CN114204971B - Iterative aggregate beam forming design and user equipment selection method - Google Patents
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Abstract
The invention discloses an iterative aggregate beam forming design and user equipment selection method, which aims at an edge intelligent system, models the problem that a central node and user equipment jointly complete a machine learning task as a distributed federal learning model, adopts an air computing technology to improve communication efficiency, but introduces an aggregate error, so that the invention reduces mean square error by optimizing a beam forming vector of the central node and a user equipment selection scheme, and improves the learning performance of the model. When the number of the selected user equipment is fixed, compared with a random user equipment selection scheme and a channel-based user equipment selection strategy, the method and the device can obtain lower aggregation errors, and further obtain better learning performance.
Description
Technical Field
The present invention relates to the field of wireless communications, and in particular, to an iterative aggregate beamforming design and user equipment selection method.
Background
With the development of wireless communication and the breakthrough of artificial intelligence technology, more and more intelligent services are being put down to the edge of wireless communication networks. If the tasks are accomplished with a traditional centralized training framework, the user equipment is required to transmit its own private data to the central node, which can present high latency and privacy exposure issues. Therefore, a new distributed learning framework called "federal learning" has been proposed by the learner. Under the framework of federal learning, the central node and the user equipment train a global model together, and in the process, only parameters of the model are required to be transmitted instead of sensitive data, so that the problem of privacy disclosure can be avoided. However, since the model parameters need to be exchanged frequently between the central node and each user equipment, and radio spectrum resources are very valuable for an edge intelligence system, how to improve communication efficiency becomes a bottleneck for federal learning development.
The conventional communication calculation separation principle needs to decode the signal at the receiving end and then calculate the signal, and although many works have been optimized for the communication principle, there is still room for improvement in the communication efficiency. For this reason, a new technology called "air computing technology" has been proposed. Different from the traditional communication calculation separation principle, the air calculation technology carries out analog modulation on signals, utilizes the waveform superposition principle and can finish calculation while transmitting. However, this technique also introduces an aggregate error due to the fading nature of the communication channel and noise, which can negatively impact the training of the model if the aggregate error is excessive. In order to reduce this aggregation error, many researchers have therefore optimized various factors including the selection scheme of the user equipment, the signal transmission power of the user equipment, the receive beamforming vector and the scaling factor of the central node, etc. There has been no study aimed at minimizing the aggregate error while considering the problem of optimizing the combination of user equipment selection scheme and beamforming vector.
And the aim of minimizing the aggregation error is achieved by jointly optimizing the beam forming vector and the user equipment selecting method. For this we propose an iterative aggregated beamforming design and user equipment selection method. When the number of the selected user equipment is fixed, the algorithm of the invention can obtain lower aggregation error compared with the original random user equipment selection scheme and the channel-based user equipment selection strategy, thereby obtaining better learning performance.
Disclosure of Invention
In view of this, the present invention aims to provide an iterative aggregate beamforming design and user equipment selection method, which models the problem that a central node and user equipment together complete a machine learning task as a distributed federal learning model for an edge intelligent system, and adopts an air computing technology to improve communication efficiency, but the technology can introduce aggregate errors, so that the present invention reduces mean square error and improves learning performance of the model by optimizing a beamforming vector of the central node and a user equipment selection scheme.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
an iterative aggregate beamforming design and user device selection method, the method comprising the steps of:
step S1, constructing a distributed federation learning model aiming at an edge intelligent system with a plurality of edge user devices and a central node;
s2, training the distributed federal learning model constructed in the step S1, wherein when the edge user equipment transmits model parameters to a central node, an air computing technology is adopted to improve communication efficiency, and then an aggregation error generated by adopting the air computing technology is determined;
s3, constructing an overall optimization problem aiming at the aggregation error determined in the step S2, and converting the overall optimization problem into a combined optimization problem;
and S4, firstly, splitting the combined optimization problem obtained in the step S3 into an aggregate beam forming optimization sub-problem and a user equipment selection optimization sub-problem, then, alternately solving the aggregate beam forming optimization sub-problem and the user equipment selection optimization sub-problem, and finally, obtaining an optimal solution through iteration.
