Fast Distributed Gradient Methods (1112.2972v4)

Published 13 Dec 2011 in cs.IT and math.IT

Abstract: We study distributed optimization problems when $N$ nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant $L$), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications $\mathcal{K}$ and the per-node gradient evaluations $k$. Our first method, Distributed Nesterov Gradient, achieves rates $O\left({\log \mathcal{K}}/{\mathcal{K}}\right)$ and $O\left({\log k}/{k}\right)$. Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of $L$ and $\mu(W)$ -- the second largest singular value of the $N \times N$ doubly stochastic weight matrix $W$. It achieves rates $O\left({1}/{\mathcal{K}^{{2-\xi}}\right)$} and $O\left({1}/{k^2}\right)$ ($\xi>0$ arbitrarily small). Further, we give with both methods explicit dependence of the convergence constants on $N$ and $W$. Simulation examples illustrate our findings.

Authors (3)

Dusan Jakovetic (47 papers)
Joao Xavier (16 papers)
Jose M. F. Moura (24 papers)

Citations (573)

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Summary

Fast Distributed Gradient Methods: Analysis and Insights

The paper "Fast Distributed Gradient Methods" by Dusan Jakovetic et al., examines the problem of distributed optimization where $N$ nodes aim to collectively minimize the sum of convex functions associated with each node. These functions are assumed to have Lipschitz continuous and bounded gradients. The researchers propose two innovative algorithms that build upon the foundational Nesterov gradient method, emphasizing improved convergence rates and practical applicability in communication-constrained networks.

Overview of Proposed Algorithms

The paper introduces two algorithms:

Distributed Nesterov Gradient (D--NG): This algorithm employs a diminishing step-size, which is crucial for achieving strict convergence to the optimal solution. It operates by iteratively averaging local node updates and utilizing a strategy inspired by Nesterov's accelerated gradient method. The D--NG algorithm is designed to exhibit rates of $O(\log \mathcal{K}/\mathcal{K})$ and $O(\log k/k)$ concerning communication rounds and gradient evaluations per node, respectively.
Distributed Nesterov Gradient with Consensus (D--NC): The D--NC algorithm requires nodes to be aware of the global Lipschitz constant $L$ and the network's connectivity properties. It integrates a consensus mechanism into the optimization process, achieving accelerated rates of $O(1/\mathcal{K}^{2-\xi})$ and $O(1/k^2)$ . The inclusion of consensus steps addresses potential misalignment in local calculations across the network.

Convergence and Performance Implications

The theoretical analysis provided in the paper underscores the enhanced convergence properties of the proposed methods. The D--NG algorithm, while effective for general scenarios, is adept for cases where global parameters like $L$ and network connectivity details are not explicitly known by the nodes. Conversely, the D--NC method leverages such global information to expedite convergence significantly.

The researchers highlight the superior network scaling and robustness of the proposed algorithms compared to existing distributed gradient methods. By leveraging Nesterov's acceleration techniques, the methods efficiently balance the computational and communication costs prevalent in distributed networks.

Network Scaling and Practical Considerations

The paper explores the critical aspect of network scalability, emphasizing how the proposed methods handle increasing node numbers and variations in network topology. The authors demonstrate that even under limited communication, the methods maintain rigorous convergence guarantees.

For practical applications, the authors suggest that the knowledge of network parameters can be obtained through distributed estimation methods, though the need for such knowledge varies between the D--NG and D--NC algorithms.

Future Directions and Theoretical Contributions

The paper opens avenues for further exploration in distributed optimization, especially in fine-tuning communication steps and exploring scenarios with dynamic network topologies. The authors suggest potential modifications, such as distributed line search strategies, to further enhance the applicability of the methods.

The research contributes significantly to understanding optimization in decentralized environments, providing foundational techniques that bridge the gap between classical centralized optimization and modern distributed requirements.

In conclusion, the paper offers a comprehensive examination of innovative distributed gradient methods, providing both theoretical insights and practical guidelines for their deployment in varied networking scenarios. These contributions serve as a foundation for advancing distributed optimization methodologies in an increasingly decentralized computational landscape.

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