Quantized distributed online composite optimization method in bandwidth-limited directed network

Abstract

The application discloses a kind of quantized distributed online compound optimization methods under bandwidth limited directed network, belong to distributed optimization technical field.The method mainly solves the problem of performance decline of existing distributed optimization algorithm under the condition of asymmetric directed network and limited communication bandwidth.The technical scheme includes: establishing directed network graph and determining node neighborhood relationship, constructing compound optimization problem model, designing the distributed online compound optimization algorithm of fusion adaptive uniform quantization and proximal gradient descent, and analyzing its dynamic regret convergence.The application breaks through the dependence on network symmetry, significantly reduces the communication overhead, and still guarantees sublinear convergence performance in bandwidth limited and unbalanced directed network, suitable for real-time optimization scenarios such as unmanned aerial vehicle formation, multi-agent cooperative control.

Description

Technical Field

[0001] This invention belongs to the field of distributed optimization technology, specifically relating to a quantitative distributed online composite optimization method for bandwidth-constrained directed networks. Background Technology

[0002] Distributed online optimization, as a key technology for dealing with dynamic environments and multi-agent collaborative decision-making, has attracted widespread attention in the Industrial Internet of Things (IIoT) fields such as smart grids, multi-robot collaboration, and wireless sensor networks. Particularly in distributed online composite optimization problems incorporating non-smooth regularization terms, this method can effectively handle structured constraints and demonstrates significant advantages in scenarios such as sparse learning, resource allocation, and real-time control.

[0003] However, in practical industrial applications, the strict constraints on communication resources and the complexity of network topologies severely limit the performance of distributed optimization algorithms. On the one hand, many real-world networks (such as drone swarms, vehicle-mounted ad hoc networks, and some industrial wireless networks) have naturally asymmetric communication links, forming directed graph topologies. Most existing distributed optimization algorithms rely on bidirectional or symmetric communication weight assumptions, making them difficult to apply directly to such unbalanced directed networks. On the other hand, inter-node communication bandwidth is often limited by hardware and power consumption, failing to support frequent exchanges of high-precision real-valued information. Traditional unquantized communication mechanisms result in high transmission overhead and latency, failing to meet real-time optimization requirements.

[0004] To reduce communication costs, existing research has proposed mechanisms such as gradient compression, quantized communication, and event triggering. However, these mechanisms still have significant shortcomings in scenarios involving directed topology and coupled composite optimization: First, existing quantization algorithms are mostly based on undirected or balanced graph designs, failing to effectively handle information inconsistency issues in asymmetric directed networks. Second, research on composite optimization frameworks that simultaneously handle dynamic loss functions and non-smooth regularization terms is still relatively weak, especially lacking theoretical and algorithmic support applicable to directed graphs under communication constraints. Third, existing methods often fail to guarantee convergence performance in complex environments where bandwidth is limited and directed network topology coexists.

[0005] Therefore, there is an urgent need for a distributed online composite optimization method that can adapt to both unbalanced directed network structures and limited bandwidth communication conditions, in order to support the pressing need of practical industrial systems for efficient and robust collaborative decision-making. Summary of the Invention

[0006] This invention aims to overcome the technical problems existing in the background art and proposes an adaptive quantization distributed online composite optimization method suitable for bandwidth-constrained unbalanced directed network environments. This method integrates adaptive quantization communication and non-smooth regularization optimization, does not rely on the symmetry of the network graph, and can guarantee sublinear dynamic regret convergence performance under limited communication bits. It meets the needs of efficient collaborative optimization in complex industrial scenarios, such as UAV formation.

[0007] The core idea of ​​this invention lies in integrating an adaptive uniform quantization mechanism with a proximal gradient descent strategy to address the performance degradation of existing distributed optimization algorithms in environments with limited communication resources and asymmetric topologies. This method constructs a directed network graph model, designs row-stochastic weight matrices to achieve asymmetric coordination between nodes, and introduces an adjustable quantizer to compress and transmit decision vectors, significantly reducing communication overhead. Simultaneously, it combines online learning and a composite optimization framework to handle dynamic loss functions and non-smooth regularization terms. Finally, through theoretical proof and experimental verification, it is ensured that the algorithm can still achieve sublinear growth in dynamic loss in finite bandwidth and unbalanced directed networks, demonstrating good convergence performance and robustness.

