Optimal and Efficient Algorithms for Decentralized Online Convex Optimization
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866912151113826304 |
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| author | Wan, Yuanyu Wei, Tong Xue, Bo Song, Mingli Zhang, Lijun |
| author_facet | Wan, Yuanyu Wei, Tong Xue, Bo Song, Mingli Zhang, Lijun |
| contents | We investigate decentralized online convex optimization (D-OCO), in which a set of local learners are required to minimize a sequence of global loss functions using only local computations and communications. Previous studies have established $O(n^{5/4}ρ^{-1/2}\sqrt{T})$ and ${O}(n^{3/2}ρ^{-1}\log T)$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners, $ρ<1$ is the spectral gap of the communication matrix, and $T$ is the time horizon. However, there exist large gaps from the existing lower bounds, i.e., $Ω(n\sqrt{T})$ for convex functions and $Ω(n)$ for strongly convex functions. To fill these gaps, in this paper, we first develop a novel D-OCO algorithm that can respectively reduce the regret bounds for convex and strongly convex functions to $\tilde{O}(nρ^{-1/4}\sqrt{T})$ and $\tilde{O}(nρ^{-1/2}\log T)$. The primary technique is to design an online accelerated gossip strategy that enjoys a faster average consensus among local learners. Furthermore, by carefully exploiting spectral properties of a specific network topology, we enhance the lower bounds for convex and strongly convex functions to $Ω(nρ^{-1/4}\sqrt{T})$ and $Ω(nρ^{-1/2}\log T)$, respectively. These results suggest that the regret of our algorithm is nearly optimal in terms of $T$, $n$, and $ρ$ for both convex and strongly convex functions. Finally, we propose a projection-free variant of our algorithm to efficiently handle practical applications with complex constraints. Our analysis reveals that the projection-free variant can achieve ${O}(nT^{3/4})$ and ${O}(nT^{2/3}(\log T)^{1/3})$ regret bounds for convex and strongly convex functions with nearly optimal $\tilde{O}(ρ^{-1/2}\sqrt{T})$ and $\tilde{O}(ρ^{-1/2}T^{1/3}(\log T)^{2/3})$ communication rounds, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09173 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal and Efficient Algorithms for Decentralized Online Convex Optimization Wan, Yuanyu Wei, Tong Xue, Bo Song, Mingli Zhang, Lijun Machine Learning We investigate decentralized online convex optimization (D-OCO), in which a set of local learners are required to minimize a sequence of global loss functions using only local computations and communications. Previous studies have established $O(n^{5/4}ρ^{-1/2}\sqrt{T})$ and ${O}(n^{3/2}ρ^{-1}\log T)$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners, $ρ<1$ is the spectral gap of the communication matrix, and $T$ is the time horizon. However, there exist large gaps from the existing lower bounds, i.e., $Ω(n\sqrt{T})$ for convex functions and $Ω(n)$ for strongly convex functions. To fill these gaps, in this paper, we first develop a novel D-OCO algorithm that can respectively reduce the regret bounds for convex and strongly convex functions to $\tilde{O}(nρ^{-1/4}\sqrt{T})$ and $\tilde{O}(nρ^{-1/2}\log T)$. The primary technique is to design an online accelerated gossip strategy that enjoys a faster average consensus among local learners. Furthermore, by carefully exploiting spectral properties of a specific network topology, we enhance the lower bounds for convex and strongly convex functions to $Ω(nρ^{-1/4}\sqrt{T})$ and $Ω(nρ^{-1/2}\log T)$, respectively. These results suggest that the regret of our algorithm is nearly optimal in terms of $T$, $n$, and $ρ$ for both convex and strongly convex functions. Finally, we propose a projection-free variant of our algorithm to efficiently handle practical applications with complex constraints. Our analysis reveals that the projection-free variant can achieve ${O}(nT^{3/4})$ and ${O}(nT^{2/3}(\log T)^{1/3})$ regret bounds for convex and strongly convex functions with nearly optimal $\tilde{O}(ρ^{-1/2}\sqrt{T})$ and $\tilde{O}(ρ^{-1/2}T^{1/3}(\log T)^{2/3})$ communication rounds, respectively. |
| title | Optimal and Efficient Algorithms for Decentralized Online Convex Optimization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2402.09173 |