MSE (Mean Square Error) parameter aggregation method and system for air computing assisted federated learning

Abstract

The invention relates to the technical field of federated learning of artificial intelligence, in particular to an MSE parameter aggregation method and system for air computing assisted federated learning, and the method comprises the steps that a model is trained through multiple rounds of iteration, and each round of iteration comprises the steps that a mobile device receives global parameters, calculates a gradient based on local part data, and uploads the gradient according to an amplification coefficient of the mobile device; after receiving the superposed signal, the base station recovers an aggregation gradient signal according to an amplification coefficient of the base station, and updates a global parameter by using the signal; and the base station minimizes a mean square error (MSE) between the recovery signal and the ideal aggregation gradient through two-layer optimization, and distributes updated parameters to the mobile equipment until training convergence. According to the method, through collaborative optimization of the amplification coefficients of the mobile device and the base station and the training data volume of the mobile device, the MSE is deeply reduced, the calculation load of the device is reduced, the calculation complexity is controlled, the universality is high, the method can be compatible with a mainstream model, and the training performance is superior to that of a traditional method.

Description

Technical Field

[0001] This invention relates to the field of federated learning technology in artificial intelligence, and in particular to a method and system for MSE parameter aggregation in computationally assisted federated learning. Background Technology

[0002] With the rapid development of mobile internet and IoT technologies, the massive amounts of data generated by mobile devices such as smartphones and sensors are growing exponentially, providing a data foundation for machine learning applications such as disaster early warning based on digital twins and health monitoring based on wearable devices. However, traditional centralized machine learning requires uploading raw local data from devices to a central server, which not only consumes a large amount of wireless communication resources but also poses a significant risk of user privacy leaks.

[0003] To address privacy and communication resource concerns, Federated Learning (FL) has emerged as a distributed model training architecture. Multiple mobile devices, coordinated by a base station (BS), collaboratively train the model by iteratively exchanging parameters, eliminating the need to upload raw data. However, the parameter iteration in federated learning involves frequent multiple access communications, leading to bottlenecks such as low spectrum efficiency and high energy consumption in scenarios with a large number of devices and limited spectrum resources.

[0004] Over-the-air computing (OTA) technology leverages the superposition characteristics of channel signals to support multiple mobile devices simultaneously transmitting parameter signals to a base station. It allows for direct summation of data on the communication signal, significantly improving the spectral efficiency of parameter aggregation and overcoming communication bottlenecks. However, in practical applications, heterogeneous channel fading and transmission noise cause deviations between the base station's received signal and the ideal aggregated signal, typically expressed as mean square error (MSE), thus reducing aggregation accuracy. Furthermore, existing technologies only optimize the signal amplification factor without considering the impact of the amount of local training data on MSE, easily leading to overfitting and excessive computational load, making it difficult to meet the requirements of high-precision training. Summary of the Invention

[0005] The purpose of this invention is to provide a method and system for MSE parameter aggregation in computationally assisted federated learning, which solves the problems of insufficient MSE reduction and high equipment computational load caused by optimizing only a single amplification factor in the existing technology. Through collaborative optimization, it minimizes the mean square error (MSE), reduces the computational load, and controls the computational complexity.

[0006] To achieve the above objectives, the present invention provides the following technical solution: According to one aspect of the present invention, a method for MSE parameter aggregation in over-the-air computing-assisted federated learning is provided, applicable to mobile edge computing networks consisting of a base station and multiple mobile devices. The method trains the model through multiple iterations, each iteration including the following steps: S1: Each mobile device receives the current global parameters from the base station and selects a portion of data from the local dataset to calculate the local gradient; S2: Each mobile device uploads its gradient to the base station according to its own amplification factor. After receiving the superimposed signal, the base station recovers the aggregated gradient signal according to its own amplification factor. S3: The base station uses the recovered aggregated gradient signal to update the global parameters according to a preset step size; S4: The base station performs two-layer optimization to minimize the mean square error (MSE) between the recovered signal and the ideal aggregation gradient; S5: The base station distributes the optimized and updated parameters and new global parameters to the mobile devices, and enters the next iteration until the training converges.

