A homomorphic encryption distributed optimization method of multi-agent system

Abstract

The application discloses a homomorphic encryption distributed optimization method of a multi-agent system and belongs to the technical field of distributed control and optimization. In view of the communication bandwidth limitation, data security demand and time-varying directed topology constraint, a distributed optimization model under a time-varying directed graph is established; a dynamic coding-decoding quantization scheme is designed to discretize continuous signals and prove an error upper bound; Paillier homomorphic encryption is introduced to ensure transmission security; a distributed optimization algorithm suitable for asymmetric directed topology is designed based on a push-sum mechanism; a system error dynamic equation is established and a convergence sufficient condition is given.

Description

Technical Field

[0001] This invention belongs to the field of distributed control and optimization technology for multi-agent systems, and specifically relates to a distributed optimization method for time-varying directed graph communication topologies under the constraints of limited communication bandwidth and security requirements, specifically a homomorphic encrypted distributed optimization method for multi-agent systems. Background Technology

[0002] Distributed optimization has wide applications in modern cyber-physical systems, including sensor networks, smart grids, distributed machine learning, and collaborative robots. In these scenarios, a group of agents collaboratively solve a global optimization problem through local communication and computation. Each agent only knows its local objective function and can only exchange information with neighboring agents.

[0003] In practical distributed systems, two key constraints significantly increase the complexity of algorithm design and implementation. First, in resource-constrained environments, communication bandwidth limitations prevent the transmission of high-precision continuous-value data, necessitating quantization of the transmitted information, which introduces errors. Second, privacy and security requirements necessitate the protection of sensitive information exchanged over potentially insecure channels.

[0004] The challenges become even more severe when the underlying communication network is modeled as a time-varying directed graph. Unlike undirected or balanced graphs where information flows symmetrically, directed graphs introduce inherent asymmetry, complicating the design of distributed algorithms.

[0005] Existing distributed optimization methods typically address bandwidth constraints, security, or directed topology individually, but rarely consider all three constraints simultaneously. Regarding push-sum protocols, the subgradient push-sum method introduced by Nedic and Olshevsky has proven convergence on time-varying directed graphs. In quantization consensus research, existing work has shown that finite-bit communication can achieve approximate convergence. In secure multi-party computation, homomorphic encryption is attractive for distributed optimization. However, the simultaneous consideration of quantization, encryption, and directed time-varying graphs remains largely unexplored. Therefore, developing a unified framework to simultaneously address these three constraints is of great significance for the theoretical development and practical applications of distributed optimization. Summary of the Invention

[0006] To achieve the above objectives, the present invention provides the following technical solution: This invention provides a homomorphic encrypted distributed optimization method for multi-agent systems, characterized by comprising the following steps: Step (1): Establish a distributed optimization mathematical model for a multi-agent system under a time-varying directed graph communication network; Step (2): Design an encoding-decoding quantization scheme to convert continuous bandwidth signals into discrete quantized data, and prove the upper bound of the quantization error by mathematical induction; Step (3): Design an encryption-decryption scheme based on Paillier homomorphic encryption technology to ensure the security of communication data between intelligent agents; Step (4): Combining encoding-decoding quantization and homomorphic encryption, design a push and distributed optimization algorithm suitable for time-varying directed graphs; Step (5): Establish the system dynamic equations by expanding the weight matrix, and analyze the propagation law of quantization error in the network by using the exponential decay property of matrix multiplication; Step (6): Derive the dynamic equation of the average state, establish the error recursion relationship using the properties of convex functions and the subgradient inequality, and prove the convergence of the system using the existing convergence theorem; Step (7): Select a specific step size form, derive the explicit upper bound of the objective function error through summation and averaging techniques, and give the trade-off between convergence rate and quantization accuracy.

[0007] Furthermore, in step (1), n ​​agents are established to collaboratively solve the following optimization problem on a time-varying directed graph:

[0008] in As decision variables, For intelligent agents The local objective function is known only to the agent; time-varying directed graph A consistent strongly connected condition exists, i.e., there exists an integer Makes any Joint diagram It is strongly connected; each function It is a convex function, and secondly, its gradient norm is uniformly bounded.

[0009] Furthermore, in step (2), the design of the encoding-decoding quantization scheme includes: Encoder design: for continuous signals Perform quantization encoding and output quantized values.

[0010] in These are the decoder's internal state variables. Let quantization function be defined as: for any ,

[0011] The length in binary bits. For quantization resolution; Decoder design: Decoding and recovering quantized signals

[0012] Quantitative error analysis: Proving that the internal variables satisfy the following through mathematical induction. Thus, the quantization error satisfies

[0013] Furthermore, in step (3), the Paillier homomorphic encryption scheme includes: Key generation: Randomly select two large prime numbers and ,satisfy ;calculate , ;choose ,calculate ,in The public key is The private key is ; Encryption function: for quantized plaintext Select random number Calculate ciphertext

[0014] Decryption function: decrypts ciphertext Calculate plaintext

[0015] Homomorphic property: The additive homomorphism is used to support the weighted summation operation in the summation algorithm.

[0016] Furthermore, in step (4), the push and distributed optimization algorithm is designed as follows: Each agent Maintain state variables Auxiliary variables Decision variables and scalar variables The update rules are as follows:

[0017]

[0018] in For intelligent agents At any moment The set of neighbors, For intelligent agents The degree of exit, for exist The subgradient at that point, To decrease the step size, satisfy and ; and This refers to the data after encoding, encryption, decryption, and decoding.

[0019] Furthermore, in step (5), the quantization error propagation analysis is based on the weight matrix method, including: Define the weight matrix Its elements are:

[0020] An explicit expression of the system's dynamics is obtained through iterative expansion:

[0021] in The product of the weight matrices. This is the quantization error vector; Utilizing the properties of uniformly strongly connected graphs, and referring to the exponential decay lemma of matrix multiplication: there exists a random vector and constant , , making

[0022] Based on the above properties, the deviation bound between the decision variable and the average state is derived, and a comprehensive error expression including initial condition error, cumulative quantization error and gradient error is established.

