Data processing system based on federated learning

Abstract

The present invention relates to a data processing system based on federated learning, comprising a central server S * , n data terminals participating in federated learning {S 1 , S 2 …S n}, a processor and a memory storing a computer program, wherein S i Indicates the i-th data terminal, where i ranges from 1 to n, S i The corresponding training data set D is stored in i , all D i The data format of each piece of data in S is the same. i The corresponding local model is L i (x),L i The input of (x) is D i The data in L i The output of S(x) is the input of G(x), and G(x) is the * The output of the maintained global model G(x) is the target prediction result. The present invention can improve the training efficiency of the prediction model.

Description

Technical Field

[0001] The present invention relates to the field of computer technology, and in particular to a data processing system based on federated learning. Background Art

[0002] A series of machine learning applications in Internet technology, such as recommendation systems, image recognition, and dialogue systems, are all based on deep learning models. The common point of these models is that they require a large amount of data for training, while the amount of data in a single Internet data terminal is often insufficient. The solution in the past was to concentrate a large amount of data in a data center to train a model. For example, if multiple data terminals need to jointly train an image classification model or a sentence sentiment classification model, they often choose to concentrate all the data in a trusted data center for data training. However, this method has the risk of leaking data information from data terminals. The currently proposed federated deep learning algorithm solves this problem to a certain extent and has a certain privacy protection capability. However, it also faces technical problems such as poor model prediction effect, non-convergence, and long training time caused by non-independent and identically distributed data. The long process of training deep learning models will greatly increase the cost of equipment, manpower, etc.; in addition, the insufficient prediction ability of the deep learning model finally obtained means that the deep learning model has no way to make correct predictions for samples that have not been seen, and the accuracy of the deep learning model is low. Therefore, how to provide a technology that can reduce the time cost of predictive model training and improve the predictive ability of the predictive model has become a technical problem that needs to be solved urgently. Summary of the invention

[0003] The purpose of the present invention is to provide a data processing system based on federated learning, which can improve the training efficiency of the prediction model.

[0004] The present invention provides a data processing system based on federated learning, characterized in that:

[0005] Including the central server S * , n data terminals participating in federated learning {S 1 , S 2 …S n}, a processor and a memory storing a computer program, wherein S i Indicates the i-th data terminal, where i ranges from 1 to n, S i The corresponding training data set D is stored in i , all D i The data format of each piece of data in S is the same. i The corresponding local model is L i (x),L i The input of (x) is D i The data in Li The output of S(x) is the input of G(x), and G(x) is the * The output of the maintained global model G(x) is the target prediction result;

[0006] When the processor executes the computer program, the following steps are implemented:

[0007] Step S1: In the first model training stage, S i Based on D i 、Current L i (x) and the current G(x) perform the preset F rounds of local model training, and update L in real time i The local model parameters p of (x) i , and obtain S i The corresponding global cumulative update parameters

[0008] Step S2: In the second model training phase, L i (x) and Send to S * , S * based on Update the global model parameter p of G(x) * , send the updated G(x) to each S i , and then return to step S1 to perform the next round of first model training phase.

[0009] Compared with the prior art, the present invention has obvious advantages and beneficial effects. By means of the above technical solution, the data processing system based on federated learning provided by the present invention can achieve considerable technical advancement and practicality, and has wide industrial utilization value, and has at least the following advantages:

[0010] The present invention divides federated learning into two model training stages. In the first model training stage, the local model of the data terminal is trained without changing the global model and only the global model update parameters are recorded. In the second model training stage, the global model is updated based on the global model update parameters, thereby improving the convergence speed of the prediction model training and thus improving the prediction model training efficiency.

[0011] The above description is only an overview of the technical solution of the present invention. In order to more clearly understand the technical means of the present invention, it can be implemented in accordance with the contents of the specification. In order to make the above and other purposes, features and advantages of the present invention more obvious and easy to understand, the following specifically cites a preferred embodiment and describes it in detail with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 A schematic diagram of a data processing system provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0013] In order to further explain the technical means and effects adopted by the present invention to achieve the predetermined purpose of the invention, the specific implementation method and effects of a data processing system based on federated learning proposed by the present invention are described in detail below in combination with the accompanying drawings and preferred embodiments.