[0007] According to an embodiment of the present invention, in step S4, the two-layer optimization specifically includes: first fixing the amplification factor of the base station, and then classifying and optimizing the amplification factor and data weight of the mobile device according to the characteristics of the mobile device; then optimizing the amplification factor of the base station to obtain the updated amplification factor and the data training amount of the mobile device.

[0008] According to one embodiment of the present invention, the characteristic classification criterion of the mobile device is whether the product of the device's maximum amplification factor, the channel factor, and the base station amplification factor is higher than its own data volume weight limit.

[0009] According to one embodiment of the present invention, the data volume weight refers to the proportion of the data volume of a single mobile device participating in training to the total data volume of all mobile devices participating in training, and the weight is set with an upper limit threshold.

[0010] According to one embodiment of the present invention, in step S4, the ideal aggregated gradient is the result of weighting the local gradients of each mobile device according to the corresponding amount of data.

[0011] According to an embodiment of the present invention, in step S1, a portion of data is selected from the local dataset, specifically the selected amount is less than or equal to the size of the dataset itself and the total amount is greater than or equal to the minimum total amount of training data.

[0012] According to one embodiment of the present invention, in step S2, the amplification factor of the mobile device is set to an upper limit threshold, and the amplification factor of the base station is greater than 0.

[0013] According to one embodiment of the present invention, in step S2, when the base station recovers the aggregated gradient signal, it needs to eliminate the interference effect of noise during transmission on the superimposed signal.

[0014] According to one embodiment of the present invention, the preset step size is a fixed value set in advance, or a dynamic value that is dynamically adjusted according to the training loss of each iteration.

[0015] On the other hand, the present invention also provides an MSE parameter aggregation system for over-the-air computing-assisted federated learning, the system comprising a base station and multiple mobile devices; The base station includes a parameter distribution module, a signal recovery module, a global update module, an MSE optimization module, and a data storage module, wherein: The parameter distribution module is configured to synchronously distribute the global model parameters for the current round, as well as the optimized and updated amplification factor and training data volume configuration parameters, to all mobile devices. The signal recovery module is configured to receive superimposed gradient signals uploaded by multiple mobile devices, recover the aggregated gradient signal based on the base station's own amplification factor, and eliminate the interference of transmission noise on the signal. The global update module is configured to update the global model parameters according to a preset step size based on the recovered aggregate gradient signal. The MSE optimization module is configured to execute a two-layer optimization algorithm to optimize the amplification factor of the mobile device, the data weight, and the amplification factor of the base station, respectively, and minimize the mean square error between the recovered signal and the ideal aggregation gradient. The data storage module is configured to store key data such as global model parameters, channel coefficients of each mobile device, amplification coefficient thresholds, and training data volume thresholds. The mobile device includes a parameter receiving module, a gradient calculation module, a signal uploading module, and a local storage module, wherein: The parameter receiving module is configured to receive global model parameters, optimized amplification coefficients, and training data volume configuration parameters sent by the base station. The gradient calculation module is configured to select a portion of data that meets the requirements from the local dataset based on the received global parameters, and calculate the local gradient. The signal uploading module is configured to upload the calculated local gradient signal to the base station according to its own amplification factor; The local storage module is configured to store local datasets, received global parameters, its own amplification factor, and training data volume configuration, among other data.

[0016] The present invention provides a method and system for MSE parameter aggregation in computationally assisted federated learning. Compared with the prior art, the beneficial effects of the present invention are as follows: 1. Achieved deep reduction in MSE. By comprehensively optimizing the amount of training data and amplification factor of mobile devices, it can accurately offset channel fading and noise interference, significantly improve the consistency between the signal received by the base station and the ideal aggregated signal, and provide high-precision aggregation parameters through model training.

