Quantized distributed adaptive online optimization method and system in bandwidth limited scenario

Abstract

The application relates to a quantized distributed adaptive online optimization method and system under a bandwidth limited scene, and belongs to the technical field of information communication. The method and the system combine a random quantizer and an adaptive gradient algorithm into a distributed online algorithm, and can realize adaptive convergence of the algorithm under the bandwidth limited scene. The technical scheme provided by the application not only overcomes the problem of bandwidth limitation in the field of distributed online optimization algorithms in the prior art, but also overcomes the problem that a learning rate significantly influences convergence of the algorithm.

Description

Technical Field

[0001] This invention belongs to the field of information and communication technology, and relates to a quantitative distributed adaptive online optimization method and system for bandwidth-constrained scenarios. Background Technology

[0002] In recent years, distributed optimization algorithms have generated considerable discussion in academia and industry, demonstrating broad application potential in various scenarios, including signal processing, resource allocation, privacy protection, source localization, and economic resource allocation. Unlike traditional centralized methods, distributed optimization algorithms involve multiple nodes that collaborate, communicate, and exchange information within the network to jointly optimize the problem. The advantages of distributed optimization algorithms lie in their ability to avoid centralizing all data to a single node, thus protecting user privacy and data security. Secondly, distributed optimization algorithms can utilize network resources more efficiently because each node can independently process data and share information, avoiding single points of failure. Furthermore, distributed optimization algorithms are better adapted to large-scale networks and dynamic environments, self-adjusting and optimizing as the network changes.

[0003] However, in the real world, many problems require real-time solutions. Therefore, the optimal solution may change over time. To address this, distributed online learning algorithms have emerged. These algorithms can learn and update models online, enabling them to quickly adapt to changes in the environment and data. The core idea of ​​online learning algorithms is to update model parameters at each time step to adapt to different environments. In a distributed scenario, each node can independently process data and update model parameters. In this way, the entire network can continuously learn and adapt to changes in the environment while maintaining high efficiency.

[0004] However, when deploying distributed online algorithms, the problem of limited communication bandwidth may be encountered, which restricts information exchange between nodes. Therefore, in order to deploy algorithms in practical application scenarios, it is necessary to consider reducing the communication volume of distributed optimization algorithms so that the algorithms can run in bandwidth-constrained scenarios. A common method to reduce node communication volume is quantization. Quantization is a method of discretizing a set of data into representative values ​​to reduce the information volume while retaining the basic characteristics of the original data. This method enables nodes to communicate in bandwidth-constrained scenarios without transmitting large amounts of original data. Currently, there are distributed algorithms for solving optimization problems in bandwidth-constrained scenarios, but these algorithms rarely consider the dynamic characteristics of the optimal solution in real-world environments. In addition, the initial step size and its update strategy can significantly affect the optimization performance of the algorithm, leading to different convergence results. The distributed online optimization problem in bandwidth-constrained scenarios still needs to be solved. Therefore, this invention proposes a quantized distributed adaptive online optimization algorithm to solve this type of problem. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a distributed adaptive online optimization method and system for bandwidth-constrained scenarios. This method and system combine a random quantizer and an adaptive gradient algorithm into a distributed online algorithm, enabling adaptive convergence of the algorithm under bandwidth-constrained conditions. The technical solution provided by this invention not only overcomes the bandwidth-constrained problem in the field of distributed online algorithms in the prior art, but also solves the problem that the learning rate has a significant impact on the convergence of the algorithm.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A quantized distributed adaptive online optimization algorithm for bandwidth-constrained scenarios is proposed. In this method, the communication between nodes can be regarded as an undirected network. To describe, by node set and edge set Composition. Definition of this invention. To represent an undirected graph The matrix. [W] ij The value ∈ [0,1] represents the influence of node i on node j, and the larger the weight, the greater the influence. When the weight is 0, the two nodes have no influence on each other. This invention uses... Let represent the neighborhood of node i. Furthermore, the weight matrix of this network must be a doubly random matrix, i.e., satisfying .

[0008] In distributed optimization algorithms, nodes At time t∈{1,2,3,…,T}, it is necessary to predict the new model based on local data and past models. Assuming a network with n nodes, the main problem studied in this invention is described by the following mathematical formula:

[0009]

[0010] Where f i,t (x t Let ) represent the local loss function at node i at time t. It is a non-empty, bounded convex set.

