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Random Fourier Approximation of the Kernel Function in Programmable Networks

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 26 Apr 2022
Accepté: 21 Jun 2022
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Introduction

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 sub-sampling 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 shift-invariant 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 large-scale 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 high-dimensional space, and then the maximum interval principle is used to realize the traffic forwarding making in the programmable network.

Related Works
Kernel Trick

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 k(x, y) with, x, yRd defines an inner product and a lifting ϕ. Therefore, the inner product between lifted data points can be calculated as: ϕ(x),ϕ(y) =k(x,y) \left\langle {\phi \left( x \right),\,\phi \left( y \right)} \right\rangle = k\left( {x,y} \right)

The cost of this convenience is that the algorithm accesses the data only through evaluations of k(x, y), or through the kernel matrix consisting of k applied to all pairs of data points.

Random Fourier Features

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.

Theorem 1. A continuous kernel k(x, y) = k(xy) on Rd is positive definite if and only if k (δ) is the Fourier transform of a non-negative measure.

If a shift variant kernel k (δ) is properly scaled, then Theorem 1 guarantees that p (ω) in Equation (2) is a proper probability distribution. k(xy)=Rdp(ω)ejωT(xy)dω=Eω(ζω(x)ζω(y)*) k\left( {x - y} \right) = \int_{{R^d}} {p\left( \omega \right){e^{j{\omega ^T}\left( {x - y} \right)}}d\omega = {E_\omega }\left( {{\zeta _\omega }\left( x \right){\zeta _\omega }\left( y \right)*} \right)}

Basically, random Fourier features approximate the integral in Equation (2) using samples drawn from p (ω). Draw L samples ω1, ω2,..., ωL samples from p(ω). k(xy)=Rdp(ω)ejωT(xy)dω=Rdp(ω)cos(ωTxωTy)dω1Li=1Lcos(ωiTxωiTy)=1Li=1Lcos(ωiTx)cos(ωiTy)+sin(ωiTx)sin(ωiTy)=1Li=1L[ cos(ωiTx),sin(ωiTx) ]T[ cos(ωiTy),sin(ωiTy) ]=z(x)Tz(y) \matrix{ {k\left( {x - y} \right)} \hfill & = \hfill & {\int_{{R^d}} {p\left( \omega \right){e^{j{\omega ^T}\left( {x - y} \right)}}d\omega } } \hfill \cr {} \hfill & = \hfill & {\int_{{R^d}} {p\left( \omega \right)\cos \left( {{\omega ^T}x - {\omega ^T}y} \right)d\omega } } \hfill \cr {} \hfill & \approx \hfill & {{1 \over L}\sum\limits_{i = 1}^L {\cos \left( {\omega _i^Tx - \omega _i^Ty} \right)} } \hfill \cr {} \hfill & = \hfill & {{1 \over L}\sum\limits_{i = 1}^L {\cos \left( {\omega _i^Tx} \right)\cos \left( {\omega _i^Ty} \right) + \sin \left( {\omega _i^Tx} \right)\sin \left( {\omega _i^Ty} \right)} } \hfill \cr {} \hfill & = \hfill & {{1 \over L}\sum\limits_{i = 1}^L {{{\left[ {\cos \left( {\omega _i^Tx} \right),\,\sin \left( {\omega _i^Tx} \right)} \right]}^T}\left[ {\cos \left( {\omega _i^Ty} \right),\,\sin \left( {\omega _i^Ty} \right)} \right]} } \hfill \cr {} \hfill & = \hfill & {z{{\left( x \right)}^T}\,z\left( y \right)} \hfill \cr } where z(x)=1L[ cos(ωiTx),sin(ωiTx) ]R2L z\left( x \right) = {1 \over {\sqrt {\rm{L}} }}\left[ {\cos \left( {\omega _{\rm{i}}^{\rm{T}}{\rm{x}}} \right),\,\sin \left( {\omega _{\rm{i}}^{\rm{T}}{\rm{x}}} \right)} \right] \in {{\rm{R}}^{2{\rm{L}}}} is an approximate non linear feature mapping.

