A federated learning method with high communication efficiency in a wireless communication scenario

Abstract

The application discloses a high-communication-efficiency federated learning method in a wireless communication scenario, and the method comprises the following steps: S1, constructing a federated learning framework from the aspects of a federated learning system in a wireless communication scenario, a training algorithm and a communication model; S2, performing convergence analysis on the training process of the constructed federated learning framework; S3, constructing an optimization problem about the federated learning framework according to the result of the convergence analysis, and solving the problem by a joint optimization method for device selection and beamforming. Based on air computing and a second-order training algorithm, the method realizes low-delay model aggregation through the waveform superposition characteristics of a channel, reduces the iteration rounds required for training through the fast convergence characteristics of the second-order algorithm, and solves the communication bottleneck problem existing in most current wireless federated learning methods. Meanwhile, the joint optimization method for the federated learning framework further improves the training accuracy.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of wireless communication and federated learning, and in particular to a federated learning method with high communication efficiency in a wireless communication scenario. BACKGROUND

[0002] Nowadays, the related technology of artificial intelligence (AI) is in a stage of rapid development and is being researched and applied in various scenarios. As a data-driven technology, its reliability and accuracy depend largely on the size and quality of the data source. However, for most enterprises, it is not easy to obtain a high-quality data set for AI model training. At the same time, data privacy issues are gradually being valued by people, and data in real-world applications also form a data island, making it increasingly difficult to gather data in the cloud for AI model training. Therefore, federated learning has become a new learning paradigm to solve these problems. A universal federated learning framework can be described as a distributed training process, and each round of iteration process can be described as the following three steps: first, the center server broadcasts the current global model parameters to all devices participating in training; then each device performs local model training based on its own data set and returns the update to the server; finally, the server aggregates all device updates and calculates the global model parameters for the next round of training. Essentially, this process is to solve a distributed optimization problem together by the server and the device, and this mode is usually called federated optimization. Unlike centralized optimization, federated optimization faces many challenges, including communication efficiency, data heterogeneity, security, system complexity, etc. Among these challenges, the communication efficiency problem is considered a core challenge in federated optimization because the communication between the server and the device is often affected by unreliable network connections, limited resources, delays, etc.

[0003] In order to deal with the communication efficiency problem in federated learning, a large number of scholars have carried out a lot of research work on it in recent years. On the one hand, reducing the communication amount in each iteration process is considered an effective method. Specifically, quantization and sparsification techniques are used to reduce the number of bits transmitted and remove redundant updates in the parameters, while some scholars use low-rank compression methods to achieve higher compression rates and maintain the quality of the model. These compression techniques have shown significant results in dealing with high-dimensional models, however, their design needs to consider the compatibility problem with the communication model. On the other hand, reducing the total number of communication rounds is considered the core goal. In some special cases (such as non-smooth objective functions, black box attacks), zero-order algorithms are widely used due to their characteristics of only using objective function values to approximate gradient information. In the case where gradient information is available, first-order algorithms such as gradient descent are more familiar to people. By increasing the amount of local computation on this basis, the total number of iterations has been proven to be effectively reduced, and a series of methods have been developed.

[0004] Due to the existence of noise interference in the wireless channel, and the characteristics of limited resources and high delay, the transmission of federated learning model parameters in the wireless channel faces more severe challenges. The existing zero-order and first-order methods mentioned above can only achieve linear convergence speed under optimal conditions, resulting in a relatively large number of iteration rounds required to achieve the required accuracy. Therefore, the second-order optimization algorithm has great potential in the wireless environment due to its quadratic convergence speed. However, the construction of the standard second-order Newton update direction requires both Hessian matrix and gradient information, and the aggregation of Hessian matrix information will bring a huge communication burden. In order to deal with this problem, many approximate distributed second-order optimization algorithms have been proposed, mainly including two categories: explicit use of Hessian information and implicit use of Hessian information. SUMMARY

[0005] In view of the communication efficiency problem existing in the prior art, the purpose of the present application is to provide a federated learning method with high communication efficiency in a wireless communication scenario, which solves the bottleneck problem of communication efficiency.