[0014] Embodiment 1

[0015] Embodiment 1 provides a data processing system based on federated learning, such as Figure 1 It should be noted that the subsequent embodiments 2 and 3 share the same data processing system framework diagram with the embodiment 1, but the specific processing processes performed by the processors are different. The data processing system based on federated learning includes a central server S * , n data terminals participating in federated learning {S 1 , S 2 …S n}, a processor and a memory storing a computer program, wherein S i Indicates the i-th data terminal, where i ranges from 1 to n, S i The corresponding training data set D is stored in i , all D i The data format of each piece of data in S is the same. i The corresponding local model is L i (x),L i The input of (x) is D i The data in L i The output of S(x) is the input of G(x), and G(x) is the * The global model maintained, the output of G(x) is the target prediction result; L i The output of (x) can be a k-dimensional vector, so the input of G(x) is also a k-dimensional vector. i The function of (x) is to extract the features of the input data x and abstract the k-dimensional features that are easier to classify. As an example, L i (x) and G(x) are neural network models. i The output features of (x) are generated based on the intermediate output layer of the neural network model. It should be noted that L i The extracted features of (x) are the features of the intermediate layer of the neural network, which further protect the security of terminal data. * The global model maintained, the output of G(x) is the target prediction result, G(x) can be set as a classifier, by i(x) The useful features abstracted are classified to obtain the final prediction results. The target prediction results can be set in the form of the target prediction results based on the specific application scenario. For example, for the problem of single written digital image recognition, the target prediction result can output a Z-dimensional vector, and the z-th dimension of the vector represents the probability that the sample is the number z, and the value range of z is 1 to Z. It can be understood that the system of the present invention can also be applicable to deep learning application scenarios such as recommendation systems, image recognition, and dialogue systems, which are not listed here one by one. It should be noted that the data terminal can specifically be a server of different companies, a mobile terminal such as a mobile phone or a computer of different users. The present invention is based on federated learning, which can avoid the leakage of user data of each data terminal. The central server is a trusted server.

[0016] When the processor executes the computer program, the following steps are implemented:

[0017] Step S1: In the first model training stage, S i Based on D i 、Current L i (x) and the current G(x) perform the preset F rounds of local model training, and update L in real time i The local model parameters p of (x) i , and obtain S i The corresponding global cumulative update parameters

[0018] Step S2: In the second model training phase, L i (x) and Send to S * , S * based on Update the global model parameter p of G(x) * , send the updated G(x) to each S i , and then return to step S1 to perform the next round of first model training phase.

[0019] The present invention divides federated learning into two model training stages. In the first model training stage, the local model of the data terminal is trained without changing the global model and only the global model update parameters are recorded. In the second model training stage, the global model is updated based on the global model update parameters, thereby improving the convergence speed of the prediction model training and thus improving the prediction model training efficiency.

[0020] As an embodiment, in the initial stage, the preset initial G(x) is sent to each S i , the randomly obtained L i (x) is sent to the corresponding S i It should be noted that different L iThe model structure of (x) is the same, but the initialized local model parameters are specifically set based on the local data, that is, the initialized model parameters can be different.

[0021] As an embodiment, the step S1 includes:

[0022] Step S11: In the f-th round of local training, obtain the f-th round of local update parameters and global update parameters

[0023] In step S11, the local update parameter of the fth round can be obtained by the following formula: and global update parameters

[0024]

[0025]

[0026] Among them, loss(L i ,D i ,G) indicates based on L i (x), G(x), and D i The loss function obtained is the model L on the i-th data terminal i (x) A loss function is used to calculate the loss by comparing the preset result generated by G(x) with the true value. The specific form of the loss function is set according to the specific application scenario. This application does not make specific restrictions on this. The loss function only needs to satisfy the requirement for p i and p * All can be derived.

[0027] Step S12: Based on and Update p i and p * Remain unchanged:

[0028]

[0029]

[0030] Among them, λ represents the preset gradient step size, and the value range of f is 1 to F. It represents the updated value after the fth round of local training.

[0031] Steps S11-S12 are used to implement the first model training phase. i Real-time update, p * Remain unchanged, but record the gradient update of G(x) and calculate Pass to S* , used to update G(x) in the first model training phase. As an embodiment, in step S2, S * based on Update the global model parameter p of G(x) * ,include:

[0032]

[0033] Among them, λ represents the preset gradient step size.