[0017] 2. Significantly reduced computational load. By limiting the acceptable range of training data for mobile devices, overfitting and resource consumption caused by excessive data are avoided, thus adapting the device resource constraints of mobile edge computing networks.

[0018] 3. It achieves controllability of computational complexity. By decomposing non-convex problems into solvable subproblems through hierarchical iterative optimization algorithms, and combining techniques such as binary search and golden section search, the computational complexity is significantly reduced with minimal performance loss, achieving a balance between practicality and efficiency.

[0019] 4. High versatility. The method can be applied to various federated learning scenarios based on aerial computing and is compatible with mainstream models such as ResNet. Tests on datasets such as CIFAR100 show that its training loss and test accuracy are superior to traditional methods such as AirFedSGD, TPC, and COP. Attached Figure Description

[0020] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a flowchart of an MSE parameter aggregation method for computationally assisted federated learning according to an embodiment of the present invention; Figure 2 This is a schematic diagram of an MSE parameter aggregation system for computationally assisted federated learning according to an embodiment of the present invention; Figure 3 This is a schematic diagram of the signal interaction between the base station and the mobile device according to an embodiment of the present invention; Figure 4 This is a schematic diagram illustrating the change in test accuracy with communication rounds compared to the comparison method in this embodiment of the invention; Figure 5 This is a schematic diagram showing the change of the training loss function with the number of communication rounds in an embodiment of the present invention, compared with the comparison method. Detailed Implementation

[0021] To facilitate a clear description of the technical solutions in the embodiments of the present invention, the terms "first" and "second" are used to distinguish identical or similar items with essentially the same function and effect. For example, the first threshold and the second threshold are merely used to distinguish different thresholds and do not limit their order. Those skilled in the art will understand that the terms "first" and "second" do not limit the quantity or execution order, and that the terms "first" and "second" are not necessarily different.

[0022] It should be noted that in this invention, the terms "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design described as "exemplary" or "for example" in this invention should not be construed as being more preferred or advantageous than other embodiments or designs. Specifically, the use of terms such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0023] In this invention, "at least one" refers to one or more, and "more than one" refers to two or more. "And / or" describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone, where A and B can be singular or plural. The character " / " generally indicates that the preceding and following related objects are in an "or" relationship. "At least one" or similar expressions refer to any combination of these items, including any combination of singular or plural items. For example, "at least one of a, b, or c" can represent: a, b, c, a combination of a and b, a combination of a and c, a combination of b and c, or a, b, and c, where a, b, and c can be single or multiple.

[0024] This invention proposes a method and system for MSE parameter aggregation in over-the-air computing-assisted federated learning, applicable to mobile edge computing networks consisting of a base station and multiple mobile devices. The method generates gradients locally on the mobile devices and uploads them to the base station for aggregation. It jointly optimizes the amplification coefficients of the mobile devices and the base station, as well as the amount of data used by the mobile devices in training, to reduce the MSE between the aggregated recovered signal and the ideal aggregated signal received by the base station, thus alleviating the computational burden on the mobile devices. The federated learning process includes multiple iterative steps, further employing a hierarchical iterative optimization algorithm to decompose the non-convex problem into solvable subproblems, significantly reducing computational complexity with minimal performance loss. Applying this invention enables collaborative optimization during distributed model training to minimize the mean squared error (MSE) of parameter aggregation, reduce the computational burden on mobile devices, and control computational complexity.

[0025] like Figure 1The diagram shows a flowchart of an MSE parameter aggregation method for over-the-air computing-assisted federated learning. This method is applicable to mobile edge computing networks consisting of a base station and multiple mobile devices. The model is trained through multiple iterations, each iteration including the following steps: S101: Each mobile device receives the current global parameters from the base station and selects a portion of data from the local dataset to calculate the local gradient; specifically, when selecting a portion of data from the local dataset, a selected quantity (denoted as...) is... (where k is the mobile device ID) is less than or equal to the size of its own dataset (denoted as k). ), and the total amount of data selected from all mobile devices Greater than or equal to the minimum total training data size (denoted as ) This ensures the training performance of federated learning and avoids poor training results due to insufficient data.

