Optimal and Efficient Algorithms for Decentralized Online Convex Optimization
Yuanyu Wan, Tong Wei, Bo Xue, Mingli Song, Lijun Zhang
TL;DR
The paper tackles decentralized online convex optimization on a network, addressing gaps between existing upper and lower bounds by introducing an accelerated gossip-based algorithm (AD-FTGL) with a blocking update that yields near-optimal regret: $\tilde{O}(n\rho^{-1/4}\sqrt{T})$ for convex and $\tilde{O}(n\rho^{-1/2}\log T)$ for strongly convex objectives. It also provides matching topology-aware lower bounds and extends the framework with a projection-free variant achieving favorable regret-communication trade-offs, including $O(nT^{3/4})$ for convex and $O(nT^{2/3}(\log T)^{1/3})$ for strongly convex cases, with nearly optimal communication rounds. The analysis hinges on improved consensus via Acc_Gossip, a blocking update mechanism, and refined spectral-gap arguments, including a tighter $O(\rho^{-1}\log n)$ bound for gossip errors. Collectively, the results demonstrate that the proposed algorithms are nearly optimal in $T$, $n$, and $\rho$ and offer practical projection-free alternatives for complex constraint sets. The study advances understanding of topology- and communication-dependent limits in D-OCO and offers tools for efficient distributed online learning under realistic communication constraints.
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 $O(n^{5/4}ρ^{-1/2}\sqrt{T})$ 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.