Privacy protection distributed dual average online optimization method and system based on state decomposition

Abstract

The invention relates to a privacy protection distributed dual average online optimization method and system based on state decomposition, and belongs to the technical field of information communication. According to the algorithm provided by the invention, state decomposition and gradient adjustment strategies are combined, so that the privacy protection capability is enhanced, and the unbalance in the directed network is eliminated at the same time. A state variable of each node is decomposed, one state variable is only updated in the node and does not communicate with the outside, and the other state variable is used for exchanging information with adjacent nodes. In addition, the weights between the variables are time-varying, and only the node itself knows its specific value, thereby further enhancing the privacy protection mechanism. According to the method, extra hidden signals are not needed, and the calculated amount is not obviously increased. Under the support of a distributed optimization theory, the algorithm provided by the invention has better effectiveness, does not need central control, can solve the actual problems in the time-varying directed network, has the characteristic of privacy protection, and has wide application prospects.

Description

Technical Field

[0001] The present invention belongs to the field of information and communication technologies, and relates to a privacy-preserving distributed dual averaging online optimization method and system based on state decomposition. Background Art

[0002] In recent years, with the growing demand for optimization problems in practical application fields such as traffic control, computer networks, and sensor networks, optimization technologies have also been widely applied in multiple disciplinary fields such as mathematics, control, and artificial intelligence. Traditional methods face many challenges when solving large-scale optimization problems, mainly due to the limitation of the computing power of the central node. The development of multi-node network optimization technologies has made distributed optimization gradually attract wide attention because of its low requirements for communication, computing, and storage resources. Up to now, researchers have designed various algorithms for different distributed optimization scenarios. Among these scenarios, the simplest case is to optimize an unconstrained objective function, while more complex cases involve the distributed optimization of a global constraint set, that is, the decision variables are restricted within a non-empty closed convex set. Existing distributed optimization algorithms can be roughly divided into two categories: consensus-based algorithms, which mainly rely on the consistency of information between nodes, and typical algorithms include asynchronous distributed gradient algorithms, stochastic mirror descent algorithms, and distributed quasimonotone subgradient algorithms, etc.; dual decomposition-based algorithms decompose the optimization problem through dual variables, and representative algorithms include distributed dual proximal gradient algorithms and distributed forward-backward Bregman splitting algorithms, etc. In practical optimization problems, since information transmission is usually unidirectional, communication networks are often unbalanced. To solve this problem, researchers have developed various algorithms, such as push-sum strategies and surplus-based strategies, etc. For example, the subgradient scaling strategy, whose key idea is to use the estimated value of the normalized Perron eigenvector of the weight matrix to adjust the subgradients of each node, enabling it to update its own variables by weighted averaging according to its own and neighbor nodes' states.

[0003] In addition, during the iterative process of the optimization algorithm, each node needs to exchange information with adjacent nodes and perform iterative calculations based on the gradient of the objective function, which often involves the problem of privacy leakage. In actual optimization tasks, nodes usually do not want their private information to be obtained by the outside world or other nodes. For example, in a healthcare system, each patient hopes to keep their personal information private and not be leaked. Therefore, it is of great practical significance to study distributed optimization algorithms with privacy protection mechanisms. In distributed optimization, a variety of privacy protection strategies have been proposed. Among them, the differential privacy strategy is one of the most common methods. Its basic principle is to add noise obeying a specific distribution to the data to prevent external observers from accurately obtaining the original data. However, due to the introduction of noise, this method may reduce the optimization accuracy while enhancing privacy security. Therefore, researchers need to balance the solution accuracy and the privacy protection level. Another common privacy protection method is data encryption, which encrypts and decrypts data during each information interaction to ensure data security. However, in a resource-constrained network environment, data encryption will significantly increase communication and computational costs, limiting its practicality. Summary of the Invention

[0004] In view of this, the purpose of the present invention is to provide a privacy-preserving distributed dual averaging online optimization method and system based on state decomposition, which is abbreviated as the SD-DDA algorithm. In this algorithm, the state variable of each node is decomposed. One of the state variables is only updated internally within the node and does not communicate with the outside, while the other state variable is used to exchange information with adjacent nodes. In addition, the weights between these variables are time-varying, and only the node itself knows their specific values, thereby further enhancing the privacy protection mechanism.