Random Fourier Features

1. Input: A positive definite shift-invariant kernel k(x, y) = k(xy)
2. Compute: Fourier transform p of the kernel k: p(ω)=12πRdejωTδk(δ)dΔ p\left( \omega \right) = {1 \over {2\pi }}\int_{{R^d}} {{e^{ - j{\omega ^T}\delta }}k\left( \delta \right)d\Delta }
3. Draw L iid ω1, ω2,..., ωLRd samples from p
4. Return: z(x)=1L[ cos(ωiTx),sin(ωiTx) ]i=1LR2L z\left( x \right) = {1 \over {\sqrt {\rm{L}} }}\left[ {\cos \left( {\omega _{\rm{i}}^{\rm{T}}{\rm{x}}} \right),\,\sin \left( {\omega _{\rm{i}}^{\rm{T}}{\rm{x}}} \right)} \right]_{{\rm{i}} = 1}^{\rm{L}} \in {{\rm{R}}^{2{\rm{L}}}}
Programmable network

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 re-program network infrastructures rather than re-building them manually [15].

The programmable network originally originated from the Software-Defined 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 cost-effective, dynamic, adaptable and manageable, making it great suitable for the high-bandwidth, 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 vendor-agnostic interface, such as OpenFlow, allows a control plane to control forwarding devices from different software and hardware vendors [17]. In recent years, Programming Protocol-Independent Packet Processors (P4) is proposed as a domain-specific 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.

Fig. 1

A typical representation of SDN architecture

Forwarding Policy Making Method based on Random Fourier Approximation of the Kernel Function in Programmable Networks

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 high-dimensional 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.

Fig. 2

Flow of proposed network traffic forwarding policy making method

Network access data preprocessing and segmentation

Network-accessed data is stored in form of text. In the preprocessing stage, each character in the network access data is preprocessed into a standard-length 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.

Model

With random Fourier transform in dual space, the abnormal flow can be mapped to Hilbert high-dimensional 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 f(x). Specifically, classify the input network access text into normal access and abnormal access with the classification function f(x).

(1) Classification model: h(x)=sign(f(x)) h\left( x \right) = sign\left( {f\left( x \right)} \right) where when f(x) is positive, the output of sign (·) is 1, and when f(x) is negative, the output is 0. f(x)=i=1NαiK(x,xi) f\left( x \right) = \sum\limits_{i = 1}^N {{\alpha _i}K\left( {x,{x_i}} \right)} where αi is the coefficient in dual space, K(x, xi) is the inner product of Hilbert space and N is the sample size. Decomposition of K(x, xi) is: K(x,xi)=ϕ(x)Tϕ(xi) {\rm{K}}\left( {{\rm{x}},\,{{\rm{x}}_{\rm{i}}}} \right) = \,\phi {\left( {\rm{x}} \right)^{\rm{T}}}\phi \left( {{{\rm{x}}_{\rm{i}}}} \right) where ϕ(x)1T[ sin(Ωx),cos(Ωx) ] \phi \left( {\rm{x}} \right){1 \over {\sqrt {\rm{T}} }}\left[ {\sin \left( {\Omega {\rm{x}}} \right),\,\cos \left( {\Omega {\rm{x}}} \right)} \right] T = M / 2, M is the dimension of Hilbert space, Ω ∈ RT×D is a random Gaussian matrix. Each element is sampled independently and identically from the normal distribution N(0,1/σ2), and σ is the bandwidth of the core. D is the length of input sample. x is the network sample to be determined, which is the i-th network sample in the training set.

ϕ(x)1T[ sin(Ωx),cos(Ωx) ] \phi \left( {\rm{x}} \right){1 \over {\sqrt {\rm{T}} }}\left[ {\sin \left( {\Omega {\rm{x}}} \right),\,\cos \left( {\Omega {\rm{x}}} \right)} \right] is the Fourier approximation of kernel function.

Obtain the model loss function L based on the maximum interval principle:

f(x) is a linear function in a high-dimensional Hilbert space, and αn yn is the n-th linear vector in the Hilbert space. w = (w1, w2, …, wn) is the linear coefficient, where wn=αny˜n {{\rm{w}}_{\rm{n}}} = {\alpha _{\rm{n}}}{{\rm{\tilde y}}_{\rm{n}}} and αn is the weight in Hilbert space and y˜n {{\rm{\tilde y}}_{\rm{n}}} represents the label of the sample.

Decompose the sample x along the normal and tangential directions of w in Hilbert space. The spacing is: 1 w [ y˜n(wx+w0) ] {1 \over {\left\| {\rm{w}} \right\|}}\left[ {{{{\rm{\tilde y}}}_{\rm{n}}}\left( {{\rm{wx}} + {{\rm{w}}_0}} \right)} \right] where w0 is the intercept of the model in Hilbert space, ||w|| is the length of the vector w. w0=1| S |iS(yi(jSαjyjK(xj,x))) {{\rm{w}}_0} = {1 \over {\left| {\rm{S}} \right|}}\sum\limits_{{\rm{i}} \in {\rm{S}}} {\left( {{{\rm{y}}_{\rm{i}}} - \left( {\sum\limits_{{\rm{j}} \in {\rm{S}}} {{\alpha _{\rm{j}}}{{\rm{y}}_{\rm{j}}}{\rm{K}}\left( {{{\rm{x}}_{\rm{j}}},\,{\rm{x}}} \right)} } \right)} \right)}

In Hilbert space, the vector on the linear decision boundary is called the support vector, S is the set of support vectors, and |S| is the number of support vectors.