[0006] In order to achieve the above purpose, the specific technical scheme adopted by the present application is:

[0007] A federated learning method with high communication efficiency in a wireless communication scenario, comprising the following specific steps:

[0008] S1, construction of a federated learning framework;

[0009] Firstly, the federated learning system composed of server, device and wireless channel is modeled, and the modeling of the model training task is completed; then the algorithm for completing the model training task is established; then the communication model between the server and the device is modeled; finally, the complete federated learning framework is obtained by integrating the three; specifically:

[0010] S11, model of wireless federated learning system is established:

[0011] The federated learning in wireless scenario is regarded as the process of completing a model training task by m single-antenna devices and a server with k antennas; the whole sample set and device set are represented by and respectively; the device stores the local data set with data size s and has where ∪ is the set operation, u i,j is the feature vector, v i,j is the corresponding label, and z i,j is used to represent the tuple of (u i,j , v i,j );

[0012] Next, the model training task is modeled as minimizing a global loss function; the local loss function of each device i is modeled as where w∈R d is the parameter vector of the model, ∑ is the summation function, the function f is used to measure the error of the model parameters, γ is the regularization parameter, |·| represents the size of the set, and ||·|| is the l2 norm; based on the local loss function, the global loss function is modeled as the average of the local loss functions of all devices in the device set:

[0013] S12, establish the algorithm for completing the model training task;

[0014] The algorithm used is described as an iterative training process, and the specific process of the tth iteration is as follows:

[0015] 1) Device selection: the server decides the device set participating in the current iteration, and records it as

[0016] 2) Global model broadcast: the server broadcasts the current global model parameter w t to the devices participating in the current training;

[0017] 3) Local model update: after receiving the global model parameter, the device i first calculates the local gradient value where f'( ) is the first derivative; then further compute the local Hessian matrix where f"( ) is the second derivative, I d is the d-dimensional identity matrix; finally, compute the local Newton update direction based on the local gradient and the local Hessian matrix where is the second partial derivative, is the first partial derivative;

[0018] 4) Model aggregation: the devices participating in this round of training transmit their local Newton update directions to the server, and the server aggregates and averages to obtain the global update direction where is the set of selected devices in the t-th round of training;

[0019] 5) Global model update: the server updates the global model parameters through the global update direction where a is the learning rate;

[0020] S13, establish a communication model between the server and the devices in the model aggregation process in S12;

[0021] First, the wireless channel between the server and the devices is regarded as a block fading channel, where each time block is further divided into d time slots to satisfy the transmission of an update direction vector; air computing is used to realize the process of model aggregation, and all devices simultaneously transmit their local update directions, and the sum operation is completed in the air through the superposition characteristics of the wireless channel;

[0022] Specifically, the air computing is expressed in the form of a nomographic function: where p t,F is the local update direction of device i in this round, is the global update direction received by the server, and ψ are preprocessing and postprocessing functions, and correspond to normalization and denormalization processing, respectively;

[0023] Before transmission, first normalize the local update direction p t,F of device i to s t,i : where

[0024] Then each item x t,F [l], l e [1, d] of the transmission signal is expressed as: x t,i [l] = b t,i s t,i [l], where bt,i ∈R is the transmission energy control factor; meanwhile, the transmission energy limit for each device is where denotes arbitrary, E() is the operation of taking the mean, P0 is the maximum transmission energy;

[0025] Let h t,i ∈C k is the channel correlation coefficient between device i and the server, y t ∈C k denotes the signal received by the server, where e t ∈C k is a Gaussian white noise vector with σ 2 size energy, and the signal-to-noise ratio (SNR) is defined as P0 / σ 2 .

[0026] After the server receives the signal, an estimated value vector is obtained through a post-processing operation, each item of which is denoted as where a t ∈C k represents the beamforming vector of the server receiving end, η t is the adjustment factor, and H is the conjugate transpose; let be the channel coefficient matrix, be the energy transmission matrix, where diag() denotes the diagonal matrix, be the signal transmission matrix, where T is the matrix transpose, E t = [e t,1 ,…, e t,d ] is the noise matrix, where e t,i is the same Gaussian white noise vector as e t , and accordingly, the total estimated value vector r t = [r t [1], …, r t [d]] is simplified as where each item of the energy transmission matrix is set as to improve the accuracy of the over-the-air computation; substituting the value of the energy transmission matrix into the total estimated value vector yields a further simplified form of r t

[0027] Finally, through the denormalization operation of the post-processing function ψ, the server obtains a global update direction in a wireless scenario where ​The results of the local update direction averaging operation for the model aggregation operation in S12 are obtained;

[0028] S14, the complete federated learning framework is obtained by integrating the wireless federated learning system model in S11, the training algorithm for the wireless federated learning model in S12, and the communication model for the model aggregation operation in the training process in S13.

