A unified compression method for distributed non-convex optimization

Abstract

This invention discloses a unified compression method for distributed nonconvex optimization in the field of compressed communication technology. It includes setting local model parameters and calculating the gradient of the local cost function based on multiple agents, where the local model parameters are local estimates of the solution to the global optimization problem by the multiple agents during iteration. The multiple agents interact through a communication network, constructing a compressor to compress the transmitted information during the interaction. The compressor includes a locally bounded compressor and a globally bounded compressor. Auxiliary variables are constructed for the multiple agents, and compressed information is recorded and transmitted through these auxiliary variables to mitigate compression errors caused by directly using the compressor. This invention updates the primal and dual variables of the local model parameters by constructing a primal-dual algorithm, making it applicable to both local compressors such as 1-bit compressors and saturable quantizers, as well as global compressors under bounded noise, effectively reducing communication burden.

Description

Technical Field

[0001] This invention relates to the field of compressed communication technology, specifically a unified compression method for distributed non-convex optimization. Background Technology

[0002] Distributed optimization has been widely applied in scenarios such as large-scale machine learning, cooperative control, multi-agent decision-making, and edge computing. Distributed optimization applications typically involve networked systems composed of n agents, each holding its own private local cost function. (Not necessarily convex), solve the following optimization problem collaboratively through network communication: (1) To achieve consistency and obtain optimal model parameters, where, These are the model parameters. To address the above problems, various distributed algorithms have been developed, ranging from convex optimization to non-convex optimization, with corresponding performance guarantees provided.

[0003] In distributed optimization, communication remains a critical factor limiting algorithm efficiency and practical deployment, especially in large-scale neural network training, real-time decision-making systems, and resource-constrained edge devices. To reduce communication burden, existing technologies have proposed various efficient communication methods, such as compressed communication, event-triggered communication, latency-resistant design, and asynchronous updates. Among these, compressed communication, due to its relatively simple implementation and its better alignment with the limited precision of digital devices, has become a widely used technique in distributed optimization. Its basic idea is to compress information before transmitting it between agents, reducing the number of communication bits and thus lowering communication costs. Based on this approach, existing technologies have researched and applied various types of compressors, including quantizers, unbiased compressors, and shrinking compressors, and these techniques have been further extended to non-convex cases.

[0004] Existing compressed communication methods primarily target global compressors, meaning that the compression error of any input vector can be controlled by a uniform form. In contrast, local compressors (especially 1-bit compressors) that satisfy error constraints only within a local region transmit only 1 bit of information for each coordinate, thus carrying extremely limited information. Theoretical analyses of these stronger compression methods typically consider convex optimization cases and require strong assumptions to guarantee convergence. For example, some studies have established convergence results under strong convex conditions and Polyak-Lojasiewicz (P-L) conditions. Although existing compressed communication methods also consider convex optimization cases, they only prove convergence without providing explicit convergence rates. In non-convex optimization cases, existing 1-bit methods are only applied to scenarios with a central server, thus neglecting consistency issues. These methods also rely on strong assumptions, such as Lipschitz continuity and homogeneity. Another representative type of local compressor is the saturated quantizer. However, many works ignore the saturation effect in their analyses, effectively treating it as a type of global compressor. For quantizers with a given saturation level or arbitrary saturation levels, current theoretical guarantees are still few and usually rely on strong convexity conditions or PL conditions. In the case of convex optimization, some studies have also established probabilistic convergence results proportional to the saturation level.

[0005] Because research on local compressors is relatively limited, it is even more difficult to handle both local and global compressors within a unified framework. Aside from work on saturated quantizers that simultaneously cover the unsaturated case, to our knowledge, only a few studies have proposed unified algorithms applicable to both local and global compressors. Moreover, these methods only apply to the case of absolute compression error and depend on the PL condition. On the other hand, for global compressors, existing research has explored unified compression models and algorithm design. For example, some studies have proposed an algorithm applicable to both unbiased and shrinking compressors; others have further considered a more general class of compressors, thus unifying unbiased, shrinking, and some biased non-shrinking compressors into a unified framework. In recent years, related research has further developed towards a more unified compressor category, enabling it to accommodate both relative and absolute compression errors.

