Distributed Online Convex Optimization with Efficient Communication: Improved Algorithm and Lower bounds

TL;DR

This work tackles distributed online convex optimization with compressed communication, addressing costly data exchange in networks of $n$ learners. It introduces Top-DOGD, a two-level blocking gossip framework that combines online gossip and projection-error compensation to improve consensus under compression while maintaining a single update per round. The authors prove improved regret bounds of $\tilde{O}(\omega^{-1/2}\rho^{-1}n\sqrt{T})$ for convex and $\tilde{O}(\omega^{-1}\rho^{-2}n\ln T)$ for strongly convex losses, and establish near-matching lower bounds $\Omega(\omega^{-1/2}\rho^{-1/4}n\sqrt{T})$ and $\Omega(\omega^{-1}\rho^{-1/2}n\ln T)$, justifying optimality with respect to both $\omega$ and $T$. The method extends to bandit feedback using classical gradient estimators, yielding improved rates in one-point and two-point settings. Together, these results significantly reduce communication requirements while preserving fast regret, enabling scalable distributed learning in bandwidth-constrained networks.

Abstract

We investigate distributed online convex optimization with compressed communication, where $n$ learners connected by a network collaboratively minimize a sequence of global loss functions using only local information and compressed data from neighbors. Prior work has established regret bounds of $O(\max\{ω^{-2}ρ^{-4}n^{1/2},ω^{-4}ρ^{-8}\}n\sqrt{T})$ and $O(\max\{ω^{-2}ρ^{-4}n^{1/2},ω^{-4}ρ^{-8}\}n\ln{T})$ for convex and strongly convex functions, respectively, where $ω\in(0,1]$ is the compression quality factor ($ω=1$ means no compression) and $ρ<1$ is the spectral gap of the communication matrix. However, these regret bounds suffer from a \emph{quadratic} or even \emph{quartic} dependence on $ω^{-1}$. Moreover, the \emph{super-linear} dependence on $n$ is also undesirable. To overcome these limitations, we propose a novel algorithm that achieves improved regret bounds of $\tilde{O}(ω^{-1/2}ρ^{-1}n\sqrt{T})$ and $\tilde{O}(ω^{-1}ρ^{-2}n\ln{T})$ for convex and strongly convex functions, respectively. The primary idea is to design a \emph{two-level blocking update framework} incorporating two novel ingredients: an online gossip strategy and an error compensation scheme, which collaborate to \emph{achieve a better consensus} among learners. Furthermore, we establish the first lower bounds for this problem, justifying the optimality of our results with respect to both $ω$ and $T$. Additionally, we consider the bandit feedback scenario, and extend our method with the classic gradient estimators to enhance existing regret bounds.

Figures (1)

• Figure 1: Example of $1$-connected cycle graph

Theorems & Definitions (20)

• Definition 3.6
• Lemma 3.7
• Lemma 3.8
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3