[0023] Furthermore, in step (6), the convergence proof includes: Derivation of the average state dynamics: due to It is a sequence of random matrices, and the average state satisfies

[0024] in , ; Using the subgradient inequality and Lipschitz continuity of convex functions, for any optimal solution... Establish the error recursive relationship for the average state:

[0025] in and Non-negative parameters that satisfy the summability condition; Verify the application conditions of existing convergence theorems: when the quantization error is... Rate decay and step size Under the assumption that the Robbins-Monro conditions are satisfied, prove the error term. Satisfying the summability condition, applying the convergence lemma yields... converged to ; Combining the decision variable bias bounds derived in step (5), prove Thus, all agents converge to the same optimal solution.

[0026] Furthermore, in step (7), the convergence rate is derived as follows: Select step size ,in It is a constant; Multiply both sides of the error recursion relation by And on from arrive Summation; We define a weighted average by utilizing the convexity of the objective function. ,but ; By using summation techniques and bias bound estimation, an explicit upper bound on the objective function error is derived:

[0027] in , , It is a positive constant that depends on the network parameters, initial conditions, and Lipschitz constant. Attached Figure Description

[0028] Figure 1 This is a basic flowchart of the present invention; Figure 2 These are the error convergence curves of the objective function under different quantization accuracies; Figure 3 This is a verification diagram of the encoder-decoder quantization scheme; Figure 4 It is the graph of decision variables converging to the optimal solution. Detailed Implementation

[0029] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0030] To facilitate understanding of this application, a full description will be provided below with reference to the accompanying drawings, which illustrate preferred embodiments of the application. However, this application can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to make the disclosure of this application more thorough and complete.

[0031] This invention proposes a homomorphic encrypted distributed optimization method for multi-agent systems, comprising the following steps: Step 1: Establish a distributed optimization mathematical model for a multi-agent system under a time-varying directed graph communication network; Step 2: Design an encoding-decoding quantization scheme to convert continuous bandwidth signals into discrete quantized data, and prove the upper bound of the quantization error through mathematical induction; Step 3: Design an encryption-decryption scheme based on Paillier homomorphic encryption technology to ensure the security of communication data between intelligent agents; Step 4: Combining encoder-decode quantization and homomorphic encryption, design a push and distributed optimization algorithm suitable for time-varying directed graphs; Step 5: Establish the system dynamic equations using the weight matrix expansion method, and analyze the propagation law of quantization error in the network using the exponential decay property of matrix multiplication; Step 6: Derive the dynamic equation of the mean state, establish the error recursion relationship using the properties of convex functions and the subgradient inequality, and prove the convergence of the system using existing convergence theorems; Step 7: Select a specific step size and derive an explicit upper bound for the objective function error using summation and averaging techniques, thus providing a trade-off between convergence rate and quantization accuracy.

[0032] Step 1: Establish a distributed optimization mathematical model To achieve distributed optimization of multi-agent systems under bandwidth-constrained and secure communication requirements, this invention first establishes a mathematical model of the optimization problem in a time-varying directed graph communication network. Considering... A network composed of intelligent agents has its communication topology modeled as a time-varying directed graph. , where the node set Represents all intelligent agents, edge set This varies over time to reflect potential communication link failures. The collective goal of the network is to solve the following optimization problem in a distributed manner:

[0033] in As a global decision variable, For intelligent agents The locally convex objective function, and only the agent Know your objective function. Assume the optimal solution set. Not empty. Define the intelligent agent. At any moment The set of incoming neighbors is This indicates that all those who can... The node that sends the message; the set of outgoing neighbors is This indicates that all receivers are capable of receiving it. The node that sends the message; its out-degree is ,express The number of outgoing neighbors. This invention assumes that the communication network satisfies consistent strong connectivity, i.e., there exists a positive integer... Such that for any initial time Time range Union diagram within It is strongly connected, which ensures that information can be fully disseminated within the network.

[0034] Step 2: Encoding-Decoding Quantization Scheme Design Due to bandwidth limitations in practical communication networks, continuous signals cannot be transmitted with infinite precision. This invention designs a dynamic encoding-decoding quantization scheme to convert continuous signals into a discrete representation of finite bits, and rigorously proves the upper bound of the quantization error. For continuous signals that need to be transmitted... The present invention defines the dynamic encoding function as follows:

[0035] in These are the encoder's internal state variables. and Given pre-designed, known initial conditions, The uniform quantization function is defined as: for any ,

[0036] in The bit length determines the quantization precision. Let be the quantization step size. The decoding function at the receiver is defined as:

[0037] This invention rigorously proves the upper bound of quantization error using mathematical induction. First, the basic steps are established: at time... Set initial conditions At this point, the equation This is true. Then, the inductive step is performed: assuming that for time... Equation If true, then for time... The encoder's internal state is The decoder output is From the inductive hypothesis, we can directly derive Therefore, through mathematical induction, for all... This is true. Based on this result, the quantization error can be expressed as:

[0038] Utilizing the basic properties of uniform quantizers Finally, a strict upper bound on the quantization error is obtained:

[0039] This upper bound indicates that the quantization error is determined by the quantization step size. Encoding length Together with the encoding parameters, they provide crucial error estimates for subsequent convergence analysis.

[0040] Step 3: Design of the Paillier Homomorphic Encryption Scheme Considering that the state information transmitted by agents in distributed optimization may contain sensitive data, and is vulnerable to eavesdropping and tampering attacks on insecure communication channels, this invention employs Paillier homomorphic encryption technology to design an encryption-decryption scheme. This scheme ensures data security while leveraging homomorphic properties to support addition operations on ciphertext. The Paillier encryption scheme comprises three core algorithms: key generation, encryption, and decryption. Key generation algorithm: The system randomly selects two large prime numbers. and Requirements To ensure the validity of the key, calculate the modulus. and Carmichael function ,in This represents the least common multiple. From the set... Randomly select generator The set is defined as That is, all of them Coprime integers. Calculate auxiliary parameters. , where the function Defined as After the key is generated, output the public key. For encryption purposes, private key For decryption purposes.