[0026] S102: Each mobile device uploads the gradient to the base station according to its own amplification factor. After receiving the superimposed signal, the base station recovers the aggregated gradient signal according to its own amplification factor. Among them, the magnification factor of the mobile device (denoted as ) Set the upper limit threshold (denoted as) This is because mobile devices have limited ability to amplify transmitted signals; the amplification factor (denoted as 'a') of the base station is greater than 0. After receiving the aggregated signal, the base station can first quantize the received signal to digitize it, and then amplify the digitized signal at any ratio. Simultaneously, when recovering the aggregated gradient signal, the base station needs to eliminate noise during transmission (denoted as 'a'). The effect of t (the number of iterations) on the interference generated by the superimposed signal is studied to improve the accuracy of the recovered signal.

[0027] S103: The base station uses the recovered aggregated gradient signal to update the global parameters according to the preset step size; The preset step size (denoted as...) The step size can be a pre-set fixed value or a dynamic value that is dynamically adjusted based on the training loss of each iteration. Setting a fixed step size is simple, while a dynamic step size can be flexibly adjusted according to the training situation to optimize training speed and effectiveness.

[0028] S104: The base station minimizes the mean square error (MSE) between the recovered signal and the ideal aggregated signal through two-layer optimization. Ideal aggregation gradient (denoted as) ) is the local gradient of each mobile device (denoted as ) The result is a weighted average based on the corresponding data volume, where the weight is the proportion of data from a single mobile device used in training to the total data from all mobile devices used in training (denoted as ). And this weight has an upper limit threshold (denoted as ). = ).

[0029] The two-layer optimization specifically includes: first fixing the amplification factor 'a' of the base station, and then classifying and optimizing the amplification factor of the mobile devices according to their characteristics. and data volume weight ; Then optimize the amplification factor 'a' of the base station to obtain the updated amplification factor (a, ) and the amount of data trained on mobile devices ( ).

[0030] The characteristic classification criteria for the mobile device are: the maximum magnification factor of the device. With channel coefficient The product of the base station amplification factor a (a・ • Whether it exceeds its own data volume weight limit Based on this, the equipment is divided into two sets: as well as , where K is the total number of mobile devices.

[0031] The specific implementation process of the two-layer optimization is as follows: Lower-level optimization (fixed a, optimization) and ): Calculate the KKT conditions and determine the Lagrange multipliers and constraints; like Calculate the threshold ; when At that time, the Lagrange multipliers are determined by the binary search method. , Class of equipment , renew, Class of equipment , renew; when hour, Class of equipment , renew, Class of equipment , renew; like , Type of equipment selection satisfy ,in , , and according to renew; Class of equipment renew.

[0032] Upper-level optimization (optimization a): calculate And sort them in ascending order, dividing variable a into K+1 intervals; Determine the minimum value of 'a' , making When a≥ When, the optimal solution is ( (for noise variance) filter The interval containing the solution and the interval to its left are further refined according to the saturation conditions of the underlying solution. Within each subinterval, the optimal solution for 'a' is found using the golden section search method, and the solution with the minimum global cost is selected. As the final amplification factor of the base station.

[0033] S105: The base station will optimize and update the parameters ( , , ) and new global parameters ( The training loss is distributed to mobile devices to enter the next iteration until the training converges (the convergence condition is that the training loss is less than the set threshold or the number of iterations reaches the preset upper limit).

[0034] like Figure 2 The diagram shows a schematic of an MSE parameter aggregation system for over-the-air computation-assisted federated learning. This system is used to implement the above method and includes a base station and multiple mobile devices.

