u(i)w) 'RSL(dI(i)- Ic N, Ic N,

where

H RS(eji))= exp(A(l- k7 (e,(i))))

2 k, (x - y) = eXp(- 2 Y) ~~ 2o

Step 2: An algorithm based on steepest descent for estimating CO at node k can be obtained that

(Pk (Pk P 8HKRSL (l() awC Step 3: Though sharing information with all its neighbors, each node k updates estimation vectorW (i) at time instant i. Since nodes are restricted by communication resources, transmitting all the information at each time is not necessary. To solve this problem, we take a compromise strategy that gives different weights to all local estimates. Therefore, the estimation at node k is a linear combination of these local

estimates

4. Brief Description of the Drawings

Fig.1 is topology over asynchronous networks Fig.2 is network topology with 20 nodes.

Fig.3 is variance of input signal and noise. Fig.4 is synchronous network in Gaussian noise environment, where (a) is desired signal, and (b) is convergence curves of d-KRSL algorithm over synchronous networks with different parameters.

Fig.5 is comparison of MSD curves over synchronous networks, where (a) is in Gaussian noise environment, (b) is in impulsive noise environment. Fig.6 is synchronous network in impulsive noise environment, where (a) is desired signal, and (b)

is convergence curves of d-KRSL algorithm over synchronous networks with different parameters. Fig.7 is asynchronous network in Gaussian noise environment, where (a) is desired signal, (b) is network MSD curves of d-KRSL algorithm for different asynchronous networks, (c) is steady-state

MSD performance of d-KRSL algorithm. Fig.8 is asynchronous network in impulsive noise environment, where (a) is desired signal, (b) is network MSD curves of d-KRSL algorithm for different asynchronous networks, (c) is steady-state

MSD performance of d-KRSL algorithm. Fig.9 is comparison of network MSD curves in the presence of Gaussian noise environment over asynchronous networks,

Fig.10 is comparison of network MSD curves in the presence of impulsive noise environment over asynchronous networks. Fig. I1is network MSD curves of d-KRSL algorithm in a fully connected graph with nodes

collaboration and a graph where the linked nodes noncollaboration.

5. Detailed Description

we verify the performance of the proposed d-KRSL algorithm in two distinct network scenarios: synchronous networks and asynchronous networks. Both of them consist of N= 20 nodes. These

two types of networks are randomly dispersed in a square area of [0, 1.2] x [0, 1.2]. The topology of synchronous networks is depicted in Fig. 2, while the topology of asynchronous networks will change during every iteration in the following related simulations.

In this work, our aim is to estimate an unknown parameter w° with size M= 5. In the following simulations, the unknown parameter 0) is set to (1/ I)(M = 5).The regression inputs{uk(i) are i.i.d. Gaussian with covariance matrices Ruk - 02,I, and o-2, is shown in the plot of Fig.3. The background noise {v. (i) is drawn independently of the regressors and i.i.d. We employ the MSD of estimated parameter errors as the performance measure.

Example 1: synchronous network We now investigate the performance of the proposed algorithm over synchronous networks. To

model the synchronous networks, we set pk = q =1to ensure a fixed topology for synchronous networks. First, as shown in Fig. 4(a), the desired signal is a random process with a zero-mean Gaussian

i.i.d. noise signal. In this scenario, the step size is p=lx 10-3, 3=0,and U = 0.4 In this simulation, we study the effect of the parameters A, o and r/ on the estimation performance of the d-KRSL. Fig. 4(b) shows that the d-KRSL algorithm has a similar performance with different

parameters. It also depicts that the d-KRSL algorithm can achieve relatively good performance in terms of the network MSD. In synchronous networks with the interference of Gaussian noise, Fig. 5(a) compares the learning

curves of the proposed algorithm (d-KRSL) and other related algorithms, such as Robust AF, DSE-

LMS, DLMSs with adaptive combiners(AC-dLMS), and diffusion LMS (d-LMS). From Fig. 5(a),

in the Gaussian noise environment over synchronous networks scenario, it depicts that the d-KRSL algorithm outperforms other algorithms. Since in the proposed algorithm, besides the kernel bandwidth, the risk-sensitive loss parameter is an extra free parameter, which is introduced to

control the shape of the performance surface. Second, we use the impulsive noise model with 3 =0. 4 , 3 i =10- 2, and k=100.In impulsive noise environment over synchronous networks, Fig. 5(b) shows that the proposed algorithm is

insensitive to impulsive noise and has better performance than other typical algorithms against impulse noise algorithms such as d-MCC (diffusion maximum correntropy criterion algorithm). This simulation compares the learning curves of MSD computed empirically and theoretically to

verify the transient analysis. The d-LMS algorithm has poor estimation performance due to the presence of impulsive noise. In Fig. 6(a), the sampled desired signals are plotted in the presence of impulsive noise with 3=

0.07 and k = 1000.Fig. 6(b) depicts that in the impulsive noise environment, d-KRSL can achieve good performance in terms of convergence rate with different parameters over synchronous networks. This simulation illustrates that the proposed algorithm is robust to impulsive noise.

Example 2: asynchronous network

In this simulation, the topologies of asynchronous networks are changing with a certain probability related to the number of idle nodes. First, the simulation is illustrated in the Gaussian noise environment over different asynchronous networks, and we report the network MSD learning

curves for three cases as follows. Case 1: 70% idle: p, =q,= 0.7. Case 2: 50% idle: p, =q,= 0.5. Case 3: 30% idle: p, =q,= 0.3.