[0005] To achieve the above object, the present invention provides the following technical solutions:

[0006] A privacy-preserving distributed dual averaging online optimization method based on state decomposition, which aims at the distributed online optimization objective with a common constraint set in a directed network, and realizes privacy protection through state decomposition based on the dual averaging strategy, including the following steps:

[0007] S1: For the distributed online optimization problem in a directed network, decompose the state variable of each node into two sub-states: an internal update sub-state and a communication sub-state, where the internal update sub-state is only iteratively updated within the node and does not participate in communication between nodes; the communication sub-state is used to exchange information with adjacent nodes, and its update depends on a time-varying weight matrix, and the time-varying weight matrix is a row-stochastic matrix, and only the node itself knows its specific value;

[0008] S2: Initialize the variables of all nodes, including the dual variable components, state variable components, and auxiliary variables;

[0009] S3: Construct a strongly connected directed network, ensuring that the weight matrix satisfies row stochasticity;

[0010] S4: At each iteration, perform the following operations: The node calculates the subgradient of its local loss function with respect to the decision variable and adjusts the gradient based on the weight relationship between the internal update substate and the communication substate; The node sends the communication substate and its associated auxiliary variables to its external neighbors; The node receives the communication substate and auxiliary variables from its internal neighbors, updates its dual variable components and auxiliary variables; Inside the node, update the internal update substate by decreasing the step size;

[0011] S5: Repeat step S4 until the preset maximum number of iterations is reached, and output the global optimal decision variable.

[0012] Furthermore, in this method, the directed network is represented as where is the set of nodes, is the set of edges, is the weight matrix; when (i, j) ∈ ε or i = j, a ij > 0, otherwise a ij = 0; In this network, node j receives information from the nodes in its in-neighbor set ; It is restricted that in this directed network, a node can only access the information of the nodes in its in-neighbor set; Assume that at time k, each node in the directed network can only access the information transmitted from its in-neighbor nodes, and the information is transmitted in the channel, and the channel environment is good, without delay, packet loss, and congestion; On this basis, each node uses the information transmitted by its in-neighbors in the channel at each time k to correspondingly complete its iteration; This communication network also conforms to the following assumptions: The communication topology is a strongly connected directed network, and its weight matrix A is row-stochastic.

[0013] Furthermore, in this method, for each node there is a set of local loss functions whose specific form is unknown; At each iteration, the node generates a corresponding loss according to the current decision variable; Therefore, the distributed optimization problem is expressed as that at each iteration k ∈ {1, 2,..., K}, each node in the network cooperatively minimizes the global objective function:

[0014]

[0015] where is non-empty, closed, and convex; x ∈ Ω is the global decision variable; Each node i can only access information about fi For any node, the global function f is unknown; the above parameters also meet the following assumptions: the upper bound of the diameter of the non - empty closed convex set Ω is M > 0; the local loss function f i,k (x) is a convex function, and its sub - gradient g i,k (x) is L - Lipschitz continuous (L > 0), that is, for there is ||g i,k (x)|| ≤ L.

[0016] Furthermore, the method specifically includes the following steps:

[0017] S1. Initialize all variables for all nodes i ∈ {1, 2, …, N}: η1 > 0, the component of the dual variable the component of the state variable the auxiliary variable y j,1 = 1; in the initialization of the projection operator is defined as where η > 0, the proximal function h(u) has the property of 1 - strong convexity and satisfies h(0) = 0, h(u) > 0 for all u ∈ Ω;

[0018] S2. Set the iteration number k of all nodes i ∈ {1, 2, …, N} to 1, and then set a maximum iteration number K as the criterion for the algorithm to stop iterative update;

[0019] S3. Randomly construct a directed network

[0020] S4. Judge whether k < K holds. If it holds, perform the update in steps S5 - S10; if not, exit the algorithm;

[0021] S5. Node i calculates the sub - gradient g on i，k where is the state of node i at time k and is the weight between them;

[0022] S6. Node i sends and A y ji y i,k to its out - neighbors

[0023] S7. After node i receives and A ij y i，k from its in - neighbors, update the component of the dual variable and the auxiliary variable y i，k, update the components of the dual variable inside the node The update rule is:

[0024]

