Heterogeneous fairness algorithm based on federated learning in intelligent transportation system

Abstract

With the development of the Intelligent Transportation System, various distributed sensors (including GPS, radar, infrared sensors) process massive data and make decisions for emergencies. Federated learning is a new distributed machine learning paradigm, in which system heterogeneity is the difficulty of fairness design. This paper designs a system heterogeneous fair federated learning algorithm (SHFF). SHFF introduces the equipment influence factor $I$ into the optimization target and dynamically adjusts the equipment proportion with other performance. By changing the global fairness parameter $\theta$ , the algorithm can control fairness according to the actual needs. Experimental results show that, compared with the popular q-FedAvg algorithm, the SHFF algorithm proposed in this paper improves the average accuracy of the Worst 10% by 26% and reduces the variance by 61%.

Keywords

Federated learning heterogeneous system intelligent transportation system distributed learning

1. Introduction

The Intelligent Transportation System (ITS) [1, 2] has many types of equipment (such as global positioning system, radar, infrared sensor, etc.) and massive privacy data, which brings network delay, energy consumption, and data security problems in the distributed system. Federated learning [3, 4] is a new distributed machine learning paradigm, which can perform collaborative learning without sharing data, which protects users’ privacy. Federated Learning (FL) opens a new way for data analysis of the intelligent transportation system.

Recently, more and more researchers focus on Federated Learning (FL) algorithms with accuracy, fairness, and convergence time. Fairness means that the performance of each equipment should be as close as possible while the average accuracy is basically unchanged, so that the global model can fit all devices as much as possible. Balakrishnan etc. [5] developed a federated learning optimization algorithm based on importance sampling and ranking algorithm. Experiments show that compared with FedAvg [6] and FedProx [7], this model can significantly shorten the overall training time without losing performance. Liu et al. [8] proposed an FL based simulation learning framework for cloud robot systems with heterogeneous sensor data. The research shows that FL can improve the efficiency and accuracy of robot imitation learning by other robots’ knowledge. Yu et al. [9] proposed a revenue-sharing scheme for FL motivators (FLI), which dynamically allocates data owners’ budgets through context-aware methods. In order to ensure fairness between users and robustness to malicious adversaries, Hu et al. [10] defined federated learning as a multi-objective optimization problem, and proposed a new algorithm FedMGDA+ algorithm, which guarantees convergence to Pareto stationary solution. Li et al. [11] proposed an extensible q-FedAvg to solve the problem of FL fairness under statistical heterogeneity.

The above algorithm only solves the fairness problem of federated learning under statistical heterogeneity, does not consider the fairness of system heterogeneity. Li et al. [12] pointed out that there are differences in equipment hardware, network connectivity, and battery power (i.e. system heterogeneity) in FL. These differences will lead to different equipment proportions with other performance optimization objectives and bring the risk of difficult to fit. A reasonable method for the heterogeneity of the system is lack of attention. Therefore, this paper optimizes the above problems.

In this paper, we propose a system heterogeneous fair federated learning algorithm (SHFF). The algorithm introduces the equipment influence factor $I_{k}$ into the optimization objective and dynamically adjusts the optimization objective’s proportion of different performance equipment. By changing the global fairness parameter $\theta$ , the algorithm can control fairness according to the actual needs. Experimental results show that, compared with the popular q-FedAvg algorithm, the SHFF algorithm proposed in this paper improves the average accuracy of the Worst 10% by 26% and reduces the variance by 61%.

2. Problem definition

2.1 System model

Vehicle type recognition [13, 14] based on sensors is one of the key technologies of ITS. In this paper, the SHFF algorithm solves the fair distribution of vehicle recognition accuracy. The basic architecture of the system is shown in Fig. 1, including a server (S) and $K$ equipment (distributed sensors [15, 16, 17]) $C=\{c_{1},c_{2},\cdots,c_{k}\}$ . The system symbol description is shown in Table 1.

Table 1
Symbol description

Symbol	Description
$S$	Server
$K$	Total number of equipment
$C=\{c_{1},c_{2},\ldots,c_{k}\}$	Clients
$T$	Loop iterations
$w^{0}$	Initial global model
$\textit{Round}=\{\textit{round}_{1},\textit{round}_{2},\ldots,\textit{round}_{% k}\}$	Local training round
$\textit{round}_{k}^{\text{Max}}$	Maximum local training round
$\textit{round}_{k}^{\text{Min}}$	Minimum local training round
$\textit{round}_{k}$	Local training round
$a_{k}$	Local training accuracy
$I=\{I_{1},I_{2},\ldots,I_{k}\}$	Equipment influence factor
$N=\{n_{1},n_{2},\ldots,n_{k}\}$	Number of local samples
$n$	Total number of samples
$n_{k}$	Number of local training samples

