A privacy-preserving distributed zeroth-order dual average online optimization method and system

Abstract

This invention relates to a privacy-preserving distributed zero-order dual averaging online optimization method and system, belonging to the field of information and communication technology. This method targets distributed online optimization objectives with a common constraint set in time-varying directed networks, adapting to situations where the objective function is unknown by using the zero-order gradient instead of calculating the true subgradient information. On one hand, this method overcomes the imbalance problem caused by time-varying directed networks by constructing a row random matrix and rescaling the zero-order gradient using auxiliary variables; on the other hand, this method ensures the privacy of normally participating nodes by employing a differential privacy strategy. The method and system provided by this invention can effectively optimize large-scale machine learning problems and have good efficiency, showing broad application prospects.

Description

Technical Field

[0001] This invention belongs to the field of information and communication technology, and relates to a privacy-preserving distributed zero-order dual average online optimization method and system. Background Technology

[0002] In recent years, advancements in communication and sensing technologies within network control systems have led to the emergence of diverse optimization tasks. Traditional centralized optimization methods typically employ a central node for computation, communication, and storage when addressing these tasks. However, this approach is susceptible to single points of failure, lacks robustness in current scenarios, and incurs significant storage costs. In contrast to centralized optimization, distributed optimization methods decompose large-scale problems into smaller, manageable ones, avoiding long-distance communication. Furthermore, distributed optimization methods reduce communication overhead, offer better privacy protection, and achieve greater robustness. This has enabled the widespread application of distributed optimization methods in various fields, such as large-scale data analysis, information control, smart grids, and online learning.

[0003] However, many practical applications of distributed optimization involve significant uncertainties, such as online prediction and estimation problems in wireless networks. These problems can be solved within the architecture of online optimization; however, the node-related functionalities may change over time, and these changes can only be captured by the nodes retrospectively. Numerous effective methods exist for online optimization problems, including early research based on distributed gradient descent, and various extensions to solve constrained problems, apply to complex networks, improve efficiency, and protect privacy.

[0004] Distributed dual averaging optimization methods have also provided many new ideas for solving online optimization problems. For online constrained optimization problems on undirected networks, researchers have invented a distributed dual averaging optimization method. However, in practical online optimization problems, information transmission is usually not bidirectional, leading to an imbalance in the communication network structure. To address this issue, several distributed dual averaging methods have emerged. These methods employ effective strategies such as push-sum strategies and surplus-based strategies to overcome the problems caused by imbalanced directed networks. Recently, many studies have further considered the more general case in time-varying directed networks. For example, a distributed dual averaging push-sum algorithm is introduced into the stochastic dual averaging method to solve online constrained optimization problems in time-varying directed networks. Unlike the push-sum protocol used in the stochastic dual averaging method, researchers have proposed a strategy involving constructing row random matrices and rescaling the zeroth-order gradient, which has shown significant effectiveness in overcoming the imbalance problem in time-varying directed networks.

[0005] Unfortunately, the methods mentioned above all involve learning the entire loss function or its subgradient, which is impossible in some practical applications, such as online source routing in data networks and adversarial black-box machine learning. This has led to the adoption of zero-order gradient methods, which approximate the gradient of the loss function using only its value. Furthermore, many distributed zero-order gradient methods have been investigated. These decentralized methods do not directly compute the true gradient but instead construct gradient estimates using the function values. This distributed zero-order method has also been extended to online optimization problems. Researchers have invented a push-sum distributed online optimization method using the zero-order gradient method, which can achieve the desired sublinear loss on time-varying directed networks.