[0029] The complete federated learning framework based on over-the-air computation and the second-order algorithm is obtained by integrating the wireless federated learning system model in S11, the training algorithm for the wireless federated learning model in S12, and the communication model for the model aggregation operation in the training process in S13. The process of the tth training iteration in the framework is as follows:

[0030] The server first selects the devices participating in the current iteration training and stores them as Then the server broadcasts the current model parameter vector w t to the devices participating in the current iteration. After receiving the parameter vector, each device i first calculates the local gradient value Then the local Hessian matrix is calculated and the local Newton update direction is calculated based on the two items Next, each device i encodes the update direction as and transmits the signal x t,i [l]=b t,i s t,i [l], l∈[1, d] through the wireless channel; the server accepts the aggregated results of the signals sent by the devices participating in the training through over-the-air computation and finally obtains the global update direction through a series of decoding operations Finally, the server updates the global model parameters according to the global update direction S2, the convergence of the training process of the proposed federated learning framework is analyzed through the second-order auxiliary function and vector decomposition.

[0031] First, the global and local gradient, global and local Hessian matrix are given to facilitate the representation form for the convergence analysis; let where The global Hessian matrix is represented as Let N t =[n1,…,n n ]∈R d×n , where n i =f'(w t , z i,j ), the global gradient value is represented as Let Let L be the sketching matrix, where L is the sketching matrix. i ∈R n×s For a special matrix where each column contains only one non-zero term representing the data held by device i, the local Hessian matrix and local gradient value are represented as follows: and

[0032] Then, a method for characterizing the global update direction is given. Auxiliary function for accuracy The optimal point is calculated based on the characteristic that the auxiliary function is a quadratic function: The result is the exact Newton update direction, where arg minφ() is the parameter that makes the function φ reach its minimum value;

[0033] Next, we will discuss the global update direction in each iteration process given in S14. Disassembled into the following form in This refers to the global update direction adopted in the given federated learning framework. This is the average result for the local update direction of the devices participating in this iteration. This is the result of averaging the local update directions of all devices without performing any device selection processing. To use the global gradient to calculate the local update direction and further obtain the result of the global update direction;

[0034] Meanwhile, the following assumptions are made: (1) The global loss function F is smooth, and the smoothness coefficient is L; (2) The global loss function F is strongly convex; (3) The local loss function F i It is twice differentiable, smooth and convex; (4) Local data estimates Less than a certain constant

[0035] Subsequently, based on the properties of the local Hessian matrix: Where U is matrix M t An orthogonal basis, where λ is a constant from 0 to 1, and L is... The concatenated matrix; and the properties of the local gradient values: Where δ is a constant between 0 and 1, and ln() is the natural logarithm; the global update direction in this federated learning framework is derived. With the accurate Newtonian update direction p * The relationship between the two with respect to the auxiliary function φ(p) is as follows: where where σ min and σ max denote the minimum and maximum singular values of the matrix, respectively;

[0036] Finally, according to the properties of the global update direction and the accurate Newton update direction, the convergence properties of the model parameter iteration Δ t = w t -w * are derived: where κ t is the condition number of H t , and the error term of each training iteration is

[0037] S3, an optimization problem of minimizing the error term is constructed according to the error term existing in each round of training in the convergence analysis result, and the optimization problem is solved by a joint optimization method for device selection and beamforming to improve the accuracy of the federated learning framework model obtained by training.

