Distributed network collaborative optimization method adopting projection gradient method and power iteration method

Abstract

The invention discloses a distributed network collaborative optimization method adopting a projection gradient method and a power iteration method. The distributed network collaborative optimization method comprises the following steps: solving an optimization variable of an actual cost function on each node in a distributed network; the optimization variables are initialized, and each node in the distributed network calculates an intermediate variable vector at the next moment through stochastic gradient descent; after each node in the distributed network receives the intermediate variable vectors obtained by all neighbor nodes, all the intermediate variable vectors form a local intermediate variable matrix according to a column ascending order; iterating the local intermediate variable matrix by adopting a power iteration method so as to project the local intermediate variable matrix to a low-rank space to obtain a low-rank matrix; and solving the optimization variable of each node in the distributed network by using a single-task combination strategy. According to the method, the low-rank matrix projected to the low-rank space can be solved with relatively low calculation complexity on the premise of not needing low-rank space base prior information, and relatively good parameter estimation performance is achieved.

Description

Technical Field

[0001] The present invention relates to the field of adaptive signal processing, and in particular to a distributed network collaborative optimization method using a projected gradient method and a power iteration method. Background Art

[0002] Online parameter estimation technology can track the non-stationary changes of the external environment through self-learning of real-time streaming data without prior information, and finally approximate the potential model parameters. The parameters to be estimated can be signal, channel parameters, etc. Classic online parameter estimation technology has been successfully applied to fields such as communication, radar, sonar, control and biomedicine. In recent years, with the rapid development of multi-agent systems, the theory of online parameter estimation in distributed networks has become an important and challenging research point in the field of signal processing today. Compared with the classical online parameter estimation theory, network nodes not only need to estimate parameters based on their own observation data, but also need to complete the global parameter estimation task through collaboration with other connected neighbor nodes in a certain associated perception area. The relevant results of distributed parameter estimation theory have played an important role in applications such as target positioning and tracking, distributed speech enhancement, spectrum perception, and distributed image processing.

[0003] A key to solving practical problems with distributed parameter estimation technology is to reasonably model the relationship between the parameter vectors to be estimated at different nodes in the network. Existing distributed parameter estimation theories only consider association methods such as sparse, joint sparse, and subspace constraints, and these models can only cover limited application scenarios. In application scenarios such as distributed speech enhancement and distributed neural networks, multiple parameter vectors to be estimated have structural similarities to a certain extent and are suitable for being modeled as low-rank relationships, that is, the matrix composed of the parameter vectors to be estimated at all nodes in the network is a low-rank matrix. By applying low-rank relationships to the design of distributed parameter estimation algorithms, the estimation effect of distributed adaptive filters can be further improved. How to design distributed adaptive filtering methods with low computational complexity and high accuracy for low-rank relationships in the context of real-time streaming data processing is an urgent problem to be studied, and they are of great significance in solving practical problems. Summary of the invention

[0004] The purpose of the present invention is to provide a distributed network collaborative optimization method using the projected gradient method and the power iteration method, which can provide a low-complexity solution method for low-rank models in distributed networks, so as to overcome the defects of existing algorithms that require prior information of the low-rank spatial basis and have high computational complexity, and has better estimation performance for online parameter estimation problems for low-rank models.

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

[0006] A distributed network collaborative optimization method using a projected gradient method and a power iteration method, the method solving the optimization variables of an actual cost function on each node in a distributed network composed of multiple nodes; the method comprising:

[0007] S1, initialize the optimization variables, and each node k in the distributed network calculates the intermediate variable vector of the next moment through stochastic gradient descent;

[0008] S2, after each node in the distributed network receives the intermediate variable vectors obtained by all its neighboring nodes, all the intermediate variable vectors are sorted in ascending order to form a local intermediate variable matrix;

[0009] S3, iterating the local intermediate variable matrix using a power iteration method, thereby projecting the local intermediate variable matrix into a low-rank space to obtain a low-rank matrix;

[0010] S4, based on the low-rank matrix obtained by projection, uses a single-task combination strategy to solve the optimization variables of each node in the distributed network.

[0011] Furthermore, the actual cost function at each node is expressed as:

[0012]

[0013] 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.

[0014] Furthermore, when the collaborative optimization method is applied to acoustic system identification in a distributed network, the loss function in the actual cost function adopts a minimum mean square error cost function:

[0015]

[0016] 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;

[0017] 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 execute S1-S4 to get w k .