[0041] Encryption algorithm: For quantized plaintext The encryptor needs to randomly select a random number. To enhance the security of the ciphertext, the ciphertext is then computed:

[0042] This encryption scheme possesses an important additive homomorphic property: for two plaintexts... and Its encrypted ciphertext satisfies This allows the receiver to directly perform a summation operation on the encrypted data without first decrypting it.

[0043] Decryption algorithm: For the received ciphertext The recipient holding the private key calculates the plaintext:

[0044] In the push-sum algorithm of this invention, each agent first processes its quantized state before transmitting its state information. and weight Encryption yields ciphertext and The encrypted message is then transmitted via a communication network. Upon receiving the encrypted message, the receiving agent decrypts it using its private key. and The data is then decoded to obtain the results used for subsequent calculations. The entire process ensures the confidentiality and integrity of the transmitted data.

[0045] Step 4: Design of Push and Distributed Optimization Algorithm To achieve distributed optimization under bandwidth-constrained and secure communication requirements, this invention organically combines the quantization scheme in step two and the encryption scheme in step three to design an encrypted quantization push-sum algorithm suitable for time-varying directed graphs. The core idea of ​​this algorithm is to utilize the push-sum mechanism to handle the asymmetry of directed graphs, while simultaneously quantizing and encrypting the data before each communication. Each agent... Four core variables need to be maintained: auxiliary state Weighted sum Used to accumulate neighbor information, decision variables As candidate solutions to the optimization problem, and weight scalar Used for normalization. The algorithm's update rule is designed as follows:

[0046]

[0047] in and Neighbors The state and weights are encoded and encrypted in a complete process before being transmitted to the agent. By intelligent agents The result obtained after decryption and decoding. This represents the subgradient of the objective function at the decision variables. For intelligent agents At any moment The out-degree. During the initialization phase, each agent will set its auxiliary state. Set as any vector, weights Unified initialization Step size sequence The classic Robbins-Monro condition must be satisfied to guarantee convergence:

[0048] This invention recommends selecting a step size. ,in This is a normal number that is adjusted based on the problem size and network parameters.

[0049] The algorithm's execution flow is divided into two phases: sender operations and receiver operations. Sender operations: at each time step... intelligent agent As the sender of information, first check its status. Each component and weight Call the encoder from step two to obtain the quantized result. and Then, the encryption algorithm from step three is called to encrypt the quantized value to obtain the ciphertext. and Finally, the encrypted ciphertext is broadcast to all outgoing neighbors via the communication network. Receiver's action: Intelligent agent. As the information receiver, from its incoming neighbor set Each neighbor in Receive ciphertext; upon receiving the ciphertext, the intelligent agent... First, the decryption algorithm in step three is called to decrypt the ciphertext using the private key to obtain the quantized value; then, the decoder in step two is called to restore the decrypted value. and After decoding is complete, the intelligent agent... The weighted sum is obtained by summing the decoded values ​​from all incoming neighbors and updating the summation algorithm. and weights ; Calculate decision variables As the current candidate solution for optimization; in Calculate the subgradient of the local objective function. Finally, the auxiliary state is updated using the subgradient descent method. This completes one iteration.

[0050] Step 5: Quantization Error Propagation Analysis To analyze how quantization errors evolve over time in a network and affect system performance, this invention employs a weight matrix expansion method to establish the system's dynamic equations and utilizes the exponential decay property of matrix multiplication to reveal the error propagation law. First, time intervals are defined. weight matrix To depict the information flow structure in time, its first Element is defined as:

[0051] This matrix has column randomness, meaning the sum of the elements in each column is 1. , represented as This reflects the conservation property of the push-sum algorithm. Define the quantization error vector. , of which Each component represents the agent. The weighted sum of the quantization errors of all incoming neighbors. According to the update rule in step four, the system dynamics after introducing quantization errors can be expressed as:

[0052] By repeatedly applying this recursive relationship, the present invention fully expands the system dynamics into a linear combination of the initial state, accumulated quantization error, and accumulated subgradient:

[0053] in Indicates from time arrive The product of the weight matrices, and To analyze the properties of this matrix product, this invention introduces a key lemma: if the communication graph sequence If a uniform strongly connected property is satisfied, then there exists a random vector. (its elements) and ),constant and attenuation factor Such that the elements of the weight matrix product satisfy:

[0054] in , This inequality shows that the product of the weight matrices converges exponentially to the rank-one matrix. This invention defines a deviation matrix. To quantify this bias, we know from the above lemma that its elements satisfy... Using this decomposition technique, the system dynamics can be rewritten as:

[0055] in This is the exponentially decaying term introduced by the deviation matrix. This expression clearly separates the consistency term (representing the average state of the network) and the deviation term (decaying at an exponential rate). Furthermore, this invention derives a similar expansion of the weight scalar and defines a lower bound for the weights. By the relevant lemma, we can obtain Based on these results, this invention establishes a deviation bound between the decision variables and the global average state:

[0056] This deviation bound reveals the pattern of how quantization error propagates and accumulates in the network through the exponential decay mechanism of the weight matrix, providing a key technical tool for convergence analysis.

[0057] Step Six: Convergence Proof To prove that the proposed encrypted quantization summation algorithm can converge the decision variables of all agents to the same optimal solution, this invention derives the evolution law of the global average state, establishes the recursive relationship of the error using the basic properties of convex functions and the subgradient inequality, and finally completes the proof by applying existing convergence theorems. Due to the weight matrix... It has columnar random properties, and the system dynamic equation in step four is left-multiplied. We can obtain:

[0058] Define the global average state Then its dynamic equation is:

[0059] in For the average quantization error, Let be the average subgradient. This equation shows that the average state is driven by both the average quantization error and the average subgradient. For any vector Calculate the average state to The distance, and unfolded as:

[0060] This invention utilizes the fundamental properties of subgradients to establish key inequalities. Consider...