[0025] S8, η k is a decreasing step size, update the state variables and values:

[0026]

[0027] S9, set k to k + 1;

[0028] S10, return the original variables at time k and

[0029] Furthermore, this method can also rewrite the algorithm by permuting two state components:

[0030] Arrange the two components of the dual variable into a column vector denoted as z k ; arrange the two components of the state variable into a column vector x k , that is

[0031]

[0032] Then this method can be reformulated as:

[0033]

[0034] where, when i = 1, 2,..., N, s i，k = 0; when i = N + 1, N + 2,..., 2N, s i,k = g i-N,k / y i-Ni-N,k , the new weight matrix M k is represented as follows:

[0035]

[0036] Due to the network structure being strongly connected and the weight matrix being a row-stochastic matrix, for the new weight matrix M k the corresponding directed network can be made strongly connected by choosing appropriate state weights such that the network structure is also strongly connected, and its weight matrix M k is also a row-stochastic matrix.

[0037] Furthermore, in this method, the following method is adopted for state decomposition:

[0038] Decompose the state \(x\) of each node i,k into two sub - states: and where the sub - state inherits the role of the original state \(x\) in the interaction between nodes. It is the only observable state value of node \(i\); while the sub - state i,k only participates in the iteration of the algorithm through interaction with ; that is, the update of the sub - state is carried out inside the node. For the adjacent nodes of node \(i\), is invisible; and on the premise of satisfying the constraint conditions and as well as the initial values of the two decomposed states can be arbitrarily selected from the set of real numbers, that is, the initial values of this algorithm do not require additional constraints;

[0039] For privacy information, the private information of the node is mainly reflected in the objective function; based on this, the gradient information of the objective function needs to be protected as private information; for an eavesdropper, whether it cannot obtain the gradient information or can only obtain an approximation of the gradient, it indicates that this algorithm can achieve privacy protection in the sense of protecting the gradient information; assuming that the eavesdropper knows the communication topology between nodes and the algorithm update rule, and can obtain the corresponding information within a certain time range \(K\), the following results can be obtained:

[0040]

[0041] For the eavesdropper, according to the algorithm update rule, it can be known that the variables and \(y\) ii，k , \(k = 1,2,\cdots,K - 1\) can be eavesdropped and calculated; however, the variables and \(g\) i，k , \(k = 1,2,\cdots,K - 1\) are unknown to external eavesdroppers; this is because 's update process is only carried out inside node \(i\) and does not interact with other nodes; in the system of equations, there are a total of \(2K - 1\) unknowns, while there are only \(K - 1\) equations. Obviously, it is impossible to uniquely determine the value of each unknown variable. Even if the eavesdropper can obtain an approximate solution using the system of equations, since the objective function \(f\) i depends on the exact gradient information rather than an approximation, that is, this algorithm can effectively protect the gradient information of node \(i\) through the state decomposition strategy.

[0042] The present invention also provides a privacy - protected distributed dual - average online optimization system based on state decomposition.

[0043] The beneficial effects of the present invention are as follows:

[0044] Aiming at the distributed constrained optimization problem on unbalanced directed networks, the present invention proposes a state decomposition distributed dual averaging algorithm, which can converge to the exact optimal solution without significantly increasing the computational burden. The row-stochastic weight matrix adopted can flexibly adapt to different network topologies. And the algorithm of the present invention can effectively overcome the imbalance problem in directed networks by adopting a gradient scaling strategy, improving the optimization effect. In addition, the strategy of state decomposition adopted by the algorithm also has the characteristic of protecting privacy.

[0045] Other advantages, objectives and features of the present invention will be described to some extent in the subsequent specification, and to some extent, will be obvious to those skilled in the art based on the study of the following text, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following specification. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be described in detail preferably with reference to the accompanying drawings, where:

[0047] Figure 1 is a schematic flow chart of the method described in the present invention;

[0048] Figure 2 is a graph showing the evolution process of the average regret of the algorithm of the present invention and the differential privacy dual averaging algorithm;

[0049] Figure 3 is a sample of the MNIST dataset in the embodiment of the present invention;

[0050] Figure 4 is the training result of the algorithm of the present invention and related algorithms using the MNIST dataset;

[0051] Figure 5 is the test result of the algorithm of the present invention and related algorithms using the MNIST dataset;

[0052] Figure 6 is a comparison of the inference results of the iDLG eavesdropper of the algorithm of the present invention and the classical distributed online optimization algorithm;

[0053] Figure 7 is the motion trajectory of a moving target at different time steps k;

[0054] Figure 8 is a comparison of the results of the algorithm of the present invention and the dual averaging algorithm using differential privacy in terms of dynamic regret. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0055] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings.