[0006] It is worth noting that in many real-world situations, communication often occurs in suboptimal environments, leading to potential security risks. Malicious nodes within the network are particularly vulnerable; they may capture information transmitted between nodes and use this information to infer the private information of participating nodes, thus exposing them to privacy breaches. Given the increasing prevalence of privacy violations in recent years, there is an urgent need to protect the privacy of each node in decentralized networks. Differential privacy, as an excellent privacy protection method, has attracted considerable interest from researchers due to its ease of implementation, verifiable security properties, and rigorous mathematical framework. However, because differential privacy strategies increase perturbation by adding noise to information transmission, a balance must be struck between protecting privacy and optimizing accuracy when using them. Summary of the Invention

[0007] In view of this, the purpose of this invention is to provide a privacy-preserving distributed zero-order dual average online optimization method and system. In this method, the zero-order gradient is used instead of calculating the true subgradient information to adapt to situations where the objective function is unknown. On the one hand, this method overcomes the imbalance problem caused by time-varying directed networks by constructing a row random matrix and rescaling the zero-order gradient using auxiliary variables; on the other hand, this method ensures the privacy of normally participating nodes by employing a differential privacy strategy.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] A privacy-preserving distributed zero-order dual average online optimization method is proposed. This method targets distributed online optimization objectives with common constraint sets in time-varying directed networks. It adapts to the situation where the objective function is unknown by using the zero-order gradient instead of calculating the true subgradient information. It overcomes the imbalance problem caused by time-varying directed networks by constructing a row random matrix and rescaling the zero-order gradient using auxiliary variables. Finally, it protects the privacy of normal participating nodes by adopting a differential privacy strategy.

[0010] Furthermore, in this method, the time-varying directed network includes:

[0011] For integer n > 1, define V = {1, 2, 3, ..., n}, and the directed network G = {V, E} consists of an edge set of ordered node pairs. It is specified that the outer neighbor set and inner neighbor set of each node i are respectively used and express;

[0012] When the network changes over time, the subscript 't' is used to represent time; G t E represents a network. t Indicates network G t edge set, and Let represent the sets of outer neighbors and inner neighbors of node i at time t, respectively; correlation matrix If (j,i)∈E t Under the conditions, then otherwise Assume that at time t, each node in the directed network can only access information transmitted from its internal neighbor nodes, and the information is transmitted in the channel with a good channel environment, without delay, packet loss, or congestion; based on this, each node uses the information transmitted by its internal neighbor in the channel at each time t to complete its iteration accordingly.

[0013] The communication network also satisfies the following assumption: when time t≥1, there exists an integer B≥1 such that the edge set The network formed is strongly connected; matrix At time t, it is a column random, that is

[0014] Furthermore, in this method, each node i∈{1,2,3,...,n} is considered to be associated with a set of locally convex loss functions. The specific form of this problem is unknown. Therefore, in this invention, the objective of each node on the network at time t∈{1,…,T} is to collaboratively optimize the following problem:

[0015]

[0016] in It is a non-empty closed convex set, and x∈Ω is a global decision vector; furthermore, at time t, each node i can only be in the decision vector Access is only possible after a decision is made. The value that each node i cannot obtain. This is the explicit form of a setup often referred to as an online and robber feedback scenario; here, no node can access it. Meanwhile, the problem model satisfies the following assumptions: 1) There exist two constants d, D > 0 (d < D) such that 2) Each local loss function is K-Lipschitz continuous on Ω, that is, for any a, b ∈ Ω, i ∈ V, all hold.

[0017] Definition of regret:

[0018] Regret is an important indicator for comprehensively evaluating the effectiveness of online optimization algorithms. The following introduces the classical individual regret of node j ∈ {1, 2, 3,..., n}: where is the local variable of x. is the optimal decision over the time horizon T. In the present invention, the true state of the node will be perturbed by random noise to achieve privacy protection for relevant nodes. Therefore, it is impossible to achieve the deterministic regret like the classical individual regret. Therefore, the present invention uses the following regret to describe the decision of node j and the error of the optimal decision over the time horizon T:

[0019] Definition of privacy protection:

[0020] In a distributed network system, the existence of malicious nodes will endanger the security of the system. These nodes can actively eavesdrop on the sensitive information exchanged between nodes and infer private data. In the present invention, although each node can only access the function value, the estimated values of the nodes are shared among neighbors, which may lead to the risk of privacy leakage. To solve this problem, the present invention adopts a differential privacy strategy to protect the loss function of the nodes. First, the concept of adjacent data sets needs to be clarified: If there are two data sets U = {u 1 ,..., u n} and U′ = {u′ 1 ,..., u′ n} that satisfy that only one data is inconsistent, then these two data sets can be called adjacent data sets.

