Distributed parameter estimation method based on online ADMM

Abstract

The invention discloses a distributed parameter estimation method based on an online ADMM, and the method comprises the steps: constructing a local optimization problem on a node through employing an actual cost function on the node for a distributed network composed of a plurality of nodes; the local optimization problem is used for solving an optimization variable of the actual cost function when a target function value reaches a minimum value; constructing a new local optimization problem on the node based on the regularization architecture and the local optimization problem on the node; and constructing an augmented Lagrange function on nodes by utilizing the new local optimization problem based on a local parameter matrix, a low-rank matrix obtained by decomposing the local parameter matrix and a Lagrange multiplier matrix, and solving the optimization variable by iteratively updating a matrix in the augmented Lagrange function. According to the distributed and online ADMM algorithm based on the Frobenius norm regular term, the parameter estimation problem of the low-rank distributed network can be solved with low calculation complexity, and high parameter estimation precision is achieved.

Description

Technical Field

[0001] The present invention relates to the field of adaptive signal processing, and in particular to a distributed parameter estimation method based on online ADMM. Background Art

[0002] Online distributed parameter estimation technology involves parameter estimation through local computation and data exchange in a distributed network. These parameters can be signal characteristics, channel parameters, etc. In such a network, nodes not only need to estimate parameters based on their own observation data, but also need to collaborate with neighboring nodes to achieve global parameter estimation. The theory of distributed parameter estimation has important applications in target positioning and tracking, distributed speech enhancement, spectrum sensing, and distributed image processing.

[0003] A key challenge is how to effectively model the relationship between the parameters to be estimated at each node in the network based on practical problems. At present, the theory of distributed parameter estimation has explored models such as sparsity, joint sparsity, and subspace constraints, but the scope of application of these models is limited. In scenarios such as distributed speech enhancement and distributed neural networks, the parameter vectors to be estimated often have structural similarities and are suitable for modeling with low-rank matrices, that is, the matrix formed by the parameter vectors to be estimated of all nodes in the network is a low-rank matrix. By incorporating low-rank relationships into the design of distributed parameter estimation algorithms, the estimation accuracy of adaptive filters can be significantly improved. Therefore, how to design a distributed adaptive filtering method with low computational complexity and high accuracy in real-time streaming data processing to deal with low-rank relationships is an important issue that needs to be studied urgently and has a significant impact on practical applications. Summary of the invention

[0004] The present invention provides a distributed parameter estimation method based on online ADMM for low-rank models in distributed networks, so as to overcome the defects of existing algorithms in requiring prior information of low-rank spatial basis and having high computational complexity.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A distributed parameter estimation method based on online ADMM, comprising:

[0007] For a distributed network composed of multiple nodes, a local optimization problem on the node is constructed using an actual cost function on the node; the local optimization problem is used to solve the optimization variable of the actual cost function when the objective function value reaches a minimum value;

[0008] Based on the regularized architecture and the local optimization problem on the node, a new local optimization problem on the node is constructed;

[0009] Utilizing the new local optimization problem, an augmented Lagrangian function is constructed on the node based on the local parameter matrix, the low-rank matrix obtained by decomposing the local parameter matrix, and the Lagrangian multiplier matrix. The optimization variable is solved by iteratively updating the local parameter matrix, the low-rank matrix, and the Lagrangian multiplier matrix in the augmented Lagrangian function.

[0010] Furthermore, the actual cost function at each node is expressed as:

[0011]

[0012] Among them, w k represents the optimization variable to be solved, G k represents the loss function, s k,n Indicates the calculation of G on node k at time n k (w k ;s k,n ), Indicates the expected value.

[0013] Furthermore, when the distributed parameter estimation method is applied to acoustic system identification in a distributed network, the loss function in the actual cost function adopts the minimum mean square error cost function:

[0014]

[0015] Among them, the symbol represents transpose, d k,n represents the observed signal, which is the input data x k,n go through The noisy observation obtained after k,n ,d k,n Satisfy the linear model z k,n represents additive noise, is the true value of the optimization variable to be solved, which represents the channel parameter in the channel identification problem;

[0016] Node k in the distributed network obtains the observation signal d at each moment k,n And the input signal x k,n , and construct the minimum mean square error cost function, and then use the distributed parameter estimation method to get w k .