[0061] The subgradient inequality is given as follows:

[0062] because It is convex, and yes exist The subgradient at that point, then

[0063] Decompose the cross terms:

[0064] Utilizing the boundedness of the subgradient And convexity, summed over all agents, yields:

[0065] This utilizes convexity. Substituting the above inequalities into the error expansion and estimating the upper bounds of each term, this invention yields the complete error recursive relationship:

[0066] Among them, higher-order error terms Including with Terms related to quantization error. And verify the following three key conditions: (1) Assume Due to the step size condition It can be known (2) item The expansion from the push and the deviation, and the deviation bound and exponential decay property established in step five, can be rigorously proven to be summable; (3) terms Includes the squared term of the step size (by...) ) and quantization error term, where the quantization error satisfies Under the conditions, it can be proven Since all three conditions are satisfied, applying existing convergence theorems yields: the average state sequence Converging to a certain optimal solution Based on the results of step five, this invention ultimately proves the decision variables of all agents. They converge to the same optimal solution, thus completing the full proof of convergence.

[0067] Step 7: Derivation of Convergence Rate To provide a quantitative convergence rate for the algorithm and reveal the trade-off between convergence performance and quantization accuracy, this invention selects a specific step size form. (in (As positive constants), and through clever summation and weighted averaging techniques, an explicit upper bound expression for the objective function error is derived. The error recursion obtained in step six is ​​then rearranged, using... By replacing and adjusting the error term using Lipschitz continuity, we obtain:

[0068] This invention employs a key summation technique: multiplying both sides of the above equation by a weighting factor. Then to from arrive Summation. The summation on the left is:

[0069] The terms on the right are summed to form a scaling series. Most terms cancel each other out, and only the first and last terms are retained, resulting in:

[0070] Note Therefore, this term is a negative contribution. This invention further utilizes Jensen's inequality for convex functions to introduce a weighted average variable. Definition:

[0071] because For convex functions, Jensen's inequality provides:

[0072] therefore:

[0073] Substituting the previous summation inequality into the equation, and carefully estimating each error term on the right side, the summation of the initial error terms is: Level, contribution The attenuation rate. Accumulation of quantization error terms: Since the single-step quantization error is as proven in step two... Combining the exponential decay factor from step five, the cumulative summation can be estimated as follows: or This depends on the coupling mode of the error. The accumulation of the subgradient term: due to the combined effect of exponential decay and the step size, the summation can be strictly controlled to... Level, contribution The convergence rate. Combining the estimates of all terms, this invention yields an explicit upper bound on the objective function error:

[0074] in For constant terms that depend on the initial error and the Lipschitz constant, , These are constant factors that contribute to the accumulation of quantization errors under different mechanisms. These constants explicitly depend on network connectivity parameters (such as decay rate). Weight lower bound Number of agents Problem Dimension and Lipschitz constant And so on. This upper bound clearly reveals the trade-off between convergence rate and quantization accuracy: when the number of iterations... When it approaches infinity, Item and If the term approaches zero, and the quantization accuracy is increased simultaneously... If this is true, then all relevant terms also tend to zero, and the algorithm converges to the exact optimal solution. With a fixed quantization precision, the dominant convergence rate is... This rate is the same as that of the classical summation algorithm in the unquantized case, proving that the quantization scheme designed in this invention does not fundamentally reduce the convergence speed while maintaining communication efficiency. Furthermore, this upper bound also guides the selection of quantization parameters: to ensure… To ensure convergence, the quantization error must meet the following requirements. That is, the quantization accuracy should increase at a superlinear rate with the number of iterations.

[0075] Lemmas and Theorems The following lemmas and theorems play a key role in the above proof: Lemma 1: If a graph sequence For all All - If the vector is uniformly strongly connected, then there must exist a random vector. Its elements are ,for There exists a constant and , making

[0076] in Indicates from time At the time The product of weighted matrices.

[0077] Lemma 2: Let the graph sequence yes - Uniformly strongly connected. Then... For all and Established.

[0078] Lemma 3: Let It is a random vector A scalar sequence. If for all have ,and , ,and

[0079] in , , ,So and It converges to a certain finite value.

[0080] Lemma 4: Given an optimization problem ,in It is continuous. Assume the solution set... Not empty. Let... It is any sequence that satisfies the following conditions, for each and :

[0081] Where the nonnegative parameters satisfy , and .So converged to one of the elements .

[0082] Theorem 1: Consider a set of... A system composed of intelligent agents, satisfying - Uniform strongly connected graphs and directed graphs based on Assumption 1 Run on. Assume all local objective functions Assumption 2 is satisfied, and the subgradient satisfies Assumption 3. Assume the quantization error is uniformly bounded and the step size satisfies the Robbins-Monro condition. Then, the induction and summation methods under the encryption-decryption scheme converge to the same optimal solution. For decision variables of all nodes .

[0083] Theorem 2: Under the conditions of Theorem 1, take , For each intelligent agent The objective function error satisfies

[0084] in , It is a positive constant that depends on the network parameters, initial conditions, and Lipschitz constant.

[0085] In one embodiment, such as Figure 1 As shown, this invention proposes a homomorphic encryption inference and distributed optimization method for multi-agent systems under encoding-decoding quantization, comprising the following steps: Establish a distributed optimization mathematical model for a multi-agent system under a time-varying directed graph communication network; Design an encoding-decoding quantization scheme to convert continuous bandwidth signals into discrete quantized data, and prove the upper bound of the quantization error through mathematical induction; An encryption-decryption scheme based on Paillier homomorphic encryption technology is designed to ensure the security of communication data between intelligent agents. Combining encoder-decode quantization and homomorphic encryption, we design a push and distributed optimization algorithm suitable for time-varying directed graphs; The system dynamic equations are established by expanding the weight matrix, and the propagation law of quantization error in the network is analyzed by utilizing the exponential decay property of matrix multiplication. The dynamic equations of the mean state are derived, and the error recursion is established using the properties of convex functions and the subgradient inequality. The convergence of the system is then proved by applying existing convergence theorems. By choosing a specific step size, an explicit upper bound for the objective function error is derived through summation and averaging techniques, giving a trade-off between convergence rate and quantization accuracy.

