A Hybrid Variance-Reduced Method for Decentralized Stochastic Non-Convex Optimization

Ran Xin, Usman Khan, Soummya Kar
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:11459-11469, 2021.

Abstract

This paper considers decentralized stochastic optimization over a network of $n$ nodes, where each node possesses a smooth non-convex local cost function and the goal of the networked nodes is to find an $\epsilon$-accurate first-order stationary point of the sum of the local costs. We focus on an online setting, where each node accesses its local cost only by means of a stochastic first-order oracle that returns a noisy version of the exact gradient. In this context, we propose a novel single-loop decentralized hybrid variance-reduced stochastic gradient method, called GT-HSGD, that outperforms the existing approaches in terms of both the oracle complexity and practical implementation. The GT-HSGD algorithm implements specialized local hybrid stochastic gradient estimators that are fused over the network to track the global gradient. Remarkably, GT-HSGD achieves a network topology-independent oracle complexity of $O(n^{-1}\epsilon^{-3})$ when the required error tolerance $\epsilon$ is small enough, leading to a linear speedup with respect to the centralized optimal online variance-reduced approaches that operate on a single node. Numerical experiments are provided to illustrate our main technical results.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-xin21a, title = {A Hybrid Variance-Reduced Method for Decentralized Stochastic Non-Convex Optimization}, author = {Xin, Ran and Khan, Usman and Kar, Soummya}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {11459--11469}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/xin21a/xin21a.pdf}, url = {https://proceedings.mlr.press/v139/xin21a.html}, abstract = {This paper considers decentralized stochastic optimization over a network of $n$ nodes, where each node possesses a smooth non-convex local cost function and the goal of the networked nodes is to find an $\epsilon$-accurate first-order stationary point of the sum of the local costs. We focus on an online setting, where each node accesses its local cost only by means of a stochastic first-order oracle that returns a noisy version of the exact gradient. In this context, we propose a novel single-loop decentralized hybrid variance-reduced stochastic gradient method, called GT-HSGD, that outperforms the existing approaches in terms of both the oracle complexity and practical implementation. The GT-HSGD algorithm implements specialized local hybrid stochastic gradient estimators that are fused over the network to track the global gradient. Remarkably, GT-HSGD achieves a network topology-independent oracle complexity of $O(n^{-1}\epsilon^{-3})$ when the required error tolerance $\epsilon$ is small enough, leading to a linear speedup with respect to the centralized optimal online variance-reduced approaches that operate on a single node. Numerical experiments are provided to illustrate our main technical results.} }
Endnote
%0 Conference Paper %T A Hybrid Variance-Reduced Method for Decentralized Stochastic Non-Convex Optimization %A Ran Xin %A Usman Khan %A Soummya Kar %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-xin21a %I PMLR %P 11459--11469 %U https://proceedings.mlr.press/v139/xin21a.html %V 139 %X This paper considers decentralized stochastic optimization over a network of $n$ nodes, where each node possesses a smooth non-convex local cost function and the goal of the networked nodes is to find an $\epsilon$-accurate first-order stationary point of the sum of the local costs. We focus on an online setting, where each node accesses its local cost only by means of a stochastic first-order oracle that returns a noisy version of the exact gradient. In this context, we propose a novel single-loop decentralized hybrid variance-reduced stochastic gradient method, called GT-HSGD, that outperforms the existing approaches in terms of both the oracle complexity and practical implementation. The GT-HSGD algorithm implements specialized local hybrid stochastic gradient estimators that are fused over the network to track the global gradient. Remarkably, GT-HSGD achieves a network topology-independent oracle complexity of $O(n^{-1}\epsilon^{-3})$ when the required error tolerance $\epsilon$ is small enough, leading to a linear speedup with respect to the centralized optimal online variance-reduced approaches that operate on a single node. Numerical experiments are provided to illustrate our main technical results.
APA
Xin, R., Khan, U. & Kar, S.. (2021). A Hybrid Variance-Reduced Method for Decentralized Stochastic Non-Convex Optimization. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:11459-11469 Available from https://proceedings.mlr.press/v139/xin21a.html.

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