[0056] Figure 1 As shown in the flowchart of the method of the present invention, in this embodiment, the privacy-preserving distributed dual-averaging online optimization method based on state decomposition technology provided by the present invention includes the following steps:

[0057] 1) Set the number of nodes n, the proximal function and initialize all variables for all nodes i ∈ {1, 2,..., N}: η0, y i，1 = 1,

[0058] 2) Set the iteration number k of all nodes i ∈ {1, 2,..., N} to 1, and the maximum iteration number K = 200.

[0059] 3) Construct a network composed of n sensors and carefully verify to maintain strong connectivity.

[0060] 4) Judge whether k ≤ K holds. If it holds, perform the updates in (5)-(11). If not, exit the algorithm.

[0061] 5) Adopt the method of random and uniform sampling to assign a probability of 40% to each edge in at each moment k, so as to form a new network, denoted as This process ensures that the time-varying directed network is strongly connected.

[0062] 6) Node i calculates the subgradient g on i，k , where is the weight between the state and of node i at time k.

[0063] 7) Node i sends and A and A ji y i,k .

[0064] 8) After node i receives and A ji y i,k from its in-neighbors, update the variables and y i,k , and update inside the node. The update rule is:

[0065]

[0066] 9) Calculate the step size value, and update the variables and values:

[0067]

[0068] 10) Set k to k + 1.

[0069] 11) Return the original variables at time k and

[0070] I. This embodiment uses a distributed classification problem of logistic regression to test the effectiveness of the proposed algorithm.

[0071] In this embodiment, it aims to solve the distributed classification problem based on logistic regression. In the experiment, n = 50 nodes communicating in the network cooperate to train a linear classifier for a set of points moving in a specific feature space. At time k, each node i is assigned v i = 50 (for all i) pairs of samples for training. Among them, a i，l ∈ {-1, 1} represents the label value, represents the position of the point. Then, a linear classification model is established using the given training samples. In particular, it is necessary to find a separated hyperplane described by a pair (y, h) ∈ Ω1×Ω2, At time k, the points where a = 1 and a i，l = 1 and a i，l = -1 are separated by the optimal hyperplane That is, for all (i, l), if then a i,l = 1, otherwise a i,l = -1. At each time k, this online classification problem can be set as the following minimization problem, where S = 1 is the regularization parameter:

[0072]

[0073] Each point moves along the circumference according to the following rules, where the radius r = 1, where represents the randomly generated center of the circle:

[0074]

[0075] At the same time, set the experimental parameters to: Other parameter settings refer to the best implementation plan. To effectively illustrate the experimental results, regret is used as a measure to compare the results.

[0076] Regret is an important metric for comprehensively evaluating the effectiveness of online optimization algorithms. The classical individual regret for node \(i\in\{1,2,\ldots,50\}\) is introduced below: where \(x\) j，k , \(j\in\{1,2,\ldots,N\}\) is the local variable of \(x\) at time \(k\). is the optimal decision over the time horizon \(K\). The objective of the present invention is to design an algorithm such that the regret is sublinear with respect to time \(K\), i.e., \(\lim\) K→∞ \(R\) j，k / \(K = 0\), When there is communication delay or noise between nodes, the timeliness and accuracy of information transmission will be affected, thus affecting the algorithm's ability to track the dynamic optimal solution and resulting in an increase in dynamic regret. Analyzing the dynamic regret in the unaffected case helps to deeply understand the performance boundary of the algorithm and provides a theoretical basis for the improvement and optimization of the algorithm. Therefore, regret is an important metric for evaluating the algorithm.

