Robust Multi-Agent Optimization: Coping with Packet-Dropping Link Failures

We study the problem of multi-agent optimization in the presence of communication failures, where agents are connected by a strongly connected communication network. Specifically, we are interested in optimizing $h(x)=\frac{1}{n}\sum_{i=1}^n h_i(x)$, where $\mathcal{V}=\{1, \ldots, n\}$ is the set of agents, and $h_i(\cdot)$ is agent $i$'s local cost function. We consider the scenario where the communication links may suffer packet-dropping failures (i.e., the sent messages are not guaranteed to be delivered in the same iteration), but each link is reliable at least once in every $B$ consecutive message transmissions. This bounded time reliability assumption is reasonable since it has been shown that with unbounded message delays, convergence is not guaranteed for reaching consensus -- a special case of the optimization problem of interest. We propose a robust distributed optimization algorithm wherein each agent updates its local estimate using slightly different routines in odd and even iterations. We show that these local estimates converge to a common optimum of $h(\cdot)$ sub-linearly at convergence rate $O(\frac{1}{\sqrt{t}})$, where $t$ is the number of iteration. Our proposed algorithm combines the Push-Sum Distributed Dual Averaging method with a robust average consensus algorithm. The main analysis challenges come from the fact that the effective communication network is time varying, and that each agent does not know the actual number of reliable outgoing links at each iteration.