[0008] To achieve the above-mentioned objectives, this invention employs the following technical method: a quantized distributed online composite optimization method for bandwidth-constrained directed networks, comprising the following steps: S1, establishing a directed network graph: based on the communication nodes of the UAV swarm, a directed network graph describing the information transmission relationship between nodes is constructed, and the in-neighbor and out-neighbor sets of each node are determined; S2, constructing a distributed online composite optimization problem model: the UAV formation control task is abstracted into a distributed online composite optimization problem, and the decision variables, local loss function, regularization term, and constraints are defined; S3, designing an adaptive quantized distributed online composite optimization algorithm framework: based on the constructed problem model, combined with the directed network graph, adaptive uniform quantizer, and near-end gradient descent technique, a distributed online composite optimization algorithm suitable for bandwidth-constrained scenarios is constructed; S4, analyzing the convergence of the algorithm: using dynamic regret as a performance index, the convergence of the proposed adaptive quantized distributed online composite optimization algorithm is analyzed.

[0009] Furthermore, S1 specifically includes the following steps: S11, obtaining each communication node from the UAV swarm; S12, establishing a non-balanced connected directed network graph. ,in Represents a directed network graph. Represents the set of communication nodes. Represents the total number of nodes. S13. Define a directed network graph, representing the set of communication edges in the graph. The communication weight matrix is ,in Represent a OK A real matrix of columns, using Representation Nodes To the node The weight of the sent information, where Representation matrix The Line 1 Column elements, whose values ​​are between 0 and 1, when Sometimes, Otherwise there are ,in Represents a node Able to direct nodes Sending information; S14, Directed network graph Communication weight matrix It is a row random matrix, i.e., matrix The sum of the elements in each row is 1; S15, in a directed network graph Below, definition For nodes The set of incoming neighbors, that is, the set of nodes that can be accessed. The set of nodes that send information is defined. For nodes The set of outgoing neighbors, i.e., the set of nodes that can receive data. The set of nodes that sent the information.

[0010] Further, S2 specifically includes the following steps: S21, defining the decision variables of the node as... ,in represent 3D real vector space; S22, definition The constraints are shared and known non-empty convex sets for all nodes, where represent 2D real vector space; S23, using Indicates the first Only nodes during round iteration Known local convex loss function, using Denotes the non-smooth convex regularization term that is known to all nodes, where Represents the set of real numbers; S24. Based on directed network graphs, the UAV formation problem is abstracted into a distributed online composite optimization problem, as follows: ,in, This represents the total number of iterations of the algorithm. This represents the total number of drone nodes.

[0011] Furthermore, S3 specifically includes the following steps: S31, parameter initialization: setting parameters. , , , , ,in It is the total number of iterations of the algorithm. yes The iteration step size of Shi Hengzheng, its value follows The increase shows a monotonic, non-increasing trend. yes The quantization parameter is between 0 and 1, and its value varies with... The increase shows a monotonic, non-increasing trend. For the iteration time, It is a quantification level parameter. It is the communication weight matrix; S32, variable initialization: setting the decision variables at the initial iteration time. Quantization intermediate value vector and weight compensation vector ,in Representative node exist The decision vector at that time, Representative node exist The quantization intermediate value vector at time, Representative node exist The weight compensation vector at that time, represent The first order identity matrix Column elements; S33, Quantization interval size setting: in the first... In each iteration, the quantization interval size vector of the quantizer is set as... ,in It is a constant used to adjust the size of the quantization interval. To represent multiplication, It is a regularization term The upper bound of the gradient, yes Iteration step size at time yes Quantization parameters at time, For the iteration time, It is a set of elements that are all 1 3D column vector; S34, for each iteration Each node performs an iterative update process; S35, output the decision vector sequence of all nodes.

