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Optimal and Efficient Algorithms for Decentralized Online Convex Optimization (2402.09173v3)

Published 14 Feb 2024 in cs.LG

Abstract: 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 $O(n{5/4}\rho{-1/2}\sqrt{T})$ and ${O}(n{3/2}\rho{-1}\log T)$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners, $\rho<1$ 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., $\Omega(n\sqrt{T})$ for convex functions and $\Omega(n)$ 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 $\tilde{O}(n\rho{-1/4}\sqrt{T})$ and $\tilde{O}(n\rho{-1/2}\log T)$. 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 $\Omega(n\rho{-1/4}\sqrt{T})$ and $\Omega(n\rho{-1/2}\log T)$, respectively. These results suggest that the regret of our algorithm is nearly optimal in terms of $T$, $n$, and $\rho$ 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 $\tilde{O}(\rho{-1/2}\sqrt{T})$ and $\tilde{O}(\rho{-1/2}T{1/3}(\log T){2/3})$ communication rounds, respectively.

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Authors (5)
  1. Yuanyu Wan (23 papers)
  2. Tong Wei (36 papers)
  3. Mingli Song (163 papers)
  4. Lijun Zhang (239 papers)
  5. Bo Xue (12 papers)
Citations (4)

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