[0017] Furthermore, constructing a local optimization problem on a node using an actual cost function on the node includes:

[0018] The objective function of the local optimization problem at each node k can be expressed as:

[0019]

[0020] In the above formula, Indicates that the solution makes the function reach the minimum value The value of represents all neighbors belonging to node k The optimized variables with solution in node l, J l (w l ) represents the actual cost function on node l, ||.|| * Represents the nuclear norm of the matrix, the matrix γ k represents the regularization parameter.

[0021] Furthermore, constructing a new local optimization problem on the node based on the regularized architecture and the local optimization problem on the node includes:

[0022]

[0023] wxya k =U k V k

[0024] In the above formula, W k express The estimated value of is called the local parameter matrix, is the neighborhood of node k The true value of the optimization variable at all nodes in The local parameter truth matrix The length is L×1, The rank of represents the neighborhood of k The trend, U k With V k W k The low-rank matrix of matrix factorization, represents the square of the Frobenius norm of the matrix, It's U k and V k The sum of squares of the Frobenius norm of , which replaces the original nuclear norm ||W k || * , and used as a regular term in the cost function to characterize the matrix W k The low-rank property of the matrix U k Dimensions: L×S k , the matrix V k The dimension is S k The value is

[0025] Furthermore, using the new local optimization problem, an augmented Lagrangian function is constructed on the node based on the local parameter matrix, the low-rank matrix obtained by decomposing the local parameter matrix, and the Lagrangian multiplier matrix, which is expressed as:

[0026]

[0027] Among them, <·,·> represents the matrix dot product, Λ k represents the Lagrange multiplier matrix, with dimension ρ represents the penalty factor;

[0028] Execute W at iteration time n=0,1,2,3... k ,U k ,V k ,Λ k Iterations, where W k ,Λ k At the initial moment, they are all zero matrices; U k ,V k At the initial moment, they are all random matrices, and each element obeys a uniform distribution in the interval (0,1).

[0029] Furthermore, the local parameter matrix W is optimized by using the stochastic gradient descent method based on the SVRG error variance reduction mechanism. k Update by column; update the local parameter matrix W of all nodes k in the entire distributed network k Afterwards, a single-task combination strategy is used to calculate the estimated value of the parameter vector of each node at time n+1.

[0030] Furthermore, the conjugate gradient method is used to transform the low-rank matrix U k Update iteratively by row.

[0031] Furthermore, the conjugate gradient method is used to transform the low-rank matrix V k Update iteratively by column.

[0032] Furthermore, using the penalty factor and the local parameter matrix W k , low-rank matrix U k , low-rank matrix V k For the Lagrange multiplier matrix Λ k to update.

[0033] A terminal device comprises a processor, a memory and a computer program stored in the memory; when the processor is executed by a computer, the distributed parameter estimation method based on online ADMM is implemented.

[0034] A computer-readable storage medium stores a computer program; when the computer program is executed by a processor, the distributed parameter estimation method based on online ADMM is implemented.

[0035] Compared with the prior art, the present invention has the following beneficial effects:

[0036] The present invention provides a distributed parameter estimation method based on online ADMM, which decomposes the original matrix into the product of two low-rank matrices, uses the square sum of the Frobenius norms of the two low-rank matrices as a regularization term, and uses the online ADMM algorithm to solve the low-rank spatial basis and parameters. In order to further reduce the computational complexity, this scheme uses the conjugate gradient method to solve the sub-optimization problem obtained when the ADMM algorithm is executed; in order to improve the accuracy of parameter estimation, this scheme uses the SVRG error variance reduction mechanism. This scheme has better solved the problems of having to rely on the prior information of the low-rank spatial basis and the high computational complexity in the prior art, and shows excellent estimation performance in the online parameter estimation problem of low-rank distributed networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 It is the overall flow chart of the present invention;

[0038] Figure 2 It is a simulation result diagram of the present invention and the comparison algorithm when white input is in a stable environment;

[0039] Figure 3 It is a simulation result diagram of the present invention and the comparison algorithm when there is colored input in a stable environment;

[0040] Figure 4 is a simulation result diagram of the present invention and the comparison algorithm when white input is in a non-stationary environment,

[0041] Figure 5 It is a simulation result diagram of the present invention and the comparison algorithm when there is colored input in a non-stationary environment. DETAILED DESCRIPTION

[0042] The problems studied by the present invention are as follows:

[0043] Consider a distributed network consisting of N nodes, where the nodes are represented by serial numbers and the real-valued cost function J on each node k is k (w k ) has the properties of first-order differentiability and strong convexity, and can be expressed as:

[0044]