[0012] Further, in step S34, the iterative update process specifically includes: S34.1, in the... In the round of iteration, nodes Make a decision and received feedback information. ,in Represents a node In the In the round of iteration, its loss function In decision vector The subgradient value at point S34.2, node S34.2. The decision vector is quantized using a uniform quantizer to obtain the quantized decision vector. ,in Represents a uniform quantization function. It is a node In the The quantization intermediate value vector in the round of iteration, It is the first The quantization interval size vector in the round iteration; S34.3, node From node Receive quantized decision vector and its quantitative decision vector By comparing the communication weight matrix and the weight compensation vector, consistency and gradient descent update operations are performed to obtain intermediate variables. The specific calculations are as follows: ,in, It is a node In the Intermediate variables in round iteration, and These are nodes and In the The decision vector in the round of iteration, It is the total number of nodes. It is a weight matrix The Line 1 Column elements, It is a node In the Quantization decision vector in round iteration, It is a node In the Quantization decision vector in round iteration, and These are nodes and In the The quantization intermediate value vector in the round of iteration, It is the first The quantization interval size vector in the round iteration, It is the first The iteration step size in a round of iteration, It is a node In the Weight compensation vector in round iteration The first in One element, It is a node In the In the round of iteration, its loss function In decision vector The subgradient value at point S34.4, node Based on the near-end projection operator for intermediate variables Perform a projection update to obtain the decision variables for the next iteration. and quantization intermediate value vector The specific calculation formula is as follows: , in, and These are nodes In the Decision variables and quantized intermediate value vectors in round iterations It is a regularization term. and They are the first The iteration step size and quantization level parameters in the round of iteration, It is a node In the Intermediate variables in round iteration, Representative vector The Euclidean norm, Indicates in the constraint set In, make the function When the minimum value is obtained Value, of which It is about Functions; S34.5, nodes The weight compensation vector is updated based on the communication weight matrix to obtain a new round of weight compensation vector. The specific calculation formula is as follows: ,in, For nodes In the The weight compensation vector in the round of iteration, It is the total number of nodes. It is a weight matrix The Line 1 Column elements, It is a node In the The weight compensation vector in the round of iteration.

[0013] Further, in step S34.2, the method for constructing the uniform quantization function is as follows: The vector to be quantized is set as... Given quantization level parameters The quantization intermediate value vector is The quantization interval size vector is ,in represent 3D real vector space, If the set represents positive integers, then the uniform quantization function... The Each component The definition is as follows: ,in, ; , , and These represent the quantization vectors respectively. Quantization vector Quantization intermediate value vector and quantization interval size vector The 1 element; according to the definition of the quantization function, when When the quantization error satisfies the following inequality: ,in, It is the dimension of the vector. Representative vector The Euclidean norm, Representative vector The infinite norm of .

[0014] Further, S4 specifically includes the following steps: S41, evaluating the convergence performance of the algorithm using an individual dynamic regret index, node The dynamic regret is: ,in, It is the total number of iterations of the algorithm. It is a node Total number of iterations The dynamic regret within, It is the total number of nodes. Indicates the first Nodes generated during round iteration The decision vector, Indicates the first Only nodes during round iteration Known local convex loss function, This represents a non-smooth convex regularization term that is known to all nodes. Indicates in the constraint set inside, when In the The optimal solution obtained when the value is minimized in the round of iterations; S42, Proof of individual dynamic regret with respect to the total number of iterations based on convex optimization theory. It exhibits sublinear growth, that is, when When it approaches positive infinity, The limit is 0.