[0035] The base station includes a parameter distribution module, a signal recovery module, a global update module, an MSE optimization module, and a data storage module, wherein: The parameter distribution module is configured to synchronously distribute the global model parameters for the current round, as well as the optimized and updated amplification factor and training data volume configuration parameters, to all mobile devices. The signal recovery module is configured to receive superimposed gradient signals uploaded by multiple mobile devices, recover the aggregated gradient signal based on the base station's own amplification factor, and eliminate the interference of transmission noise on the signal. The global update module is configured to update the global model parameters according to a preset step size based on the recovered aggregate gradient signal. The MSE optimization module is configured to execute a two-layer optimization algorithm to optimize the amplification factor of the mobile device, the data weight, and the amplification factor of the base station, respectively, and minimize the mean square error between the recovered signal and the ideal aggregation gradient. The data storage module is configured to store key data such as global model parameters, channel coefficients of each mobile device, amplification coefficient thresholds, and training data volume thresholds. The mobile device includes a parameter receiving module, a gradient calculation module, a signal uploading module, and a local storage module, wherein: The parameter receiving module is configured to receive global model parameters, optimized amplification coefficients, and training data volume configuration parameters sent by the base station. The gradient calculation module is configured to select a portion of data that meets the requirements from the local dataset based on the received global parameters, and calculate the local gradient. The signal uploading module is configured to upload the calculated local gradient signal to the base station according to its own amplification factor; The local storage module is configured to store local datasets, received global parameters, its own amplification factor, and training data volume configuration, among other data.

[0036] The parameter receiving module is configured to receive global model parameters sent by the base station. Optimized magnification factor ( , ) and training data volume configuration parameters ( , ); The gradient calculation module is configured to calculate the gradient based on the received global parameters. From local dataset Selected from Data ( ≤ and ), through the loss function: ,in Calculate the local gradient by differentiating the index of the selected data point. ; The signal upload module is configured to use its own amplification factor. The calculated local gradient signal Simultaneously uploaded to the base station, signal superposition and transmission are achieved by relying on over-the-air computing technology; The local storage module is configured to store local datasets. Received global parameters Its own magnification factor Training data volume configuration ( , Local gradient calculation results Data such as...

[0037] like Figure 3The diagram illustrates the signal interaction between a base station and mobile devices. It includes one base station and three mobile devices, each connected to the base station via a wireless signal link, forming the core interaction unit of the mobile edge computing network. First, the base station synchronously sends the global model parameter vector (parameter vector w) for the current round to all mobile devices. Then, each mobile device calculates its local gradient based on its local dataset (training data s1, s2, s3), generating gradient information ∇1, ∇2, and ∇3, which is transmitted from each mobile device to the base station. Finally, relying on over-the-air computing technology, the gradient signals from multiple mobile devices naturally superimpose in the channel. The base station directly receives the superimposed signals and performs aggregation processing, merging the gradient information through over-the-air computing to complete a single round of information interaction.

[0038] Example 1: The federated edge learning system includes a base station and =20 mobile devices. Local dataset for each mobile device. The values ​​were randomly generated as 3979, 3974, 3985, 3933, 4026, 3984, 3972, 3961, 3991, 3986, 4051, 3972, 3921, 3991, 3983, 3937, 3958, 4058, 4033, and 4051.

[0039] The training task is to train a ResNet-18 model on the CIFAR100 dataset. The minimum total data volume threshold required for training is [not specified]. . , . =1,2,..., Noise variance iteration step size .

[0040] The comparative methods for performance verification include AirFedSGD (optimizing only the transmission signal format), TPC (optimizing only the transmission power), and COP (optimizing only the allocation of computing resources).

[0041] Step 1: Each mobile device selects a portion of data from its local dataset for gradient updates.

[0042] With the first Taking the first mobile device as an example, the first Each mobile device selected multiple [items] from the local dataset. Data points are updated locally, among which, Also known as the first Data size of each mobile device , This represents the size of the local dataset on the k-th mobile device. This represents the total number of mobile devices. To ensure the performance of federated learning, the total number of data points K participating in the training should be greater than or equal to a threshold. ,Right now .

