Single Point-Based Distributed Zeroth-Order Optimization with a Non-Convex Stochastic Objective Function

Elissa Mhanna, Mohamad Assaad
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:24701-24719, 2023.

Abstract

Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus. However, it is a first-order (FO) method that requires knowledge of the gradient, which is not always possible in practice. In this work, we introduce a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique. We prove that this new technique converges with a single noisy function query at a time in the non-convex setting. We then establish a convergence rate of $O(\frac{1}{\sqrt[3]{K}})$ after a number of iterations K, which competes with that of $O(\frac{1}{\sqrt[4]{K}})$ of its centralized counterparts. Finally, a numerical example validates our theoretical results.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-mhanna23a, title = {Single Point-Based Distributed Zeroth-Order Optimization with a Non-Convex Stochastic Objective Function}, author = {Mhanna, Elissa and Assaad, Mohamad}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {24701--24719}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/mhanna23a/mhanna23a.pdf}, url = {https://proceedings.mlr.press/v202/mhanna23a.html}, abstract = {Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus. However, it is a first-order (FO) method that requires knowledge of the gradient, which is not always possible in practice. In this work, we introduce a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique. We prove that this new technique converges with a single noisy function query at a time in the non-convex setting. We then establish a convergence rate of $O(\frac{1}{\sqrt[3]{K}})$ after a number of iterations K, which competes with that of $O(\frac{1}{\sqrt[4]{K}})$ of its centralized counterparts. Finally, a numerical example validates our theoretical results.} }
Endnote
%0 Conference Paper %T Single Point-Based Distributed Zeroth-Order Optimization with a Non-Convex Stochastic Objective Function %A Elissa Mhanna %A Mohamad Assaad %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-mhanna23a %I PMLR %P 24701--24719 %U https://proceedings.mlr.press/v202/mhanna23a.html %V 202 %X Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus. However, it is a first-order (FO) method that requires knowledge of the gradient, which is not always possible in practice. In this work, we introduce a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique. We prove that this new technique converges with a single noisy function query at a time in the non-convex setting. We then establish a convergence rate of $O(\frac{1}{\sqrt[3]{K}})$ after a number of iterations K, which competes with that of $O(\frac{1}{\sqrt[4]{K}})$ of its centralized counterparts. Finally, a numerical example validates our theoretical results.
APA
Mhanna, E. & Assaad, M.. (2023). Single Point-Based Distributed Zeroth-Order Optimization with a Non-Convex Stochastic Objective Function. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:24701-24719 Available from https://proceedings.mlr.press/v202/mhanna23a.html.

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