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Register a new userProvably Accelerated Decentralized Gradient Method Over Unbalanced Directed Graphs
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Added by
Zhuoqing Song
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this work, we consider the decentralized optimization problem in which a network of $n$ agents, each possessing a smooth and convex objective function, wish to collaboratively minimize the average of all the objective functions through peer-to-peer communication in a directed graph. To solve the problem, we propose two accelerated Push-DIGing methods termed APD and APD-SC for minimizing non-strongly convex objective functions and strongly convex ones, respectively. We show that APD and APD-SC respectively converge at the rates $Oleft(frac{1}{k^2}right)$ and $Oleft(left(1 - Csqrt{frac{mu}{L}}right)^kright)$ up to constant factors depending only on the mixing matrix. To the best of our knowledge, APD and APD-SC are the first decentralized methods to achieve provable acceleration over unbalanced directed graphs. Numerical experiments demonstrate the effectiveness of both methods.
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In this paper, we propose two communication-efficient algorithms for decentralized optimization over a multi-agent network with general directed network topology. In the first part, we consider a novel communication-efficient gradient tracking based method, termed Compressed Push-Pull (CPP), which combines the Push-Pull method with communication compression. We show that CPP is applicable to a general class of unbiased compression operators and achieves linear convergence for strongly convex and smooth objective functions. In the second part, we propose a broadcast-like version of CPP (B-CPP), which also achieves linear convergence rate under the same conditions for the objective functions. B-CPP can be applied in an asynchronous broadcast setting and further reduce communication costs compared to CPP. Numerical experiments complement the theoretical analysis and confirm the effectiveness of the proposed methods.
Decentralized optimization over time-varying graphs has been increasingly common in modern machine learning with massive data stored on millions of mobile devices, such as in federated learning. This paper revisits the widely used accelerated gradient tracking and extends it to time-varying graphs. We prove the $O((frac{gamma}{1-sigma_{gamma}})^2sqrt{frac{L}{epsilon}})$ and $O((frac{gamma}{1-sigma_{gamma}})^{1.5}sqrt{frac{L}{mu}}logfrac{1}{epsilon})$ complexities for the practical single loop accelerated gradient tracking over time-varying graphs when the problems are nonstrongly convex and strongly convex, respectively, where $gamma$ and $sigma_{gamma}$ are two common constants charactering the network connectivity, $epsilon$ is the desired precision, and $L$ and $mu$ are the smoothness and strong convexity constants, respectively. Our complexities improve significantly over the ones of $O(frac{1}{epsilon^{5/7}})$ and $O((frac{L}{mu})^{5/7}frac{1}{(1-sigma)^{1.5}}logfrac{1}{epsilon})$, respectively, which were proved in the original literature only for static graphs, where $frac{1}{1-sigma}$ equals $frac{gamma}{1-sigma_{gamma}}$ when the network is time-invariant. When combining with a multiple consensus subroutine, the dependence on the network connectivity constants can be further improved to $O(1)$ and $O(frac{gamma}{1-sigma_{gamma}})$ for the computation and communication complexities, respectively. When the network is static, by employing the Chebyshev acceleration, our complexities exactly match the lower bounds without hiding any poly-logarithmic factor for both nonstrongly convex and strongly convex problems.
In this paper, we propose Push-SAGA, a decentralized stochastic first-order method for finite-sum minimization over a directed network of nodes. Push-SAGA combines node-level variance reduction to remove the uncertainty caused by stochastic gradients, network-level gradient tracking to address the distributed nature of the data, and push-sum consensus to tackle the challenge of directed communication links. We show that Push-SAGA achieves linear convergence to the exact solution for smooth and strongly convex problems and is thus the first linearly-convergent stochastic algorithm over arbitrary strongly connected directed graphs. We also characterize the regimes in which Push-SAGA achieves a linear speed-up compared to its centralized counterpart and achieves a network-independent convergence rate. We illustrate the behavior and convergence properties of Push-SAGA with the help of numerical experiments on strongly convex and non-convex problems.
The present work introduces the hybrid consensus alternating direction method of multipliers (H-CADMM), a novel framework for optimization over networks which unifies existing distributed optimization approaches, including the centralized and the decentralized consensus ADMM. H-CADMM provides a flexible tool that leverages the underlying graph topology in order to achieve a desirable sweet-spot between node-to-node communication overhead and rate of convergence -- thereby alleviating known limitations of both C-CADMM and D-CADMM. A rigorous analysis of the novel method establishes linear convergence rate, and also guides the choice of parameters to optimize this rate. The novel hybrid update rules of H-CADMM lend themselves to in-network acceleration that is shown to effect considerable -- and essentially free-of-charge -- performance boost over the fully decentralized ADMM. Comprehensive numerical tests validate the analysis and showcase the potential of the method in tackling efficiently, widely useful learning tasks.
We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method, thereby providing a systematic way for deriving several well-known decentralized algorithms including EXTRA arXiv:1404.6264 and SSDA arXiv:1702.08704. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower bounds. We provide experimental results that demonstrate the effectiveness of the proposed algorithm on highly ill-conditioned problems.
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