[0043] No. Gradient of mobile devices The updated formula is as follows: (1) in, It is in the The parameter vector broadcast in the next iteration; The loss function for the training task is defined as follows: (2) in, Indicates the first The index of the selected data points.

[0044] Step 2: Each mobile device is magnified according to its own magnification factor. The generated gradient data is simultaneously uploaded to the base station.

[0045] Due to the superposition characteristics of the channel, the base station can directly receive the signal after the data uploaded by each mobile device is superimposed, and the base station receives it with an amplification factor. The signal is amplified. Then, using over-the-air computing technology, the recovered signal is retrieved at the base station. for: (3) in, Let the noise vector be an element that follows a distribution. , This refers to the channel coefficient between each mobile device and the base station. Meanwhile, the ability of each mobile device to amplify the transmitted signal is limited, i.e., the amplification factor. There is an upper limit, denoted as However, the amplification factor of the received signal at the base station is... There are no restrictions because after the base station receives the aggregated signal, it first digitizes the received signal through quantization; then, the base station amplifies the digitized signal at any ratio.

[0046] Due to noise signal Channel coefficients between multiple mobile devices Unevenness, in signal recovery With the ideal aggregation gradient Distortion exists between them. Ideal recovery signal. It can be represented as: (4) If we use mean squared error (MSE) to measure and The distortion between them, the mean square error (MSE) is expressed as: = (5) in, To express expectations, for the sake of subsequent description, [the following will be used]. Defined as , , It is an indicator function. For the MSE defined in formula (5), if =0, then It will be zero if it is zero, otherwise it will be 1. Because when When =0, the first The mobile device did not actually participate in gradient aggregation, so the relevant MSE will not be calculated. Gradient aggregation distortion caused by mobile devices.

[0047] Step 3: In federated learning, the base station utilizes the recovered signal (The signal is) (approximate value), update the parameter vector as follows: : (6) in, It is a predefined step size.

[0048] For federated edge learning systems based on over-the-air computing, since the MSE of the aggregated signal is closely related to the training loss function value, the following MSE optimization problem needs to be solved at the base station to minimize the amplification factor and the amount of training data in order to update... , , : Question 1: (7) (8) (9) (10) definition: (11) It can be known that: (12) Problem 1 can be transformed into a lower-level problem (Problem 2) and a higher-level problem (Problem 3): Question 2: (7) (13) (14) Question 3: (10) By solving the lower-level problem through the following steps, a new amplification factor can be obtained. and the amount of training data (by (presented in the form of)

[0049] Sub-step 31: Calculate its KKT conditions.

[0050] (15) (16) (17) (18) (19) (20) (twenty one) (twenty two) (twenty three) (twenty four) in , , and They are respectively with constraints , , and The relevant non-negative Lagrange multipliers, λ, are Lagrange multipliers related to the constraints in (18).

[0051] At this point, the mobile device is divided into two parts: Define a set ,as well as Define a set .

[0052] If the judgment is If yes, proceed to sub-step 32; otherwise, proceed to sub-step 33.

[0053] Sub-step 32: Calculation (25) If Then use the binary search method to find the expression (26). , recorded as .

[0054] (26) for ,renew and Specifically: (27) (28) And for ,renew and Specifically: (29) (30) And if ,for ,renew and Specifically: (31) (32) And for Then update and Specifically: (33) (34) Sub-step 33: For ,renew and .choose To make it meet the following requirements: (35) Here , Then select (36) And for Then update and Specifically: (37) (38) The underlying problem (Problem 2) has now been solved, and the amplification factor... and the amount of training data The update has been implemented.

[0055] The following addresses the upper-level problem (Problem 3) in order to update... .