[0015] Compared with existing technologies, the beneficial effects of this invention are as follows: 1. This invention overcomes the limitation of communication graph symmetry, significantly improving the applicability of the algorithm in unbalanced directed networks. Traditional distributed optimization algorithms typically require the network to have symmetric or full-duplex communication capabilities, greatly limiting their application in practical scenarios with naturally directed topologies, such as UAV swarms and the Industrial Internet of Things. This invention, by introducing row-random weight matrices and an asymmetric coordination mechanism between nodes, achieves effective information aggregation and gradient collaboration in directed graphs, without relying on the assumptions of graph balance or symmetry, thereby significantly expanding the application scope of distributed online composite optimization algorithms. This characteristic is particularly suitable for complex network environments such as edge computing and multi-agent collaborative control, effectively improving the adaptability and practicality of the system under non-ideal communication conditions.

[0016] 2. By introducing an adaptive uniform quantization mechanism, this invention significantly reduces communication overhead between multiple nodes and improves deployment capabilities in bandwidth-constrained environments. In practical industrial applications, communication bandwidth is often constrained by hardware costs, energy limitations, and signal interference, making traditional continuous-value communication methods difficult to meet real-time optimization requirements. The adaptive quantization strategy employed in this invention can compress high-dimensional decision vectors with finite bits and dynamically adjust the quantization interval and precision, greatly reducing data transmission volume while ensuring information validity. This mechanism not only alleviates network congestion but also significantly reduces communication latency and energy consumption, providing a feasible collaborative optimization solution for resource-constrained distributed systems (such as wireless sensor networks and lightweight UAV swarms).

[0017] 3. This invention maintains excellent convergence performance and dynamic adaptability even in complex environments with limited communication and asymmetric networks. By integrating proximal gradient descent with an online learning mechanism and incorporating a theoretically proven upper bound on sublinear dynamic regret, this method maintains stable optimization performance even when faced with dynamic loss functions and non-smooth regularization terms. Mathematically, the algorithm rigorously guarantees asymptotic convergence of the overall optimization process even under the influence of quantization errors and asymmetric topology, demonstrating strong robustness and reliability. Simulation results further demonstrate that the algorithm effectively controls average dynamic regret at different quantization levels, making it particularly suitable for industrial scenarios with high requirements for optimization accuracy and real-time performance, such as real-time resource scheduling and distributed machine learning. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0019] Figure 1 This is a schematic diagram of the implementation steps of the method of the present invention.

[0020] Figure 2 This is a directed network graph containing 8 nodes provided for an embodiment of the present invention.

[0021] Figure 3 This is an example of the average dynamic regret effect of the algorithm under different quantization levels provided in the embodiments of the present invention. Detailed Implementation

[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] like Figure 1 As shown, this invention provides a quantized distributed online composite optimization method for bandwidth-constrained directed networks. In large-scale network systems with asymmetric inter-node communication topology and limited communication bandwidth, this method combines an adaptive uniform quantization mechanism with a near-end gradient descent algorithm to construct a distributed online composite optimization framework. This framework enables multiple network nodes to collaboratively solve and dynamically optimize a global composite objective function containing a common regularization term online, thereby reducing communication overhead while ensuring algorithm convergence performance and optimization accuracy.

[0024] This method specifically includes the following steps: Step 1, establish a directed network graph, that is, based on the communication nodes of the UAV cluster, construct a directed network graph describing the information transmission relationship between nodes, and determine the set of in-neighbors and out-neighbors of each node.

[0025] Step 1 specifically includes: Step 11, obtaining each communication node from the drone swarm; Step 12, establishing a non-equilibrium connected directed network graph to characterize drone interactions. ,in Represents a directed network graph. Represents the set of communication nodes. Represents the total number of nodes. Represents the set of communication edges in a directed network graph; Step 13: Define the directed network graph. The communication weight matrix is ,in Represent a OK A real matrix of columns, using Representation Nodes To the node The weight of the sent information, where Representation matrix The Line 1 Column elements, whose values ​​are between 0 and 1, when Sometimes, Otherwise there are ,in Represents a node Can be nodes Sending information; Step 14, Directed network graph Communication weight matrix It is a row random matrix, i.e., matrix The sum of each row of elements is 1; Step 15: In the directed network graph Below, definition For nodes The set of incoming neighbors, that is, the set of nodes that can be accessed. The set of nodes that send information is defined. For nodes The set of outgoing neighbors, i.e., the set of nodes that can receive data. The set of nodes that sent the information.