[0077] Meanwhile, the problem model needs to satisfy the following assumptions: The upper bound of the diameter of the non - empty closed convex set \(\Omega\) is \(M>0\); the local loss function \(f\) i，k (\(x\)) is a convex function, and its sub - gradient \(g\) i，k (\(x\)) is \(L\) - Lipschitz continuous (\(L>0\)), i.e., for there is \(\|g\) i，k (\(x\))\(\|\leq L\). In this example, the experimental results are as follows:

[0078] (1) Comparison of the effects of the algorithm of the present invention with other related algorithms

[0079] In this part, the convergence performance of the algorithm of the present invention and the dual - averaging algorithm using differential privacy within the time span of \(K = 200\) is highlighted. The results are as Figure 2 shown. The regret curves of the algorithm of the present invention and the dual - averaging algorithm using differential privacy are similar, both tending to be stable over time and sublinear with respect to \(K\). This result indicates that the algorithm of the present invention can be well applied to time - varying directed networks. Even considering the key aspects of privacy protection, the algorithm of the present invention can asymptotically achieve a sublinear convergence rate. The effectiveness and feasibility of the algorithm of the present invention are verified.

[0080] (2) Effectiveness of the algorithm of the present invention in machine learning problems

[0081] To further verify the effectiveness of the algorithm of the present invention in large - scale machine learning, the MNIST dataset is used in this embodiment to train a distributed logistic regression classification model. As Figure 3As shown, the dataset contains 13,007 samples, of which 12,000 samples are randomly selected for training and the remaining 1,007 are for testing. Each sample is converted into a vector with dimension d = 784 for model training. Figure 4 Figure 4 shows the change trends of the residuals (i.e., the expected squared error) of the algorithm of the present invention and other related algorithms (differential privacy dual averaging algorithm, push-sum dual averaging algorithm, consensus online algorithm) during the training process. The change of the residual with the number of iterations can intuitively reflect the convergence speed of the algorithm. Although the convergence rates of the four algorithms are different, as the number of iterations increases, the residuals all eventually tend to be stable, indicating that they can all approximate the optimal solution within a small number of iterations, thus verifying the convergence of the algorithm of the present invention. In addition, Figure 5 Figure 5 shows the classification accuracies of the four algorithms. The classification capabilities of each algorithm can be intuitively evaluated from the experimental results. It can be observed that as the iteration progresses, the classification accuracies of all algorithms eventually tend to be stable and approach 1, indicating that the four algorithms can all achieve high classification accuracies in this task.

[0082] (3) Effectiveness of the algorithm of the present invention in terms of privacy protection

[0083] In this part, the privacy protection characteristics of the algorithm of the present invention are further verified, that is, whether the private cost functions of the participating nodes can be retained. Here, the worst privacy risk is considered, that is, the external eavesdropper not only knows the network structure and the general update rules, but also can obtain the information shared within the network. It is assumed that the cost function remains unchanged and the algorithm uses deterministic gradients for updates. Figure 6 Figure 6 shows the inference results of the eavesdropper iDLG under the classical distributed algorithms and the algorithm proposed in the present invention. From the experimental results, in the classical distributed online algorithm without using any privacy protection strategy, the eavesdropper can easily deduce the results; while under the algorithm of the present invention, the eavesdropper cannot deduce the private information of the nodes. That is to say, without taking any privacy protection measures, the external eavesdropper can easily deduce the private data, thus threatening the system security, while under the constraints of this algorithm, the eavesdropper cannot accurately recover the original data. This result clearly proves the effectiveness of this algorithm in terms of privacy protection.

[0084] Second, this embodiment uses the dynamic target tracking problem to test the effectiveness of the proposed algorithm.

[0085] The dynamic target tracking problem refers to the cooperation of multiple sensors to track multiple moving targets. In practical applications, the communication network composed of sensors is often unbalanced, which is consistent with the algorithm research background proposed in this invention. In this embodiment, it is assumed that the communication network consists of 5 nodes, and the connection relationship between these nodes is directed and time-varying. At the same time, it is ensured that the entire network always remains connected, and the strategy for constructing the network structure is the same as in the previous example. All nodes work together to track 3 moving targets. At time step k, the position of the moving target can be expressed as: Specifically:

[0086]