[0006] In summary, theoretical research results on local compressors remain relatively limited, and research on unified processing of local and global compressors is even more insufficient. Specifically, existing technologies still lack corresponding theoretical guarantees in the following aspects: First, in the non-convex case, existing distributed non-convex optimization compression methods are not applicable to typical local compressors such as 1-bit compressors or saturated quantizers; second, in the non-convex case, existing distributed non-convex optimization compression methods cannot be simultaneously applied to local and global compressors; third, a unified algorithm that is applicable to both local compressors and global compressors with both relative and absolute compression errors is still lacking. Summary of the Invention

[0007] The purpose of this invention is to provide a unified compression method for distributed non-convex optimization. By constructing a local compressor and a global compressor, iteratively performing compression of transmitted information, reducing compression error, and updating the original and dual variables of the local model parameters until convergence, a solution to the global optimization problem can be obtained. It is applicable to local compressors such as 1-bit compressors and saturated quantizers, as well as global compressors under bounded noise, and can effectively reduce the communication burden.

[0008] To achieve the above objectives, the present invention provides the following technical solution: A unified compression method for distributed nonconvex optimization, applied to a distributed network composed of multiple agents, includes the following steps: S1: Setting local model parameters based on multiple agents And calculate the gradient of the local cost function. Wherein, the local model parameters are the local estimates of the solution to the global optimization problem by multiple agents i at the k-th iteration; S2: Multiple intelligent agents interact with each other through a communication network. During the interaction, a compressor is constructed to compress the transmitted information. The compressor includes a locally bounded compressor and a globally bounded compressor. S3: Construct auxiliary variables for multiple agents, record and transmit compression information through auxiliary variables to alleviate compression errors caused by directly using the compressor. The auxiliary variables include the first auxiliary variable and the second auxiliary variable. S4: Construct the primal and dual variables of the primal-dual algorithm to update the local model parameters, and iteratively execute steps S2 to S4 until convergence, to obtain the solution to the global optimization problem.

[0009] As a further aspect of the present invention: by iteratively performing compression of transmitted information, reducing compression error, and updating the original and dual variables of the local model parameters until convergence, the solution to the global optimization problem can be obtained. This solution is applicable to both local compressors such as 1-bit compressors and saturated quantizers, as well as global compressors under bounded noise, and can effectively reduce the communication burden.

[0010] As a further aspect of the present invention: multiple agents i are used to calculate the gradient of the local cost function. .

[0011] As a further aspect of the present invention: a locally bounded compressor satisfies that the compression error of the locally bounded compressor is bounded only when the input vector is located within a preset local region; a globally bounded compressor satisfies that the expected compression error of any input vector is bounded, and allows for the existence of additional bounded noise.

[0012] As a further aspect of the present invention: the compression error of the locally bounded compressor is bounded only when the input vector is located within a preset local region, including: Constructing a locally bounded compressor ,satisfy: ,in, , , p, r, and δ represent the first compression factor, the second compression factor, and the third compression factor, respectively. Indicates a local area.

[0013] As a further aspect of the present invention: the locally bounded compressor includes a 1-bit compressor and a saturable quantizer; A globally bounded compressor includes a compressor that simultaneously has relative compression error, absolute compression error, and additional bounded noise.

[0014] As a further aspect of the present invention: the compression error of the locally bounded compressor is bounded only when the input vector is located within a preset local region, further comprising: Constructing a 1-bit compressor for: ,in, To quantify the level, It is a constant. , For a 1-bit compressor, the first compression factor of the 1-bit compressor Second compression coefficient Third compression factor ; The number of bits required for a 1-bit compressor to transmit a vector is: .

[0015] As a further aspect of the present invention: the compression error of the locally bounded compressor is bounded only when the input vector is located within a preset local region, further comprising: Constructing a saturable quantizer for: , The number of bits required by a saturation quantizer to transmit a vector for: ,in, Indicates saturation level. To quantize the step size, and These are the floor and floor operators, 1. d Let d be a vector of all 1s. The first compression factor of the saturable quantizer Second compression factor and the third compression factor , .

[0016] As a further aspect of the present invention: the locally bounded compressor includes a global compressor.