[0026] Step 2: Construct a distributed online composite optimization problem model, which involves abstracting the UAV formation control task into a distributed online composite optimization problem, defining the decision variables, local loss functions, regularization terms, and constraints, and forming a mathematical model that can be used for algorithm design.

[0027] Step 2 specifically includes: Step 21, defining the decision variables for the nodes as follows: ,in represent 2D real vector space; Step 22, Define The constraints are shared and known non-empty convex sets for all nodes, where represent 23. Real vector space; Step 23, using Indicates the first Only nodes during round iteration Known local convex loss function, using Denotes the non-smooth convex regularization term that is known to all nodes, where Represents the set of real numbers; Step 24: Based on the directed network graph described in step S1, and combined with steps S21 to S23, the UAV formation problem is abstracted into a distributed online composite optimization problem, as follows: ,in, This represents the total number of iterations of the algorithm. This represents the total number of drone nodes.

[0028] Step 3: Design an adaptive quantization distributed online composite optimization algorithm framework. Based on the constructed problem model, combine directed network topology, adaptive uniform quantizer and near-end gradient descent technique to construct a distributed online composite optimization algorithm suitable for bandwidth-constrained scenarios, and realize collaborative optimization of multiple nodes under limited communication conditions.

[0029] Step 3 specifically includes: Step 31, Parameter Initialization: Setting parameters. , , , , ,in This is the total number of iterations of the algorithm, and its value is a positive integer. yes The iteration step size of Shi Hengzheng, its value follows The increase is monotonically non-increasing. yes The quantization parameter at time has a value between 0 and 1, and varies with time. The increase is monotonically non-increasing. For the iteration time, This is the quantization level parameter, and its value is a positive integer. Directed network graph The communication weight matrix.

[0030] Step 32, Variable Initialization: Set the decision variables for the initial iteration. Quantization intermediate value vector and weight compensation vector ,in Representative node exist The decision vector at that time, Representative node exist The quantization intermediate value vector at time, Representative node exist The weight compensation vector at that time, represent The first order identity matrix Column elements.

[0031] Step 33, Quantization interval size setting: In the first step... In each iteration, the quantization interval size vector of the quantizer is set as... ,in It is a constant used to adjust the size of the quantization interval. To represent multiplication, It is a regularization term The upper bound of the gradient, yes Iteration step size at time yes Quantization parameters at time, For the iteration time, It is a set of elements that are all 1 Dimensional column vector.

[0032] Step 34, Iterative Update Process: For each iteration Each node performs the following steps.

[0033] Step 34.1, in the... In the round of iteration, nodes Make a decision and received feedback information. ,in Represents a node In the In the round of iteration, its loss function In decision vector The subgradient value at the node; it should be noted that decision feedback information cannot be predicted in advance, but can only be obtained at the node. The decision was revealed after it was made.

[0034] Step 34.2, Node The decision vector is quantized using a uniform quantizer to obtain the quantized decision vector. ,in Represents a uniform quantization function. It is a node In the The quantization intermediate value vector in the round of iteration, It is the first The quantization interval size vector in the round iteration.

[0035] The uniform quantization function is constructed as follows: The vector to be quantized is set as... Given quantization level parameters The quantization intermediate value vector is The quantization interval size vector is ,in represent 3D real vector space, If the set represents positive integers, then the uniform quantization function... The Each component The definition is as follows: ,in, ; , , and These represent the quantization vectors respectively. Quantization vector Quantization intermediate value vector and quantization interval size vector The Each element.

[0036] According to the definition of the quantization function, when When the quantization error satisfies the following inequality: ,in, It is the dimension of the vector. Representative vector The Euclidean norm, Representative vector The infinite norm of .

