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

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(maxω-2ρ-4n1/2,ω-4ρ-8n√T) and O(maxω-2ρ-4n1/2,ω-4ρ-8nlnT) for convex and strongly convex functions, respectively, where ω∈(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 quadratic or even quartic dependence on ω-1. Moreover, the super-linear dependence on n is also undesirable. To overcome these limitations, we propose a novel algorithm that achieves improved regret bounds of tildeO(ω-1/2ρ-1n√T) and tildeO(ω-1ρ-2nlnT) for convex and strongly convex functions, respectively. The primary idea is to design a two-level blocking update framework incorporating two novel ingredients: an online gossip strategy and an error compensation scheme, which collaborate to 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.