[0045] In the above formula, w k represents the optimization variable to be solved, G k represents a loss function with first-order differentiable and strongly convex properties, s k,n Indicates the calculation of G on node k at time n k (w k ;sk,n ), Indicates the expected value of the parameter. In different problems and scenarios, G k (w k ;s k,n ) has different expressions. For example, in the most commonly used and typical minimum mean square error cost function, The symbol represents transpose, d k,n represents the expected signal, which is the input data x k,n Through the linear system The noisy observation obtained after that is x k,n ,d k,n Satisfy the linear model z k,n represents additive noise, and the superscript * is used to emphasize that the relevant quantity is the true value of the parameter to be estimated. In the minimum mean square error cost function, the random data s k,n Corresponding to {x k,n ,d k,n}. will also is called the true value of the optimization variable to be solved, which is the cost function J k (w k ). In different problems, Has a specific meaning, for example, in the channel identification problem, is the true value of the channel parameter.

[0046] The present invention considers the following parameter vector model:

[0047] The true value of the optimized variable with a length of L×1 at each node in the entire network Arrange the columns into an L×N dimensional matrix W ＊ , and W ＊ has a low rank property. Therefore, the neighborhood of each node k The local parameter truth matrix composed of the optimization variable truth values ​​of all nodes It also has the low-rank property, namely:

[0048]

[0049] In the above formula, represents all neighbors belonging to node k The true value of the optimization variable to be solved in, The dimension is A real-valued matrix with the symbol express The trend of local reference The rank of Right now Using the low-rank model, the objective function of the local optimization problem on each node k can be expressed as:

[0050]

[0051] In the above formula, Indicates that the solution makes the function reach the minimum value The value of represents all neighbors belonging to node k The parameter vector within, ||.|| * Represents the nuclear norm of the matrix, which is used to constrain the low-rank matrix. γ k represents the regularization parameter.

[0052] See also Figure 1 , a distributed parameter estimation method based on online ADMM, comprising the following steps:

[0053] S1, based on the regularized architecture and the original local optimization problem on node k, construct a new local optimization problem on node k:

[0054]

[0055] wxya k =U k V k

[0056] In the above formula, W k express The estimated value of is called the local parameter matrix, U k With V k W k The low-rank matrix of matrix factorization, represents the square of the Frobenius norm of the matrix, It's U k and V k The sum of squares of the Frobenius norm of , which replaces the original nuclear norm ||W k || * , and as a regular term of the cost function, to characterize the matrix W k The low-rank property of the matrix U k Dimensions: L×S k , the matrix V k The dimension is In this scheme, S k The value is That is U k All column vectors or V k All row vectors of can be considered as W k A set of bases.

[0057] Next, the online ADMM method is used to estimate the parameter vector w k The process is S2-S6, and the specific steps are as follows:

[0058] S2, using the new local optimization problem, based on the local parameter matrix W k , local parameter matrix W k The low-rank matrix U obtained by decomposition k With V k And the Lagrangian multiplier matrix constructs the augmented Lagrangian function L at node k ρ (W k ,U k ,V k ,Λ k ), by changing the W in the function k ,U k ,V k ,Λ k Iterative update to solve the optimization variable w k .

[0059]

[0060] Among them, <·,·> represents the matrix dot product, that is, the elements of all corresponding positions of the two matrices are directly multiplied and then summed to obtain a scalar, Λ k represents the Lagrange multiplier matrix, with dimension ρ represents the penalty factor. Next, at the iteration time n=0, 1, 2, 3, ..., perform W according to steps S3-S6. k ,U k ,V k ,Λ k Iterations, where W k ,Λ k At the initial moment, they are all zero matrices, that is, U k ,V k At the initial moment, they are all random matrices, and each element obeys a uniform distribution in the interval (0,1).

[0061] S3, iterative update W k :

[0062] S301, using the stochastic gradient descent method based on the SVRG error variance reduction mechanism to calculate W k Update by column:

[0063] In L ρ (W k ,U k ,V k ,Λ k ) about W kThe sub-optimization problem can be expressed as:

[0064]

[0065] In the above formula, W k,n+1 Indicates the time n+1 W k The updated value of U k,n ,V k,n ,Λ k.n Represents U at time n respectively k ,V k ,Λ k According to the estimated value of W k The optimization variables of each node in the sub-optimization problem are unrelated, that is, W k The column vectors are uncorrelated, so k Update by column. Therefore, for W k No. Column vector The optimization problem can be expressed as (where W k The index of the parameter vector on the node l in the middle):