[0018] Furthermore, the optimization variables are initialized, and each node k in the distributed network calculates the intermediate variable vector at the next moment by stochastic gradient descent, including:

[0019]

[0020] In the above formula, ψ k,n+1 represents the intermediate variable vector at time n+1, w k,n represents the optimization variable w to be solved at time n k The value of Indicates w k Seeking guidance, Represents the calculation loss function G k (w k ;s k,n ) k and calculate its derivative at w k =w k,n The derivative value of a point; μ k >0 is a non-negative step size parameter.

[0021] Furthermore, the local intermediate variable matrix is ​​iterated by a power iteration method, so as to project the local intermediate variable matrix into a low-rank space to obtain a low-rank matrix, including:

[0022] When the number of iterations j=1, the iterations are performed according to S301-S305 until (2) in S305 is satisfied for the first time:

[0023] S301, node k generates an approximate low-rank matrix in the jth iteration at time n+1

[0024]

[0025] In the above formula, Ψ k,n+1 Represents the local intermediate variable matrix, and the parameter superscript (j) indicates that the current corresponding iteration is the jth. and They represent Ψ at time n+1 respectively. k,n+1 The j-1th largest singular value of and the corresponding left and right singular vectors, Indicates the number of iterations;

[0026] S302, node k initializes a random vector of length L×1 As the initial value for subsequent power iterations, each element of All obey the uniform distribution on the interval (0,1);

[0027] S303, iterative update where t max is the number of iterations:

[0028]

[0029] S304, calculate Ψ k,n+1 The j-th largest singular value of And the corresponding left and right singular vectors and

[0030]

[0031] In the above formula, ‖·‖2 represents the operation of calculating the two-norm of a vector;

[0032] S305, using iterative singular value decomposition method to estimate matrix rank, the steps are as follows:

[0033] (1) Using the j singular values ​​estimated for each node k at time n+1 Constructing variables

[0034]

[0035] in, represents the neighborhood of node k, express The momentum;

[0036] (2) When the conditions are met for the first time When , stop the iteration and let And use the j singular values ​​obtained in the first j iterations And the corresponding left and right singular vectors and Construct the projected low-rank matrix as follows:

[0037]

[0038] The threshold τ k ∈[0,1].

[0039] Furthermore, the method of solving the optimization variables of each node in the distributed network by using a single-task combination strategy includes:

[0040]

[0041] In the above formula, the superscript(l) is the node k-neighborhood The neighbor nodes in is the low-rank matrix of node l at time n+1, w k,n+1 represents the estimated value of the optimization variable of each node k in the distributed network at time n+1, Representation Matrix The parameter vector of the k nodes estimated in, Indicates that k nodes are in the matrix The column index value in Represents taking the matrix The Column; a k Represents the elements of the static combination matrix A, wherein the static combination matrix includes the average rule, the Laplace rule, the relative degree rule, and the maximum degree rule.

[0042] 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 network collaborative optimization method using the projected gradient method and the power iteration method is implemented.

[0043] A computer-readable storage medium stores a computer program; when the computer program is executed by a processor, the distributed network collaborative optimization method using a projected gradient method and a power iteration method is implemented.

[0044] Compared with the prior art, the present invention has the following beneficial effects:

[0045] In this scheme, a distributed network collaborative optimization method using the projected gradient method and the power iteration method is designed. In the problem of adaptive parameter estimation of distributed networks with low-rank characteristics, the proposed method can adaptively estimate the low-rank spatial basis with a lower amount of computation, thereby eliminating the need to use prior information of the low-rank spatial basis and ensuring the low computational complexity of the designed algorithm. This can solve the problem of the existing technology having to rely on prior information of the low-rank spatial basis and the high computational complexity of the existing algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 It is the overall flow chart of the present invention;

[0047] Figure 2 It is a simulation result diagram of the present invention and the comparison algorithm when white input is in a stable environment;

[0048] Figure 3 It is a simulation result diagram of the present invention and the comparison algorithm when there is colored input in a non-stationary environment;

[0049] Figure 4 It is a simulation result diagram of the present invention under different step lengths;

[0050] Figure 5 It is a simulation result diagram of the present invention under different preset rank values. DETAILED DESCRIPTION

[0051] The problems studied by the present invention are as follows:

[0052] 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:

[0053]

[0054] 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 ;s k,n ), Indicates the expected value of the parameter.