[0086] In step one, to achieve distributed optimization of a multi-agent system under bandwidth-constrained and secure communication requirements, this invention first establishes a mathematical model of the optimization problem under a time-varying directed graph communication network. Considering... A network composed of intelligent agents has its communication topology modeled as a time-varying directed graph. , where the node set Represents all intelligent agents, edge set This varies over time to reflect potential communication link failures. The collective goal of the network is to solve the following optimization problem in a distributed manner: (2) in As a global decision variable, For intelligent agents The locally convex objective function, and only the agent Know your objective function. Assume the optimal solution set. Not empty. Define the intelligent agent. At any moment The set of incoming neighbors is This indicates that all those who can... The node that sends the message; the set of outgoing neighbors is This indicates that all receivers are capable of receiving it. The node of information; out-degree is ,express The number of outgoing neighbors. This invention assumes that the communication network satisfies consistent strong connectivity, i.e., there exists a positive integer... Such that for any initial time Time range Union diagram within It is strongly connected, which ensures that information can be fully disseminated within the network.

[0087] In step two, due to bandwidth limitations in actual communication networks, continuous signals cannot be transmitted with infinite precision. This invention designs a dynamic encoding-decoding quantization scheme to convert continuous signals into a discrete representation of finite bits, and rigorously proves the upper bound of the quantization error. For the continuous signals that need to be transmitted... The present invention defines the dynamic encoding function as follows: (3) in These are the encoder's internal state variables. and For pre-designed known encoding parameters, The uniform quantization function is defined as follows: (4) in The binary bit length determines the quantization precision. Let be the quantization step size. The decoding function at the receiver is defined as: (5) This invention rigorously proves the upper bound of quantization error using mathematical induction. First, the basic steps are established: At any time, set initial conditions At this point, the equation holds true. Then, proceed with the inductive step: assuming that for... Time, Equation If it is established, then for At time t, the encoder's internal state is The decoder output is By inductive hypothesis It can be directly derived Therefore, through mathematical induction, For all This is true. Based on this result, the quantization error... It can be represented as: (6) Utilizing the basic properties of uniform quantizers Finally, a strict upper bound on the quantization error is obtained: (7) This upper bound indicates that the quantization error is determined by the quantization step size. Encoding length and encoding parameters This joint decision provides crucial error estimation for subsequent convergence analysis.

[0088] In step three, considering that the state information transmitted by agents in distributed optimization may contain sensitive data and is vulnerable to eavesdropping and tampering attacks on insecure communication channels, this invention employs Paillier homomorphic encryption technology to design an encryption-decryption scheme. This scheme ensures data security while leveraging homomorphic properties to support addition operations on ciphertext. The Paillier encryption scheme comprises three core algorithms: key generation, encryption, and decryption. Key generation algorithm: The system randomly selects two large prime numbers. and Requirements To ensure the validity of the key, calculate the modulus. and Carmichael function ,in This represents the least common multiple. From the set... Randomly select generator The set is defined as That is, all of them Coprime integers. Calculate auxiliary parameters. ,in The function is defined as After the key is generated, output the public key. For encryption purposes, private key For decryption purposes.

[0089] Encryption Algorithm: For the quantized plaintext m, the encryptor needs to randomly select a random number. To enhance the security of the ciphertext, the ciphertext is then computed: (8) This encryption scheme possesses an important additive homomorphic property: for two plaintexts... and Its encrypted ciphertext satisfies This allows the receiver to directly perform a summation operation on the encrypted data without first decrypting it.

[0090] Decryption algorithm: For the received ciphertext The recipient holding the private key calculates the plaintext: (9) In the push-sum algorithm of this invention, each agent Before transmitting its state information, the quantized state is first processed. and weight Encryption yields ciphertext and The encrypted message is then transmitted via a communication network. The receiving intelligent agent... After receiving the ciphertext, decrypt it using the private key to obtain... and Then, after decoding, we get and Used for subsequent push and calculation. The entire process ensures the confidentiality and integrity of the transmitted data.

[0091] In step four, to achieve distributed optimization under bandwidth-constrained and secure communication requirements, this invention organically combines the quantization scheme from step two and the encryption scheme from step three to design an encrypted quantization push-sum algorithm suitable for time-varying directed graphs. The core idea of ​​this algorithm is to utilize the push-sum mechanism to handle the asymmetry of directed graphs, while simultaneously quantizing and encrypting the data before each communication. Each agent... Four core variables need to be maintained: auxiliary state Weighted sum Used to accumulate neighbor information, decision variables As candidate solutions to the optimization problem (with the goal of converging to the optimal solution), and weight scalar Used for normalization. The algorithm's update rule is designed as follows:

[0092]

[0093]

[0094] (10) in and Neighbors The state and weights are encoded and encrypted in a complete process before being transmitted to the agent. By intelligent agents The result obtained after decryption and decoding. For the objective function In decision variables The subgradient at that point, For intelligent agents exist The time-out of time. During the initialization phase, each agent... Auxiliary state Set as any vector (selectable based on problem characteristics), weights Initialize to 1. Step size sequence. The classic Robbins-Monro condition must be satisfied to guarantee convergence: (11) This invention recommends selecting a step size. ,in This is a constant adjusted according to the problem size and network parameters. The specific execution flow of the algorithm is divided into two stages: sender operation and receiver operation. Sender operation: at each time step... intelligent agent As the sender of information, first check its status. Each component and weight Call the encoder from step three to obtain the quantized result. and Then, the encryption algorithm from step two is called to encrypt the quantized value to obtain the ciphertext. and Finally, the encrypted ciphertext and Broadcast to all outgoing neighbors via the communication network. Receiver action: Intelligent agent. As the information receiver, from its incoming neighbor set Each neighbor in Receive ciphertext and Upon receiving the encrypted message, the intelligent agent... First, the decryption algorithm from step two is called, and the ciphertext is decrypted using the private key. and Then, the decoder from step three is invoked to restore the decrypted value. and After decoding is complete, the intelligent agent... The weighted sum is obtained by summing the decoded values ​​from all incoming neighbors and applying a summation algorithm. and weights ; Calculate decision variables As the current candidate solution for optimization; in Calculate the local objective function subgradient Finally, the auxiliary state is updated using the subgradient descent method. This completes one iteration.