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: L(α)=12ijαiαjyiyjK(xi,xj) {\rm{L}}\left( \alpha \right) = - {1 \over 2}\sum\limits_{\rm{i}} {\sum\limits_{\rm{j}} {{\alpha _{\rm{i}}}\,{\alpha _{\rm{j}}}\,{{\rm{y}}_{\rm{i}}}\,{{\rm{y}}_{\rm{j}}}\,{\rm{K}}\left( {{{\rm{x}}_{\rm{i}}},\,{{\rm{x}}_{\rm{j}}}} \right)} } where αn is the weight of the n-th sample in the Hilbert space, and y˜n {{\rm{\tilde y}}_{\rm{n}}} is the label of the n-th sample.

Model training

The standard-length 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: L(α)α=ijy˜iy˜iK(xi,xj)αα+L(α)α \matrix{ {{{\partial {\rm{L}}\left( \alpha \right)} \over \alpha } = \sum\limits_{\rm{i}} {\sum\limits_{\rm{j}} {{{{\rm{\tilde y}}}_{\rm{i}}}\,{{{\rm{\tilde y}}}_{\rm{i}}}\,{\rm{K}}\left( {{{\rm{x}}_{\rm{i}}},\,{{\rm{x}}_{\rm{j}}}} \right)} } } \hfill \cr {\alpha \leftarrow \alpha + {{\partial {\rm{L}}\left( \alpha \right)} \over \alpha }} \hfill \cr } where α is the set of all αn. αn is the weight of the n-th sample in Hilbert space, and y˜n {{\rm{\tilde y}}_{\rm{n}}} is the label of the n-th sample.

Traffic forwarding policy making

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.

Experiments

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].

Experimental configuration and data preparation

This experiment was run on a 4-core Intel(R) i7-4720HQ-CPU @2601Mhz laptop with 16GB of memory. The data set used is obtained from the real-world programmable network, of which 20,000 are normal access records and 5,000 are offensive access records.

Data preprocessing

An example of a network access in real-world programmable network is shown in Figure 3.

Fig. 3

An example of a network access in real-world programmable network

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.

Model construction and training

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.

Experimental results

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 η, the ratio TPR as recall can be calculated, and the ratio FPR that tested as false by mistake can be calculated. FPR=FPFP+TNTPR=TPTP+FN \matrix{ {FPR = {{FP} \over {FP + TN}}} \hfill \cr {TPR = {{TP} \over {TP + FN}}} \hfill \cr }

Each threshold η, corresponding to the coordinates of (FPR, TPR), forms a series of coordinate points, and these points constitute the ROC. Area under the curve (AUC) stands for the area under the ROC curve [25].

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.

Fig. 4

Experimental result

Conclusions

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 high-dimensional 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 real-time performance of the method in practical application in large-scale programmable networks.

Fig. 1

A typical representation of SDN architecture
A typical representation of SDN architecture

Fig. 2

Flow of proposed network traffic forwarding policy making method
Flow of proposed network traffic forwarding policy making method

Fig. 3

An example of a network access in real-world programmable network
An example of a network access in real-world programmable network

Fig. 4

Experimental result
Experimental result

Random Fourier Features

1. Input: A positive definite shift-invariant kernel k(x, y) = k(xy)
2. Compute: Fourier transform p of the kernel k: p(ω)=12πRdejωTδk(δ)dΔ p\left( \omega \right) = {1 \over {2\pi }}\int_{{R^d}} {{e^{ - j{\omega ^T}\delta }}k\left( \delta \right)d\Delta }
3. Draw L iid ω1, ω2,..., ωLRd samples from p
4. Return: z(x)=1L[ cos(ωiTx),sin(ωiTx) ]i=1LR2L z\left( x \right) = {1 \over {\sqrt {\rm{L}} }}\left[ {\cos \left( {\omega _{\rm{i}}^{\rm{T}}{\rm{x}}} \right),\,\sin \left( {\omega _{\rm{i}}^{\rm{T}}{\rm{x}}} \right)} \right]_{{\rm{i}} = 1}^{\rm{L}} \in {{\rm{R}}^{2{\rm{L}}}}

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