A Compressed Gradient Tracking Method for Decentralized Optimization With Linear Convergence

Liao, Yiwei; Li, Zhuorui; Huang, Kun; Pu, Shi

doi:10.1109/tac.2022.3180695

Cited by 20 publications

(5 citation statements)

References 44 publications

(85 reference statements)

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“…It is worthwhile to point out that the proposed LSGT and MUST methods, to the best of our knowledge, are the first stochastic GT algorithm with multiple local SGD updates for decentralized learning. In the future, we plan to extend this framework to time-varying communication network topologies [35] and that with compression [36].…”

Section: Discussionmentioning

confidence: 99%

“…so that the coefficient of the last term in the RHS of ( 35) is negative and the term can be removed. In addition, we have used the properties of λ 2 w < 1 and E ≥ 1 to obtain bounds for coefficients of the 3rd and 4th terms in the RHS of (36). Finally, after dividing T γE 4 on both sides of (36), we obtain the results in Theorem 1.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Gradient and Variable Tracking with Multiple Local SGD for Decentralized Non-Convex Learning

Ge¹,

Chang²

2023

Preprint

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Stochastic distributed optimization methods that solve an optimization problem over a multi-agent network have played an important role in a variety of large-scale signal processing and machine leaning applications. Among the existing methods, the gradient tracking (GT) method is found robust against the variance between agents' local data distribution, in contrast to the distributed stochastic gradient descent (SGD) methods which have a slowed convergence speed when the agents have heterogeneous data distributions. However, the GT method can be communication expensive due to the need of a large number of iterations for convergence. In this paper, we intend to reduce the communication cost of the GT method by integrating it with the local SGD technique. Specifically, we propose a new local stochastic GT (LSGT) algorithm where, within each communication round, the agents perform multiple SGD updates locally. Theoretically, we build the convergence conditions of the LSGT algorithm and show that it can have an improved convergence rate of O(1/ √ ET ), where E is the number of local SGD updates and T is the number of communication rounds. We further extend the LSGT algorithm to solve a more complex learning problem which has linearly coupled variables inside the objective function. Experiment results demonstrate that the proposed algorithms have significantly improved convergence speed even under heterogeneous data distribution.

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Section: Discussionmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Gradient and Variable Tracking with Multiple Local SGD for Decentralized Non-Convex Learning

Ge¹,

Chang²

2023

Preprint

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“…In recent years, various techniques have been developed to reduce communication costs [27,31]. They are extensively incorporated into centralized optimization methods [1,23,30] and decentralized methods [8,9,18,20,33]. This motivates us to provide an extension of D-ASCGD by combining it with communication compressed method, which reads as follows.…”

Section: Compressed D-ascgd Methodsmentioning

confidence: 99%

“…For vector x ∈ R d or z ∈ R p , the class of compressors satisfying (28) or ( 29) is broad, such as random quantization [23,26], sparsification [15,19], the norm-sign compressor [18,33]. For matrix y ∈ R d×p , we may construct the dp-length vector by stacking up the columns of y and then implement predetermined compressors to compress the new constructed vector.…”

Section: And Compute Function Valuesmentioning

confidence: 99%

Numerical Methods for Distributed Stochastic Compositional Optimization Problems with Aggregative Structure

Zhao¹,

Liu²

2022

Preprint

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The paper studies the distributed stochastic compositional optimization problems over networks, where all the agents' inner-level function is the sum of each agent's private expectation function. Focusing on the aggregative structure of the inner-level function, we employ the hybrid variance reduction method to obtain the information on each agent's private expectation function, and apply the dynamic consensus mechanism to track the information on each agent's inner-level function. Then by combining with the standard distributed stochastic gradient descent method, we propose a distributed aggregative stochastic compositional gradient descent method. When the objective function is smooth, the proposed method achieves the optimal convergence rate O K −1/2 . We further combine the proposed method with the communication compression and propose the communication compressed variant distributed aggregative stochastic compositional gradient descent method. The compressed variant of the proposed method maintains the optimal convergence rate O K −1/2 . Simulated experiments on decentralized reinforcement learning verify the effectiveness of the proposed methods.

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“…For another example, let f i (x) := (1/|S i |) ζj ∈Si F (x, ζ j ) denote an empirical risk function, where S i is the local dataset of agent i. Under this setting, the gradient estimation of f i (x) can incur noise from various sources such as sampling and discretization [15]. The second condition in (2) holds for many popular problems such as linear regression, smooth SVMs, logistic regression, and softmax regression [16].…”

Section: Introductionmentioning

confidence: 99%

Improving the Transient Times for Distributed Stochastic Gradient Methods

Huang

2022

IEEE Trans. Automat. Contr.

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We consider the distributed optimization problem where n agents , each possessing a local cost function, collaboratively minimize the average of the n cost functions over a connected network. Assuming stochastic gradient information is available, we study a distributed stochastic gradient algorithm, called exact diffusion with adaptive stepsizes (EDAS) adapted from the Exact Diffusion method [1] and NIDS [2] and perform a non-asymptotic convergence analysis. We not only show that EDAS asymptotically achieves the same network independent convergence rate as centralized stochastic gradient descent (SGD) for minimizing strongly convex and smooth objective functions, but also characterize the transient time needed for the algorithm to approach the asymptotic convergence rate, which behaves as, where 1 − λ 2 stands for the spectral gap of the mixing matrix. To the best of our knowledge, EDAS achieves the shortest transient time when the average of the n cost functions is strongly convex and each cost function is smooth. Numerical simulations further corroborate and strengthen the obtained theoretical results.

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A Compressed Gradient Tracking Method for Decentralized Optimization With Linear Convergence

Cited by 20 publications

References 44 publications

Gradient and Variable Tracking with Multiple Local SGD for Decentralized Non-Convex Learning

Gradient and Variable Tracking with Multiple Local SGD for Decentralized Non-Convex Learning

Numerical Methods for Distributed Stochastic Compositional Optimization Problems with Aggregative Structure

Improving the Transient Times for Distributed Stochastic Gradient Methods

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