[0095] In step five, to analyze how quantization errors evolve over time in the network and affect system performance, this invention employs a weight matrix expansion method to establish the system's dynamic equations and utilizes the exponential decay property of matrix multiplication to reveal the error propagation law. First, define... weight matrix To depict moments The information flow structure, its first Element is defined as: (12) This matrix has the property of column randomness, meaning that the sum of the elements in each column is 1, and is represented as follows: This reflects the conservation property of the push-sum algorithm. Define the quantization error vector. , of which Each component Represents intelligent agents In the The weighted sum of the quantization errors of all incoming neighbors received by the dimension. According to the update rule in step four, the system dynamics after introducing quantization errors can be expressed as: (13) By repeatedly applying this recursive relationship, the present invention fully expands the system dynamics into a linear combination of the initial state, accumulated quantization error, and accumulated subgradient:

[0096] (14) in Indicates from time arrive The weight matrix is ​​a product of consecutive matrices. To analyze the properties of this matrix product, this invention introduces a key lemma: if the communication graph sequence If a uniform strongly connected property is satisfied, then there exists a random vector. (its elements) and ),constant and attenuation factor Such that the elements of the weight matrix product satisfy: (15) in and This inequality shows that the product of the weight matrices grows at an exponential rate. Converging to matrix This invention defines a deviation matrix. To quantify this bias, we know from the above lemma that its elements satisfy... Using this decomposition technique, the system dynamics can be rewritten as: (16) This expression clearly separates the consistency terms. (Representing the network average state) and the bias term (decaying at an exponential rate). Furthermore, this invention derives a weight scalar. A similar expansion, and a lower bound for the weights is defined. ε, from the relevant lemma, can be obtained Based on these results, this invention establishes decision variables. Compared with the global average state Deviation limits between:

[0097]

[0098] (17) This deviation bound reveals the pattern of how quantization error propagates and accumulates in the network through the exponential decay mechanism of the weight matrix, providing a key technical tool for convergence analysis.

[0099] In step six, to prove that the encrypted quantization summation algorithm proposed in this invention can make the decision variables of all agents converge to the same optimal solution, this invention derives the evolution law of the global average state, establishes the recursive relationship of the error using the basic properties of convex functions and the subgradient inequality, and finally completes the proof by applying existing convergence theorems. Due to the weight matrix... It has column random properties Left-multiply the system dynamic equations in step four We can obtain: (18) Define the global average state Then its dynamic equation is: (19) This equation shows that the average state is driven by both the average quantization error and the average subgradient. For any vector Calculate the average state to The distance, and unfolded as:

[0100]

[0101]

[0102] (20) This invention utilizes the fundamental properties of subgradients to establish key inequalities. Consider...

[0103] (twenty one) The subgradient inequality is given as follows: (twenty two) because It is convex, and yes exist The subgradient at that point, then

[0104] (twenty three) For cross terms Decompose:

[0105] Utilizing the boundedness of the subgradient and convexity Summing over all agents yields: (25) It utilizes convexity. Substituting the above inequalities into the error expansion and estimating the upper bounds of each term, this invention yields the complete error recursive relationship:

[0106] (26) The higher-order error term includes... Terms related to quantization error. And verify the following three key conditions: (1) Assume Due to the step size condition It can be known (2) Item from The expansion is based on the deviation bound and exponential decay property established in step five. It can be rigorously proven (3) The term includes the square of the step size. (Depend on ) and quantization error term, where the quantization error satisfies Under the conditions, it can be proven Since all three conditions are satisfied, applying existing convergence theorems yields: the average state sequence Converging to a certain optimal solution z*∈Z*. Combined with the results of step five. This invention ultimately proves the decision variables of all agents. Converging to the same optimal solution This completes the proof of convergence.

[0107] In step seven, to provide the quantitative convergence rate of the algorithm and reveal the trade-off between convergence performance and quantization accuracy, this invention selects a specific step size. (in (As positive constants), and through clever summation and weighted averaging techniques, an explicit upper bound expression for the objective function error is derived. The error recursion obtained in step six is ​​then rearranged, using... replace By adjusting the error term using Lipschitz continuity, we can obtain: (27) This invention employs a key summation technique: multiplying both sides of the above equation by a weighting factor. Then to From 1 to Summation. The summation on the left is: (28) The main item on the right After summing to form a stretching series, most terms cancel each other out, and only the first and last terms are retained, resulting in:

[0108] (29) Note Therefore, this term is a negative contribution. This invention further utilizes Jensen's inequality for convex functions to introduce a weighted average variable. Definition: (30) because For convex functions, Jensen's inequality provides: (31) therefore: (32) Substituting the previous summation inequality into the equation, and carefully estimating each error term on the right side, the summation of the initial error terms is: Level, contribution The attenuation rate. Accumulation of quantization error terms: Since the single-step quantization error is as proven in step two... Combined with the exponential decay factor in step five Cumulative summation It can be estimated as or This depends on how the error is coupled. The accumulation of the subgradient term: due to exponential decay... and step length The combined effect of the two, summation It can be strictly controlled as Level, contribution The convergence rate. Combining the estimates of all terms, this invention yields an explicit upper bound on the objective function error: (33) in For constant terms that depend on the initial error and the Lipschitz constant, , , These are constant factors that contribute to the accumulation of quantization errors under different mechanisms. These constants explicitly depend on network connectivity parameters (such as decay rate). Lower Boundary Number of agents Problem Dimension and Lipschitz constant And so on. This upper bound clearly reveals the trade-off between convergence rate and quantization accuracy: as the number of iterations τ approaches infinity, Item and If the term approaches zero, and the quantization accuracy is increased simultaneously... Then all The correlation terms also tend to zero, and the algorithm converges to the exact optimal solution. Under a fixed quantization precision... At that time, the dominant convergence rate is This rate is the same as that of the classical summation algorithm in the unquantized case, proving that the quantization scheme designed in this invention does not fundamentally reduce the convergence speed while maintaining communication efficiency. Furthermore, this upper bound also guides the selection of quantization parameters: to ensure… To ensure convergence, the quantization error must meet the following requirements. That is, the quantization accuracy should increase at a superlinear rate with the number of iterations.