[0034] Embodiment 2

[0035] Embodiment 2 provides a federated learning data processing system based on soft clustering, such as Figure 1 As shown, including the central server S * , n data terminals participating in federated learning {S 1 , S 2 …S n}, a processor and a memory storing a computer program, wherein S i Indicates the i-th data terminal, where i ranges from 1 to n, S i The corresponding training data set D is stored in i , all D i The data format of each piece of data in S is the same. i The corresponding local model is L i (x),L i The input of (x) is D i The data in S * It also includes n cluster lists, T i Indicates S i The corresponding cluster list, T i =(T i1 ,T i2 …T iC ), where T ic Indicates S i The probability of belonging to the cth category, where the value of c ranges from 1 to C, and C is the preset number of cluster categories; L i The output of (x) can be a k-dimensional vector, L i The function of (x) is to extract the features of the input data x and abstract the k-dimensional features that are easier to classify. i (x) is the neural network model, L i The output features of (x) are generated based on the intermediate output layer of the neural network model. It should be noted that L i(x) The extracted features are the features of the intermediate layer of the neural network, which further protects the security of the terminal data. The present invention is based on federated learning and can avoid the leakage of user data of each data terminal. The central server is a trusted server.

[0036] When the processor executes the computer program, the following steps are implemented:

[0037] Step C1: In the first model training phase, S i Based on D i , Current L i (x) and the current G(x) perform the preset F rounds of local model training, and update L in real time i The local model parameters p of (x) i , and obtain S i The corresponding local cumulative update parameters

[0038] Step C2: In the second model training phase, L i (x) and Send to S * , S * based on T i Update L i (x)p i , the updated L i (x) is sent to the corresponding S i , and then return to step S1 to perform the next round of first model training phase.

[0039] The system of the present invention classifies terminals with close distribution into one cluster through a cluster list, thereby reducing the interference caused by non-independent and identically distributed data, reducing the time cost of training the prediction model and improving the accuracy of the prediction model. As an embodiment, when the processor executes the computer program, the following steps are implemented:

[0040] Step S10: When the global training round X=m*a, a is the preset clustering round, m is a positive integer greater than or equal to 1, S * Updates the cluster list based on the current model training data.

[0041] It should be noted that each execution of the second model training phase represents a global training. In addition, the existing clustering-based federated learning algorithms often only perform clustering once, and after a clustering, each cluster is trained separately. The algorithm described in the present invention supports re-clustering after a period of clustering, avoiding the contingency of unsatisfactory clustering results in the first clustering.

[0042] As an example, in the initial stage of model training, each T icThe initial value of can be set to 1 / C, which means that at the beginning of model training, the probability that each data terminal belongs to each cluster is the same. Taking C = 4 as an example, each T i will be initialized to

[0043] As an embodiment, in step S10, S * Update the cluster list based on the current model training data, including:

[0044] Step S101: Based on the current L i (x) Get each S i The corresponding cluster input vector v i :

[0045]

[0046] in, Indicates D i The jth data in the table, j ranges from 1 to Bi, Bi represents D i The number of data items in ;

[0047] Step S102: All v i As input to the preset soft clustering algorithm, update the cluster list.

[0048] The preset soft clustering algorithm may be a soft clustering EM (Expectation Maximization Algorithm) algorithm, and the EM algorithm is not described in detail here. i Classifying them into one cluster reduces the interference caused by non-independent and identically distributed data.

[0049] It should be noted that the clustering object of the clustering federated learning algorithm based on clustering in the prior art is the parameters of the model. Often, clustering is performed based on the parameters of the model after several rounds of initial training. The number of model parameters is quite large, which will bring a great time cost to the clustering algorithm. The training of the model itself is a process with considerable chance. The same set of data may have multiple models with different parameters. It is not reasonable to cluster directly. However, the clustering object in the present invention is an L i (x), compared with the number of model parameters, the sample dimension of the clustering of the present invention is much lower, so it can converge faster. It can be seen that the object of the clustering algorithm described in the present invention is L i (x) output, limit L i The diversity of (x) is similar to the input format of the unified clustering algorithm and can play a positive role in clustering.