[0066]

[0067] In the above formula, Indicates the time n+1 W k No. Column vector w l , and symbol Represents the quantity that belongs to node l and needs to be transferred to node k later; column vector and Represents the matrix Q at time n k,n and Λ k,n No. Column, matrix Q k,n Defined as Q k,n =U k,n V k,n .symbol represents the optimal vector belonging to node k and estimated node l To reduce the estimation error, we use the stochastic gradient descent method to update The error variance reduction mechanism was introduced when , and the update was obtained The algorithm comprises the following steps:

[0068] (1) When n≤P, the stochastic gradient descent method is executed:

[0069]

[0070] Among them, the parameter P represents the window length, which is a positive integer, and the non-negative parameter μ l,k Indicates the step length. Indicates w l Derivation is to calculate the gradient of the function. Since the local cost function J cannot be obtained in practice l (w l ), so the local loss function G is used when executing the algorithm. l (w l,n ;s l,n ) to approximate J l (w l ) and perform gradient calculation, Represents the calculation of the local loss function G l (w ,n ;s ,n ) l and calculate its derivative at w l =w l,n The derivative value at the point.

[0071] (2) When n>P, calculate The mod function returns the value of the division operation n divided by The remainder of Indicates the cycle length; at this time, if t==1 or n==P+1 (the symbol == is a logical equal operation, judging whether the two ends of the symbol are equal), execute:

[0072] e=e+1,

[0073]

[0074] Among them, e represents the number of cycles, and its initial value is 0. Indicates the e-th period in the variance reduction mechanism The estimated value of express The average gradient calculated using a window of length P at the e-th cycle.

[0075] (3) Calculate the n+1 time Updated value of:

[0076]

[0077] Among them, the symbol It means "to all".

[0078] S302, using S301 to update the local parameter matrix W of all nodes k in the entire network k After that, the single-task combination strategy commonly used in distributed diffusion algorithms is used to calculate the estimated value of the parameter vector of each node at time n+1:

[0079]

[0080] In the above formula, a kl represents the kth row and lth column element of the static combination matrix A, that is, [A] (k,l) =a kl , its value depends on the selected static combination matrix. Commonly used static combination matrices include average rule, Laplace rule, relative degree rule, maximum degree rule, etc.

[0081] S303, updated and W k,n+1 :

[0082]

[0083] in, Indicates that w calculated in S302 is l,n+1 Passed to node k, symbol Indicates that all The parameter vector w of the node l,n+1 The matrix formed.

[0084] S4, using the conjugate gradient method to calculate U k Iterate and update row by row:

[0085] In L ρ (W k ,U k ,V k ,Λ k ), about the iteration U k The sub-optimization problem can be expressed as:

[0086]

[0087] In the above formula, U k,n+1 Indicates the time U at n+1 k The updated value of U k The optimization variables of each node in the sub-optimization problem are unrelated, that is, U k The row vectors are independent of each other. k Updates are performed row by row. k The mth row vector of The optimization problem is expressed as:

[0088]

[0089] In the above formula, Indicates the time U at n+1 k The mth row vector of The updated value of and Respectively represent Λ k,n and W k,n+1 The mth row vector of Represents the square of the 2-norm of the vector. The steps to solve the optimization problem using the conjugate gradient method are as follows:

[0090] (1) Yes Initialize and set the initial value of the residual vector to Set the initial value of the search direction to g0 = -d0;

[0091] (2) Iterate j=0, 1, ..., IterNum-1; IterNum is the total number of iterations;

[0092] (2-1) Calculate step size α j : in is the identity matrix;

[0093] (2-2) Update solution

[0094] (2-3) Update the residual vector d j+1 :

[0095] (2-4) Check convergence: If it satisfies make And stop the iteration; where ε represents the set threshold;

[0096] (2-5) Update the scalar value parameter β j :

[0097] (2-6) Update search direction g j+1 :g j+1 =-d j+1 +β j g j ;

[0098] (3) The result obtained after the last iteration As n+1 moment The estimated value of

[0099] (4) Update U k,n+1 : in[·] (m,:) Represents the mth row of the matrix.