[0038] S31, optimization problem construction;

[0039] According to the goal of minimizing the error term ∈' of each iteration to improve the accuracy of the trained model, the system optimization problem is constructed as:

[0040] S32, joint optimization method for device selection and beamforming vector a t ;

[0041] First, the initialization of system optimization is completed; the current channel coefficient {h t,i} is determined, the initial temperature coefficient T (0) , the temperature attenuation coefficient ρ, the system optimization iteration number K, the penalty coefficient θ, and the threshold value ξ are set; and the device selection set is initialized as

[0042] In the kth iteration process of system optimization, first, the neighbor set of the current device selection set is generated For each device selection set in , substitute it into the original system optimization problem to obtain a new optimization problem for the beamforming vector Let and rank(A) = 1, by matrix lifting and the properties of convex function difference, the problem is transformed into stA≥0,Tr(A)>0,Tr(AQ i )≥1 Where <·,·> is the matrix inner product, and θ||A j ||2 is||A j The subgradient of ||, Tr() represents the trace of the matrix, A j Let A0 be the iteration variable; initialize A0 to any positive definite matrix, and solve the problem iteratively until |Tr(A j )-||A j ||2|<ξ, ultimately we get The corresponding optimal beamforming vector

[0043] According to probability distribution From Mid-sampling yields a new set of device choices. Where exp() is the exponential function and J() is the system optimization problem. The objective function value; and the sampled device selection set Set as the set for the new iteration Simultaneously update the temperature parameter T (k+1) ←ρT (k) ;

[0044] The device selection set output after the final K rounds of iterations The beamforming vector and its corresponding beamforming vector are the results of system optimization.

[0045] The beneficial effects of this invention are:

[0046] This invention proposes a highly efficient federated learning method for wireless communication scenarios. By fully utilizing the loss function information through a second-order algorithm and leveraging the superposition characteristics of wireless channels, it simultaneously reduces the number of communication rounds and communication latency, greatly alleviating the communication bottleneck problem in existing federated learning frameworks. Furthermore, to address the error terms generated during training, this invention proposes a joint optimization method for device selection and receiver beamforming vectors, effectively improving the accuracy of model training. Attached Figure Description

[0047] Figure 1 This is a diagram of the federated learning framework in this invention;

[0048] Figure 2 This is an experimental comparison diagram between the present invention and two other wireless federated learning methods. Detailed Implementation

[0049] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. Obviously, the examples listed are only for explaining the present invention and are not intended to limit the scope of the invention.

[0050] Referring to Figure 1 The high-communication-efficiency federated learning method in a wireless communication scenario provided by the present application is a wireless federated learning method based on aerial computing and a second-order optimization algorithm, and comprises the following steps:

[0051] S1, construction of a federated learning framework;

[0052] S11, model establishment of a wireless federated learning system:

[0053] The federated learning in a wireless scenario is regarded as a process in which m single-antenna devices and a server equipped with k antennas jointly complete a model training task; and the overall sample set and device set are represented by and respectively; the device stores a local data set with a data size of s and has where ∪ is a set union operation, u i,j is a feature vector, v i,j is a corresponding label, and z i,j is used to represent the tuple (u i,j , v i,j );

[0054] Next, the model training task is modeled as minimizing a global loss function; the local loss function of each device i is modeled as where w ∈ R d is a parameter vector of the model, ∑ is a summation function, the function f is used to measure the error of the model parameters, γ is a regularization parameter, |·| represents the size of a set, and ||·|| is an l2 norm; based on the local loss function, the global loss function is modeled as the average of the local loss functions of all devices in the device set:

[0055] S12, establishment of an algorithm for completing the model training task;

[0056] The algorithm to be used is described as an iterative training process, and the specific process of the tth iteration is as follows:

[0057] 1) device selection: the server decides the device set participating in the current iteration, and records it as

[0058] 2) global model broadcast: the server broadcasts the current global model parameter w t to the devices participating in the current training;

[0059] 3) Local model update: After device i receives the global model parameters, it first calculates the local gradient value according to the local loss function where f'( ) is the first derivative; then further calculates the local Hessian matrix where f"( ) is the second derivative, I d is the d-dimensional identity matrix; finally, the local Newton update direction is calculated according to the local gradient and the local Hessian matrix where is the second partial derivative, is the first partial derivative;

[0060] 4) Model aggregation: The devices participating in this round of training transmit their local Newton update directions to the server, and the server aggregates and averages to obtain the global update direction where is the set of selected devices in the t-th round of training;

[0061] 5) Global model update: The server updates the global model parameters through the global update direction where a is the learning rate;

[0062] S13, establish a communication model between the server and the devices in the model aggregation process in S12;

[0063] First, the wireless channel between the server and the devices is regarded as a block fading channel, where each time block is further divided into d time slots to satisfy the transmission of an update direction vector; air computing is used to realize the model aggregation process, and all devices simultaneously transmit their local update directions, and the sum operation is completed in the air through the superposition characteristics of the wireless channel;