[0087] Where, and represent the position and velocity of target l at time k respectively, ρ l represents the phase of target l, γ l represents the angular frequency, υ l represents the amplitude. The numbers 1 and 2 represent exponents respectively, corresponding to the values of the vector in two different dimensions. Let and randomly generate the measurement matrix Subsequently, each node uses the measurement value to track the moving target. When x ∈ [-2, 2] 6 the target tracking problem can be expressed as optimizing the following objective function:

[0088] In the simulation experiment, the random variables ρ l and υ l follow a uniform distribution in the intervals [0, π] and [0, 2] respectively. The sampling frequency is set to 100Hz, and the decreasing step size is set to Figure 7 shows the motion trajectories of the moving targets at different time steps k, where and represent the positions of the moving targets in time step k. The figure also gives the state values of node 1 in the first, third, and fifth dimensions at time step k From Figure 7 it can be seen that the tracking trajectory of node 1 basically coincides with the actual motion trajectories of the three moving targets, indicating that the algorithm of this invention can effectively achieve target tracking. In addition, from Figure 8 it can be observed that both the algorithm of this invention and the dual averaging algorithm using differential privacy show a sublinear trend in dynamic regret, which further verifies the effectiveness of the algorithm of this invention.

[0089] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified without departing from the purpose and scope of the present technical solutions, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A privacy-preserving distributed dual averaging online optimization method based on state decomposition, characterized in that: This method aims at the distributed online optimization objective with a common constraint set in a directed network. Based on the dual averaging strategy, it achieves privacy protection through state decomposition, including the following steps: S1: For the distributed online optimization problem in a directed network, the state variable of each node is decomposed into two sub-states: an internal update sub-state and a communication sub-state. The internal update sub-state is only iteratively updated within the node and does not participate in communication between nodes. The communication sub-state is used to exchange information with adjacent nodes, and its update depends on a time-varying weight matrix, which is a row-stochastic matrix and only the node itself knows its specific value. S2: Initialize the variables of all nodes, including dual variable components, state variable components, and auxiliary variables. S3: Construct a strongly connected directed network to ensure that the weight matrix satisfies row-stochasticity. S4: At each iteration, perform the following operations: The node calculates the sub-gradient of its local loss function with respect to the decision variable and adjusts the gradient based on the weight relationship between the internal update sub-state and the communication sub-state. The node sends the communication sub-state and its associated auxiliary variable to external neighbors. The node receives the communication sub-state and auxiliary variable from internal neighbors, updates its dual variable component and auxiliary variable. The internal update sub-state is updated within the node with a decreasing step size. S5: Repeat step S4 until the preset maximum number of iterations is reached, and output the global optimal decision variable.

2. The privacy-preserving distributed dual averaging online optimization method based on state decomposition according to claim 1, wherein: In this method, the directed network is represented as where is the node set, is the edge set, is the weight matrix; When \((i, j)\in\varepsilon\) or \(i = j\), \(a\) ij \(> 0\), otherwise \(a\) ij \(= 0\); In this network, node \(j\) receives information from the nodes in its in-neighbor set ; It is restricted that in this directed network, a node can only access the information of the nodes in its in-neighbor set; Assume that at time \(k\), each node in the directed network can only access the information transmitted from its in-neighbor nodes, and the information is transmitted in the channel, and the channel environment is good, without delay, packet loss and congestion; On this basis, each node uses the information transmitted by its in-neighbors in the channel at each time \(k\) to complete its iteration accordingly; This communication network also conforms to the following assumptions: The communication topology is a strongly connected directed network, and its weight matrix \(A\) is row-stochastic.

3. A privacy-preserving distributed dual averaging online optimization method based on state decomposition according to claim 2, characterized in that: In this method, for each node there is a set of local loss functions whose specific form is unknown; in each iteration, the node generates a corresponding loss according to the current decision variable; thus, the distributed optimization problem is expressed as, in each iteration \(k\in\{1,2,\ldots,K\}\), each node in the network minimizes the global objective function in a cooperative manner: where is non - empty, closed, and convex; \(x\in\Omega\) is the global decision variable; each node \(i\) can only access information about \(f\) i and the global function \(f\) is unknown to any node; the above parameters also satisfy the following assumptions: the upper bound of the diameter of the non - empty closed convex set \(\Omega\) is \(M\gt0\); the local loss function \(f\) i，k (x) is a convex function and its sub - gradient \(g\) i，k (x) is \(L\) - Lipschitz continuous (\(L\gt0\)), that is, for \(x\in\Omega\), we have \(\left\lVert g\right.\) i，k (x)\(\left.\right\rVert\leq L\).