Federated learning learns the generated data on the local equipment and communicates with the server regularly to achieve global optimization. The heterogeneity of systems among equipment (i.e., the number of samples between equipment, training rounds per unit time, the accuracy of local training, etc.) will lead to differences in local equipment models. If only minimized the optimization objective of q-FFL without considering the system [12], the global optimization model is challenging to fit all equipment. Therefore, we design the equipment influence factor $I_{k}$ as:

$\displaystyle I_{k}=\{a_{k},n_{k},\text{round}_{k}\}$ (1)

The equipment influence factor $I_{k}$ is determined by accuracy, sample number and local training rounds. According to the influence of these three parameters, the equipment can be divided into High-intelligence equipment and Low-intelligence equipment, as shown in Table 2.

During the training process, the equipment uploads the influence factor $I_{k}$ and the local model to the server. The server aggregates the local model based on influence factors, as shown in Fig. 1.

Table 2

Equipment classification

Higher-intelligent equipment	Lower-intelligent equipment
Larger sample number	Less sample size
More local training rounds	Less local training rounds
Higher local training accuracy	Lower local training accuracy

Figure 1.

System model.

SHFF improves the fairness between equipment by dynamically applying influence factors on them with different intelligence levels. The lower the intelligence of the equipment, the smaller the influence on the overall situation in the optimization goal. Therefore, a larger equipment influence factor enhances the proportion of the equipment in the global optimization objective. The equipment influence factors are continually changing in the training process to achieve dynamic adjustment and increase the influence of low-intelligence equipment in the optimization goal. If equipment with higher intelligence and the global average accuracy rate does not fluctuate much, the algorithm can improve the accuracy of equipment with lower intelligence, reduce variance between equipment, and enhance fairness between equipment.

2.2 Optimization objective

$\displaystyle{\rm{\bf OPT}}=\mathop{\min}\limits_{w}F_{I_{k}}(w)=\sum\limits_{% k=1}^{K}{\frac{n_{k}}{(I_{k}+1)\times n}F_{k}^{I_{k}+1}(w)}$ (2) $\displaystyle I_{k}=[((\alpha\times A_{k}+1)+(\beta\times S_{k}+1)+(\gamma% \times E_{k}+1))\times\theta]$ (3) $\displaystyle\alpha+\beta+\gamma=1$ (4)

Equation (2) gives the optimization objective, where $K$ represents the total number of equipment, $w$ represents the training model, $n_{k}$ represents the number of local samples, $n$ represents the total number of samples, and $F_{k}(w)$ represents the local experience risk of the $k^{\text{th}}$ equipment. Equation (3) gives the calculation formula of the equipment influence factor $I_{k}$ , where $A_{k}$ represents the local calculation accuracy of the $k^{\text{th}}$ equipment, $S_{k}$ represents the local sample number of the $k^{\text{th}}$ equipment, and $E_{k}$ represents the local training rounds of the $k^{\text{th}}$ equipment in unit time. In Section 2.3, the specific calculation formulas of $A_{k}$ , $S_{k}$ and $E_{k}$ are given. Equation (4) is the constraint on the weight. $\alpha$ , $\beta$ and $\gamma$ respectively represent the weight of $A_{k}$ , $S_{k}$ and $E_{k}$ in the equipment influence factor, and the sum of $\alpha$ , $\beta$ and $\gamma$ is equal to 1. $\theta$ is the global fairness parameter according to the actual needs. The selection of $\theta$ value is described in Section 3.2.

In this paper, a larger equipment influence factor improves the low intelligent equipment for the low intelligent equipment in the optimization target. This method makes the low intelligent equipment dominate the global training model, to strengthen its local accuracy to fit all the equipment. The definition of $I_{k}$ in Section 2.3 helps to multi-objective normalization.

2.3 Multi-objective normalization

System heterogeneity parameters (such as local training accuracy, sample number, and local training rounds per unit time) are negative indicators. Taking the accuracy of local training as an example, the higher the accuracy of local training is, the higher the intelligence of equipment is, and the smaller the influence factor of equipment is required. The main parameters ( $A_{k}$ , $S_{k}$ , $S_{k}$ ) in $I_{k}$ are a multi-objective normalization technique based on SAW (simple additive weight) [18, 19], such as Eqs (5)–(7). $A_{k}$ is the local calculation accuracy scale value, where $a_{k}^{\text{Max}}$ is the maximum value of the local training accuracy, $a_{k}^{\text{Min}}$ is the minimum value of the local training accuracy, and $a_{k}$ is the actual accuracy obtained by the $k^{\text{th}}$ equipment local training. In this paper, $a_{k}^{\text{Max}}=1$ , $a_{k}^{\text{Min}}=0$ .