[0037] Step 34.3, Node From node Receive quantized decision vector and its quantitative decision vector A comparison is made, and based on this comparison, a consistency and gradient descent update operation is performed by combining the communication weight matrix and the weight compensation vector to obtain the intermediate variables. The specific calculations are as follows: ,in, It is a node In the Intermediate variables in round iteration, and These are nodes and In the The decision vector in the round of iteration, It is the total number of nodes. It is a weight matrix The Line 1 Column elements, It is a node In the Quantization decision vector in round iteration, It is a node In the Quantization decision vector in round iteration, and These are nodes and In the The quantization intermediate value vector in the round of iteration, It is the first The quantization interval size vector in the round iteration, It is the first The iteration step size in a round of iteration, It is a node In the Weight compensation vector in round iteration The first in One element, It is a node In the In the round of iteration, its loss function In decision vector The subgradient value at that point.

[0038] Step 34.4, Node Based on the near-end projection operator for intermediate variables Perform projection updates under regularization constraints to obtain the decision variables for the next round of iterations. and quantization intermediate value vector The specific calculation formula is as follows: , in, and These are nodes In the Decision variables and quantized intermediate value vectors in round iterations It is a regularization term. and They are the first The iteration step size and quantization level parameters in the round of iteration, It is a node In the Intermediate variables in round iteration, Representative vector The Euclidean norm, Indicates in the constraint set In, make the function When the minimum value is obtained Value, of which It is about The function.

[0039] Step 34.5, Node The weight compensation vector is updated based on the communication weight matrix to obtain a new round of weight compensation vector. The specific calculation formula is as follows: ,in, node In the The weight compensation vector in the round of iteration, It is the total number of nodes. It is a weight matrix The Line 1 Column elements, It is a node In the The weight compensation vector in the round of iteration.

[0040] Step 35: Output the decision vector sequence of all nodes.

[0041] Step 4: Analyze the convergence of the algorithm. Using dynamic regret as the performance index, perform convergence analysis on the proposed adaptive quantization distributed online composite optimization algorithm to verify its effectiveness and performance upper bound under unbalanced directed networks.

[0042] Step 4 specifically includes: Step 41, using the individual dynamic regret index to evaluate the convergence performance of the algorithm, node... The dynamic regret is: ,in, It is the total number of iterations of the algorithm. It is a node Total number of iterations The dynamic regret within, It is the total number of nodes. This indicates that within the algorithm framework described in step S3, at the... Nodes generated during round iteration The decision vector, Indicates the first Only nodes during round iteration Known local convex loss function, This represents a non-smooth convex regularization term that is known to all nodes. Indicates in the constraint set inside, when In the The optimal solution is obtained when the minimum value is taken in each iteration.

[0043] Step 42: Based on convex optimization theory, prove that the individual dynamic regret defined in Step 41 is related to the total number of iterations. It exhibits sublinear growth, that is, when When it approaches positive infinity, The limit is 0, which shows that the algorithm framework described in this method is effective.

[0044] In this implementation case, the online distributed LASSO problem is used to test the effectiveness of the proposed method and the impact of different quantization level parameters on the algorithm.

[0045] In this example, the present invention applies an adaptive quantization distributed online composite optimization algorithm to the online distributed LASSO problem. In this problem, all nodes need to cooperate to minimize the following composite objective function: , in, It is a node exist The feature matrix at time step, node exist The response vector at time t, It is a regularization parameter. Representative vector The Euclidean norm, Representative vector The 1 norm, represent OK A real matrix of columns, represent A real vector space of dimension , where It is a positive integer, and The value ranges from 1 to Between. Response vector The definition is as follows: ,in, It is a predefined vector, defined as: when hour, Otherwise, it is 0. It is the floor function. Represents a node exist The noise that is constantly subjected to represent A real vector space. In At any given moment, at each node Will receive and Then all nodes need to work together to minimize the composite objective function.