Fig. 7(a) shows the presence of Gaussian noise environment, with 5= 0 and 3 =0.4 .Fig. 7(b) shows the learning curves of the d-KRSL algorithm averaged over 100 independent Monte Carlo

runs. The parameters of d-KRSL are: = 3, a= 2, and r/ = 0.001. It can be seen clearly that in

different asynchronous networks suffused with Gaussian noise, the proposed algorithm can achieve stable estimation performance in different cases. The lower varying probability of topology, the

better performance d-KRSL achieves. The corresponding steady-state MSD for the d-KRSL algorithm, which is obtained by averaging 100 samples of the respective MSD curves, is plotted in Fig. 7(c).

Second, as Fig. 8(a) shows, we utilize the impulsive noise model by ,= 10-1, = 0.07, and k = 100 settings. Fig. 8(b) shows learning curves of the d-KRSL algorithm inan impulsive noise environment over different asynchronous networks. This simulation is performed over 100 independent trials. We report the network MSD learning curves in three different cases as follows. Case 1: 70% idle: p, =q,= 0.7. Case 2: 50% idle: p, =q,= 0.5. Case 3: 30% idle: p, =q,= 0.3.

The parameters= 5, a= 3, and r/ = 0.007 are included in the figures. It can be seen that in

asynchronous networks with impulsive noise, d-KRSL can also achieve stable estimation

performance in different cases. Fig. 8(c) shows the corresponding steady-state MSD for d-KRSL algorithms. Example 3: compare with other algorithms in Asynchronous networks

As a final illustrative example for the behavior of the d-KRSL algorithm in comparison with other algorithms is considered here. We consider the network consisted of N= 20 nodes, and the topology is changing with a certain probability q,, = 0.8. We present a comparison of the proposed d-KRS

algorithm with several other related algorithms. Fig. 9 shows the MSD curves of the algorithms mentioned above in the presence of Gaussian noise environment over asynchronous networks. From Fig. 9, we can see that the d-KRSL algorithm

can achieve better estimation performance than DSE-LMS, d-MCC, and d-LMP algorithms. In addition, d-KRSL and d-LMS have good performance when the measurement noise is Gaussian. The step sizes of all algorithms are chosen after many experiments to ensure the convergence speed,

and other parameters for each algorithm are experimentally selected to achieve desirable performance. Furthermore, the simulation is implemented in the presence of an impulsive noise environment

over asynchronous networks. In this simulation, we set 6 = 0.15, o = 0.6 ,and k = 1000. From the simulation results shown in Fig. 10, it is observed that the proposed d-KRSL algorithm has a relatively smaller MSD than other algorithms. The LMS algorithm is very sensitive to large outliers

and fluctuates between great extremes. The d-KRSL has a relatively more convex performance surface and more uniform contours than the LMS, which results in better performance and converges smoothly in the presence of impulsive noise environment over asynchronous networks.

As shown in Figs. 9 and 10, this simulation demonstrates that the proposed d-KRSL algorithm can achieve a better estimation performance in asynchronous networks with dynamic topology an maintains the robustness to both Gaussian noise and impulsive noise.

Fig. 11 illustrates that the collaboration strategy outperforms the noncollaboration one. Because cooperation among nodes is generally beneficial. When nodes act individually, their performance is limited by the noise power level at their location. In this way, some nodes can perform significantly better than other nodes. On the other hand, when nodes cooperate with their neighbors and share information during the adaptation process, we will see that performance can be improved across the fully connected graph.

Editorial Note 2020103326 There is 2 pages of Claims only.

Claims (1)

1. The claims defining the invention are as follows: A robust diffusion kernel risk-sensitive loss (d-KRSL) algorithm for asynchronous networks

   (1). System Model Step 1:Consider a connected network with N nodes. The sensor at each node k takes a scalar measurement d(i), and {d(i),u.(i)} are the realization of some zero-mean random process { dk(iu)(i)} 'where u.(i)is an 1xM regression vector. The date { d(i),u.(i)} are collected by all the nodes in the network. The relationship between dk(i)andu.(i)satisfies the following linear model: dk(i)=Uk(i)wO +Vk(i) Step 2: In order to obtain an intermediate estimate, a step-size parameters is introduced to control changes in the estimation from one iteration to the next. To imitate the behavior of asynchronous networks, we assume that step sizes is changing with a certain possibility over time.

   , > 0,with probabilitypk to, withprobability1-Pk

   Step 3: Given the topology of asynchronous networks, only the local estimator O(i) will be transmitted between node k and all its neighbors. This step corresponds to the fourth part in Fig. 1. Step 4: To obtain a new estimate, each node gathers its own intermediate estimate from all its neighbors.

   Ic N,

   To imitate the behavior of asynchronous networks, we assume that combination coefficients between nodes are changing with a certain possibility over time. The combination coefficients {c,k(i)}are distributed as follows:

   - rC,k > 0,with probability qk o, withprobability 1-q, for alllE N \{k}, which are i.i.d. for different i . At each iteration, node k adjusts its own weightCkk by

   leN [k

   This constraint ensures

   ZIFNk 1k

   (2). Diffusion Kernel Risk-sensitive Loss Algorithm Description Step 1: The cost function can be defined as follows:

   J KRSL () CHRSL (el(i))= CkH RSL(d 1 (i)-u1 (i)w) I N, I N,

   where 1 H RSL(e(i))- exp(A(l- k,(e,(i))))

   2 2o

   Step 2: An algorithm based on steepest descent for estimating o at node k can be obtained that

   P8HKRSL ()-e (Pk(Pk aC Step 3: Though sharing information with all its neighbors, each node k updates estimation vector w.(i)at time instant i. Since nodes are restricted by communication resources, transmitting all the information at each time is not necessary. To solve this problem, we take a compromise strategy that gives different weights to all local estimates. Therefore, the estimation at node k is a linear combination of these local estimates.

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