Decentralized Online Exp-Concave Optimization

Decentralized Online Convex Optimization (D-OCO) is a fundamental framework in which a set of local agents aims to minimize a sequence of global loss functions using only local computations and communications. While recent work has established near-optimal regret guarantees for convex and strongly convex losses, the case of exp-concave losses remains an open challenge: specifically, whether logarithmic regret is achievable. In this paper, we answer this question affirmatively by proposing Decentralized Online Newton Step (D-ONS), which adapts the idea of blocking and accelerated gossip to the Online Newton Step algorithm in a non-trivial manner. Concretely, our algorithm achieves a regret bound of $O\left(Nd(1-σ_2)^{-1/2}\,\log^2(T) \right)$ where $N$ is the number of agents, $d$ is the dimension, $T$ is the horizon, and $1-σ_2$ represents the spectral gap of the network. To our knowledge, this is the first logarithmic regret bound for decentralized online exp-concave optimization. We further establish a nearly matching lower bound of $Ω\left(Nd (1-σ_2)^{-1/2}\,\log(T/N) \right)$ showing the near optimality of our upper bound. Finally, we extend our algorithm to decentralized online linear regression in which the feasible domain is unbounded and features and labels are locally observed. By gossiping covariance and label-feature statistics, we also achieve logarithmic regret.