[0011] Regret is a key performance metric for distributed online optimization algorithms, assessing their adaptability to the constantly changing nature of the optimization problem. Static regret evaluates the algorithm assuming the optimal model remains constant throughout the optimization process; however, it may not fully capture the algorithm's performance under dynamic and ever-changing real-world conditions. Dynamic regret refers to the change of the optimal model over time in online optimization. In this case, dynamic regret becomes a more suitable metric for evaluating algorithms in broader applications, as shown below:

[0012]

[0013] in Right now This is the optimal solution at time t.

[0014] Random quantizers are used in networks for communication between nodes to reduce traffic. The random quantizer Q... t (x i,t ) is used for quantizing vectors For time t∈{1,2,3,…,T}, the random quantizer needs to satisfy the following condition:

[0015] (1)

[0016] (2) in This represents the quantization resolution, which depends on the quantization level k. t And dimension d.

[0017] Random quantizers include k-order quantizers, probabilistic time-varying quantizers, etc. The quantization part of this invention can use any random quantizer; in subsequent simulation experiments, this invention uses a probabilistic quantizer as an example. For a given vector e = [e1, e2, ..., e...] n ], where the dimension is n, and this invention is represented as The quantization operation of a number can be represented as:

[0018]

[0019] in, and The specific calculation formulas are as follows: and k t This represents the quantization level at time t. At time t, the algorithm selects one of two calculation formulas based on the generated random number.

[0020] Furthermore, the optimization problem and cost function need to satisfy the following assumptions:

[0021] (1) It is a fixed, undirected, connected graph. Its weight matrix W is a doubly random matrix.

[0022] (2) For any node At time t∈{1,2,3,…,T}, there exists a constant 0. <G ∞ <+∞ satisfies ||g i,t (x;ξ i,t )|| ∞ ≤G ∞ Additionally, fi,t (·) is a convex function and satisfies first-order L... x -Lipschitz continuity, and second-order G x -Lipschitz continuity.

[0023] (3) Let be a non-empty, bounded convex set, and for... There exists a constant 0 < γ ∞ For values ​​less than +∞, there are ||xy|| ∞ ≤γ ∞ .

[0024] Furthermore, this method specifically includes the following steps:

[0025] S1. Initialize hyperparameters: β 1,t =0.9 t β2 = 0.999, β3 = 0.9 G0 = 0.001, n = 50;

[0026] S2, For all nodes Initialization parameter: m i,0 =v i,0 =x i,1 =0, Among them 1 p Let p represent a p-dimensional unit vector, assuming p = 10;

[0027] S3, All nodes The number of iterations t is set to 1, and a maximum number of iterations T = 1000 is set as the criterion for stopping the algorithm's iterative updates;

[0028] S4. Determine if t≤T holds true. If true, then apply the following to all nodes. Perform updates S5-S10. If the condition is not met (i.e., when t=T), the algorithm returns the local average variable from T iterations.

[0029] S5. Decision Variable Quantification: At time t∈{1,2,3,…,T}, for nodes… decision variable x i,t Quantification yields Q t (x i,t ), where Q t For random quantizers;

[0030] S6. Gradient Calculation: At time t∈{1,2,3,…,T}, node… Using quantified decision variables Q t (x i,t The gradient is calculated using the following expression:

[0031] g i,t =g i,t (Q t (x i,t ),ξ i,t )

[0032] Where ξ i,t and g i,t These are partial data samples and gradients at node i at time t, respectively.

[0033] S7. First and Second Order Momentum Estimation: Next, the first order momentum is estimated using the following formula:

[0034] m i,t =β 1,t m i,t-1 +(1-β 1,t )g i,t

[0035] Where β 1,t The first-order momentum m used to balance at time t-1 i,t-1 and gradient g at time t i,t , and β 1,t It decreases as t increases. The second-order momentum estimate is obtained using the following formula:

[0036] v i,t =β2v i,t-1 +(1-β2)g i,t ⊙g i,t

[0037] Similar to the first-order momentum estimation, the second-order momentum estimation uses the second-order momentum v at equilibrium time t-1 of β2. i,t-1 The dot product of the gradient elements at time t and g i,t ⊙g i,t Furthermore, to ensure that the Q-DADAM algorithm converges to the optimal solution, it needs to use the following formula to apply v. i,t renew:

[0038]

[0039] Where β3∈[0,1);

[0040] S8. Consensus Step: In the consensus step, updates are performed using the following formula:

[0041]

[0042] Where z i,t By node i and its neighborhood The nodes are obtained by weighting the model, where the weight between node i and node j is represented by [W]. ij express;

[0043] S9. Decision Variable Generation: Next, the vector... Convert to a diagonal matrix In constrained scenarios, projection operations are used to map the results to... The expression for calculation is as follows:

[0044]

[0045] Where α t This represents the step size, while m t and It is momentum with historical gradient information;

[0046] S10. Set t to t+1. Return the original variable x at time t. i,t ,m i,t-1 ,v i,t-1 ,

[0047] This invention also provides a quantitative distributed adaptive online optimization system for bandwidth-constrained scenarios.