[0064] Specifically, the air computing is expressed in the form of a nomographic function: where p t,i is the local update direction of device i in this round, is the global update direction received by the server, and ψ are preprocessing and postprocessing functions, respectively corresponding to normalization and denormalization processing;

[0065] Before transmission, first normalize the local update direction p t,F of device i to s t,i : where

[0066] Then each item x t,F [l], l ∈ [1, d] of the transmission signal is expressed as: x t,F[l] = b t,i s t,i [l], where b t,i ∈ R is the transmission energy control factor; meanwhile, the transmission energy limit for each device is where denotes any, E() is the operation of taking the mean, P0 is the maximum transmission energy;

[0067] Let h t,i ∈ C k be the channel correlation coefficient between device i and the server, and let y t ∈ C k be the signal received by the server, where e t ∈ C k is a Gaussian white noise vector with σ 2 size energy, and the signal-to-noise ratio (SNR) is defined as P0 / σ 2 .

[0068] After the server receives the signal, an estimated value vector is obtained through a post-processing operation, each item of which is denoted as where a t ∈ C k represents the beamforming vector of the server receiving end, η t is an adjustment factor, and H is the conjugate transpose; let be the channel coefficient matrix, be the energy transmission matrix, where diag() denotes a diagonal matrix, be the signal transmission matrix, where T is the matrix transpose, E t = [e t,1 ,…, e t,d ] is the noise matrix, where e t,F is the same Gaussian white noise vector as e t , and accordingly, the total estimated value vector r t = [r t [1],…, r t [d]] is simplified as where each item of the energy transmission matrix is set as to improve the accuracy of the aerial computation; substituting the value of the energy transmission matrix into the total estimated value vector yields a further simplified form of r t

[0069] ​Finally, the server gets the global update direction in the wireless scenario by the denormalization operation of the post-processing function ψ where is the result of the average of the local update directions in the model aggregation operation in S12;

[0070] S14, integrate to get the complete federated learning framework;

[0071] Integrate the wireless federated learning system model in S11, the training algorithm for the wireless federated learning model in S12, and the communication model for the model aggregation operation in the training process in S13 to get the complete federated learning framework based on air computing and second-order algorithm. The process of the t-th round of training iteration in this framework is as follows:

[0072] The server first selects the devices participating in the current iteration training and stores them as Then the server broadcasts the current model parameter vector w t to the devices participating in the current iteration. After receiving the parameter vector, each device i first calculates the local gradient value Then the local Hessian matrix is calculated and the local Newton update direction is calculated based on the two items Next, each device i encodes the update direction as and transmits the signal x t,F [l]=b t,i s t,i [l],l∈[1,d] through the wireless channel; the server accepts the aggregated result of the signals sent by the devices participating in the training at the same time through air computing and finally gets the global update direction through a series of decoding operations Finally, the server updates the global model parameters according to the global update direction in the current training S2, analyze the convergence of the training process of the proposed federated learning framework through second-order auxiliary functions and vector decomposition;

[0073] First, give the global and local gradient, global and local Hessian matrix to facilitate the convergence analysis of the expression form; let where The global Hessian matrix is represented as Let N t =[n1,…,n n ]∈R d×n , where n i =f′(w t ,zi,j If ), then the global gradient value is expressed as make Let L be the sketching matrix, where L is the sketching matrix. i ∈R n×s For a special matrix where each column contains only one non-zero term representing the data held by device i, the local Hessian matrix and local gradient value are represented as follows: and

[0074] Then, a method for characterizing the global update direction is given. Auxiliary function for accuracy The optimal point is calculated based on the characteristic that the auxiliary function is a quadratic function: The result is the exact Newton update direction, where arg minφ() is the parameter that makes the function φ reach its minimum value;

[0075] Next, we will discuss the global update direction in each iteration process given in S14. Disassembled into the following form in This refers to the global update direction adopted in the given federated learning framework. This is the average result for the local update direction of the devices participating in this iteration. This is the result of averaging the local update directions of all devices without performing any device selection processing. To use the global gradient to calculate the local update direction and further obtain the result of the global update direction;