[0046] In this simulation experiment, such as Figure 2 The algorithm for adaptive quantization distributed online composite optimization is performed on the directed network graph containing 8 nodes, with the algorithm parameters set as follows: , , . Figure 3 The average dynamic regret performance of the algorithm in the test problem is presented under different quantization levels. Experimental results show that the method of the present invention can still maintain good performance under limited communication bandwidth conditions, and the convergence performance of the algorithm is further improved with the increase of the quantization level parameter, thus verifying its effectiveness and robustness.

[0047] The above description is merely an example and illustration of the concept of the present invention. Those skilled in the art can make various modifications or additions to the specific embodiments described or use similar methods to replace them, as long as they do not deviate from the concept of the invention or exceed the scope defined in this specification, they should all fall within the protection scope of the present invention.

Claims

1. A quantized distributed online composite optimization method for bandwidth-constrained directed networks, characterized in that, Includes the following steps: S1. Establish a directed network graph: Based on the communication nodes of the UAV cluster, construct a directed network graph describing the information transmission relationship between nodes, and determine the set of in-neighbors and out-neighbors for each node. S2. Construct a distributed online composite optimization problem model: Abstract the UAV formation control task into a distributed online composite optimization problem, and clarify the decision variables, local loss function, regularization term and constraints. S3. Design of an adaptive quantization distributed online composite optimization algorithm framework: Based on the constructed problem model, a distributed online composite optimization algorithm suitable for bandwidth-constrained scenarios is constructed by combining a directed network graph, an adaptive uniform quantizer, and a near-end gradient descent technique. S4. Analysis of algorithm convergence: Using dynamic regret as a performance index, the convergence of the proposed adaptive quantization distributed online composite optimization algorithm is analyzed. S3 specifically includes the following steps: S31. Parameter Initialization: Set parameters , , , , ,in It is the total number of iterations of the algorithm; yes The iteration step size of Shi Hengzheng, its value follows The increase shows a monotonic, non-increasing trend; yes The quantization parameter is between 0 and 1, and its value varies with... The increase shows a monotonic, non-increasing trend. The iteration time; It is a quantitative level parameter; It is the communication weight matrix; S32. Variable Initialization: Set the decision variables for the initial iteration. Quantization intermediate value vector and weight compensation vector ,in Representative node exist The decision vector at that time, Representative node exist The quantization intermediate value vector at time, Representative node exist The weight compensation vector at that time, represent The first order identity matrix Column elements; S33, Quantization interval size setting: In the... In each iteration, the quantization interval size vector of the quantizer is set as... ,in It is a constant used to adjust the size of the quantization interval. To represent multiplication, It is a regularization term The upper bound of the gradient, yes Iteration step size at time yes Quantization parameters at time, For the iteration time, It is a set of elements that are all 1 3D column vector; S34. For each iteration Each node performs an iterative update process; S35. Output the decision vector sequence of all nodes; In step S34, the iterative update process specifically includes: S34.1, in the In the round of iteration, nodes Make a decision and received feedback information. ,in Represents a node In the In the round of iteration, its loss function In decision vector The subgradient value at that point; S34.2, Node The decision vector is quantized using a uniform quantizer to obtain the quantized decision vector. ,in Represents a uniform quantization function. It is a node In the The quantization intermediate value vector in the round of iteration, It is the first The quantization interval size vector in the round iteration; S34.3, Node From node Receive quantized decision vector and its quantitative decision vector By comparing the communication weight matrix and the weight compensation vector, consistency and gradient descent update operations are performed to obtain intermediate variables. The specific calculations are as follows: ， in, It is a node In the Intermediate variables in round iteration, and These are nodes and In the The decision vector in the round of iteration, It is the total number of nodes. It is a weight matrix The Line 1 Column elements, It is a node In the Quantization decision vector in round iteration, It is a node In the Quantization decision vector in round iteration, and These are nodes and In the The quantization intermediate value vector in the round of iteration, It is the first The quantization interval size vector in the round iteration, It is the first The iteration step size in a round of iteration, It is a node In the Weight compensation vector in round iteration The first in One element, It is a node In the In the round of iteration, its loss function In decision vector The subgradient value at that point; S34.4, Node Based on the near-end projection operator for intermediate variables Perform a projection update to obtain the decision variables for the next iteration. and quantization intermediate value vector The specific calculation formula is as follows: ， in, and These are nodes In the Decision variables and quantized intermediate value vectors in round iterations It is a regularization term. and They are the first The iteration step size and quantization level parameters in the round of iteration, It is a node In the Intermediate variables in round iteration, Representative vector The Euclidean norm, Indicates in the constraint set In, make the function When the minimum value is obtained Value, of which It is about The function; S34.5, Node The weight compensation vector is updated based on the communication weight matrix to obtain a new round of weight compensation vector. The specific calculation formula is as follows: ， in, For nodes In the The weight compensation vector in the round of iteration, It is the total number of nodes. It is a weight matrix The Line 1 Column elements, It is a node In the Weight compensation vector in round iteration; In step S34.2, the method for constructing the uniform quantization function is as follows: Let the vector to be quantized be Given quantization level parameters The quantization intermediate value vector is The quantization interval size vector is ,in represent 3D real vector space, If the set represents positive integers, then the uniform quantization function... The Each component The definition is as follows: ， in, ; , , and These represent the quantization vectors respectively. Quantization vector Quantization intermediate value vector and quantization interval size vector The One element; According to the definition of the quantization function, when When the quantization error satisfies the following inequality: ， in, It is the dimension of the vector. Representative vector The Euclidean norm, Representative vector The infinite norm of .