Further, the step S1 specifically includes:
in an edge intelligence system, there are K edge user devices equipped with only one antenna and one edge user device equipped with N r A central node of the root antenna, where N r Much smaller than K;
the channel between the edge user device k and the central node obeys the same complex gaussian distribution with unit power, i.e
Definition of the definitionFor a set of all user devices, each edge user device k has its own local data setAll local data sets->Combining into a global dataset +.>
Building global data setsAverage loss function->Wherein, is a parameter of the global model, Q is a loss function of the global model, (x) n ,y n ) Is a data sample;
constructing an updating formula of local model parameters of kth edge user equipment:
constructing a central node, and updating a formula of global model parameters:
in the formulas (1) and (2), the superscript i is the round number, μ is the learning rate,is the k-th user equipment average loss function +.>Gradient of (2), wherein->
Further, in the step S2, when the edge ue transmits the model parameters to the central node, an air computing technology is adopted to improve the communication efficiency, which specifically includes:
first, in the training of the ith round, the symbol vector transmitted by the edge user equipment k is defined asAssuming normalization with unit variance, the expression: -is:>
then, the normalized symbol vector is subjected to analog modulation and then is subjected to transmission coefficient b k Pre-coding, then, superposing signals in the air, and multiplying the signals with a beam forming vector of a central node;
finally, the amplification is performed by a scaling factor eta.
Further, in the step S2, the determining an aggregate error generated by using the over-the-air computing technology specifically includes:
definition at training timeTo be selected for the subset of edge user devicesThen the expression of the ideal signal that the central node expects is:
because of the fading and noise of the wireless channel, the signal actually received by the central node is expressed as:
in the formula (3) and the formula (4), s k Is thatShorthand, w i Is w i Shorthand, g k G is g k Shorthand, h k Is the channel vector from user equipment k to the central node, b k Is the transmission coefficient of user equipment k, m is the beam forming vector of the central node, n is the additive white gaussian noise and obeys independent co-distribution, i.e. +.>The superscript H denotes a conjugate transpose;
according to the formula (3) and the formula (4), the aggregate error generated by the air computing technology is obtained, and the expression is:
further, the step S3 specifically includes:
and (2) adopting a mean square error to measure the aggregation error determined in the step (S2), and eliminating errors related to channel fading in the aggregation error, and simultaneously reducing errors related to noise as much as possible, wherein the specific expression is as follows:
in the formula (7), σ represents the standard deviation of noise, σ 2 Representing the variance of the noise;
assuming that the number of edge user devices per round selection is fixedWherein (1)>Is an edge user device subset->If the number of elements in the overall optimization problem is equal to or greater than the number of elements in the overall optimization problem, the decision variables of the overall optimization problem include: scaling factor eta, transmission coefficient b k Edge user device subset->And a beamforming vector m, the overall optimization problem is expressed as:
transmission coefficient b k The design is as follows:
the scaling factor η is designed as:
in formula (10), P is the maximum allowed transmit power for each edge user device;
substituting equation (10) back to equation (8 a) can yield our optimization objective asSince there are m in the numerator denominator, the combination optimization problem can be equivalently:
s.t.||m||＝1 (11b)
further, in the step S4, the combination optimization problem obtained in the step S3 is split into an aggregate beamforming optimization sub-problem, which specifically includes:
given edge user device subsetThen, the combinatorial optimization problem translates into:
s.t.||m||＝1 (12b)
define a semi-definite matrix M =mm H The sub-problem (12) translates into:
s.t.Tr(M)＝1 (13b)
Rank(M)＝1 (13c)
by introducing a difference of convexity function, formula (13 c) is rewritten Tr (M) - |M| 2 =0, reintroducing the auxiliary variable τ, then the sub-problem is translated into:
Tr(M)-||M|| 2 ＝0 (14c)
Tr(M)＝1 (14d)
wherein,the value of τ is [ τ ] low ,τ up ]Within (2), and τ low ＝0，
By giving τ, the problem (14) is transformed into a form that can be solved by the dichotomy, expressed as:
