Kernel techniques are among the most widely used and influential tools, and have great impact on almost all fields of statistics and machine learning [1, 2]. Their versatility stems from the function class associated with the kernel called reproducing kernel Hilbert space, which achieve great success in complex relationship modeling [3, 4]. Kernel techniques are very powerful because of their modeling capabilities, but they always computationally expensive as a cost in flexibility [1, 5]. Some methods have been proposed to alleviate this computational bottleneck, such as sketching [6, 7], nystrom and subsampling methods [8, 9, 10], and random Fourier features (RFF) [11]. Among these methods, RFF is considered as arguably the most influential and simplest approximation scheme [5]. By an explicit kernel expansion, RFF are very efficient to approximate shiftinvariant kernels [11]. Much lower runtime complexity could be achieved with the use of efficient linear models. As one of the key methods to kernelize algorithms with linear models, they are applied in various methods successfully, especially in machine learning [12].
Because these methods can overcome the shortcomings of machine learning methods that require a large number of labeled sample, they are effective to be applied to the practical areas where samples are difficult to obtain [13]. Network traffic forwarding policy making is one such practical application, and it is widely concerned in the programmable networks [14]. The programmable network brings great convenience to network management, but it also brings some problems. The most important is the security of the programmable network, which is a major challenge for the largescale commercial use of the programmable network [15, 16]. Since the control plane is the core part in the SDN network, once the control plane is maliciously violated or attacked, it will affect the entire network, or even cause the entire network to be paralyzed [17, 18]. Therefore, anomaly detection of network behavior is an indispensable link in the construction of networks. However, with the development of network technology, the attack methods of network intrusion become more and more complex and hidden, and also make it more and more difficult to detect [19]. Therefore, how to make traffic forwarding policy in programmable networks has always been one of the key tasks of network security.
In the conventional network traffic forwarding policy making method, the abnormal traffic is often blocked by filtering out illegal strings in the rule base through regular matching, but this method is based on the human maintenance of the rule base, and most of the rules are based on regular matching technology, which needs to be formulated by professionals and maintained in real time [20]. Moreover, when the number of rules is large, the detection is often affected by the performance of the hardware device.
Different from conventional methods, the application of machine learning in network traffic forwarding policy making can reduce the technical risk caused by manual maintenance of the rule base, increase the hit rate of judging abnormal traffic, and reduce the investment cost of hardware equipment [20, 21]. The existing machine learning methods for traffic forwarding policy making are mainly divided into unsupervised learning methods and supervised learning methods [21, 22]. Although unsupervised learning does not need to label abnormal sample data, it requires a large number of samples for training, and the model effect is not as good as supervised learning methods [21, 23]. The supervised learning methods produce models with good adaptability by clustering abnormal sample data, but the common supervised learning methods also have the disadvantage of insufficient interpretability [21]. Therefore, how to overcome the shortcomings of the existing technology is an urgent problem to be solved in the network security technology field.
To solve the above problems, this paper applies kernel based Random Fourier approximation to the field of network security, and proposes a network traffic forwarding policy making method based on random Fourier approximation of the kernel function in programmable networks. Through random Fourier transform, the traffic forwarding features are mapped to the Hilbert highdimensional space, and then the maximum interval principle is used to realize the traffic forwarding making in the programmable network.
Kernel trick is a concept which makes much quicker and easier to compute for classification task, when the data is not linearly separable [2]. For algorithms, it is a simple method to generate features that only rely on the inner product between pairs of input points. It depends on the observation that any positive definite function
The cost of this convenience is that the algorithm accesses the data only through evaluations of
Random Fourier features make use of sine/cosine as their basis function to approximate the kernel and make use of an advantage of Theorem 1 to approximate the kernel [11]. Compared with the traditional kernel trick, the random Fourier feature could effectively reduce the computational and storage costs.
If a shift variant kernel
Basically, random Fourier features approximate the integral in Equation (2) using samples drawn from
Random Fourier Features
1. Input: A positive definite shiftinvariant kernel 
2. Compute: Fourier transform 
3. Draw L iid 
4. Return:

Due to the increasing complexity of the network, it becomes more and more difficult for network managers to configure the network according to predefined policies, and make the network reconfiguration. Programmable network is proposed for these problems. A programmable network is one in which the behaviors of network devices and flow control are managed by software, which makes independent operation from network hardware. Programmable networks allow network engineers to reprogram network infrastructures rather than rebuilding them manually [15].
The programmable network originally originated from the SoftwareDefined Networking (SDN) technology proposed by the clean state research group of Stanford University in the United States in 2006. Since then, SDN technology has received great attention from the industry, and a series of related applications have been proposed, which has greatly promoted the innovation and development of the network [15, 17]. SDN is an emerging architecture, which is costeffective, dynamic, adaptable and manageable, making it great suitable for the highbandwidth, dynamic nature of applications at present. In SDN, the control plane is separate from the forwarding plane physically, meanwhile one control plane controls multiple forwarding devices. Although forwarding devices can be programmed in a number of ways, having a open, common and vendoragnostic interface, such as OpenFlow, allows a control plane to control forwarding devices from different software and hardware vendors [17]. In recent years, Programming ProtocolIndependent Packet Processors (P4) is proposed as a domainspecific language to program the data plane of programmable switches. P4 works with SDN control protocols such as OpenFlow. And P4 defines the programmable switches’ data plane behavior by primitives, independent of the underlying architecture.