[0017] As a further aspect of the present invention: multiple intelligent agents interact with each other through a communication network, and during the interaction process, a compressor is constructed to compress the transmitted information, including: Construct a global compressor, where the global bounded compressor includes a sparsification compressor and a norm-signed compressor; Sparse compressor The construction is as follows: , where t j This is the coordinate index of the j-th component of vector x with the largest absolute value; [x]t j The component value corresponding to the coordinate [x]; .

[0018] The number of bits required by a sparsity compressor to transmit a vector for: Where b1=32, corresponding to a 32-bit floating-point number, is the first compression factor of the sparsity compressor. Second compression factor and the third compression factor , ; Norm-signed compressor The construction is as follows: ,in, For element-wise sign functions; The number of bits required for a norm-symmetric compressor to transmit a vector for: The first compression factor of the norm-symmetric compressor Second compression factor and the third compression factor , .

[0019] As a further aspect of the present invention: multiple intelligent agents interact with each other through a communication network, and during the interaction process, a compressor is constructed to compress the transmitted information, including: Construct a globally bounded compressor as follows: ,in, , r5 and δ5 are the second and third compression coefficients of the globally bounded compressor, respectively. For absolute error, For the compressor Expectation of internal randomness; Alternatively, construct a k-bit unbiased quantizer resistant to bounded noise. The k-bit unbiased quantizer is: ,in, Bounded noise, For Hadamard products, For absolute values, the second compression factor of the k-bit unbiased quantizer, satisfying the global bounded compressor assumption, is: The third compression factor is: , ; A k-bit unbiased quantizer requires the following number of bits to transmit a vector. for: ; Alternatively, construct a Rand-k random sparse compressor resistant to bounded noise. The Rand-k random sparse compressor is: Where, r j These are randomly selected and unique coordinate indices. For the first Given standard basis vectors, the second compression coefficient of the Rand-k random sparse compressor, which satisfies the global bounded compressor assumption, is: The third compression factor is: , The Rand-k random sparse compressor requires a certain number of bits to transmit a vector. for: ; Alternatively, construct a uniform quantizer resistant to bounded noise. A uniform quantizer that is resistant to bounded noise is: Where i = 5, 6, or 7, It is a bounded noise term and satisfies , The second compression coefficient of the uniform quantizer, which is used to ensure unbiased random vectors and is resistant to bounded noise, satisfies the global bounded compressor assumption. The third compression factor is: , ; The number of bits required to transmit a vector by a uniform quantizer resistant to bounded noise. for: .

[0020] As a further aspect of the present invention: the communication network is a graph function. The Laplace operator matrix of the graph function is denoted as L.

[0021] As a further aspect of the present invention: compared with the locally bounded compressor, the globally bounded compressor allows for randomness and additional bounded noise, and the noise can be incorporated into the constant term C for unified processing.

[0022] As a further aspect of the present invention: constructing auxiliary variables for multiple agents, including: The first auxiliary variable used to reduce compression error is constructed as follows: ,in, As the first auxiliary variable, Let be the first auxiliary variable of the i-th agent in the (k-1)th iteration. These are all algorithm parameters. For use in communication; ; The second auxiliary variable constructed to reduce communication overhead is as follows: ,in, As the second auxiliary variable, Let be the second auxiliary variable of the i-th agent in the (k-1)th iteration. Let L be the (i,j)th element of the Laplace matrix L of the communication network. The first auxiliary variable is used for the i-th agent in the k-th iteration. The second auxiliary variable is also used to record the information of the first auxiliary variable, so that the agents only need to transmit the compressed variable.

[0023] As a further aspect of the present invention: constructing the primal and dual variables for updating local model parameters using the primal-dual algorithm, including: Constructing the original variable update algorithm: Update the original variables of the local model parameters. in, It is the original variable of the i-th agent in the (k+1)-th iteration. These are all algorithm parameters. Let (i,j) be the (i,j)th element of the Laplace matrix L of the communication network. The dual variable corresponding to the original variable. for ; Constructing a dual variable update algorithm: Update the dual variable of the local model parameters, where, Let be the dual variable of the i-th agent in the (k+1)-th iteration. Let be the dual variable of the i-th agent in the k-th iteration. This is the second auxiliary variable for the i-th agent in the k-th iteration.