$\displaystyle A_{k}=\left\{{\begin{array}[]{ll}\frac{a_{k}^{\text{Max}}-a_{k}}% {a_{k}^{\text{Max}}-a_{k}^{\text{Min}}},&\text{if }a_{k}^{\text{Max}}-a_{k}^{% \text{Min}}\neq 0\\ 1,&\text{if }a_{k}^{\text{Max}}-a_{k}^{\text{Min}}=0\\ \end{array}}\right.$ (5)

Similarly, the number of samples and the number of local training rounds per unit time are negative indicators. $S_{k}$ and $E_{k}$ are used to represent the scale value of the sample number and the scale value of local training rounds per unit time. The specific formula is as follows:

$\displaystyle S_{k}=\left\{{\begin{array}[]{ll}\frac{n_{k}^{\text{Max}}-n_{k}}% {n_{k}^{\text{Max}}-n_{k}^{\text{Min}}},&\text{if }n_{k}^{\text{Max}}-n_{k}^{% \text{Min}}\neq 0\\ 1,&\text{if }n_{k}^{\text{Max}}-n_{k}^{\text{Min}}=0\\ \end{array}}\right.$ (6) $\displaystyle E_{k}=\left\{{\begin{array}[]{ll}\frac{\textit{round}_{k}^{\text% {Max}}-\textit{round}_{k}}{\textit{round}_{k}^{\text{Max}}-\textit{round}_{k}^% {\text{Min}}},&\text{if }\textit{round}_{k}^{\text{Max}}-\textit{round}_{k}^{% \text{Min}}\neq 0\\ 1,&\text{if }\textit{round}_{k}^{\text{Max}}-\textit{round}_{k}^{\text{Min}}=0% \\ \end{array}}\right.$ (7)

We use $n_{k}^{\text{Max}}$ and $n_{k}^{\text{Min}}$ to represent the maximum and minimum values of the number of samples respectively. In this paper, $n_{k}^{\text{Max}}$ is the total number of global models, $n_{k}^{\text{Min}}=0$ , and $n_{k}$ is the number of local models of the $k^{\text{th}}$ equipment. $\textit{round}_{k}^{\text{Max}}$ is the maximum local training round in unit time, $\textit{round}_{k}^{\text{Min}}$ is the minimum local training round in unit time, and $\textit{round}_{k}$ is the local training round in the $k^{\text{th}}$ equipment unit time.

3. SHFF algorithm

3.1 SHFF algorithm

In the SHFF algorithm, the number of equipment, iterations, samples, initial accuracy and other parameters are initialized firstly. And we propose an initial equipment influence factor $I_{k}$ to each equipment through global initialization. At the beginning of the k-round iteration, the server randomly selects a training equipment subset $S_{t}$ and sends the global training model $w_{0}$ to each equipment in the equipment subset. The equipment (distributed sensor) uses the received global training model to train. After the training, Eq. (2) is used to update the equipment influence factor $I_{k}$ and the training results are sent back to the server. The server aggregates the received training models to get the global training model of the current round and iterate the above process circularly until the training, as shown in Algorithm 1:

Algorithm 1 SHFF
Input: $K,E,T,1/L,\eta,\omega^{0},n_{k},n,a_{k},\textit{round}_{k},k=1,\cdots,m$
1: for $t=0,\cdots,T-1$ do
2: Server selects a subset $S_{t}$ of $K$ devices at random (each device $k$ is chosen with prob. $p_{k}$ )
3: Server send $\omega^{t}$ to all selected devices
4: Each selected device $k$ updates $\omega^{t}$ for $\textit{round}_{k}$ epochs of SGD on $F_{k}$ with step-size $\eta$ to obtain $\overline{\omega}_{k}^{t+1}$
5: Each selected device $k$ computes: $\displaystyle A_{k}=\frac{a_{k}^{\text{Max}}-a_{k}}{a_{k}^{\text{Max}}-a_{k}^{% \text{Min}}}$ $\displaystyle S_{k}=\frac{n_{k}^{\text{Max}}-n_{k}}{n_{k}^{\text{Max}}-n_{k}^{% \text{Min}}}$ $\displaystyle E_{k}=\frac{\textit{round}_{k}^{\text{Max}}-\textit{round}_{k}}{% \textit{round}_{k}^{\text{Max}}-\textit{round}_{k}^{\text{Min}}}$ $\displaystyle I_{k}=\left\lfloor{((\alpha\times A_{k}+1)+(\beta\times S_{k}+1)% +(\gamma\times E_{k}+1))\times\theta}\right\rfloor$ $\displaystyle\Delta\omega_{k}^{t}=\omega^{t}-\overline{\omega}_{k}^{t+1}$ $\displaystyle\Delta_{k}^{t}=F_{k}^{I_{k}}(\omega^{t})\Delta\omega_{k}^{t}$ $\displaystyle h_{k}^{t}=I_{k}F_{k}^{I_{k}-1}(\omega^{t})\left\\|{\Delta\omega_{% k}^{t}}\right\\|^{2}+LF_{k}^{I_{k}}(\omega^{t})$
6: Each selected device $k$ sends $I_{k}$ , $\Delta_{k}^{t}$ and $h_{k}^{t}$ back to the server
7: Server updated $\omega^{t+1}$ as: $\displaystyle\omega^{t+1}=\omega^{t}-\frac{\sum\nolimits_{k\in S_{t}}{\Delta_{% k}^{t}}}{\sum\nolimits_{k\in S_{t}}{h_{k}^{t}}}$
8: end for