[0048] The beneficial effects of this invention are as follows:

[0049] This invention primarily addresses distributed online optimization problems in bandwidth-constrained scenarios, where the network of nodes forms an undirected graph. In most existing distributed online optimization algorithms, information exchange between nodes is transmitted through the undirected network. When the amount of data transmitted is too large, it leads to excessive network load, especially when network bandwidth is limited. To address these problems and challenges, this invention proposes an effective quantized distributed adaptive online optimization algorithm—the Q-DADAM algorithm. This algorithm uses a randomized quantizer to process the information transmitted in the network, thereby reducing communication traffic in bandwidth-constrained scenarios. Furthermore, to mitigate the significant impact of the step size and its update strategy on the algorithm and enhance its convergence, this invention employs first-order and second-order adaptive momentum estimation methods.

[0050] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0051] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0052] Figure 1 This is a flowchart illustrating the method described in this invention;

[0053] Figure 2 To compare the algorithm of this invention with other algorithms in the problem of dynamic sparse signal repair;

[0054] Figure 3 To demonstrate the effectiveness of the algorithm of this invention at different quantization levels in the problem of dynamic sparse signal repair;

[0055] Figure 4 For online logistic regression problems, the algorithm of this invention is compared with other algorithms;

[0056] Figure 5 The algorithm of this invention is demonstrated to perform well in online logistic regression problems with different numbers of nodes.

[0057] Figure 6 To compare the algorithm of this invention with other algorithms in multi-class classification problems;

[0058] Figure 7 In the target tracking problem, the algorithm of this invention tracks the trajectory graph;

[0059] Figure 8 This paper compares the algorithm of this invention with other algorithms in the target tracking problem. Detailed Implementation

[0060] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings.

[0061] Figure 1 The flowchart of the method described in this invention is shown in the figure. The quantitative distributed adaptive online optimization method for bandwidth-constrained scenarios provided by this invention specifically includes the following steps:

[0062] (1) Initialize hyperparameters: β 1,t =0.9 t β2 = 0.999, β3 = 0.9 G0 = 0.001, n = 50;

[0063] (2) For all nodes Initialization parameter: m i,0 =v i,0 =x i,1 =0, Among them 1 p Let p represent a p-dimensional unit vector, assuming p = 10;

[0064] (3) All nodes The number of iterations t is set to 1, and a maximum number of iterations T = 1000 is set as the criterion for stopping the algorithm's iterative updates;

[0065] (4) Determine whether t≤T holds true. If it does, then check all nodes. Perform updates (5)-(10). If this is not true, i.e., when t=T, the algorithm returns the local average variable after T iterations.

[0066] (5) Decision variable quantification: At time t∈{1,2,3,...,T}, for node... decision variable x i,t Quantification yields Q t (x i,t ), where Q t For random quantizers;

[0067] (6) Gradient calculation: At time t∈{1,2,3,…,T}, node Using quantified decision variables Q t (x i,t The gradient is calculated using the following expression:

[0068] g i,t =g i,t (Q t (x i,t ),ξ i,t )

[0069] Where ξ i,t and g i,t These are partial data samples and gradients at node i at time t, respectively.

[0070] (7) First and second momentum estimates: Next, the first momentum estimate is performed using the following formula:

[0071] m i,t =β 1,t m i,t-1 +(1-β 1,t )g i,t

[0072] Where β 1,t The first-order momentum m used to balance at time t-1 i,t-1 and gradient g at time t i,t , and β 1,t It decreases as t increases. The second-order momentum estimate is obtained using the following formula:

[0073] v i,t =β2v i,t-1 +(1-β2)g i,t ⊙g i,t

[0074] Similar to the first-order momentum estimation, the second-order momentum estimation uses the second-order momentum v at equilibrium time t-1 of β2. i,t-1 The dot product of the gradient elements at time t and g i,t ⊙g i,t Furthermore, to ensure that the Q-DADAM algorithm converges to the optimal solution, it needs to use the following formula to apply v. i,t renew:

[0075]

[0076] Where β3∈[0,1);

[0077] (8) Consensus steps: Update using the following formula:

[0078]