[0024] As a further aspect of the present invention: selecting a compressor that satisfies the locally bounded compressor assumption. and Where the constant C = 100. Compressor and The globally bounded compressor assumption is satisfied, and tests are conducted under both noise-free and bounded-noise conditions, with the upper noise bound set to 100. The remaining compressor parameters are set as follows: Δ=1, In the case of k=1500000, and In the case of k=4.

[0025] Compared with the prior art, the beneficial effects of the present invention are: 1. This invention updates the original and dual variables of the local model parameters by constructing an original-dual algorithm. It is applicable to local compressors such as 1-bit compressors and saturated quantizers, as well as global compressors under bounded noise, and can effectively reduce the communication burden.

[0026] 2. By constructing a first auxiliary variable and a second auxiliary variable, the present invention can record and transmit compression information through the auxiliary variables, thereby mitigating compression errors caused by directly using the compressor and improving compression accuracy. Attached Figure Description

[0027] Figure 1 This is a diagram illustrating the method steps of the present invention; Figure 2 The curves showing the change of perplexity with the number of iterations under different compressors in this invention are shown. Figure 3 The curves showing the variation of perplexity with the number of communication bits under different compressors in this invention are shown. Detailed Implementation

[0028] 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.

[0029] Example: Please see Figures 1-3 This embodiment provides a unified compression method for distributed non-convex optimization, applied to a distributed network composed of multiple agents, including the following steps: S1: Setting local model parameters based on multiple agents And calculate the gradient of the local cost function. Wherein, the local model parameters are the local estimates of the solution to the global optimization problem by multiple agents i at the k-th iteration; S2: Multiple agents interact through a communication network, and a compressor is constructed during the interaction to compress the transmitted information. The compressor includes locally bounded compressors and globally bounded compressors. By iteratively executing the compression of transmitted information, reducing compression errors, and updating the original and dual variables of the local model parameters until convergence, it can obtain the solution to the global optimization problem. It is applicable to local compressors such as 1-bit compressors and saturated quantizers, as well as global compressors under bounded noise, thereby effectively reducing the communication burden.

[0030] In this embodiment, the locally bounded compressor includes a 1-bit compressor and a saturable quantizer.

[0031] A locally bounded compressor satisfies the following condition: the compression error of the locally bounded compressor is bounded only when the input vector lies within a predetermined local region. Constructing a locally bounded compressor... ,satisfy: ,in, , , p, r, and δ represent the first compression factor, the second compression factor, and the third compression factor, respectively. Indicates a local area; A globally bounded compressor satisfies the following: for any input vector, the expected compression error is bounded, and additional bounded noise is allowed.

[0032] In this embodiment, the compression error of the locally bounded compressor is bounded only when the input vector is located within a preset local region, and further includes: Constructing a 1-bit compressor for: ,in, To quantify the level, It is a constant. , For a 1-bit compressor, the first compression factor of the 1-bit compressor Second compression coefficient Third compression factor ; The number of bits required for a 1-bit compressor to transmit a vector is: .

[0033] In this embodiment, the compression error of the locally bounded compressor is bounded only when the input vector is located within a preset local region, and further includes: Constructing a saturable quantizer for: , The number of bits required by a saturation quantizer to transmit a vector for: ,in, Indicates saturation level. To quantize the step size, and These are the floor and floor operators, 1. d Let d be a vector of all 1s. The first compression factor of the saturable quantizer Second compression factor and the third compression factor , .

[0034] In this embodiment, the locally bounded compressor includes the global compressor.

[0035] Construct a global compressor, where the global bounded compressor includes a sparsification compressor and a norm-signed compressor; Sparse compressor The construction is as follows: , where t j This is the coordinate index of the j-th component of vector x with the largest absolute value; [x]t j The component value corresponding to the coordinate [x]; .

[0036] The number of bits required by a sparsity compressor to transmit a vector for: Where b1=32, corresponding to a 32-bit floating-point number, is the first compression factor of the sparsity compressor. Second compression factor and the third compression factor , ; Norm-signed compressor The construction is as follows: ,in, For element-wise sign functions; The number of bits required for a norm-symmetric compressor to transmit a vector for: The first compression factor of the norm-symmetric compressor Second compression factor and the third compression factor , .