[0021] Definition of differential privacy:

[0022] For any two adjacent data sets U and U′, if the randomized algorithm B satisfies the algorithm B is said to have ∈-differential privacy. Where ∈ > 0 is the differential privacy level, and S is an arbitrary subset of the result set of algorithm B.

[0023] It's easy to see from this definition that slight changes in the private dataset will not cause significant differences in the final result of randomization algorithm B. A smaller ∈ indicates higher privacy protection for nodes; in this sense, ∈ should be minimized by increasing the variance of the noise, but this may lead to a decrease in the algorithm's accuracy.

[0024] This invention expresses the sensitivity of randomization algorithm B at time t as follows: Among them, Adj(U t U t ′) represents the neighboring dataset U t and U t The adjacency relationship at time t. The definition of sensitivity further clarifies that, while maintaining the same level of privacy, sensitivity is directly proportional to the amplitude of the noise. Therefore, the amplitude of random noise can be adjusted by limiting sensitivity to ensure differential privacy.

[0025] Furthermore, the method specifically includes the following steps:

[0026] S1. Initialize all variables of all nodes i∈{1,2,3,...,n}: α0>0, exist During initialization, the projection operator will be... Defined as Where α > 0, Meanwhile, φ(u) is a proximal function with 1-strong convexity, satisfying φ(0)=0, φ(u>0, u∈Ω;

[0027] S2. Set the iteration count t of all nodes i∈{1,2,3,…,n} to 1, and set a maximum iteration count T as the criterion for stopping the algorithm's iteration update;

[0028] S3. Randomly construct a directed time-varying network G t ;

[0029] S4. Determine if t≤T is true. If true, proceed to steps S5-S12 for updating. If false, exit the algorithm.

[0030] S5. Calculate the zeroth gradient (sub-gradient estimate) according to the following formula: Among the disturbances It is randomly and uniformly generated on the unit sphere, where δ∈(0,dξ] represents the exploration parameter and ξ∈(0,1) represents the shrinkage coefficient; since these perturbated points may be moved out of Ω, the above equation is defined on the set (1-ξ)Ω. The feasibility of the perturbation can be guaranteed, that is, for any x∈(1-ξ)Ω, Both are true;

[0031] S6. Generate a random noise vector that follows a Laplace distribution. Right now Where L(·) is the Laplace distribution function, which can generate noise that conforms to the Laplace distribution, b t These are adjustable parameters;

[0032] S7, Dual Variables Perform random vector The perturbation is used to obtain the dual variable with the added perturbation.

[0033] S8, node i will and Transmitted to its outer neighbor node

[0034] S9, Node i receives data from its internal neighbor. Transmitted and Then, update the auxiliary variable using the two received variables. and dual variables in

[0035] S10, Update variables Value: Where α t >0 is a decay step size;

[0036] S11. Set t to t+1;

[0037] S12, Return the original variables at time t.

[0038] The present invention also provides a privacy-preserving distributed zero-order dual average online optimization system employing the above-described method.

[0039] The beneficial effects of this invention are as follows:

[0040] This invention addresses the distributed online optimization problem with a common constraint set in time-varying directed networks by using the zero-order gradient instead of calculating the true subgradient information to adapt to situations where the objective function is unknown. On one hand, this method overcomes the imbalance problem caused by time-varying directed networks by constructing a row random matrix and rescaling the zero-order gradient using auxiliary variables. On the other hand, this method ensures the privacy of normally participating nodes by employing a differential privacy strategy.

[0041] 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

[0042] 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:

[0043] Figure 1 This is a flowchart illustrating the method described in this invention;

[0044] Figure 2 This diagram illustrates the evolution of the average regret of the algorithm of this invention and other related algorithms.