[0050] As an embodiment, in the initial stage, the randomly obtained L i (x) is sent to the corresponding S i It should be noted that different L i The model structure of (x) is the same, but the initialized local model parameters are specifically set based on the local data, that is, the initialized model parameters can be different.

[0051] As an embodiment, the step C1 includes:

[0052] Step C11: In the f-th round of local training, obtain the f-th round of local update parameters

[0053]

[0054] Among them, loss(L i ,D i ,G) indicates based on L i (x), G(x), and D i The loss function obtained;

[0055] Step S12: Based on Update p i ,

[0056]

[0057]

[0058] Among them, λ represents the preset gradient step size, and the value range of f is 1 to F. It represents the updated value after the fth round of local training.

[0059] As an embodiment, in step S2, L is updated based on the following formula: i (x)p i :

[0060]

[0061] Among them, T ir Indicates S i The probability of belonging to the rth category, T qr Indicates S q The probability of belonging to the rth category, q ranges from 1 to n, and q≠i, Indicates the S q The corresponding local cumulative update parameters. It should be noted that through T ir *T qrThe degree of association of the qth data terminal with the ith data terminal when the qth data terminal belongs to the rth category can be calculated, and then multiplied by The image data can be further obtained. The above formula can be used to update p based on the influence of all other data terminals on the i-th data terminal belonging to the r-th category. i , thereby updating L i (x).

[0062] It should be noted that the existing clustering-based federated learning algorithms all use hard clustering methods. The final result is that there is no correlation between the two clusters. Training a model independently is not conducive to making full use of all data, and the model accuracy is low and the model convergence is slow. The soft clustering method adopted by the present invention is different from S i The models above accept each other's gradient updates with probability weights, and each model has the opportunity to be exposed to updates brought by all data, which improves the convergence speed and accuracy of the model.

[0063] Embodiment 1 mainly divides the model training into two stages to perform prediction model training, thereby improving the prediction model training efficiency. Embodiment 2 introduces soft clustering, by soft clustering the data terminals and updating the local model based on the mutual response between the data of the data terminals, thereby improving the model convergence speed and accuracy. However, it can be understood that Embodiment 1 and Embodiment 2 can be combined to obtain Embodiment 3, thereby improving the accuracy and training efficiency of the prediction model.

[0064] Embodiment 3

[0065] Embodiment 3 provides a federated learning data processing system based on soft clustering, such as Figure 1 As shown, including the central server S * , n data terminals participating in federated learning {S 1 , S 2 …S n}, a processor and a memory storing a computer program. i Indicates the i-th data terminal, where i ranges from 1 to n, S i The corresponding training data set D is stored in i , all D i The data format of each piece of data in S is the same. i The corresponding local model is L i (x),L i The input of (x) is D i The data in, so L i The input structure of (x) is similar to D i The data format is the same as in L i The output of (x) is the input of G(x), L iThe output of (x) can be a k-dimensional vector, so the input of G(x) is also a k-dimensional vector. i The function of (x) is to extract the features of the input data x and abstract the k-dimensional features that are easier to classify. It should be noted that L i The extracted features of (x) are the features of the intermediate layer of the neural network, which further protect the security of terminal data. * The global model maintained, the output of G(x) is the target prediction result, G(x) can be set as a classifier, by i (x) The useful features abstracted are classified to obtain the final prediction result. The target prediction result can be set in the form of the target prediction result based on the specific application scenario. For example, for the problem of single written digital image recognition, the target prediction result can output a Z-dimensional vector, and the z-th dimension of the vector represents the probability that the sample is the number z, and the value range of z is 1 to Z. It can be understood that the system of the present invention can also be applied to deep learning application scenarios such as recommendation systems, image recognition, and dialogue systems, which are not listed here one by one. S * It also includes n cluster lists, T i Indicates S i The corresponding cluster list, T i =(T i1 ,T i2 …T iC ), where T ic Indicates S i The probability of belonging to the cth category, where the value range of c is 1 to C, and C is the preset number of cluster categories. The present invention is based on federated learning and can avoid leakage of user data of each data terminal. The central server is a trusted server.