[0079] Where z i,t By node i and its neighborhood The nodes are obtained by weighting the model, where the weight between node i and node j is represented by [W]. ij express;

[0080] (9) Generation of decision variables: Next, the vector Convert to a diagonal matrix In constrained scenarios, projection operations are used to map the results to... The expression for calculation is as follows:

[0081]

[0082] Where α t This represents the step size, while m t and It is momentum with historical gradient information;

[0083] (10) Set t to t+1. Return the original variable x at time t. i,t ,m i,t-1 ,v i,t-1 ,

[0084] I. In this embodiment, the dynamic sparse signal repair problem, a popular problem in distributed online optimization, is used to test the effectiveness of the proposed method and the impact of different quantization levels on the algorithm.

[0085] In this example, the present invention applies the Q-DADAM algorithm to the problem of dynamic sparse signal recovery. In this problem, each node... We need to work together to minimize the following loss function:

[0086]

[0087] in Represents the predictive model. This is the observation matrix, and σ is used as a constant regularization parameter. i,t Defined as:

[0088] z i,t =C i,t s t +v i,t

[0089] in Indicates noise. This is the truth value of a dynamically sparse signal. At time t∈{1,2,…,T}, each node... Will receive z i,t and C i,t These nodes need to work together to repair the signal. t .

[0090] In this simulation experiment, the network consists of n = 50 nodes. Each node... The process involves 8 3D data points from t=1 to t=500. The parameters in the Q-DADAM algorithm are set as follows: α=1, β1=0.9, β2=0.999, β3=0.9, G0=0.001, The parameter settings for other algorithms are similar to those for the Q-DADAM algorithm. Figure 2 The dynamic limitations of various distributed online optimization algorithms on this problem are demonstrated. Even with the addition of a quantization mechanism, the Q-DADAM algorithm still exhibits performance close to that of the GTADADM and DADAM algorithms. This result shows that the Q-DADAM algorithm still possesses excellent performance in the field of distributed online optimization. Furthermore, to investigate the impact of different quantization levels on the convergence performance of the Q-DADAM algorithm, this invention tests the Q-DADAM algorithm on this problem using different quantization levels while keeping other parameters consistent. Figure 3 The results show that in the dynamic sparse signal repair problem, the higher the quantization level, the faster the algorithm converges.

[0091] II. In this embodiment, the present invention uses the logistic regression problem, a popular problem in distributed online optimization, to test the effectiveness of the proposed algorithm and the impact of different numbers of nodes on the algorithm.

[0092] In this example, the present invention applies the Q-DADAM algorithm to dynamic binary classification and multi-class classification problems. Assume there is a network with multiple nodes, where each node at t∈{1,2,…,T}… You will get These data are represented as And with labels The task objective is to train a program by The hyperplane represented by the given formula can be expressed mathematically as follows: when Sometimes, when Sometimes, This online classification problem can be represented as:

[0093]

[0094] Where C is the regularization parameter. Furthermore, each point moves along the circular path according to the following formula:

[0095]

[0096] in The center of the circle is randomly generated.

[0097] In the simulation experiment, there is a network consisting of n=50 nodes. Each node i will receive m at each time t∈{1,2,…,1000}. i There are 10 data points, all located on a circle with radius r = 1. In this simulation, the parameters of each algorithm are the same as in previous simulations. For example... Figure 4 As shown, the results indicate that the performance of the Q-DADAM algorithm is close to that of the DSGD and DADAM algorithms. This result demonstrates that the Q-DADAM algorithm has the ability to handle dynamic classification tasks. Furthermore, to investigate the impact of the number of nodes in the Q-DADAM algorithm on its convergence performance, this invention uses different numbers of nodes while ensuring that other parameters remain consistent. The results are as follows... Figure 5 As shown, the convergence speed of the Q-DADAM algorithm slows down as n increases.

[0098] Furthermore, this invention uses the MNIST dataset to test the algorithm's multi-class classification ability in offline learning. The MNIST dataset consists of handwritten digits with an image size of 28×28 pixels, and has ten categories from 0 to 9. The training set contains 6000 data samples, and the test set contains 1000 data samples. This invention uses a network consisting of n=50 nodes. Each node... Having 120 data samples, and using y i,j , Let y represent this. i,j ∈{1,2,...,10} corresponds to 10 labels, z i,j Representing features. In this simulation experiment, the parameters of each algorithm are the same as in previous simulation experiments. For example... Figure 6As shown in the test dataset, the results indicate that even with the addition of a random quantizer, the performance of the Q-DADAM algorithm is still close to that of the GTADAM and DADAM algorithms, demonstrating that the Q-DADAM algorithm can handle large-scale machine learning problems.

