Distributed Online Optimization With Long-Term Constraints

In this article, we consider distributed online convex optimization problems, where the distributed system consists of various computing units connected through a time-varying communication graph. In each time step, each computing unit selects a constrained vector, experiences a loss equal to an arbitrary convex function evaluated at this vector, and may communicate to its neighbors in the graph. The objective is to minimize the system-wide loss accumulated over time. We propose a decentralized algorithm with regret and cumulative constraint violation inO(Tmax c,1-c)andO(T1-c/2), respectively, for anyc∈ (0,1), whereTis the time horizon. When the loss functions are strongly convex, we establish improved regret and constraint violation upper bounds inO(log (T))andO(√Tlog (T)). These regret scalings match those obtained by state-of-the-art algorithms and fundamental limits in the corresponding centralized online optimization problem (for both convex and strongly convex loss functions). In the case of bandit feedback, the proposed algorithms achieve a regret and constraint violation inO(Tmax c,1-c/3)andO(T1-c/2)for anyc∈ (0,1). We numerically illustrate the performance of our algorithms for the particular case of distributed online regularized linear regression problems on synthetic and real data.