[0066] When the processor executes the computer program, the following steps are implemented:

[0067] Step E1: In the first model training phase, S i Based on D i , Current L i (x) and the current G(x) perform the preset F rounds of local model training, and update L in real time i The local model parameters p of (x) i , and obtain S i The corresponding local cumulative update parameters And global cumulative update parameters

[0068] Step E2: In the second model training phase, L i (x), and Send to S * , S* based on Update the global model parameter p of G(x) * ,based on T i Update L i (x)p i , the updated G(x) and the updated L i (x) is sent to the corresponding S i , and then return to step S1 to perform the next round of first model training phase.

[0069] The system described in the present invention classifies terminals with similar distribution into one cluster through a cluster list, reduces the interference caused by non-independent and identically distributed data, improves the accuracy of the prediction model while reducing the time cost of prediction model training, and further divides the model training into two stages, in which the local model is updated in the first stage and the global model is updated in the second stage, thereby improving the prediction model training speed.

[0070] As an example, in the initial stage of model training, each T ic The initial value of can be set to 1 / C, which means that at the beginning of model training, the probability that each data terminal belongs to each cluster is the same. Taking C = 4 as an example, each T i will be initialized to

[0071] As an embodiment, when the processor executes the computer program, the following steps are implemented:

[0072] Step E10: When the global training round X=m*a, a is the preset clustering round, m is a positive integer greater than or equal to 1, S * Updates the cluster list based on the current model training data.

[0073] It should be noted that each execution of the second model training phase represents a global training. In addition, the existing clustering-based federated learning algorithms often only perform clustering once, and after a clustering, each cluster is trained separately. The algorithm described in the present invention supports re-clustering after a period of clustering, avoiding the contingency of unsatisfactory clustering results in the first clustering.

[0074] As an embodiment, in step E10, S * Update the cluster list based on the current model training data, including:

[0075] Step E101: Based on the current L i (x) Get each S i The corresponding cluster input vector v i :

[0076]

[0077] in, Indicates D i The jth data in the table, j ranges from 1 to Bi, Bi represents D i The number of data items in ;

[0078] Step E102: Set all v i As input to the preset soft clustering algorithm, update the cluster list.

[0079] The preset soft clustering algorithm may be a soft clustering EM algorithm, and the algorithm of the EM algorithm is not described in detail here. i Classifying them into one cluster reduces the interference caused by non-independent and identically distributed data.

[0080] It should be noted that the clustering object of the clustering federated learning algorithm based on clustering in the prior art is the parameters of the model. Often, clustering is performed based on the parameters of the model after several rounds of initial training. The number of model parameters is quite large, which will bring a great time cost to the clustering algorithm. The training of the model itself is a process with considerable chance. The same set of data may have multiple models with different parameters. It is not reasonable to cluster directly. However, the clustering object in the present invention is an L i (x), the sample dimension of the clustering of the present invention is much lower than the number of model parameters, so it can converge faster. In addition, all models of the present invention share G(x), which also limits L i (x) diversity, it can be seen that the object of the clustering algorithm described in the present invention is L i (x) output, limit L i The diversity of (x) is similar to the input format of the unified clustering algorithm and can play a positive role in clustering.

[0081] As an embodiment, in the initial stage, the preset initial G(x) is sent to each S i , the randomly obtained L i (x) is sent to the corresponding S i It should be noted that different L i The model structure of (x) is the same, but the initialized local model parameters are specifically set based on the local data, that is, the initialized model parameters can be different.

[0082] As an embodiment, the step E1 includes:

[0083] Step E11: In the f-th round of local training, obtain the f-th round of local update parameters and global update parameters

[0084]

[0085]

[0086] Among them, loss(L i ,D i ,G) indicates based on L i (x), G(x), and D i The loss function obtained is the model L on the i-th data terminal i (x) A loss function is used to calculate the loss by comparing the preset result generated by G(x) with the true value. The specific form of the loss function is set according to the specific application scenario. This application does not make specific restrictions on this. The loss function only needs to satisfy the requirement for p i and p * All can be derived.

[0087] Step E12: Based on and Update p i , and p * Remain unchanged:

[0088]

[0089]

[0090]

[0091] Among them, λ represents the preset gradient step size, and the value range of f is 1 to F. Represents the updated value after the fth round of local training It represents the updated value after the fth round of local training.