[0056] Sub-step 34: Interval partitioning and cutoff point determination. Calculation. (39) Sort in ascending order And accordingly, the variables Divided into each interval At the same time, calculate the minimum. value , so that: (40) when When, the optimal solution is .

[0057] Sub-step 35: Interval filtering and refinement.

[0058] Sure Location of the interval and only Optimization is performed within the interval. Within each interval, the interval is further refined based on the saturation condition of the lower-level solution, while maintaining the set... constant.

[0059] Sub-step 36: Interval optimization and global optimal selection.

[0060] Within each subinterval, the cost function is related to For a convex function, the optimal solution is obtained using the golden section search method. Compare the optimal solutions across all subintervals and select the one with the lowest cost as the global optimal solution. .

[0061] The magnification factor at this base station It has also been updated.

[0062] In this example, the default parameters are as follows.

[0063] This invention compares the performance of the proposed DSC (Data Size Control) method with that of the AirFed SGD method, Transmission Power Control (TPC) method, and Computation Optimal Policy (COP) method. The comparison results are attached. Figure 4 , Figure 5 As shown. Figure 4 and Figure 5 The performance of the proposed method and its comparison methods in training ResNet-18 on the CIAFR100 dataset is demonstrated. It can be seen that as the number of communication rounds T increases, the proposed method consistently outperforms the comparison algorithm in terms of test accuracy and training loss. This verifies that reducing MSE in gradient aggregation can help improve model training performance.

[0064] This invention discloses a method and system for MSE parameter aggregation applied to computationally assisted federated learning (CEL). It specifically optimizes the amplification factor and training data volume jointly, balancing MSE, computational burden, and complexity through a hierarchical iterative algorithm, filling a technical gap in multi-objective collaborative optimization in CEL. The invention proposes a joint optimization combination of the amplification factor and the amount of training data involved, rather than simply superimposing existing optimization objects. Based on the signal transmission characteristics of CEL, it identifies the coupled influence of these two factors on MSE and computational burden, forming a precise optimization logic. The hierarchical iterative optimization algorithm is not readily apparent; it decomposes the non-convex problem into solvable subproblems, overcoming the high complexity bottleneck of traditional exhaustive or global search methods. It significantly improves computational efficiency with minimal performance loss, and the adaptability of this algorithm to the core optimization objective is unprecedented in existing technologies. Applying this invention can significantly reduce the MSE between the base station received signal and the ideal aggregated signal, improving signal consistency. Simultaneously, it reduces the computational burden on mobile devices, adapting to the resource constraints of mobile edge computing networks. The hierarchical iterative algorithm significantly reduces computational complexity, balancing practicality and efficiency, achieving collaborative optimization of performance, burden, and complexity compared to traditional methods.

[0065] Although the invention has been described herein in conjunction with various embodiments, those skilled in the art will understand and implement other variations of the disclosed embodiments by reviewing the accompanying drawings, disclosure, and other materials. In this specification, the word "comprising" does not exclude other components or steps, and "a" or "an" does not exclude multiple components. A single processor or other unit can implement several functions listed in the specification. While certain measures are described in different embodiments, this does not mean that these measures cannot be combined to produce good results.

[0066] Although the invention has been described in conjunction with specific features and embodiments, it is obvious that various modifications and combinations can be made therein without departing from the spirit and scope of the invention. Accordingly, this specification and drawings are merely illustrative of the invention and are considered to cover any and all modifications, variations, combinations, or equivalents within the scope of the invention. Clearly, those skilled in the art can make various alterations and modifications to the invention without departing from its spirit and scope. Thus, if such modifications and modifications fall within the scope of the invention and its equivalents, the invention is also intended to include such modifications and modifications.