[0037] In this embodiment, multiple agents interact with each other through a communication network. During the interaction process, a compressor is constructed to compress the transmitted information, including: Construct a globally bounded compressor as follows: ,in, , r5 and δ5 are the second and third compression coefficients of the globally bounded compressor, respectively. For absolute error, For the compressor Expectation of internal randomness; Alternatively, construct a k-bit unbiased quantizer resistant to bounded noise. The k-bit unbiased quantizer is: ,in, Bounded noise, For Hadamard products, For absolute values, the second compression factor of the k-bit unbiased quantizer, satisfying the global bounded compressor assumption, is: The third compression factor is: , ; A k-bit unbiased quantizer requires the following number of bits to transmit a vector. for: ; Alternatively, construct a Rand-k random sparse compressor resistant to bounded noise. The Rand-k random sparse compressor is: Where, r j These are randomly selected and unique coordinate indices. For the first Given standard basis vectors, the second compression coefficient of the Rand-k random sparse compressor, which satisfies the global bounded compressor assumption, is: The third compression factor is: , The Rand-k random sparse compressor requires a certain number of bits to transmit a vector. for: ; Alternatively, construct a uniform quantizer resistant to bounded noise. A uniform quantizer that is resistant to bounded noise is: Where i = 5, 6, or 7, It is a bounded noise term and satisfies , The second compression coefficient of the uniform quantizer, which is used to ensure unbiased random vectors and is resistant to bounded noise, satisfies the global bounded compressor assumption. The third compression factor is: , ; The number of bits required to transmit a vector by a uniform quantizer resistant to bounded noise. for: .

[0038] In this embodiment, the communication network is a graph function: The Laplace operator matrix of the graph function is denoted as L.

[0039] S3: Construct auxiliary variables for multiple agents, record and transmit compression information through auxiliary variables to alleviate compression errors caused by directly using the compressor. The auxiliary variables include the first auxiliary variable and the second auxiliary variable.

[0040] Construct auxiliary variables for multiple agents, including: The first auxiliary variable used to reduce compression error is constructed as follows: ,in, As the first auxiliary variable, Let be the first auxiliary variable of the i-th agent in the (k-1)th iteration. These are all algorithm parameters. For use in communication; ; The second auxiliary variable constructed to reduce communication overhead is as follows: ,in, As the second auxiliary variable, Let be the second auxiliary variable of the i-th agent in the (k-1)th iteration. Let L be the (i,j)th element of the Laplace matrix L of the communication network. The first auxiliary variable is used for the i-th agent in the k-th iteration. The second auxiliary variable is also used to record the information of the first auxiliary variable, so that the agents only need to transmit the compressed variable.

[0041] S4: Construct the primal and dual variables of the primal-dual algorithm to update the local model parameters, and iteratively execute steps S2 to S4 until convergence, to obtain the solution to the global optimization problem.

[0042] include: Constructing the original variable update algorithm: Update the original variables of the local model parameters. in, It is the original variable of the i-th agent in the (k+1)-th iteration. These are all algorithm parameters. Let (i,j) be the (i,j)th element of the Laplace matrix L of the communication network. The dual variable corresponding to the original variable. for ; Constructing a dual variable update algorithm: Update the dual variables of the local model parameters; where, Let be the dual variable of the i-th agent in the (k+1)-th iteration. Let be the dual variable of the i-th agent in the k-th iteration. This is the second auxiliary variable for the i-th agent in the k-th iteration.

[0043] A globally bounded compressor includes a compressor that simultaneously has relative compression error, absolute compression error, and additional bounded noise. Compared with a locally bounded compressor, a globally bounded compressor allows for randomness and additional bounded noise, and the noise can be uniformly processed in the constant term C.

[0044] To facilitate understanding of the specific embodiments of the present invention, it should be noted that: The proposed method is validated using distributed training of a Long Short-Term Memory (LSTM) network language model as an example. Due to the large number of parameters in this model, communication overhead during training is significant, making it suitable for verifying the effectiveness of the proposed compression method. The experiment uses the Penn Treebank dataset, with half of the training data evenly distributed among n agents.