Tr(M)＝1 (15c)
wherein for the problem (15), if the rank of the solution of the problem is 1, then at the current τ, the problem (14) is viable, otherwise, it is not viable;
the convex function M 2 Performing first-order Taylor expansion to obtain M 2 ≥Real(Tr(vv H M), wherein v is M t-1 The feature value of the largest feature vector of (2), t-1 represents the iteration number;
then, the solution of the problem (15) is obtained by iteratively solving the following equation:
Tr(M)＝1 (16c)
further, in the step S4, splitting the combined optimization problem obtained in the step S3 into the user equipment selection optimization sub-problem specifically includes:
simplifying the combined optimization problem to determine a subset of user equipments by giving a beamforming vector mThe user equipment selection optimization sub-problem is expressed as:
further, in the step S4, the alternately solving the aggregate beamforming optimization sub-problem and the user equipment selection optimization sub-problem, and finally, obtaining an optimal solution through iteration specifically includes:
step S401, initializing, first, starting fromS of the individual devices are randomly selected as user equipment subset +.>
Step S402, performing aggregate beam forming design, when obtaining the channel vector h k Given a subset of user devicesThen, the optimal beam forming vector m under the condition is obtained, and the specific steps are as follows:
first, initialize M 0 ，τ low ＝0，When τ is up -τ low >Let τ= (τ) at δ up +τ low )/2；
Then solve the problem (16) until convergence;
if Rank (M) >1, then problem (14) is not feasible at the current τ condition,
then let τ up ＝τ；
If Rank (M) =1,
then the problem (14) is feasible at the current τ and let τ be low ＝τ；
Step S403, user equipment selection optimization is performed, and given the wave beam forming vector m, the optimal user equipment subset under the condition is obtainedThe method comprises the following specific steps:
first, toThe projection m is calculated by each user equipment k H h k ||；
Then, these projection values are arranged in descending order;
finally, the first S arranged user equipment are taken as
Step S404, circularly executing the step S402 and the step S403 until reaching a convergence condition;
step S405, when the subset of user equipmentsWhen no change is made, the convergence condition is reached, the circulation is terminated
The beneficial effects of the invention are as follows:
the invention provides an iterative aggregation beam forming design and user equipment selection method, when the number of the selected user equipment is fixed, the algorithm of the invention can obtain lower aggregation error compared with the original random user equipment selection scheme and the channel-based user equipment selection strategy, thereby obtaining better learning performance.
Drawings
FIG. 1 is a schematic illustration of the Federal learning system model provided in example 1;
fig. 2 is a pseudo code of the iterative beamforming design and user equipment selection method provided in embodiment 1;
fig. 3 is a schematic diagram of the aggregation error based on the total number of four devices provided in embodiment 1, wherein fig. a is a schematic diagram when the number of antennas is 4; figure b is a schematic diagram of the number of antennas at 8; figure c is a schematic diagram of the number of antennas at 12; figure d is a schematic diagram of the number of antennas 16;
FIG. 4 is a probability distribution function diagram of aggregate errors in four cases provided in example 1, where FIG. a is N r Schematic when=4, k=20; FIG. b is N r Schematic when=4, k=80; FIG. c is N r Schematic when=4, k=140; FIG. d is N r Schematic diagram when=4, k=200;
FIG. 5 is a graph showing the accuracy of the test on the dataset for the method of example 1, wherein FIG. a is the test result on the MNIST10 dataset; panel b is the test results on the CIFAR10 dataset.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments of the present invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Example 1
Referring to fig. 1-5, the present embodiment provides an iterative aggregate beamforming design and user equipment selection method, including the steps of:
step S1, constructing a distributed federation learning model aiming at an edge intelligent system with a plurality of edge user devices and a central node, wherein the step specifically comprises the following steps:
system model as shown in fig. 1, in an edge intelligence system, there are K edge user equipments equipped with only one antenna and one edge user equipment equipped with N r A central node of the root antenna, where N r Much smaller than K.
