Distributed Online Convex Optimization with Nonseparable Costs and Constraints

TL;DR

This paper introduces a distributed online primal-dual belief consensus algorithm for nonseparable convex optimization with coupled constraints, achieving sublinear regret and constraint violation bounds, thus advancing online distributed control methods.

Contribution

It proposes a novel belief-sharing protocol that handles nonseparable costs and constraints, breaking the $O(T^{3/4})$ barrier for cumulative constraint violation in distributed online convex optimization.

Findings

Achieves $O(T^{1/2})$ bounds for regret and constraint violation.

Eliminates coupling between primal disagreement and dual violation.

Matches the lower bound of online constrained convex optimization.

Abstract

This paper studies distributed online convex optimization with time-varying coupled constraints, motivated by distributed online control in network systems. Most prior work assumes a separability condition: the global objective and coupled constraint functions are sums of local costs and individual constraints. In contrast, we study a group of agents, networked via a communication graph, that collectively select actions to minimize a sequence of nonseparable global cost functions and to satisfy nonseparable long-term constraints based on full-information feedback and intra-agent communication. We propose a distributed online primal-dual belief consensus algorithm, where each agent maintains and updates a local belief of the global collective decisions, which are repeatedly exchanged with neighboring agents. Unlike the previous consensus primal-dual algorithms under separability that ask agents to only communicate their local decisions, our belief-sharing protocol eliminates coupling between the primal consensus disagreement and the dual constraint violation, yielding sublinear regret and cumulative constraint violation (CCV) bounds, both in $O({T}^{1/2})$, where $T$ denotes the time horizon. Such a result breaks the long-standing $O(T^{3/4})$ barrier for CCV and matches the lower bound of online constrained convex optimization, indicating the online learning efficiency at the cost of communication overhead.