[0055] 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 T represents transpose, d k,n represents the observed 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 channel parameter.

[0056] The present invention considers the following parameter vector model: the true value of the optimization variable to be solved with a length of L×1 at each node in the entire distributed 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 truth values ​​of all parameter vectors in It also has the low-rank property, namely:

[0057]

[0058] In the above formula, represents all neighbors belonging to node k The true value of the parameter vector in The dimension is A real-valued matrix with the symbol express The potential, local parameter truth matrix The rank of Right now

[0059] Using the low-rank model, the local optimization problem on each node k can be It is expressed as:

[0060]

[0061] In the above formula, Express the solution so that When w reaches the minimum value k The value of , st represents the expression of the constraint parameter vector, Indicates w k The set of constraints in the local optimization problem is defined as follows:

[0062]

[0063] Among them, R L represents a real-valued vector of length L×1, symbol is the set of all nodes in the neighborhood of node k except node k, Represents the parameter vector w to be estimated k A local parameter matrix (except w k In addition, other columns The true value of the parameter ).

[0064] like Figure 1 As shown, the present invention provides a distributed network collaborative optimization method using a projected gradient method and a power iteration method, comprising the following steps:

[0065] S1, initialize w k,n , set w at time n = 0 k,0 is a zero vector. Each node k in the distributed network calculates the intermediate variable vector ψ at time n+1 by stochastic gradient descent. k,n+1 , as shown below:

[0066]

[0067] In the above formula, w k,n represents time n w k The value of Indicates w k Derivation, that is, calculating the gradient of the local cost function; since the real local cost function J cannot be obtained in practice k (w k ), so the local loss function G is used when executing the algorithm k (w k,n ;s k,n ) instead of J k (w k ), Represents the calculation of the local loss function G k (w k ;s k,n ) k and calculate its derivative at w k =w k,n The derivative value of a point; μ k >0 is a non-negative step size parameter.

[0068] S2, the neighboring nodes of each node k in the distributed network pass the intermediate variables obtained by the node to node k, and node k constructs a local intermediate variable matrix Ψ by sorting all the intermediate variable vectors from the neighboring nodes in ascending order. k,n+1 :

[0069]

[0070] S3, the power iteration method, is an eigenvalue iterative estimation algorithm. For diagonalizable matrices, it multiplies the matrix by a random vector in each iteration and normalizes it, eventually converging to the main eigenvector and main eigenvalue of the matrix. This scheme adopts a power iteration variant algorithm extended to general matrices.

[0071] Yes k,n+1 conduct Iterations (where Express After the approximation of k,n+1 Projecting to the low-rank space to obtain a low-rank matrix, the specific process of this scheme using the power iteration method is as follows:

[0072] When the number of iterations j=1, the iterations are performed according to S301-S305 until (2) in step S305 is satisfied for the first time:

[0073] S301, node k generates an approximate low-rank matrix in the jth iteration at time n+1

[0074]

[0075] In the above formula, the superscript parameter (j) indicates the current The corresponding iteration is the jth, and They represent Ψ at time n+1 respectively. k,n+1 The j-1th largest singular value and the corresponding left and right singular vectors are calculated as follows.

[0076] S302, node k initializes a random vector of length L×1 As the initial value for subsequent power iterations, each element of All obey the uniform distribution on the interval (0,1), that is,

[0077] S303, iterative update where t max is the number of iterations:

[0078]

[0079] S304, calculate Ψ k,n+1 The j-th largest singular value of And the corresponding left and right singular vectors and

[0080]

[0081] In the above formula, ‖·‖2 represents the operation of calculating the binary norm of a vector.

[0082] S305, using iterative singular value decomposition method to estimate matrix rank, the steps are as follows:

[0083] (1) Using the j singular values ​​estimated for each node k at time n+1 Constructing variables

[0084]

[0085] (2) When the conditions are met for the first time When , stop the iteration and let the estimated value of rank And use the j singular values ​​obtained in the first j iterations And the corresponding left and right singular vectors and Construct the projected low-rank matrix as follows:

[0086]

[0087] The threshold τ k ∈[0,1].