[0100] S5, using the conjugate gradient method to calculate V k Update iteratively by column:

[0101] In L ρ (W k ,Uk ,V k ,Λ k ), about the iteration V k The sub-optimization problem can be expressed as:

[0102]

[0103] In the above formula, V k,n+1 Indicates V at time n+1 k Update value of. According to the iteration V k The irrelevant properties of each node in the sub-optimization problem, that is, V k The column vectors are uncorrelated. k Update by column. About V k No. Column vector The optimization problem is expressed as:

[0104]

[0105] In the above formula, Indicates V at time n+1 k No. Column vector The updated value of and Denote the matrix Λ k,n and W k,n+1 No. The steps to solve the optimization problem using the conjugate gradient method are as follows:

[0106] (1) Yes Perform initialization and set the initial value of the residual to:

[0107]

[0108] Set the initial value of the search direction to g0 = -d0;

[0109] (2) Iterate j=0,1,…,IterNum-1:

[0110] (2-1) Calculate step size α j :

[0111] (2-2) Update solution

[0112] (2-3) Update the residual vector d j+1 :

[0113] (2-4) Check convergence: If it satisfies make And stop iterating;

[0114] (2-5) Update the scalar value parameter β j :

[0115] (2-6) Update search direction g j+1 :g j+1 =-d j+1 +β j g j ;

[0116] (3) The result obtained after the last iteration As n+1 moment The estimated value of

[0117]

[0118] (4) Update V k,n+1 : in Represents the matrix List.

[0119] S6, iterative update Λ k :

[0120] Λ k,n+1 =Λ k,n +ρ(W k,n+1 -U k,n+1 V k,n+1 )

[0121] In the above formula, Λ k,n+1 represents the n+1 time Λ k The updated value of .

[0122] Example 1

[0123] This embodiment is an application in stationary system identification. The distributed network consists of N = 16 nodes. The observed signal of the node satisfies the linear model: Input data x k,n and noise z k,n are generated by a zero-mean Gaussian distribution. The entire network nodes are divided into two groups: g1 = {1, 3, 5, ..., 15} and g2 = {2, 4, 6, ..., 16}. The parameter vector of each group is generated as follows:

[0124]

[0125] Among them, c1, c2 are generated by standard Gaussian distribution, α kIt is generated by a uniform distribution U(0,1) on the interval (0,1). The mean square error cost function is used as the loss function. Now consider using two different inputs x k,n Under the condition of , the estimated performance of the proposed algorithm and the comparison algorithm is given. k,n There are two settings for: white input and colored input. The covariance matrix of colored input is And satisfy where λ max (.) represents the maximum eigenvalue of the matrix, λ min (.) represents the smallest eigenvalue of the matrix, represents the variance of node k, is a covariance matrix.

[0126] The results of white input and colored input are as follows Figure 2 and Figure 3 As shown in the figure. D-ADMM-LR(SVRG-4) and D-ADMM-LR(SVRG-20) respectively indicate that the proposed algorithm sets the parameters P=4 and P=20 in the variance reduction step. D-ADMM-LR indicates the proposed algorithm using only stochastic gradient descent (without using the variance reduction mechanism), and C-ADMM-LR indicates the corresponding centralized algorithm. C-Subspace indicates the centralized algorithm based on subspace constraints proposed in the existing literature, which additionally utilizes the α k The prior information of D-Subspace represents the distributed version of the algorithm C-Subspace. Figure 2 and Figure 3 It can be found that the D-ADMM-LR (SVRG-20) algorithm proposed in this invention does not require prior information α k By comparing D-ADMM-LR(SVRG-20) with D-ADMM-LR, it can be proved that the introduction of variance reduction mechanism can further improve the estimation performance. Figure 2 White input and Figure 3 The results in the case of colored input show that the convergence speed of all algorithms is slower in the case of colored input than in the case of white input.

[0127] Example 2

[0128] This embodiment is an application in non-stationary system identification. The nodes of the entire network are divided into four groups: C1 = {1, 5, 9, 13}, C2 = {2, 6, 10, 14}, C3 = {3, 7, 11, 15}, C4 = {4, 8, 12, 16}. The non-stationary system is divided into three stages, namely: n = 1-2000, n = 2001-4000 and n = 4001-6000. In these three stages, the rank of the system is 4, but the target parameter vector is different. Also consider x k,n The estimated performance of the proposed algorithm and the comparison algorithm is given for both white input and colored input. k,n The two settings are the same as those in Example 1.

[0129] The results of white input and colored input are as follows Figure 4 and Figure 5 As shown. Figure 4 and Figure 5 It can be seen that Figure 2 and Figure 3 The conclusions drawn from the analysis of the proposed algorithm in a stationary environment also apply to Figure 4 and Figure 5 In addition, by observing that in the In the non-stationary system, D-ADMM-LR, D-ADMM-LR(SVRG-4) and D-ADMM-LR(SVRG-20) can achieve good estimation performance, which can further verify the effectiveness of the proposed algorithm.