In Figure 1, a typical representation of SDN architecture contains three layers, including the application layer, the control layer and the infrastructure layer. These three layers communicate with southbound and northbound application programming interfaces (APIs). The application layer includes the functions organizations use or typical network applications. This can contain load balancing, firewalls or intrusion detection systems. The control layer represents the centralized SDN controller software that acts as the brain of SDN. This controller resides on a server, and it manages policies and traffic flows throughout the networks. The infrastructure layer is made up of the physical switches in the network. And these switches forward the network traffic to their destinations.
For the difficulty of obtaining samples in the field of network traffic model recognition, this paper combines the characteristics and advantages of kernel trick and random Fourier features, and proposes a network traffic forwarding policy making method based on Random Fourier approximation of the kernel function that can be applied to programmable networks. Through random Fourier transform, the traffic forwarding features are mapped to the Hilbert highdimensional space, and then the maximum interval principle is used to detect the adversarial samples to realize traffic model detection. The flow of proposed method is illustrated in Figure 2, including four steps: network access data preprocessing and segmentation, model construction, model training and traffic forwarding policy making.
Networkaccessed data is stored in form of text. In the preprocessing stage, each character in the network access data is preprocessed into a standardlength digital string as a sample set. Specifically, each character in the network access data is mapped to the corresponding number format according to the dictionary table, and the maximum length limit of the network access text is the maximum length of the text trained by the model. For the ones that don’t reach the maximum length, use 0 to complete it. And for the ones that exceed the set maximum length, only the text to the maximum length is intercepted. The label of normal network access data is marked as 1, and the label of abnormal network access data is marked as 0. In the data set segmentation stage, the preprocessed sample data is divided into a training set and a test set according to a certain proportion.
With random Fourier transform in dual space, the abnormal flow can be mapped to Hilbert highdimensional space. Then, in Hilbert space, adversarial examples can be detected with the maximum interval principle. The model of the proposed method is a discriminative model, and its objective is the classification function
(1) Classification model:
Obtain the model loss function L based on the maximum interval principle:
Decompose the sample x along the normal and tangential directions of
In Hilbert space, the vector on the linear decision boundary is called the support vector,
The goal of the model is to maximize the above spacing. This function is a quadratic programming function. According to the convex optimization theory, it is transformed into the dual space for processing.
Maximize the above spacing as minimizing the loss function L:
The standardlength digital string in the training set is used as the model input, and the label marked by the sample is used as the output. The gradient descent method is used for training to obtain the model parameters, and the test set is used for testing.
Solving the loss function with gradient descent algorithm:
After preprocessing the network access data, input it into the trained model for traffic model recognition. In detail, when the model output is 1, it is judged that the network access data is in normal traffic, and when the model output is 0, the network access data is in traffic to be controlled. That is, if the model classification result is determined to be normal, it is determined that the network access data is normal, while if the model classification result is determined as to be controlled, the network control plane should compute a forwarding policy for the traffic.
In order to test the effect of network traffic forwarding policy making method proposed in programmable networks, the network access data in the programmable network is used as the data set for verification. 25,000 access data were collected, of which 20,000 were normal access data and 5,000 were abnormal access data. The data set is divided into training set and test set according to the ratio of 8:2. Receiver operating characteristic curve (ROC) was used to evaluate the performance of the model on the test set [24].
This experiment was run on a 4core Intel(R) i74720HQCPU @2601Mhz laptop with 16GB of memory. The data set used is obtained from the realworld programmable network, of which 20,000 are normal access records and 5,000 are offensive access records.
An example of a network access in realworld programmable network is shown in Figure 3.
The maximum length limit of the network access text is the maximum length of the text trained by the model. Fill with 0 to the one less than the maximum length and intercept the one exceeding the set maximum length. Finally, label the data set, marking the label of normal network access data as 1, while the label of abnormal network access data as 0. Divide the data set into training set and test set according to 8:2.
Create a model object class and set the stopping error to 1e–6. Input the training set into the model and minimize the loss function to get the optimal parameters.
There are four outcomes of binary classifiers as follows:
¥1. True positive (TP): The classifier judges it to be an abnormal access, which is actually an abnormal access.
¥2. False positive (FP): The classifier judges it to be an abnormal access, which is actually a normal access。
¥3. True negative (TN): The classifier judges it to be a normal access, which is actually a normal access。
¥4. False negative (FN): The classifier judges it to be a normal access, which is actually an abnormal access.
Given a rejection threshold
Each threshold η, corresponding to the coordinates of (
In the programmable network data set, ROC curve and AUC are shown in Figure 4. AUC=0.9984 and the trend of the ROC curve also indicates that the proposed method could accurately identify the network attacks to achieve the expected goal.
Combining the characteristics and advantages of kernel techniques and random Fourier features, this paper proposes an application of a network traffic forwarding policy making method based on random Fourier approximation of kernel function in programmable networks to realize traffic forwarding policy making to improve the security of programmable networks. The proposed method maps traffic forwarding features to Hilbert highdimensional space through random Fourier transform, and then uses the principle of maximum interval to detect adversarial samples. The proposed method can effectively overcome the shortcomings of machine learning methods that require a large number of labeled sample pairs. Moreover, compared with the traditional kernel function method, it can improve the algorithm efficiency from square efficiency to linear efficiency. Through experimental verification, the proposed method can accurately making traffic forwarding policy in the programmable network with good stability. The next step is to verify the realtime performance of the method in practical application in largescale programmable networks.
Random Fourier Features
1. Input: A positive definite shiftinvariant kernel 
2. Compute: Fourier transform 
3. Draw L iid 
4. Return:

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platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long shortterm memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSRAHP model A study of local smoothnessinformed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H_{2}N_{2} atmosphere Algorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Crossborder Ecommerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model ChineseEnglish Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model