[0088] S4, calculated iteratively for each node in the network A single-task combination strategy is used to calculate the estimated value w of the parameter vector of each node in the distributed network at time n+1. k,n+1 :

[0089]

[0090] In the above formula, Representation Matrix The parameter vector of the k nodes estimated in, Indicates that k nodes are in the matrix The column index value in Represents taking the matrix The Column; a k Represents the elements of the static combination matrix A, that is, [A] (l,k) =a lk , 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.

[0091] In the system identification problem, w k It can correspond to the channel parameter vector in the time-frequency domain. Accordingly, solve w k Corresponding to the estimated system channel; the acoustic channel in the air has the characteristic of low rank, so in the application of acoustic system identification in distributed network, the distributed adaptive parameter estimation method for low rank model proposed in the present invention can be used for channel estimation. When applied, the node needs to obtain the observation signal d at each moment k,n And the input signal x k,n , and construct the cost function based on the minimum mean square error criterion, and then through the above method, that is, execute S1-S4, to obtain w k .

[0092] The effectiveness of the algorithm is verified by simulation experiments on stationary and non-stationary system identification.

[0093] Example 1

[0094] This embodiment is an application of the proposed algorithm 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, namely: g1 = {1, 2, ..., 8} and g2 = {9, 10, ..., 16}, and the parameter vector of each group is generated according to the following formula:

[0095]

[0096] Among them, the basis vector and Generated by a standard Gaussian distribution, β k It obeys the uniform distribution U(0,1). The mean square error cost function is used as the loss function.

[0097] The results are as follows Figure 2 As shown, DPI-T(A) and DPI-E(A) represent the proposed algorithms using the rank truth and rank estimation strategies, respectively, and CPI-T and CPI-E represent their corresponding centralized algorithms, and DPI-T(I) and DPI-E(I) represent the proposed algorithms without the single-task combination strategy. Figure 2 It can be seen that compared with the proposed algorithm using the true rank value, the proposed algorithm using the rank estimation strategy can still achieve better performance. The parameter estimation performance of the proposed algorithms DPI-T(A) and DPI-E(A) using the single-task combination strategy is also better than that of the algorithm without the single-task combination strategy, which further proves the effectiveness of the single-task combination strategy.

[0098] Example 2

[0099] This embodiment is an application of the proposed algorithm in the identification of a non-stationary system that varies over time. The distributed network consists of N = 16 nodes. The observed signal of the node satisfies the linear model: Input data x k,n is a colored input, whose covariance matrix 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. The noise z k,n Generated by a zero-mean Gaussian distribution. Weight coefficient vector The non-stationary system is divided into three stages: the matrix W corresponding to the system at n = 1-1000 and n = 2001-3000 ★Satisfying the low-rank model, the matrix W corresponding to the system at n = 1001-2000 ★ is full rank.

[0100] The results are as follows Figure 3 As shown. Figure 3 It can be seen that the proposed algorithms DPI-T(A) and DPI-E(A) can achieve good performance when the system satisfies the low-rank model. When the system is a full-rank model, the performance of DPI-T(A) and DPI-E(A) will decline. When the system re-satisfies the low-rank model at n=2001-3000, the performance of the proposed algorithm is significantly improved. This experiment verifies that the proposed algorithm has good tracking ability and is effective for low-rank models.

[0101] Example 3

[0102] Consider the step size μ k The effect of the proposed algorithm DPI-E(A) on the performance in the application of stable system identification. The setting of the stable system is the same as that in Example 2. The results are shown in Figure 4 As shown. Figure 4 It can be seen that the trade-off between steady-state performance and convergence speed is determined by the step size μ k Specifically, increasing the step size can speed up the convergence but reduce the steady-state performance; conversely, decreasing the step size can improve the accuracy of the approximation but slow down the convergence.

[0103] Example 4

[0104] Consider the impact of the preset rank value on the performance of the proposed algorithm DPI-T(A) in the application of stationary system identification. The setting of the stationary system is the same as that in Example 2, that is, the rank of the low-rank system is 2. The results are as follows Figure 5 As shown. Figure 5 It can be seen that when the algorithm presets the rank value R k The performance is the worst when it is 1, because information is lost in the process of constructing the low-rank matrix, resulting in the preset rank value R k = 1 for all R k In addition, the default R k =2 will achieve the best performance. k When the value of increases between 2 and 10, the algorithm performance will gradually decrease.