4. A privacy-preserving distributed dual averaging online optimization method based on state decomposition according to claim 3, characterized in that: This method specifically includes the following steps: S1. Initialize all variables for all nodes \(i\in\{1,2,\ldots,N\}\): \(\eta_1 > 0\), the components of the dual variable the components of the state variable the auxiliary variable \(y\) j，1 \(= 1\); In the initialization of define the projection operator as where \(\eta>0\), the proximal function \(h(u)\) has the 1 - strong convex property and satisfies \(h(0)=0\), \(h(u)>0\) for all \(u\in\Omega\); S2. Set the iteration number k of all nodes i ∈ {1, 2,..., N} to 1, and then set a maximum iteration number K as the criterion for the algorithm to stop iterative update. S3. Randomly construct a directed network S4. Judge whether k < K holds. If it holds, perform the update in steps S5 - S10. If not, exit the algorithm. S5. Node i calculates the subgradient g on i，k , where is the state of node i at time k and is the weight between S6. Node i sends to its external neighbors send and A ji y i，k ; S7. After node i receives and A ij y i，k from its inner neighbors, it updates the dual variable component and the auxiliary variable y i,k . Inside the node, it updates the component of the dual variable using the update rule: S8, η k is a decreasing step size to update the state variables and values: S9. Set k to k + 1. S10. Return the original variables at time k and 5. A privacy-preserving distributed dual averaging online optimization method based on state decomposition according to claim 4, characterized in that: This method can also rewrite the algorithm by permuting the two state components: Arrange the two components of the dual variable into a column vector and denote it as z k ; arrange the two components of the state variable into a column vector x k , that is Then this method can be reformulated as: where, when i = 1, 2, …, N, s i，k = 0; when i = N + 1, N + 2, …, 2N, s i，k = g i-N，k / y i-Ni-N，k . The new weight matrix M k is represented as follows: Due to the network structure is strongly connected and the weight matrix is a row-stochastic matrix, for the new weight matrix M k the corresponding directed network can, by choosing appropriate state weights make the network structure also be strongly connected, and its weight matrix M k is also a row-stochastic matrix.

6. A privacy-preserving distributed dual averaging online optimization method based on state decomposition according to claim 5, characterized in that: In this method, the following method is used for state decomposition: Decompose the state x of each node i,k into two sub-states: and where the sub-state inherits the role of the original state x in the interaction between nodes i，k and it is the only observable state value of node i; while the sub-state only participates in the iteration of the algorithm through interaction with that is, the update of the sub-state is carried out inside the node and is invisible to the adjacent nodes of node i; and on the premise of satisfying the constraint conditions and as well as the initial values of the two decomposed states can be arbitrarily selected from the set of real numbers, that is, the initial values of this algorithm do not require additional constraints; For private information, the private information of a node is mainly reflected in the objective function. Based on this, the gradient information of the objective function needs to be protected as private information. For an eavesdropper, whether it cannot obtain the gradient information or can only obtain an approximation of the gradient indicates that this algorithm can achieve privacy protection in the sense of protecting the gradient information. Assume that the eavesdropper knows the communication topology between nodes and the algorithm update rule, and can obtain the corresponding information within a certain time range K, then the following results can be obtained: For an eavesdropper, according to the algorithm update rule, it can be known that the variables and y ii,k , k = 1, 2, …, K - 1 can be eavesdropped and calculated; however, the variables and g i，k , k = 1, 2, …, K - 1 are unknown to external eavesdroppers; this is because the update process of is only carried out inside node i and does not interact with other nodes; in the system of equations, there are a total of 2K - 1 unknowns, while there are only K - 1 equations, obviously it is impossible to uniquely determine the value of each unknown variable. Even if the eavesdropper can obtain an approximate solution using the system of equations, since the objective function f i depends on accurate gradient information rather than approximate values, that is, this algorithm can effectively protect the gradient information of node i through the state decomposition strategy.

7. A privacy-preserving distributed dual averaging online optimization system based on state decomposition, characterized in that: This system adopts the method described in any one of claims 1 to 6.

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