The flowchart of the algorithm SHFF is shown in Fig. 2.

Figure 2.

The flowchart of the algorithm SHFF.

3.2 Selection of

\theta

The global fairness parameters $\theta$ can be dynamically adjusted to fit the data set according to the expected fairness in practice – the bigger, the fairer. However, the average accuracy will also be damaged, so that this paper will seek a balance between accuracy and fairness.

Training a series of target families with different $\theta$ values increases training costs. We rely on the gradient descent method to scale the optimization objective OPT, as shown in Eq. (8).

$\displaystyle L_{\theta}(w)=Lf(w)^{\theta}+\theta f(w)^{\theta-1}||\nabla f(w)% ||^{2}$ (8)

The step size is inversely proportional to the Lipchitz constant of the function gradient, which is selected by grid search. However, with the change of $\theta$ value, the Lipchitz constant also changes accordingly, resulting in the search space explosion [20, 21, 22]. We select the Lipchitz constant at $\theta=0$ as the constant local gradient of the optimization objective family. In this paper, the step size based dynamically adjusts to avoid manual adjustment for each $\theta$ .

4. Experiment

4.1 Data set introduction

The data set used in this experiment is the vehicle data set from DARPA/Ixos sensitive project [23]. The data set consists of acoustic, seismic, and infrared sensor data collected by a distributed network of 23 sensors. Each sensor is modeled as equipment with $n$ instances on each equipment, with 43695 cases. Each instance has a 100-dimensional feature description (including acoustic, seismic, and infrared features). A linear support vector machine (SVM) model [24, 25, 26] predicts two types of military vehicles, AAV and DW, for 23 equipment. The experimental environment is Windows 10 system, GEFORCE RTX 2080 Super GPU, TensorFlow 1.14, python 3.7.

4.2 Isomorphic system

Figure 3 shows the fairness distribution between the SHFF algorithm, FedAvg (uniform sampling) algorithm, and q-FedAvg algorithm without applying system heterogeneity. The horizontal axis represents the accuracy, and the vertical axis represents the number of equipment that achieve the corresponding accuracy. The accuracy distribution of the SHFF algorithm is more centralized than that of FedAvg and q-FedAvg without applying system heterogeneity. It shows that the accuracy distribution of the SHFF algorithm is fairer than the above two algorithms.

Table 3 shows the average accuracy, the Worst 10% equipment accuracy, the Best 10% equipment accuracy, and variance of the data statistics of SHFF, FedAvg, and q-FedAvg algorithms in the case of 5 random shuffles of data sets without applying system heterogeneity. The global final average accuracy and the average accuracy of the Best 10% equipment fluctuate little. The average accuracy and variance of the Worst 10% of equipment are significantly improved compared with FedAvg and q-FedAvg algorithms. Compared with q-FedAvg algorithm, the average accuracy of SHFF algorithm on Worst 10% equipment is improved by 4.7%, and the variance is reduced by 35%.