The channel between user equipment k and the central node obeys the same complex gaussian distribution with unit power, i.e
The set of all user devices is denoted asEach user device k has its own local data set +.>The global data set is formed by combining the local data sets>These user devices together implement an intelligent application.
In general, the goal of an intelligent learning task is to find an optimal set of model parameters w o So that the loss functionMinimum, wherein->Is a parameter of the model, +.>Is global dataset +.>The average loss function on the upper, defined as +.>Where Q is a loss function, (x) n ,y n ) Is a data sample.
In the present embodiment, for privacy protection and delay reduction, a distributed learning framework of federal learning is employed.
Step S2, training the distributed federal learning model constructed in the step S1, wherein when the edge user equipment transmits model parameters to a central node, an air computing technology is adopted to improve communication efficiency, and then an aggregation error generated by adopting the air computing technology is determined, and the step specifically comprises the following steps:
each round of federal learning can be divided into three phases: 1) Selecting a subset among all user devices to participate in the training of the round; 2) The selected user equipment updates local model parameters according to the local data set of the user equipment, and then the parameters are uploaded to the central node; 3) The central node updates the global model based on the received parameters.
The specific update formulas of the local model parameters of the kth user equipment and the global model parameters of the central node are respectively as follows:
where the superscript i is the round number, μ is the learning rate,is the gradient of the average loss function
When the user equipment transmits the local model to the central node, the air computing technology is also applied in the embodiment to improve the communication efficiency, which specifically comprises the following steps: in the ith round, the symbol vector transmitted by user equipment k may be defined asAnd assuming normalization with unit variance, i.e. +.>Then these signals are analog modulated and then transmitted by the transmission coefficient b k Pre-coding is carried out; these signals are then superimposed over the air and multiplied by the beam forming vector of the central node; finally, the amplification is carried out by a scaling factor eta.
In this embodiment, the channel is assumed to be a block fading model, which remains unchanged during each round of transmission parameters. For the sake of convenience of presentation,w i and g k The d-th element of (2) may be abbreviated as s k ，w i And g k 。
Considering that wireless communication resources are limited in edge intelligence systems, only a fraction of the user devices may transmit their local model to the central node for updating the global model in a particular round.
Representing the selected subset of user devices asThe ideal signal that the central node expects can be expressed as:
because of the fading and noise of the wireless channel, the signal actually received by the central node can be expressed as:
wherein h is k Is the channel vector from user equipment k to the central node, b k Is the transmission coefficient of user equipment k, m is the beam forming vector of the central node, n is additive white gaussian noise and obeys independent co-distribution, i.e
Thus, the aggregate error produced by the over-the-air computing technique can be expressed as:
step S3, constructing an overall optimization problem aiming at the aggregation error determined in the step S2, and converting the overall optimization problem into a combined optimization problem, wherein the step specifically comprises the following steps:
the aggregate error formula in step S2 may be measured in terms of Mean Square Error (MSE), since this error consists of two parts, one being a channel fading related error and the other being a noise related error. If this aggregate error is too large, the learning performance of the model is seriously impaired.
Therefore, to reduce the MSE, the present embodiment may eliminate the error of the channel fading correlation, while reducing the error of the noise correlation as much as possible. Thus, although the resulting MSE is not optimal, if the error associated with the channel fading is dominant and the central node is equipped with multiple antennas, the resulting MSE is still very close to the minimum.