[0045] Figure 3 This is a sample of the MNIST dataset in an embodiment of the present invention;

[0046] Figure 4 The algorithm of this invention uses the MNIST dataset at different b t The training results are as follows;

[0047] Figure 5 The algorithm of this invention uses the MNIST dataset at different b t The test results are as follows;

[0048] Figure 6 This paper compares the inference results of the algorithm of this invention with those of the classic distributed online optimization method iDLG eavesdropper. Detailed Implementation

[0049] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings.

[0050] Figure 1 As shown in the flowchart of the method described in this invention, in this embodiment, the privacy-preserving distributed zero-order dual average online optimization method provided by this invention includes the following steps:

[0051] 1) Set the number of nodes n=50, the sample dimension m=2, and the proximal function. And initialize all variables for all nodes i∈{1,2,3,…,50}: α0=1 / 20,

[0052] 2) Set the iteration count t for all nodes i∈{1,2,3,...,50} to 1, and the maximum iteration count T=200.

[0053] 3) Construct a network of n=50 sensors and carefully verify it to maintain strong connectivity.

[0054] 4) Determine if t≤T is true. If true, perform updates (5)-(12). If false, exit the algorithm.

[0055] 5) Using random and uniform sampling methods, at each time t is Each edge in the network is assigned a 40% probability, thus forming a new network, denoted as . This process guarantees time-varying directed networks Maintain strong connections at the specified B value.

[0056] 6) Calculate the zeroth gradient (sub-gradient estimate) using the following formula: Among the disturbances It is randomly and uniformly generated on a unit sphere. d = 2.

[0057] 7) Generate a random noise vector that follows a Laplace distribution. Right now Where b t =5.

[0058] 8) Dual variables Perform random vector The perturbation is used to obtain the dual variable with the added perturbation.

[0059] 9) Node i will and Transmitted to its outer neighbor node And receive from its inner neighbors Transmitted and Then, update the auxiliary variable using the two received variables. and dual variables in,

[0060] 10) Calculate The value of and update the variable Value:

[0061] 11) Set t to t+1.

[0062] 12) Return the original variable at time t.

[0063] In this embodiment, a distributed classification problem based on logistic regression is used to test the effectiveness of the proposed algorithm. In the experiment, n=50 nodes communicating in the network collaboratively train a linear classifier for a set of points moving in a specific feature space. At time t≥1, each node i is assigned v. i =50 (for all i) pairs of samples Training is conducted. Among them, a i,l ∈{-1,1} represents the label value. This represents the location of a point. Then, a linear classification model is built using the given training samples. Specifically, it is necessary to find a separating hyperplane described by a pair of (y,h)∈Ω1×Ω2. in When t≥1, a i,l =1 and a i,l The point with a value of -1 is the optimal hyperplane. Separate. That is, for all (i, l), if Then a i,l =1, otherwise a i,l =-1. At each time t≥1, the online classification problem can be set as the following minimization problem, where K=1 is the regularization parameter:

[0064]

[0065] Each point The rules for circular motion are as follows, where the radius r = 1, where... Indicates the randomly generated center of the circle:

[0066]

[0067] Meanwhile, the experimental parameters were set as follows: noise parameter b t =5, other parameter settings refer to the best implementation scheme. To effectively illustrate the experimental results, we use (Average regret) is used as a measure of the comparison results.

[0068] In this embodiment, the experimental results are as follows:

[0069] (1) Comparison of the performance of the algorithm of this invention with other related algorithms

[0070] This section focuses on demonstrating the convergence performance of the algorithm of this invention and the decentralized online optimization algorithm, specifically the decentralized double-average push algorithm, over a time span of T=200. The results are as follows... Figure 2As shown, the results indicate that the algorithm of the present invention is well applicable to time-varying directed networks, and even considering the critical aspects of privacy protection, the algorithm of the present invention can progressively achieve a sublinear convergence rate similar to the decentralized double average push algorithm.