[0130] The above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit them. Although the present application has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. These modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present application, and should all be included in the protection scope of the present application.

Claims

1. A distributed parameter estimation method based on online ADMM, characterized in that: include: For a distributed network consisting of multiple nodes, the actual cost function on the node is used to construct the local optimization problem on the node; The local optimization problem is used to solve the optimization variable of the actual cost function when the objective function value reaches the minimum value; Based on the regularized architecture and the local optimization problem on the node, a new local optimization problem on the node is constructed; Utilizing the new local optimization problem, an augmented Lagrangian function is constructed on the node based on the local parameter matrix, the low-rank matrix obtained by decomposing the local parameter matrix, and the Lagrangian multiplier matrix. The optimization variable is solved by iteratively updating the local parameter matrix, the low-rank matrix, and the Lagrangian multiplier matrix in the augmented Lagrangian function.

2. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: The actual cost function at each node is expressed as: Among them, w k represents the optimization variable to be solved, G k represents the loss function, s k,n Indicates the calculation of G on node k at time n k (w k ;s k,n ), Indicates the expected value.

3. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: When the distributed parameter estimation method is applied to acoustic system identification in a distributed network, the loss function in the actual cost function adopts the minimum mean square error cost function: Among them, the symbol represents transpose, d k,n represents the observation signal, which is the input data x k,n go through The noisy observation obtained after k,n ,d k,n Satisfy the linear model z k,n represents additive noise, is the true value of the optimization variable to be solved, which represents the channel parameter in the channel identification problem; Node k in the distributed network obtains the observation signal d at each moment k,n And the input signal x k,n , and construct the minimum mean square error cost function, and then use the distributed parameter estimation method to get w k .

4. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: The method of constructing a local optimization problem on a node by using an actual cost function on the node includes: The objective function of the local optimization problem at each node k can be expressed as: In the above formula, Indicates that the solution makes the function reach the minimum value The value of represents all neighbors belonging to node k The optimized variables with solution for the node l in represents the actual cost function on node l, ||.|| * Represents the nuclear norm of the matrix, the matrix γ k represents the regularization parameter.

5. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: The method of constructing a new local optimization problem on the node based on the regularized architecture and the local optimization problem on the node includes: s.t.W k =U k V k In the above formula, W k W k The estimated value of is called the local parameter matrix, is the neighborhood of node k The true value of the optimization variable at all nodes in The local parameter truth matrix The length is L×1, The rank of represents the neighborhood of k The trend, U k With V k W k The low-rank matrix of matrix factorization, represents the square of the Frobenius norm of the matrix, It's U k and V k The sum of squares of the Frobenius norm of , which replaces the original nuclear norm ||W k || * , and used as a regular term in the cost function to characterize the matrix W k The low-rank property of the matrix U k Dimensions: L×S k , the matrix V k The dimension is S k The value is r k * .

6. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: Using the new local optimization problem, an augmented Lagrangian function is constructed on the node based on the local parameter matrix, the low-rank matrix obtained by decomposing the local parameter matrix, and the Lagrangian multiplier matrix, which is expressed as: Among them, <·,·> represents the matrix dot product, Λ k represents the Lagrange multiplier matrix, with dimension ρ represents the penalty factor; Execute W at iteration time n=0,1,2,3... k ,U k ,V k ,Λ k Iterations, where W k ,Λ k At the initial moment, they are all zero matrices; U k ,V k At the initial moment, they are all random matrices, and each element obeys a uniform distribution in the interval (0,1).

7. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: The local parameter matrix W is optimized by using the stochastic gradient descent method based on the SVRG error variance reduction mechanism. k Update by column; update the local parameter matrix W of all nodes k in the entire distributed network k Afterwards, a single-task combination strategy is used to calculate the estimated value of the parameter vector of each node at time n+1.

8. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: The conjugate gradient method is used to transform the low-rank matrix U k 、V k Update iteratively by row.

9. The distributed parameter estimation method based on online ADMM according to claim 1, characterized in that: Using the penalty factor and the local parameter matrix W k , low-rank matrix U k , low-rank matrix V k For the Lagrange multiplier matrix Λ k to update.

10. A computer-readable storage medium, wherein a computer program is stored in the medium; characterized in that: When the computer program is executed by the processor, the distributed parameter estimation method based on online ADMM is implemented.

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