[0045] The experimental parameters are set as follows: Model dimensions The backpropagation length is truncated to 80, and the number of agents is n=10. The agents connect with each other via a probability of 0.4. Communication is achieved using random graphs. Algorithm performance is evaluated using perplexity, which is defined as an exponential form of the loss function.

[0046] Since existing methods cannot handle locally bounded compressors in the distributed non-convex case, while the existing RCPP algorithm is applicable to globally bounded compressors, this section compares the proposed algorithm with the RCPP algorithm and the corresponding uncompressed algorithm, and tests the proposed method under different compressor configurations. Specifically, a compressor that satisfies the assumption of a locally bounded compressor is selected. and Where the constant C = 100. Compressor and The globally bounded compressor assumption is satisfied, and tests are conducted under both noise-free and bounded-noise conditions, with the upper noise bound set to 100. The remaining compressor parameters are set as follows: Δ=1, In the case of k=1500000, and In the figure, k=4. The experimental groups are denoted as "1-bit", "sat", "top-k", "top-kn", "4-bit", and "4-bit-n" respectively; the curve corresponding to the RCPP algorithm is marked with "R-" in front; the uncompressed method is denoted as "uncomp".

[0047] Depend on Figure 2It can be seen that the proposed method achieves good training results under various compressors. For globally bounded compressors, the proposed algorithm still exhibits good robustness in the presence of bounded noise. Figure 3 It can be seen that, compared with no compression algorithm, the introduction of compression mechanism can significantly improve communication efficiency, and the communication efficiency improvement brought by 1-bit compressor is particularly obvious.

[0048] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A unified compression method for distributed non-convex optimization, characterized in that, Applied to a distributed network consisting of multiple agents, the following steps are included: S1: Set local model parameters and calculate the gradient of the local cost function based on multiple agents, where the local model parameters are the local estimates of the solution to the global optimization problem by multiple agents during iteration; S2: Multiple intelligent agents interact with each other through a communication network. During the interaction, a compressor is constructed to compress the transmitted information. The compressor includes a locally bounded compressor and a globally bounded compressor. S3: Construct auxiliary variables for multiple agents, record and transmit compression information through auxiliary variables to alleviate compression errors caused by directly using the compressor. The auxiliary variables include the first auxiliary variable and the second auxiliary variable. S4: Construct the primal and dual variables of the primal-dual algorithm to update the local model parameters, and iteratively execute steps S2 to S4 until convergence, to obtain the solution to the global optimization problem.

2. The unified compression method for distributed non-convex optimization of claim 1, wherein, Locally bounded compressors satisfy the condition that the compression error is bounded only when the input vector is located within a predefined local region; globally bounded compressors satisfy the condition that the expected compression error is bounded for any input vector and allow for additional bounded noise.

3. The unified compression method for distributed non-convex optimization of claim 2, wherein, The compression error of a locally bounded compressor is bounded only when the input vector lies within a preset local region, including: Constructing a locally bounded compressor satisfies: ,in, , , p, r, and δ represent the first compression factor, the second compression factor, and the third compression factor, respectively. Indicates a local area.

4. The unified compression method for distributed non-convex optimization according to claim 2, characterized in that, Locally bounded compressors include 1-bit compressors and saturable quantizers; A globally bounded compressor includes a compressor that simultaneously has relative compression error, absolute compression error, and additional bounded noise.

5. The unified compression method for distributed non-convex optimization of claim 2, wherein, The compression error of a locally bounded compressor is bounded only when the input vector lies within a preset local region, and also includes: Constructing a 1-bit compressor is: wherein, is a quantization level, is a constant, , is a 1-bit compressor, a first compression coefficient of the 1-bit compressor , a second compression coefficient , a third compression coefficient ; The number of bits required for a 1-bit compressor to transmit a vector is: 。 6. The unified compression method for distributed non-convex optimization of claim 5, wherein, The compression error of a locally bounded compressor is bounded only when the input vector lies within a preset local region, and also includes: Constructing a saturable quantizer for: ， The number of bits required for a saturated quantizer to transmit a vector For: ,in, Indicates saturation level. To quantize the step size, and These are the floor and floor operators, 1. d Let d be a vector of all 1s. First compression coefficient of a saturating quantizer Second compression coefficient and third compression coefficient , .