[0092] Steps E11 and E12 are used to implement the first model training phase. i Real-time update, p * Remain unchanged, but record the gradient update of G(x) and calculate Pass to S * , used to update G(x) in the first model training phase. As an embodiment, in step S2, S * based on Update the global model parameter p of G(x) * ,include:

[0093]

[0094] As an embodiment, in step E2, L is updated based on the following formula: i (x)p i :

[0095]

[0096] Among them, T ir Indicates S i The probability of belonging to the rth category, T qr Indicates S q The probability of belonging to the rth category, q ranges from 1 to n, and q≠i, Indicates the S q The corresponding local cumulative update parameters. It should be noted that through T ir *T qr The degree of association of the qth data terminal with the ith data terminal when the qth data terminal belongs to the rth category can be calculated, and then multiplied by The image data can be further obtained. The above formula can be used to update p based on the influence of all other data terminals on the i-th data terminal belonging to the r-th category. i , thereby updating L i (x).

[0097] It should be noted that the existing clustering-based federated learning algorithms all use hard clustering methods. The final result is that there is no correlation between the two clusters. Training a model independently is not conducive to making full use of all data, and the model accuracy is low and the model convergence is slow. The soft clustering method adopted by the present invention is different from S i The models above accept each other's gradient updates with probability weights, and each model has the opportunity to be exposed to updates brought by all data, which improves the convergence speed and accuracy of the model.

[0098] It should be noted that some exemplary embodiments are described as processes or methods depicted as flow charts. Although the flow charts describe the steps as sequential processes, many of the steps can be implemented in parallel, concurrently or simultaneously. In addition, the order of the steps can be rearranged. The process can be terminated when its operation is completed, but it can also have additional steps not included in the accompanying drawings. The process can correspond to a method, function, procedure, subroutine, subprogram, etc.

[0099] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical contents disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent change and modification made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention still falls within the scope of the technical solution of the present invention.

Claims

1. A data processing system based on federated learning, characterized in that: Including the central server S * , n data terminals participating in federated learning {S 1 , S 2 …S n }, a processor and a memory storing a computer program, wherein S i Indicates the i-th data terminal, where i ranges from 1 to n, S i The corresponding training data set D is stored in i , all D i The data format of each piece of data in S is the same. i The corresponding local model is L i (x),L i The input of (x) is D i The data in L i The output of S(x) is the input of G(x), and G(x) is the * The output of the maintained global model G(x) is the target prediction result; When the processor executes the computer program, the following steps are implemented: Step S1: In the first model training stage, S i Based on D i 、Current L i (x) and the current G(x) perform the preset F rounds of local model training, and update L in real time i The local model parameters p of (x) i , and obtain S i The corresponding global cumulative update parameters Step S2: In the second model training phase, L i (x) and Send to S * , S * based on Update the global model parameter p of G(x) * , send the updated G(x) to each S i , then return to step S1 to perform the next round of first model training phase; The step S1 comprises: Step S11: In the f-th round of local training, obtain the f-th round of local update parameters and global update parameters Step S12: Based on and Update p i and p * Remain unchanged: Among them, λ represents the preset gradient step size, and the value range of f is 1 to f. It represents the updated value after the fth round of local training. In step S11, the local update parameter of the fth round is obtained by the following formula: and global update parameters Among them, loss(L i ,D i ,G) indicates based on L i (x), G(x), and D i The loss function obtained; In the step S2, S * based on Update the global model parameter p of G(x) * ,include: Among them, λ represents the preset gradient step size.

2. The system according to claim 1, characterized in that In the initial stage, the preset initial G(x) is sent to each S i , the randomly obtained L i (x) is sent to the corresponding S i .

3. The system according to claim 1, characterized in that L i (x) and G(x) are neural network models.

4. The system according to claim 3, characterized in that L i The output features of (x) are generated based on the intermediate output layer of the neural network model.

5. The system according to claim 1, characterized in that G(x) is set as a classifier, and the target prediction result is a Z-dimensional vector. The z-th dimension of the vector represents the probability that the sample is the output result of the z-th class, and the value range of z is 1 to Z.

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