[0109] The following lemmas will be used in the following proof: Lemma 1: If a graph sequence For all If all vectors are uniformly strongly connected, then there must exist a random vector. Its elements are ,for There exists a constant and , making ,in Indicates from time At the time The product of weighted matrices. We can choose... and .

[0110] Lemma 2: Let the graph sequence It is uniformly strongly connected. Therefore...

[0111] satisfy .

[0112] Lemma 3: Let It is a random vector A scalar sequence. If for all have ,and ,and ,So

[0113] Lemma 4: Given an optimization problem ,in It is continuous. Assume the solution set... Not empty. Let... It is any sequence that satisfies the following conditions, for each and :

[0114] Where the nonnegative parameter satisfy , and .So converged to one of the elements .

[0115] Lemma 5:

[0116]

[0117]

[0118] Proof: By checking equation (10), it is easy to see that for all

[0119] (34) From this we can obtain

[0120]

[0121] (35) Furthermore, due to each matrix Both are column random matrices, therefore we have .

[0122] In addition, equation (35) shows (36) From equations (35) and (36), we obtain that for all

[0123] (37) If we define for

[0124] Then we have the element-wise decay bound. (38) Where the constant Therefore, from relation (37), we can obtain that for : (39) Therefore, for We have (40) Similarly, represented using the same structure

[0125]

[0126]

[0127] (41) From (40) and (41), for each and all We got therefore,

[0128]

[0129] Note that part of the denominator of the fraction equals Multiply The Lines and. According to The definition of this and has a lower bound. This led to

[0130] Therefore, for all and ,

[0131]

[0132]

[0133] From (36), we have From the two relations above, we can conclude that for all and ,

[0134] Therefore, based on the above lemma, the following theorem can be used to obtain sufficient conditions for all agents' decision variables to converge to the same optimal solution, as well as the convergence rate of the algorithm.

[0135] In order to prove the theorem later, we make the following assumptions: Assumption 1: Each agent knows its out-neighbor set. .

[0136] Assumption 2: Each function It is convex.

[0137] Assumption 3: Each function The subgradient has a uniformly bounded norm. Specifically, there exists a constant. Makes any , Each subgradient satisfy:

[0138] Theorem 1: Consider a set of... A system composed of several agents, in a directed graph that satisfies uniform strong connectivity and Assumption 1. Run on. Assume all locally time-varying cost functions Satisfies Assumption 2, and the subgradient Assumption 3 is satisfied. Assumption

[0139] Therefore, the Push-Sum method converges to the same optimal solution under the encryption-decryption scheme. For the state of all nodes .

[0140] prove:

[0141]

[0142]

[0143]

[0144] therefore

[0145]

[0146] The dynamic equation with errors can be rewritten as: in It is the adjacency matrix that was defined earlier. Let... From equation (7), we can obtain in If the step size satisfies assumption 3, the above inequality means

[0147] (45) because

[0148] It can be concluded that It can be obtained from equation (44) (46) make Let be any vector. From relation (46), we can obtain that for all ,

[0149] (47) Next, consider

[0150] (48) (49) because It is convex, and yes exist The subgradient at that point, then

[0151]

[0152] use Lipschitz continuity therefore, (50) Substitute (49) and (50) into (48), and use achievable (51) Note Combining (47) and (51), we can obtain

[0153] because ,but . ,but Furthermore, based on the assumption we have , , , and the results obtained in the previous proof Therefore, the condition of Lemma 4 is satisfied. Let Obviously, the sequence converged to .

[0154] Theorem 2: Under the conditions of Theorem 1, take , ,in It is a constant. For each agent We have in

[0155]

[0156] prove:

[0157] (52) because It is a continuation of Lipschitz, we have (53) Substitute (53) into (52),

[0158] make Substitute (54) into the equation.

[0159] Multiply both sides by And on Summing, we get

[0160]

[0161] use , eliminate and divide by ,get:

[0162]

[0163]

[0164] because It is convex, and ,therefore (55) From equation (48), for Summing, we get

[0165] (56) Combining (55) and (56), we can conclude that Theorem 2 is proved.

[0166] Experimental verification This paper experimentally verifies the theoretical convergence result of the induction algorithm with encoder-decoder quantization and demonstrates the impact of quantization accuracy on convergence performance. The objective function to be optimized is:

[0167] Experimental setup includes There are 1 intelligent agents, each intelligent agent in Maintain decision variables, among which The algorithm runs for a maximum of 5000 iterations, using a decreasing step size. Parameter vector and scalar Randomly generated from the standard normal distribution.

[0168] Figure 2This demonstrates the objective function error of 100 nodes in two-dimensional space. As the number of iterations changes, the three curves correspond to different quantization precision parameters. The results show that in the initial stage (the first 500 iterations), all schemes exhibit rapid convergence, with the error decreasing from [previous value]. The magnitude dropped rapidly. It is worth noting that... A larger value corresponds to higher quantization accuracy, resulting in faster convergence speed and smaller steady-state error. (Yellow dotted line) represents the fastest decay, followed by... (Orange dashed line) and (Solid blue line). The observed convergence rate is consistent with the theoretical analysis, showing the predicted convergence rate. attenuation.