[0076] Meanwhile, the following assumptions are made: (1) The global loss function F is smooth, and the smoothness coefficient is L; (2) The global loss function F is strongly convex; (3) The local loss function F i It is twice differentiable, smooth and convex; (4) Local data estimates Less than a certain constant

[0077] Subsequently, based on the properties of the local Hessian matrix: Where U is matrix M t An orthogonal basis, where λ is a constant from 0 to 1, and L is... The concatenated matrix; and the properties of the local gradient values: where δ is a constant between 0 and 1, ln() is the natural logarithm; and the global update direction in the federated learning framework is derived with the accurate Newton update direction p * The relationship about the auxiliary function φ(p) is as follows: where where σ min and σ max represent the minimum and maximum singular values of the matrix respectively;

[0078] Finally, according to the properties of the global update direction and the accurate Newton update direction, the convergence property of the model parameter iteration Δ t = w t -w * is derived: where κ t is the condition number of H t , and the error term of each training iteration is

[0079] S3, an optimization problem of minimizing the error term existing in each round of training in the convergence analysis result is constructed, and the optimization problem is solved by a joint optimization method for device selection and beamforming to improve the accuracy of the model obtained by training.

[0080] S31, problem construction;

[0081] According to the goal of minimizing the error term ∈' in each iteration to improve the accuracy of the trained model, the system optimization problem is constructed as:

[0082] S32, joint optimization method for device selection t and beamforming vector a t,F ;

[0083] First, the initialization of system optimization is completed; the current channel coefficient {h (0)} is determined, the initial temperature coefficient T i is set, the temperature attenuation coefficient ρ, the number of system optimization iterations K, the penalty coefficient θ, and the threshold value ξ are determined; and the device selection set is initialized as

[0084] In the kth iteration process of system optimization, first, the neighbor set of the current device selection set is generated For each device selection set in Substituting this into the original system optimization problem yields a new optimization problem for the beamforming vector. make And rank(A) = 1, This problem is transformed using the properties of matrix lifting and the difference of convex functions. stA≥0,Tr(A)>0,Tr(AQ i )≥1 Where <·,·> is the matrix inner product, and θ||A j ||2 is||A j The subgradient of ||, Tr() represents the trace of the matrix, A j Let A0 be the iteration variable; initialize A0 to any positive definite matrix, and solve the problem iteratively until |Tr(A j )-||A j ||2|<ξ, ultimately we get The corresponding optimal beamforming vector

[0085] According to probability distribution From Mid-sampling yields a new set of device choices. Where exp() is the exponential function and J() is the system optimization problem. The objective function value; and the sampled device selection set Set as the set for the new iteration Simultaneously update the temperature parameter T (k+1) ←ρT (k) ;

[0086] The device selection set output after the final K rounds of iterations The beamforming vector and its corresponding beamforming vector are the results of system optimization.

[0087] To verify the effectiveness of this invention, it was compared with two existing wireless federated learning methods. Method 1 is a federated learning method based on the FedAvg algorithm and over-the-air computation, while Method 2 is a federated learning method based on the Fedsplit algorithm and over-the-air computation. Experiments were conducted on four different datasets: Covtype, a9a, w8a, and phishing. The experimental results are as follows: Figure 2 As shown, the horizontal axis represents the number of communication rounds, and the vertical axis represents the loss function value. It can be seen that this invention only requires about 10 communication rounds to reduce the loss function value to a low level (10...). -5 10 -3, 0.02, 0.002), while the loss function of the comparative method still maintains a high level (0.1, 0.01, 0.05, 0.1) after 50 rounds of communication rounds. The application accelerates the convergence speed of training by using the second-order information of the loss function, thereby effectively reducing the required communication rounds of training and improving the communication efficiency of federated learning in a wireless environment.