[0105] 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 network collaborative optimization method using a projected gradient method and a power iteration method, characterized in that: The method solves the optimization variables of the actual cost function on each node in a distributed network composed of multiple nodes; the method comprises: S1, initialize the optimization variables, and each node k in the distributed network calculates the intermediate variable vector of the next moment through stochastic gradient descent; S2, after each node in the distributed network receives the intermediate variable vectors obtained by all its neighboring nodes, all the intermediate variable vectors are sorted in ascending order to form a local intermediate variable matrix; S3, iterating the local intermediate variable matrix using a power iteration method, thereby projecting the local intermediate variable matrix into a low-rank space to obtain a low-rank matrix; S4, based on the low-rank matrix obtained by projection, uses a single-task combination strategy to solve the optimization variables of each node in the distributed network.

2. The distributed network collaborative optimization method using the projected gradient method and the power iteration method according to claim 1 is 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 network collaborative optimization method using the projected gradient method and the power iteration method according to claim 1 is characterized in that: When the collaborative optimization 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; 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 execute S1-S4 to get w k .

4. The distributed network collaborative optimization method using the projected gradient method and the power iteration method according to claim 1 is characterized in that: The optimization variables are initialized, and each node k in the distributed network calculates the intermediate variable vector at the next moment by stochastic gradient descent, including: In the above formula, ψ k,n+1 represents the intermediate variable vector at time n+1, w k,n represents the optimization variable w to be solved at time n k The value of Indicates w k Seeking guidance, Represents the calculation loss function G k (w k ;s k,n ) k and calculate its derivative at w k =w k,n The derivative value of a point; μ k >0 is a non-negative step size parameter.

5. The distributed network collaborative optimization method using the projected gradient method and the power iteration method according to claim 1 is characterized in that: The local intermediate variable matrix is ​​iterated by using a power iteration method, so as to project the local intermediate variable matrix into a low-rank space to obtain a low-rank matrix, including: When the number of iterations j=1, the iterations are performed according to S301-S305 until (2) in S305 is satisfied for the first time: S301, node k generates an approximate low-rank matrix in the jth iteration at time n+1 In the above formula, Ψ k,n+1 Represents the local intermediate variable matrix, and the parameter superscript (j) indicates that the current corresponding iteration is the jth. and They represent Ψ at time n+1 respectively. k,n+1 The j-1th largest singular value of and the corresponding left and right singular vectors, Indicates the number of iterations; S302, node k initializes a random vector of length L×1 As the initial value for subsequent power iterations, each element of All obey the uniform distribution on the interval (0,1); S303, iterative update where t max is the number of iterations: S304, calculate Ψ k,n+1 The j-th largest singular value of And the corresponding left and right singular vectors and In the above formula, ‖·‖2 represents the operation of calculating the two-norm of a vector; S305, using iterative singular value decomposition method to estimate matrix rank, the steps are as follows: (1) Using the j singular values ​​estimated for each node k at time n+1 Constructing variables in, represents the neighborhood of node k, express The momentum; (2) When the conditions are met for the first time When , stop the iteration and let And use the j singular values ​​obtained in the first j iterations And the corresponding left and right singular vectors and Construct the projected low-rank matrix as follows: The threshold τ k ∈[0,1].

6. The distributed network collaborative optimization method using the projected gradient method and the power iteration method according to claim 1 is characterized in that: The method of solving the optimization variables of each node in the distributed network by using a single-task combination strategy includes: In the above formula, the superscript is the node k-neighborhood The neighbor nodes in For Node The low-rank matrix at time n+1, w k,n+1 represents the estimated value of the optimization variable of each node k in the distributed network at time n+1, Representation Matrix The parameter vector of the k nodes estimated in, Indicates that k nodes are in the matrix The column index value in Represents taking the matrix The List; Represents the elements of the static combination matrix A, wherein the static combination matrix includes the average rule, the Laplace rule, the relative degree rule, and the maximum degree rule.

7. A terminal device, comprising a processor, a memory and a computer program stored in the memory; characterized in that: When the processor is executed by a computer, it implements the distributed network collaborative optimization method using the projected gradient method and the power iteration method according to any one of claims 1 to 6.

8. A computer-readable storage medium, wherein a computer program is stored in the medium; characterized in that: When the computer program is executed by a processor, a distributed network collaborative optimization method using a projected gradient method and a power iteration method according to any one of claims 1 to 6 is implemented.

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