[0169] Figure 3 This demonstrates the performance of the encoder-decoder quantization scheme, with the following parameters: The blue curve marked with a circle represents the original signal. The red curve marked with a box shows the quantized decoded signal. It can be observed that, despite quantization, the decoded signal still closely tracks the original signal, and the quantization error remains within [the specified range]. Within the bounds. The discrete-time representation clearly proves that the step size is... The uniform quantizer successfully reconstructed the signal while satisfying the theoretical error bound. This verifies the effectiveness of the proposed quantization scheme in bandwidth-constrained communication in distributed networks.

[0170] Figure 4 The results show that the decision variables of all agents converge to the neighborhood of the optimal solution, and the error stabilizes at around 0.11-0.12 after 500 iterations, depending on the quantization accuracy. These results validate our theoretical predictions: convergence was achieved despite the use of quantized communication, and the observed... The rate is consistent with theoretical findings, higher quantization accuracy improves performance, and the algorithm is effectively extended to 100-node networks with time-varying directed graphs.

[0171] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A homomorphic encrypted distributed optimization method for multi-agent systems, characterized in that, Includes the following steps: Step (1): Establish a distributed optimization mathematical model for a multi-agent system under a time-varying directed graph communication network; Step (2): Design an encoding-decoding quantization scheme to convert continuous bandwidth signals into discrete quantized data, and prove the upper bound of the quantization error by mathematical induction; Step (3): Design an encryption-decryption scheme based on Paillier homomorphic encryption technology to ensure the security of communication data between intelligent agents; Step (4): Combining encoding-decoding quantization and homomorphic encryption, design a push and distributed optimization algorithm suitable for time-varying directed graphs; Step (5): Establish the system dynamic equations by expanding the weight matrix, and analyze the propagation law of quantization error in the network by using the exponential decay property of matrix multiplication; Step (6): Derive the dynamic equation of the average state, establish the error recursion relationship using the properties of convex functions and the subgradient inequality, and prove the convergence of the system using the existing convergence theorem; Step (7): Select a specific step size form, derive the explicit upper bound of the objective function error through summation and averaging techniques, and give the trade-off between convergence rate and quantization accuracy.

2. The homomorphic encrypted distributed optimization method for a multi-agent system as described in claim 1, characterized in that, In step (1), n ​​agents are established to collaboratively solve the following optimization problem on a time-varying directed graph: in As decision variables, For intelligent agents The local objective function is known only to the agent; time-varying directed graph A consistent strongly connected condition exists, i.e., there exists an integer Makes any Joint diagram It is strongly connected; each function It is a convex function, and secondly, its gradient norm is uniformly bounded.

3. The homomorphic encrypted distributed optimization method for a multi-agent system as described in claim 1, characterized in that, In step (2), the design of the encoding-decoding quantization scheme includes: Encoder design: for continuous signals Perform quantization encoding and output quantized values. in These are the decoder's internal state variables. Let quantization function be defined as: for any , The length in binary bits. For quantization resolution; Decoder design: Decoding and recovering quantized signals Quantitative error analysis: Proving that the internal variables satisfy the following through mathematical induction. Thus, the quantization error satisfies 。 4. The homomorphic encrypted distributed optimization method for a multi-agent system as described in claim 1, characterized in that, In step (3), the Paillier homomorphic encryption scheme includes: Key generation: Randomly select two large prime numbers and ,satisfy ;calculate , ;choose ,calculate ,in The public key is The private key is ; Encryption function: for quantized plaintext Choose a random number Calculate ciphertext Decryption function: decrypts ciphertext Calculate plaintext Homomorphic property: The additive homomorphism is used to support the weighted summation operation in the summation algorithm.

5. The homomorphic encrypted distributed optimization method for a multi-agent system as described in claim 1, characterized in that, In step (4), the push and distributed optimization algorithm is designed as follows: Each agent Maintain state variables Auxiliary variables Decision variables and scalar variables The update rules are as follows: in For intelligent agents At any moment The set of neighbors, For intelligent agents The degree of exit, for exist The subgradient at that point, To decrease the step size, satisfy and ; and This refers to the data after encoding, encryption, decryption, and decoding.

6. The homomorphic encrypted distributed optimization method for a multi-agent system as described in claim 1, characterized in that, In step (5), the quantization error propagation analysis is based on the weight matrix method, including: Define the weight matrix Its elements are: An explicit expression of the system's dynamics is obtained through iterative expansion: in The product of the weight matrices. This is the quantization error vector; Utilizing the properties of uniformly strongly connected graphs, and referring to the exponential decay lemma of matrix multiplication: there exists a random vector and constant , , making Based on the above properties, the deviation bound between the decision variable and the average state is derived, and a comprehensive error expression including initial condition error, cumulative quantization error and gradient error is established.

7. The homomorphic encrypted distributed optimization method for a multi-agent system as described in claim 1, characterized in that, In step (6), the convergence proof includes: Derivation of the average state dynamics: due to It is a sequence of random matrices, and the average state satisfies in , ; Using the subgradient inequality and Lipschitz continuity of convex functions, for any optimal solution... Establish the error recursive relationship for the average state: in and Non-negative parameters that satisfy the summability condition; Verify the application conditions of existing convergence theorems: when the quantization error is... Rate decay and step size Under the assumption that the Robbins-Monro conditions are satisfied, prove the error term. Satisfying the summability condition, applying the convergence lemma yields... converged to ; Combining the decision variable bias bounds derived in step (5), prove Thus, all agents converge to the same optimal solution.

8. The homomorphic encrypted distributed optimization method for a multi-agent system as described in claim 1, characterized in that, In step (7), the convergence rate is derived as follows: Select step size ,in It is a constant; Multiply both sides of the error recursion relation by And on from arrive Summation; We define a weighted average by utilizing the convexity of the objective function. ,but ； By using summation techniques and bias bound estimation, an explicit upper bound on the objective function error is derived: in , , It is a positive constant that depends on the network parameters, initial conditions, and Lipschitz constant.

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