[0071] (2) The effectiveness of the algorithm of this invention in machine learning problems

[0072] To further verify the effectiveness of the algorithm in large-scale machine learning, this embodiment uses the MNIST dataset to train a distributed logistic regression classification model. Figure 3 The dataset sample shown contains 13,007 samples of the digits 1 and 2, including... There are 1007 (randomly selected) training samples and 1007 test samples (the rest). Each sample is transformed into a vector of d = 784 dimensions. If sample c... i,l If the number is 1, then a i,l =-1; if sample c i,l If the number is 2, then a i,l =+1. Then, the training and testing results are as follows: Figure 4 and Figure 5 As shown, the algorithm of this invention can effectively optimize large-scale machine learning problems.

[0073] (3) The effectiveness of the algorithm of this invention in terms of privacy protection

[0074] This section further verifies the privacy protection characteristics of the algorithm of this invention, specifically whether the private cost functions of participating nodes can be preserved. Here, we consider the worst-case scenario, assuming that an attacker can know the constraint set Ω, all knowledge of the network, shared information in the network, and general update rules. Furthermore, we assume that the cost function is not a time-varying function, and that the algorithm of this invention uses a deterministic gradient g. t i Update. iDLG eavesdropper's inference results regarding LetNet are as follows: Figure 6 As shown in the figure. Compared with distributed online optimization methods, this result demonstrates the privacy-preserving characteristics of the algorithm of this invention.

[0075] 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 method for privacy-preserving distributed zeroth-order dual averaging online optimization, the method comprising: This method is designed for distributed online optimization objectives with a common set of constraints in time-varying directed networks. It adapts to situations where the objective function is unknown by using the zeroth gradient instead of computing the true subgradient information. By constructing a row random matrix and rescaling the zero-order gradient using auxiliary variables, the imbalance problem caused by time-varying directed networks is overcome. The privacy of normally participating nodes is protected by employing a differential privacy strategy; The time-varying directed network includes: for integers ,definition Directed network edge set of ordered node pairs Specify, each node The outer neighbor set and the inner neighbor set are respectively used and express; Using subscripts when the network changes over time To indicate time; Represents a network. Represents network edge set, and Representing nodes respectively At any moment The set of external and internal neighbors; correlation matrix If satisfied Under the conditions, then ,otherwise ; Assuming in At any given moment, each node in the directed network can only access information transmitted from its internal neighbors, and this information is transmitted through a channel with a good environment, free from delays, packet loss, and congestion. Based on this, each node utilizes each moment... The information transmitted by its neighbors within the channel is used to complete its iteration accordingly; The network also meets the following assumptions: at time... At that time, there exists an integer This makes the edge set The network formed is strongly connected; matrix At any moment It is a random sequence, that is... ; The goal of the nodes on the network at each moment is to cooperatively optimize the following problem: ， where is a non-empty closed convex set, is a global decision vector; in addition, at time, each node can only access the value of after the decision vector is determined, that is, each node cannot obtain in explicit form, where no node can access ; meanwhile, the problem satisfies the following assumptions: 1) There exist two constants d, D, 0 < d < D, such that ; 2) Each local loss function is continuous on , that is, for any , holds; This method specifically includes the following steps: S1, for all nodes Initialize all variables: , , , ;exist During initialization, the projection operator will be... Defined as ,in , ,at the same time It is a proximal function with 1-strong convexity, satisfying ; S2, 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; S3. Randomly construct a directed time-varying network. ; S4, judging whether it is true, if true, updating of steps S5-S12 is performed, if not true, the algorithm is exited; S5. Compute the subgradient estimate according to the following formula: where the perturbation is randomly and uniformly generated on the unit sphere, denotes the exploration parameter, denotes the shrinkage coefficient; S6. Generate a random noise vector that follows a Laplace distribution. ,Right now ,in Given a Laplace distribution function, noise that conforms to a Laplace distribution can be generated. These are adjustable parameters; S7, Dual Variables Perform random vector The perturbation is used to obtain the dual variable with the added perturbation. ; S8, node will and its outer neighbor nodes ; S9, Node Receive from its inner neighbors Transmitted and Then, update the auxiliary variable using the two received variables. and dual variables : ,in , ; S10, Update variables Value: ,in It is a decay step size; S11, to set to ; S12, return original variable of time .

2. A privacy-preserving distributed zero-order dual average online optimization system employing the method described in claim 1.

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