2. The quantized distributed online composite optimization method for bandwidth-constrained directed networks according to claim 1, characterized in that, S1 specifically includes the following steps: S11. Obtain each communication node from the drone swarm; S12. Construct a non-balanced connected directed network graph. ,in Represents a directed network graph. Represents the set of communication nodes. Represents the total number of nodes. This represents the set of communication edges in a directed network graph; S13. Define a directed network graph. The communication weight matrix is ,in Represent a OK A real matrix of columns, using Representation Nodes To the node The weight of the sent information, where Representation matrix The Line 1 Column elements, whose values ​​are between 0 and 1, when Sometimes, Otherwise there are ,in Represents a node To the node Send a message; S14, Directed Network Graph Communication weight matrix It is a row random matrix, i.e., matrix The sum of the elements in each row is 1; S15, In directed network graphs Below, definition For nodes The set of incoming neighbors, that is, the set of nodes that can be accessed. The set of nodes that send information is defined. For nodes The set of outgoing neighbors, i.e., the set of nodes that can receive data. The set of nodes that sent the information.

3. The quantized distributed online composite optimization method for bandwidth-constrained directed networks according to claim 1, characterized in that, S2 specifically includes the following steps: S21. Define the decision variables for the nodes as follows: ,in represent 3D real vector space; S22, Definition The constraints are shared and known non-empty convex sets for all nodes, where represent 3D real vector space; S23, Use Indicates the first Only nodes during round iteration Known local convex loss function, using Denotes the non-smooth convex regularization term that is known to all nodes, where Represents the set of real numbers; S24. Based on directed network graphs, the UAV formation problem is abstracted into a distributed online composite optimization problem, as follows: ， in, This represents the total number of iterations of the algorithm. This represents the total number of drone nodes.

4. The quantized distributed online composite optimization method for bandwidth-constrained directed networks according to claim 1, characterized in that, S4 specifically includes the following steps: S41. Use the individual dynamic regret index to evaluate the convergence performance of the algorithm, node The dynamic regret is: ， in, It is the total number of iterations of the algorithm. It is a node Total number of iterations The dynamic regret within, It is the total number of nodes. Indicates the first Nodes generated during round iteration The decision vector, Indicates the first Only nodes during round iteration Known local convex loss function, This represents a non-smooth convex regularization term that is known to all nodes. Indicates in the constraint set inside, when In the The optimal solution obtained by taking the minimum value in each iteration; S42. Prove, based on convex optimization theory, that individual dynamic regret is related to the total number of iterations. It exhibits sublinear growth, that is, when When it approaches positive infinity, The limit is 0.

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