Claims

1. A federated learning method with high communication efficiency in a wireless communication scenario, characterized in that, The method comprises the following specific steps: S1, construction of a federated learning framework First, a federated learning system composed of a server, a device and a wireless channel is modeled, and modeling of a model training task is completed; then, an algorithm for completing the model training task is established; subsequently, a communication model between the server and the device is modeled; finally, the three are integrated to obtain a complete federated learning framework; S2, convergence analysis of the training process of the federated learning framework proposed in step S1 is performed by means of a second-order auxiliary function and vector decomposition; S3, an optimization problem for minimizing an error term existing in each round of training in the convergence analysis result is constructed, and the optimization problem of the error term is solved by a joint optimization method for device selection and beamforming to improve the accuracy of the federated learning framework model obtained by training; wherein: The step S1 specifically comprises: S11, modeling of a wireless federated learning system The federated learning conducted in the wireless scenario is regarded as a process in which m single-antenna devices and a server equipped with k antennas jointly complete a model training task; and and are used to represent the overall sample set and the device set respectively; the device stores a local data set with a data size of s , and has where ∪ is a set union operation, u i,j is a feature vector, v i,j is a corresponding label, and a tuple of (u i,j , v i,j ) is represented by z i,j . Next, the task of training the model is modeled as minimizing a global loss function; the local loss function for each device i is modeled as where w e R d is the parameter vector of the model,∑ is the summation function, the function f is used to measure the error of the model parameters, γ is a regularization parameter, | · | represents the size of the set, and || · || is the l2 norm; based on the local loss function, the global loss function is modeled as the average of the local loss functions of all devices in the device set: S12, establishment of an algorithm for completing a model training task The algorithm adopted is described as an iterative training process, and the specific process of the tth round of iteration is as follows: 1) Device Selection: The server decides the set of devices to participate in this round of iteration and notes it as 2) Global model broadcast: the server broadcasts the current global model parameters w t to the devices participating in the current round of training; 3) Local model update: After device i receives the global model parameters, it first calculates the local gradient value according to the local loss function where f'( ) is the first derivative; then further calculates the local Hessian matrix where f''( ) is the second derivative, I d is a d-dimensional identity matrix; finally, the local Newton update direction is calculated according to the local gradient and the local Hessian matrix where is the second-order partial derivative, is the first-order partial derivative; 4) Model aggregation: the devices participating in this round of training transmit their local Newton update directions to the server, and the server aggregates the average to obtain the global update direction where is the set of devices selected in the tthround of training process; 5) Global model update: the server updates the global model parameters by global update direction where a is the learning rate; S13, establishment of a communication model between the server and the device in the model aggregation process in S12 First, the wireless channel between the server and the device is regarded as a block fading channel, wherein each time block is further divided into d time slots to satisfy the transmission of an update direction vector; air computing is adopted to realize the process of model aggregation, and all devices simultaneously transmit their local update directions, and the sum operation is completed in the air through the superposition characteristics of the wireless channel; In particular, the aerial computation is expressed in the form of a nomographic function: where p t,i is the local Newton update direction for device i at this round, is the global update direction accepted by the server, and ψ are pre-processing and post-processing functions, and correspond to normalization and denormalization, respectively; Before transmission, first the local update direction p of device i is t,i Normalization preprocessing for s t,i : where Each term x t,i [l], l ∈ [1, d] is expressed as: x t,i [l] = b t,i s t,i [l], where b t,i ∈ R is the transmission energy control factor; meanwhile, the transmission energy limit of each device is where denotes any, E() is the operation of taking the mean, P0 is the maximum transmission energy; h t,i ∈C k is the channel correlation coefficient between device i and the server, and y t ∈C k is the signal received at the server. where e t ∈C k is a Gaussian white noise vector with σ 2 size energy, while the signal-to-noise ratio (SNR) is defined as P0 / σ 2 ; After receiving the signal, the server obtains an estimated value vector by a post-processing operation, each item of which is expressed as where a t ∈C k represents a beamforming vector of the server receiving end, η t is an adjustment factor, and H is a conjugate transpose; let be a channel coefficient matrix, be an energy transmission matrix, where diag() represents a diagonal matrix, be a signal transmission matrix, where T is a matrix transpose, and E t =[e t,1 ,…,e t,d ] is a noise matrix, where e t,i is a Gaussian white noise vector identical to e t , according to which the total estimated value vector r t =[r t [1],…,r t [d]] is simplified as where each item of the energy transmission matrix is set as to improve the accuracy of the aerial calculation; and the value of the energy transmission matrix is substituted into the total estimated value vector to obtain a further simplified form of r t ​ Finally, the server gets a global update direction in a wireless scenario by a denormalization