The above conditions and expression of MSE are expressed as:
in order to guarantee learning performance of the model, it is necessary to reduce the value of MSE as much as possible,
assuming that the number of user equipments per round selection is fixedWherein->Is a subset of user equipments->The number of elements in the matrix. The decision variables of the optimization problem include the scaling factor eta, the transmission coefficient b k User equipment subset->And a beamforming vector m. Since these variables are independent of noise n, so the optimization objective only needs to be reduced to eta m 2 Then the overall optimization problem can be expressed as:
according to the existing literature, the transmission coefficient b k Can be directly designed as follows:
then the scaling factor η may be designed as:
where P is the maximum allowed transmit power per user equipment.
Substituting equation (10) back to equation (8 a) can yield our optimization objective asBecause the numerator denominators all have the value of m, the overall optimization problem can be converted into:
s.t.||m||＝1 (11b)
it is evident that the above problem is a combinatorial optimization problem and a straightforward approach is to traverse all possible choices of user equipment, but when the total number of devices K is large, the computational effort can be very large. There is therefore a need to invent a new and viable way to reduce MSE.
Step S4, firstly, splitting the combined optimization problem obtained in the step S3 into an aggregate beam forming optimization sub-problem and a user equipment selection optimization sub-problem, then, alternately solving the aggregate beam forming optimization sub-problem and the user equipment selection optimization sub-problem, and finally, obtaining an optimal solution through iteration, wherein the step specifically comprises the following steps:
in the aggregate beamforming optimization sub-problem, a given subset of user devicesThe combinatorial optimization problem translates into:
s.t.||m||＝1 (12b)
this is a non-convex optimization problem due to the existence of constraints.
Defining a semi-definite matrix m=mm H The sub-problem (12) can be translated into:
s.t.Tr(M)＝1 (13b)
Rank(M)＝1 (13c)
to guarantee the above-mentioned Rank-one constraint (Rank (M) =1), the present embodiment introduces a convex difference function, thus, (Rank (M) =can be used 1) rewriting into Tr (M) - ||M| 2 =0, reintroducing the auxiliary variable τ, this sub-problem can be translated into:
Tr(M)-||M|| 2 ＝0 (14c)
Tr(M)＝1 (14d)
easily demonstratedTo ensure constraint->The value of τ should be at [ τ ] low ,τ up ]Within (1), where τ low ＝0，/>
Since the objective function contains only one variable, this can be solved by the dichotomy. In the dichotomy, the objective function of each step is fixed, and only the feasibility problem needs to be solved. Given τ, the above problem becomes a verification problem. This problem is viable if the solution rank of the following problem is 1, otherwise it is not viable.
Tr(M)＝1 (15c)
In addition, the convex function M 2 Performing first-order Taylor expansion to obtain M 2 ≥Real(Tr(vv H M), where v is M t-1 T-1 represents the number of iterations.
Thus, the solution of problem (15) can be obtained by iteratively solving the following equation:
Tr(M)＝1 (16c)
in the user equipment selection optimization sub-problem, the beamforming vector m is given, so that the problem is simplified to determining a subset of user equipmentThe sub-problem can be expressed as:
this problem can be translated into finding the projection value m with the first S largest values H h k User equipment of |. Thus only need to pairThe subset of user devices can be found by calculating the projection value for each user device k in (a)>
Through the above analysis, the specific steps of the algorithm can be generalized as:
(1) Initialization of
First fromRandom in personal devicesS are selected as subsets of user equipments +.>
(2) Executing loops
The algorithm alternately solves the aggregate beam forming design sub-problem and the user equipment selection sub-problem, and continuously iterates, and converges to an approximately optimal solution, so that the step (3) and the step (4) are executed in a circulating manner until a convergence condition is reached.
(3) Aggregate beamforming design
When obtaining the channel vector h k Given a subset of user devicesThen we can get the optimal beamforming vector m under this condition, which comprises the following steps:
first initialize M 0 ，τ low ＝0，When τ is up -τ low >Let τ= (τ) at δ up +τ low ) 2, then solving the problem (16) until convergence; if Rank (M)>1, then problem (14) is not feasible at the current τ, let τ up =τ; if Rank (M) =1, then problem (14) is feasible at the current τ condition, and let τ be low ＝τ。
(4) User equipment selection optimization
Given the beamforming vector m, we can derive the optimal subset of user equipments under this conditionThe method comprises the following specific steps:
first toThe projection m is calculated by each user equipment k H h k I; then, the projection values are arranged in a descending order; taking the arranged front partS user equipments as +.>
(5) Terminating the cycle
When a subset of user devicesWhen no change is made, the convergence condition is reached, and the cycle is terminated.