Table 3
SHFF, Fedavg, and q-FedAvg data statistics (isomorphic system)

Heterogeneity	Objective	Average	Worst 10%	Best 10%	Variance
epo_0	uniform	86.2% $\pm$ 0.7%	46.8% $\pm$ 1.7%	94.1% $\pm$ 0.9%	239 $\pm$ 21
	q-FedAvg	87.3% $\pm$ 0.7%	71.5% $\pm$ 1.5%	94.1% $\pm$ 1.3%	39 $\pm$ 8
	SHFF	84.8% $\pm$ 0.9%	74.9% $\pm$ 3.3%	91.8% $\pm$ 0.9%	25 $\pm$ 6

Table 4

Data statistics under four heterogeneous systems (epo_7–epo_10)

Heterogeneity	Objective	Average	Worst 10%	Best 10%	Variance
epo_7	uniform	86.3% $\pm$ 0.3%	47.0% $\pm$ 2.1%	93.8% $\pm$ 0.8%	247 $\pm$ 25
	q-FedAvg	86.6% $\pm$ 0.4%	53.9% $\pm$ 2.1%	93.4% $\pm$ 0.8%	151 $\pm$ 30
	SHFF	83.7% $\pm$ 0.7%	60.9% $\pm$ 8%	90% $\pm$ 0.6%	93 $\pm$ 57
epo_8	uniform	86.4% $\pm$ 0.3%	46.2% $\pm$ 2%	93.7% $\pm$ 0.9%	245 $\pm$ 20
	q-FedAvg	86.8% $\pm$ 0.3%	55.2% $\pm$ 1.3%	93.3% $\pm$ 0.8%	142 $\pm$ 20
	SHFF	85.3% $\pm$ 0.5%	66.0% $\pm$ 3.2%	91.6% $\pm$ 0.5%	52 $\pm$ 15
epo_9	uniform	86.5% $\pm$ 0.3%	48.4% $\pm$ 3.7%	94.2% $\pm$ 0.3%	225 $\pm$ 39
	q-FedAvg	86.6% $\pm$ 0.7%	56.4% $\pm$ 4.7%	93.8% $\pm$ 0.7%	133 $\pm$ 47
	SHFF	85.5% $\pm$ 0.8%	71.0% $\pm$ 4.5%	91.7% $\pm$ 1.6%	36 $\pm$ 19
epo_10	uniform	86.5% $\pm$ 0.2%	46.4% $\pm$ 1.1%	94.5% $\pm$ 0.8%	248 $\pm$ 20
	q-FedAvg	86.5% $\pm$ 0.5%	53.5% $\pm$ 2.1%	93.4% $\pm$ 0.5%	150 $\pm$ 31
	SHFF	84.6% $\pm$ 0.6%	67.0% $\pm$ 7%	90.7% $\pm$ 0.9%	58 $\pm$ 36

Figure 3.

Accuracy distribution of SHFF and FedAvg (isomorphic system).

4.3 Heterogeneous system

Figure 4 shows the comparison of fairness distribution of the SHFF algorithm, FedAvg algorithm, and q-FedAvg algorithm when system heterogeneity is applied. In this paper, the heterogeneity of the system changes the local training grounds. Figure 4a–d shows the fairness distribution of the local training round epo_7 to epo_10 in unit time. The accuracy distribution of the SHFF algorithm is more centralized than that of FedAvg and q-FedAvg.

Figure 4.

Accuracy distribution of SHFF, FedAvg, and q-FedAvg (heterogeneous system).

Table 4 shows the average accuracy, the Worst 10% equipment accuracy, the Best 10% equipment accuracy, and variance of the data statistics of SHFF, FedAvg, and q-FedAvg algorithms in the case of 5 random shuffles of data sets when applying system heterogeneity. The accuracy of the Best 10% equipment and the global final average accuracy fluctuates little. The accuracy of the Worst 10% of equipment is significantly improved, and the variance between equipment and equipment reduce dramatically. And with the gradual increase of the heterogeneity of the system, the accuracy of the Worst 10% of equipment is improved more obviously, and the variance decrease more obviously. When the local training round is set to epo_10, compared with the FedAvg algorithm, the average accuracy of the Worst 10% of equipment is 45% improved, and the variance is 76% reduced. Compared with the q-FedAvg algorithm, the average accuracy of the Worst 10% of equipment is 26% improved, and the variance is 61% reduced.

5. Conclusion

The intelligent transportation system processes the massive data generated and make timely decisions for emergencies. Federated learning is a new distributed machine learning paradigm, but it is difficult for heterogeneity system fairness design. In this paper, we design a system heterogeneous fair federated learning algorithm (SHFF) for Federated learning fairness. The experimental results show that SHFF better promotes fairness between heterogeneous equipment. In the future, we will focus on the differences in network resources in heterogeneous systems, such as the network transmission performance optimization of federated learning.

Footnotes

Acknowledgments

The authors acknowledge Research on Key Technologies of intelligent manufacturing management based on digital twin technology, key R & D project in industrial field of Jilin Province (20200401076GX) (2020.1.1-2022.12.31).

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