operation of the post-processing function ψ wherein is the result of the averaging of the local update directions in the model aggregation operation in S12 S14, integration of a complete federated learning framework The wireless federated learning system model in S11, the training algorithm for the wireless federated learning model in S12, and the communication model for the model aggregation operation in the training process in S13 are integrated to obtain a complete federated learning framework based on air computing and a second-order algorithm, and the process of the tth round of training iteration in the framework is as follows: The server first selects the devices participating in the current round of iterative training and stores them as Then the server broadcasts the current model parameter vector w t to the devices participating in the current round of iterative training; after receiving the parameter vector, each device i first calculates the local gradient value Then the local Hessian matrix is calculated and the local Newton update direction is calculated based on the two items Next, each device i encodes the update direction as where and transmits the signal x t,i [l] = b t,i s t,i [l], l∈[1, d] through the wireless channel; the server aggregates the results of the signals sent by the devices participating in the training through over-the-air computation and finally obtains the global update direction through a series of decoding operations Finally, the server updates the global model parameters according to the global update direction to complete the current round of training The step S2 specifically comprises: First, the global and local gradient, global and local Hessian matrix are given to facilitate the convergence analysis of the form of expression; Let Where The global Hessian matrix is expressed as Let N t = [n1,..., n n ] ∈ R d×n , where n i = f'(w t , z i,j ), the global gradient value is expressed as Let be a sketching matrix, where L i ∈ R n×s is a special matrix with only one non-zero item in each column and the non-zero item represents the data held by device i, the local Hessian matrix and the local gradient value are expressed as And Subsequently, an auxiliary function is given to characterize the accuracy of the global update direction The optimal point of this auxiliary function is computed according to the property that it is a quadratic function: The resulting direction is an exact Newton update direction, where argminφ() is the argument that minimizes the function φ;​ Next, the global update direction in each round of iteration process given in S14 is disassembled into the following form wherein is the global update direction adopted in the given federated learning framework, is the average result of the local update directions of the devices participating in the current round of iteration, is the result of averaging the local update directions of all devices without device selection processing, is the result of using the global gradient to calculate the local update direction and further obtain the global update direction; At the same time, the following assumptions are made: (1) the global loss function F is smooth with a smoothness coefficient L; (2) the global loss function F is strongly convex; (3) the local loss function F i is twice differentiable, smooth and convex; (4) the local data estimate is less than a certain constant Subsequently, according to the properties of the local Hessian matrix: where U is the orthogonal basis of the matrix M t , λ is a constant between 0 and 1, and L is the spliced matrix; and the properties of the local gradient value: where δ is a constant between 0 and 1, and ln() is the natural logarithm; the global update direction in the present federated learning framework is calculated as p * The relationship between the auxiliary function φ(p) and the exact Newton update direction p where where σ min and σ max respectively represent the minimum and maximum singular values of the matrix; Finally, according to the properties of the global update direction and the accurate Newton update direction, the convergence property of the model parameter iteration k = w t - w * is obtained: where K t is the condition number of H t , and the error term of each training iteration is 2. The federated learning method with high communication efficiency in a wireless communication scenario according to claim 1, characterized in that, The step S3 specifically comprises: S31, optimization problem construction According to the goal of minimizing the error term e' at each iteration to improve the accuracy of the trained model, the system optimization problem is constructed as: S32, device selection and beamforming vector a t joint optimization method First, initialization of system optimization is completed; current channel coefficient {h t,i} is determined, initial temperature coefficient T (0) , temperature decay coefficient ρ, system optimization iteration round number K, penalty coefficient θ, threshold value ξ are set; and the device selection set is initialized as In the kth round of iteration of the system optimization, first generate the set of neighbor sets of the current device selection set For each device selection set in the set of neighbor sets Substitute it into the original system optimization problem to obtain a new optimization problem for the beamforming vector Let and rank(A) = 1, Convert the problem into where <·,> is the matrix inner product, is the subgradient of ||A j ||, and Tr() represents the trace of the matrix, A j is a cyclic iteration variable; A0 is initialized as an arbitrary positive definite matrix, and the problem is solved cyclically until |Tr(A j )-||A j ||2| < ξ, and finally the optimal beamforming vector corresponding to According to the probability distribution from the new device selection set is sampled where exp() is an exponential function, J() is an objective function value of a system optimization problem and the sampled device selection set is set as a set of a new round of iteration The temperature parameter T is updated at the same time (k+1) ← ρT (k) ; The device selection set output after the final K iterations And its corresponding beamforming vector is the result of system optimization.

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