In order to verify the performance advantage of the above method, this embodiment provides a specific example flow, which specifically includes:
(1) Experimental parameter setting
In order to prove the effectiveness of the method of the embodiment, several most advanced algorithms at present are selected for comparison. For the problem of minimizing MSE, a comparison algorithm is a random user equipment selection scheme, which uses a convex difference function to optimize the beam forming vector, and the subset of user equipment is generated by random selection, which is represented by the random user equipment selection scheme in the figure; the second comparison algorithm is called channel-based user equipment selection strategy, which directly selects a subset of user equipments involved in the update based on the channel characteristics of the user equipments, and optimizes the beamforming vector with a convex difference function, represented in the figure by the channel-based user equipment selection strategy.
(2) Effect of the algorithm of the present invention on reducing the mean square error
To fully illustrate the effectiveness of the algorithm of the present invention, the present embodiment provides that the number of user equipments participating in the update per round under the experimental condition is 10, the maximum transmission power P allowed by each user equipment is set to 0dB, the total number of equipments K is gradually increased from 20 to 200, and the present embodiment performs an experiment for four cases where the number of antennas of the central node is 4,8, 12 and 16, respectively. The experimental results are shown in fig. 3, and it can be seen that the algorithm is significantly better than the random user equipment selection scheme. And at N r The method of this embodiment also has advantages over channel-based user equipment selection strategies when smaller, but with N r Is increased by (a)This advantage is continuously reduced.
(3) Probability distribution function of aggregate error
In order to more intuitively display the difference of the aggregation error value distribution of different methods, the embodiment is shown in N r In the four cases of=4 and k= 20,80,140,200, 50 independent experiments were performed, and then a probability distribution function diagram of the aggregation error was plotted, as shown in fig. 4. It can be seen from the figure that the random ue selection scheme performs much lower than the other two, and the performance of the method of this embodiment is slightly better than the channel-based ue selection strategy, although the advantages are diminishing as K increases.
(4) Learning performance
In this embodiment, two most classical machine learning data sets, namely MNIST10 and CIFAR10, are selected, and in order to be closer to the actual application scenario, it is further assumed that the data sets on each user device are not subject to independent and uniform distribution. The MNIST10 dataset contains ten digital black and white handwritten digital pictures of 0 to 9 digits, for which the embodiment uses a multi-layer perceptron neural network. CIFAR10 contains a color picture of class 10 objects and is therefore more difficult to train than MNIST10, so ResNet18 neural networks are used for this dataset. For both neural networks, the present embodiment employs a small batch random gradient descent technique for updating, with a batch size set to 16 and a learning rate set to 0.01. To conveniently represent the impact of the aggregate error on learning performance, this embodiment models the aggregate error as the number of model retransmissions, p=1-exp (-aMSE/σ) 2 ) Wherein a is set to 2.
In the experiment, the total number of the user equipment is set to be K=100, 10 user equipment are selected to participate in updating in each round, and the number of antennas of the central node is 16. MSE/sigma derived from the method of the present embodiment, channel-based user equipment selection strategy and random user equipment selection scheme 2 0.2904,0.3127 and 0.5734, respectively. As shown in fig. 5, on MNIST10 dataset, except for followingBesides the poor performance of the selection scheme of the machine user equipment, the other two algorithms have good performance and small difference. The random user equipment selection scheme performs still worst on the CIFAR10 dataset, while the other two algorithms perform because of the resulting MSE/σ 2 The values differ very little so their performance is still very close. Therefore, the method reduces the aggregation error and is helpful to improving the learning performance of the model.