[0099] Third, in this embodiment, the present invention uses the gradient tracking problem, a popular problem in distributed online optimization, to test the effectiveness of the proposed algorithm.

[0100] This invention applies the Q-DADAM algorithm to the target tracking problem to demonstrate its performance in variable environments. Assume a network of n=6 nodes collaboratively tracking three time-varying signals. Each signal is represented as... Where a∈{1,2,3}, its definition is as follows:

[0101]

[0102] in, Indicates the position of target a. κ represents the velocity of target a at time t. a Indicates amplitude, ω a ν represents angular frequency. a This is the phase of target a. At each time step t∈{1,2,…,6000}, each node i∈{1,2,…,6} is measured through the model. Observed y a,t ,in It is a randomly generated measurement matrix. All nodes need to collaborate to minimize the following global loss function:

[0103]

[0104] In the simulation experiment, the amplitude is κ. a =0.5a, while the frequency ω a and phase ν a Derived from uniform distributions U(0,π) and Generation. In this simulation experiment, the parameters of each algorithm are the same as in the previous simulation experiments. Figure 7 The figure shows the performance of the Q-DADAM algorithm in dynamic online tracking problems, demonstrating its effectiveness in tracking three target trajectories. Furthermore, as... Figure 8 As shown, even with the presence of an information quantization mechanism, the Q-DADAM algorithm exhibits the same dynamic regret reduction trend as the DADAM and DSGD algorithms.

[0105] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications should be covered within the scope of the claims of the present invention.

Claims

1. A quantitative distributed adaptive online optimization method for bandwidth-constrained scenarios, characterized in that: This method combines a random quantizer and an adaptive gradient algorithm into a distributed online algorithm, enabling adaptive convergence of the algorithm in bandwidth-constrained scenarios. The method specifically includes the following steps: S1. Initialize hyperparameters: ; S2, For all nodes Initialization parameters: ,in express 1-dimensional unit vector; S3, All nodes Number of iterations Set it to 1, and then set a maximum number of iterations. As a criterion for stopping the algorithm's iterative updates; S4, Judgment Whether it is true or false, if true, then for all nodes If an update from S5 to S10 fails, that is, when... When the algorithm returns Local average variable in the next iteration ; S5. Decision variable quantification: at time... For nodes Decision variables Quantification ,in For random quantizers; S6. Gradient Calculation: At time... ,node Using quantified decision variables The gradient is expressed as follows: in and They are time points node Partial data samples and gradients; S7. First and Second Order Momentum Estimation: First order momentum estimation is performed using the following formula: in Used for balance First-order momentum at time and Time gradient ,and along with As it increases, it decreases, and the second-order momentum estimate is obtained using the following formula: Similar to first-order momentum estimation, second-order momentum estimation uses... balance Second momentum at time and Time gradient element dot product To ensure the algorithm converges to the optimal solution, the following formula is used: renew: ,in ; S8. Consensus Step: In this step, the following formula is used for updating: in By node Its neighborhood The node model is weighted to obtain the value, where the node... and Weights between express; S9. Decision Variable Generation: Generating vectors Convert to a diagonal matrix In constrained scenarios, use projection operations to map the results to... The calculation expression is as follows: in Indicates the step size, while and It is momentum carrying historical gradient information; S10, will Set as ,return The original variables of time .

2. The quantized distributed adaptive online optimization method for bandwidth-constrained scenarios according to claim 1, characterized in that: In this method, communication between nodes is viewed as an undirected network, using... To describe, by node set and edge set Composition; Definition To represent an undirected graph The matrix, The value represents the node For nodes The greater the weight, the greater the impact. When the weight is 1... At that time, these two nodes do not affect each other; using Represents a node Furthermore, the weight matrix of this network must be a doubly random matrix, i.e., satisfy the following conditions: .

3. The quantized distributed adaptive online optimization method for bandwidth-constrained scenarios according to claim 2, characterized in that: In distributed optimization algorithms, nodes At any moment It is necessary to predict new models based on local data and past models; assuming a... For a network of nodes, the problem of this method can be described by the following mathematical formula: in Indicates at time node The local loss function, Let be a non-empty, bounded convex set; this method uses dynamic regret to evaluate the algorithm, as shown below: in ,Right now for The optimal solution at time t.

4. A quantitative distributed adaptive online optimization system for bandwidth-constrained scenarios, characterized in that: The system employs the method as described in any one of claims 1 to 3.

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