7. The unified compression method for distributed non-convex optimization according to claim 6, characterized in that, Multiple intelligent agents interact with each other through a communication network. During the interaction, a compressor is constructed to compress the transmitted information, including: Construct a global compressor, where the global bounded compressor includes a sparsification compressor and a norm-signed compressor; Sparse compressor configured to: , where t j Let [x]t be the coordinate index of the j-th component with the largest absolute value in vector x. j The component value corresponding to the coordinate [x] For the first ... }express The standard base in; Number of bits needed for a sparse compressor to transmit a vector For: Where b1=32, corresponding to a 32-bit floating-point number, is the first compression factor of the sparsity compressor. Second compression factor and the third compression factor , ; A norm-sign compressor is configured to: wherein is an element-wise sign function; The number of bits required for a norm-symmetric compressor to transmit a vector for: , a first compression coefficient of the norm-symbol compressor , a second compression coefficient , and a third compression coefficient , .

8. The unified compression method for distributed non-convex optimization of claim 7, wherein, Multiple intelligent agents interact through a communication network. During the interaction, a compressor is constructed to compress the transmitted information, including: Construct a globally bounded compressor as follows: wherein, , r5and δ5are respectively a second compression coefficient and a third compression coefficient of the global bounded compressor, is the absolute error, is the expectation of the internal randomness of the compressor . Alternatively, construct a k-bit unbiased quantizer resistant to bounded noise. The k-bit unbiased quantizer is: wherein, is a bounded noise, is a Hadamard product, is an absolute value, the second compression coefficient of a k-bit unbiased quantizer satisfying the global bounded compressor assumption is: , the third compression coefficient is: , ; k-bit unbiased quantizer for: ； Alternatively, construct a Rand-k random sparse compressor resistant to bounded noise. The Rand-k random sparse compressor is: Where, r j These are randomly selected and unique coordinate indices. For the first Given standard basis vectors, the second compression coefficient of the Rand-k random sparse compressor, which satisfies the global bounded compressor assumption, is: The third compression factor is: , The number of bits required for a rand-k random sparse compressor to transmit a vector For: ； Alternatively, construct a uniform quantizer resistant to bounded noise. A uniform quantizer that is resistant to bounded noise is: Where i = 5, 6, or 7, It is a bounded noise term and satisfies , The second compression coefficient of the uniform quantizer, which is used to ensure unbiased random vectors and is resistant to bounded noise, satisfies the global bounded compressor assumption. The third compression factor is: , ; Number of bits required for transmission of a vector by a uniform quantizer against bounded noise For: 。 9. The unified compression method for distributed non-convex optimization of claim 8, wherein, Construct auxiliary variables for multiple agents, including: The first auxiliary variable used to reduce compression error is constructed as follows: wherein, is a first auxiliary variable, is a first auxiliary variable of the i-th agent at the k-1th iteration, are algorithm parameters, is a compression variable for communication; ； The second auxiliary variable constructed to reduce communication overhead is as follows: ,in, As the second auxiliary variable, It is the second auxiliary variable of the i-th agent in the (k-1)th iteration. Let L be the (i,j)th element of the Laplace matrix L of the communication network. The first auxiliary variable is used for the i-th agent in the k-th iteration. The second auxiliary variable is also used to record the information of the first auxiliary variable, so that the agents only need to transmit the compressed variable.

10. The unified compression method for distributed non-convex optimization of claim 9, wherein, Construct the primal and dual variables for updating the local model parameters using the primal-dual algorithm, including: Constructing the original variable update algorithm: Update the original variables of the local model parameters. in, It is the original variable of the i-th agent in the (k+1)-th iteration. These are all algorithm parameters. Let (i,j) be the (i,j)th element of the Laplace matrix L of the communication network. The dual variable corresponding to the original variable. for ; Constructing a dual variable update algorithm: Update the dual variable of the local model parameters, where, Let be the dual variable of the i-th agent in the (k+1)-th iteration. Let be the dual variable of the i-th agent in the k-th iteration. This is the second auxiliary variable for the i-th agent in the k-th iteration.

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