In summary, when the number of the selected ues is fixed, the method of the embodiment can obtain a lower aggregation error compared with the random ue selection scheme and the channel-based ue selection policy, thereby obtaining better learning performance.
The present invention is not described in detail in the present application, and is well known to those skilled in the art.
The foregoing describes in detail preferred embodiments of the present invention. It should be understood that numerous modifications and variations can be made in accordance with the concepts of the invention by one of ordinary skill in the art without undue burden. Therefore, all technical solutions which can be obtained by logic analysis, reasoning or limited experiments based on the prior art by the person skilled in the art according to the inventive concept shall be within the scope of protection defined by the claims.
Claims (2)
1. An iterative aggregate beamforming design and user device selection method, the method comprising the steps of:
step S1, constructing a distributed federation learning model aiming at an edge intelligent system with a plurality of edge user devices and a central node;
s2, training the distributed federal learning model constructed in the step S1, wherein when the edge user equipment transmits model parameters to a central node, an air computing technology is adopted to improve communication efficiency, and then an aggregation error generated by adopting the air computing technology is determined;
s3, constructing an overall optimization problem aiming at the aggregation error determined in the step S2, and converting the overall optimization problem into a combined optimization problem;
step S4, firstly, splitting the combined optimization problem obtained in the step S3 into an aggregate beam forming optimization sub-problem and a user equipment selection optimization sub-problem, then, alternately solving the aggregate beam forming optimization sub-problem and the user equipment selection optimization sub-problem, and finally, obtaining an optimal solution through iteration;
the step S1 specifically includes:
in an edge intelligence system, there are K edge user devices equipped with only one antenna and one edge user device equipped with N r A central node of the root antenna, where N r Much smaller than K;
the channel between the edge user device k and the central node obeys the same complex gaussian distribution with unit power, i.e
Definition of the definitionFor a set of all edge user devices, each edge user device k has its own local data setAll local data sets->Combining into a global dataset +.>
Building global data setsAverage loss function->Wherein, is a parameter of the global model, Q is a loss function of the global model, (x) n ,y n ) Is a data sample;
constructing an updating formula of local model parameters of kth edge user equipment:
constructing a central node, and updating a formula of global model parameters:
in the formulas (1) and (2), the superscript i is the round number, μ is the learning rate,is the k-th edge user equipment average loss function +.>Gradient of (2), wherein->
In the step S2, when the edge ue transmits the model parameters to the central node, an air computing technology is adopted to improve the communication efficiency, which specifically includes:
first, in the training of the ith round, the symbol vector transmitted by the edge user equipment k is defined asAssuming normalization with unit variance, the expression is: />
Then, the normalized symbol vector is subjected toAfter analog modulation, the transmission coefficient b is used for k Pre-coding, and then, superposing signals subjected to normalization, analog modulation and pre-coding in the air, and multiplying the signals with a beam forming vector of a central node;
finally, amplifying by a scaling factor eta;
in the step S2, the determining an aggregate error generated by using the over-the-air computing technology specifically includes:
definition at training timeFor the selected subset of edge user equipments>D element->For example, then, the center node wants to receive the corresponding +.>The expression of the ideal signal of (2) is:
because of the fading and noise of the wireless channel, the signal actually received by the central node is expressed as:
in the formula (3) and the formula (4), s k Is thatD element->Shorthand, h k Is the channel vector from the edge user device k to the central node b k Is the transmission coefficient of the edge user equipment k, m is the beam forming vector of the central node, n is the additive white gaussian noise and obeys independent co-distribution, i.e. +.>The superscript H denotes a conjugate transpose;
according to the formula (3) and the formula (4), the aggregate error generated by the air computing technology is obtained, and the expression is:
the step S3 specifically includes:
and (2) adopting a mean square error to measure the aggregation error determined in the step (S2), and eliminating errors related to channel fading in the aggregation error, and simultaneously reducing errors related to noise as much as possible, wherein the specific expression is as follows:
in the formula (7), σ represents the standard deviation of noise, σ 2 Representing the variance of the noise;
assuming that the number of edge user devices per round selection is fixedWherein (1)>Is a selected subset of edge user devices +.>If the number of elements in the overall optimization problem is equal to or greater than the number of elements in the overall optimization problem, the decision variables of the overall optimization problem include: scaling factor eta, transmission coefficient b k Edge user device subset->And a beamforming vector m, the overall optimization problem is expressed as:
transmission coefficient b k The design is as follows:
the scaling factor η is designed as:
in formula (10), P is the maximum allowed transmit power for each edge user device;
substituting the formula (10) back to the formula (8 a) to obtain the optimization target
Since there are m in the numerator denominator, the combination optimization problem can be equivalently:
s.t.||m||＝1 (11b)
in the step S4, splitting the combination optimization problem obtained in the step S3 into an aggregate beamforming optimization sub-problem specifically includes:
given a selected subset of edge user devicesThen, the combinatorial optimization problem translates into:
s.t.||m||＝1 (12b)
define aSemi-definite matrix m=mm H The sub-problem (12) translates into:
s.t.Tr(M)＝1 (13b)
Rank(M)＝1 (13c)
by introducing a difference of convexity function, formula (13 c) is rewritten Tr (M) - |M| 2 =0, reintroducing the auxiliary variable τ, then the sub-problem is translated into:
Tr(M)-||M|| 2 ＝0 (14c)
Tr(M)＝1 (14d)
wherein,since the constraint (14 b) is to be satisfied, the value of τ is limited to [ τ ] low ,τ up ]Within (2), and τ low ＝0，/>
By giving τ, the problem (14) is transformed into a form that can be solved by the dichotomy, expressed as:
Tr(M)＝1 (15c)
wherein for the problem (15), if the rank of the solution of the problem is 1, then at the current τ, the problem (14) is viable, otherwise, it is not viable;
the convex function M 2 Performing first-order Taylor expansion to obtain M 2 ≥Real(Tr(vv H M), wherein v is M t-1 The feature value of the largest feature vector of (2), t-1 represents the iteration number;
then, the solution of the problem (15) is obtained by iteratively solving the following equation:
Tr(M)＝1 (16c)
splitting the combined optimization problem obtained in the step S3 into user equipment selection optimization sub-problems, wherein the method specifically comprises the following steps:
simplifying the combined optimization problem to determine a selected subset of edge user devices by giving a beamforming vector mThe user equipment selection optimization sub-problem is expressed as:
2. the iterative aggregate beamforming design and user equipment selection method according to claim 1, wherein in said step S4, said alternately solving said aggregate beamforming optimization sub-problem and user equipment selection optimization sub-problem, and finally, obtaining an optimal solution by iteration, specifically comprises:
step S401, initializing, first, starting fromRandomly selecting S of the individual devices as a selected subset of edge user devices +.>
Step S402, performing aggregate beam forming design, when obtaining the channel vector h k After the selected edge user equipment subset S is given, the optimal beamforming vector m under the condition is obtained, and the specific steps are as follows:
first, initialize M 0 ，τ low ＝0，When τ is up -τ low >Let τ= (τ) at δ up +τ low )/2；
Then solve the problem (16) until convergence;
if Rank (M) >1, then problem (14) is not feasible at the current τ condition,
then let τ up ＝τ；
If Rank (M) =1,
then the problem (14) is feasible at the current τ and let τ be low ＝τ；
Step S403, optimizing user equipment selection, and giving beam forming vector m, obtaining the optimal selected edge user equipment subset under the conditionThe method comprises the following specific steps:
first, toThe projection m is calculated by each edge user equipment k H h k ||；
Then, these projection values are arranged in descending order;
finally, the first S edge user equipments after arrangement are taken as
Step S404, circularly executing the step S402 and the step S403 until reaching a convergence condition;
step S405, when the selected edge user equipment subsetWhen no change is made, the convergence condition is reached, and the cycle is terminated.
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