Claims

1. A method for MSE parameter aggregation in over-the-air computing-assisted federated learning, applicable to mobile edge computing networks consisting of one base station and multiple mobile devices, characterized by multi-round iterative model training, Each iteration includes the following steps: S1: Each mobile device receives the current global parameters from the base station and selects a portion of data from the local dataset to calculate the local gradient; S2: Each mobile device uploads its gradient to the base station according to its own amplification factor. After receiving the superimposed signal, the base station recovers the aggregated gradient signal according to its own amplification factor. S3: The base station uses the recovered aggregated gradient signal to update the global parameters according to a preset step size; S4: The base station performs two-layer optimization to minimize the mean square error (MSE) between the recovered signal and the ideal aggregation gradient; S5: The base station distributes the optimized and updated parameters and new global parameters to the mobile devices, and enters the next iteration until the training converges.

2. The method according to claim 1, characterized in that, In step S4, the two-layer optimization specifically includes: first fixing the amplification factor of the base station, and then classifying and optimizing the amplification factor and data weight of the mobile device according to the characteristics of the mobile device; then optimizing the amplification factor of the base station to obtain the updated amplification factor and the data training amount of the mobile device.

3. The method according to claim 2, characterized in that, The characteristic classification criterion for the mobile device is whether the product of the device's maximum amplification factor, channel factor, and base station amplification factor is higher than its own data volume weight limit.

4. The method according to claim 2, characterized in that, The data volume weight refers to the proportion of the data volume of a single mobile device participating in training to the total data volume of all mobile devices participating in training, and this weight has an upper limit threshold.

5. The method according to claim 1, characterized in that, In step S4, the ideal aggregated gradient is the result of weighting the local gradients of each mobile device according to the corresponding amount of data.

6. The method according to claim 1, characterized in that, In step S1, a portion of the data is selected from the local dataset, specifically, the selected amount is less than or equal to the size of the dataset itself and the total amount is greater than or equal to the minimum total amount of training data.

7. The method according to claim 1, characterized in that, In step S2, the amplification factor of the mobile device is set to an upper limit threshold, and the amplification factor of the base station is greater than 0.

8. The method according to claim 1, characterized in that, In step S2, when the base station recovers the aggregated gradient signal, it needs to eliminate the interference of noise during transmission on the superimposed signal.

9. The method according to claim 1, characterized in that, The preset step size is either a fixed value set in advance, or a dynamic value that is dynamically adjusted according to the training loss of each iteration.

10. An MSE parameter aggregation system for computationally assisted federated learning in the air, for implementing the method described in any one of claims 1 to 9, characterized in that, The system includes a base station and multiple mobile devices; The base station includes a parameter distribution module, a signal recovery module, a global update module, an MSE optimization module, and a data storage module, wherein: The parameter distribution module is configured to synchronously distribute the global model parameters for the current round, as well as the optimized and updated amplification factor and training data volume configuration parameters, to all mobile devices. The signal recovery module is configured to receive superimposed gradient signals uploaded by multiple mobile devices, recover the aggregated gradient signal based on the base station's own amplification factor, and eliminate the interference of transmission noise on the signal. The global update module is configured to update the global model parameters according to a preset step size based on the recovered aggregate gradient signal. The MSE optimization module is configured to execute a two-layer optimization algorithm to optimize the amplification factor of the mobile device, the data weight, and the amplification factor of the base station, respectively, and minimize the mean square error between the recovered signal and the ideal aggregation gradient. The data storage module is configured to store key data such as global model parameters, channel coefficients of each mobile device, amplification coefficient thresholds, and training data volume thresholds. The mobile device includes a parameter receiving module, a gradient calculation module, a signal uploading module, and a local storage module, wherein: The parameter receiving module is configured to receive global model parameters, optimized amplification coefficients, and training data volume configuration parameters sent by the base station. The gradient calculation module is configured to select a portion of data that meets the requirements from the local dataset based on the received global parameters, and calculate the local gradient. The signal uploading module is configured to upload the calculated local gradient signal to the base station according to its own amplification factor; The local storage module is configured to store local datasets, received global parameters, its own amplification factor, and training data volume configuration, among other data.

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