ZONE: Zeroth-Order Nonconvex Multiagent Optimization Over Networks

Abstract

In this paper we consider distributed optimization problems over a multi-agent network, where each agent can only partially evaluate the objective function, and it is allowed to exchange messages with its immediate neighbors. Differently from all existing works on distributed optimization, our focus is given to optimizing a class of non-convex problems, and under the challenging setting where each agent can only access the zeroth-order information (i.e., the functional values) of its local functions. For different types of network topologies such as undirected connected networks or star networks, we develop efficient distributed algorithms and rigorously analyze their convergence and rate of convergence (to the set of stationary solutions). Numerical results are provided to demonstrate the efficiency of the proposed algorithms. Furthermore, for given J independent samples of {(φ j , ξ j )} J j=1 , we defineḠ µ (z, φ, ξ) as the sample average:ḠLemma 1 [34, Lemma 4.2] Suppose that Assumption A holds true for function ψ : R Q → R. Then we have the followingwhereσ 2 := 2Q[K 2 + σ 2 + µ 2L2 Q].

Distributed Zero-Order Algorithms for Nonconvex Multi-Agent optimization

“…In this subsection, we compare our algorithms and results with existing literature on distributed zero-order optimization, specifically [33,34,35,36].…”

“…1. References [34,35,36] discuss convex problems, while [33] and our work focus on nonconvex problems.…”

“…In terms of the assumptions on the noisy function queries, [36] and our work consider a noise-free case. [33] considers stochastic queries but two function evaluations are available for each random sample. [34] considers an online setting but two online function evaluations are available for each time step.…”

“…[35] assumes that independent noises are added on each query of function values. We expect that our proposed Algorithm 1 can be generalized to the setting used in [33] with heavier mathematical exposition. Extensions to general stochastic cases remain our ongoing work.…”

“…3. In terms of the approach to reach consensus among agents, our algorithms are similar to [35,36], where some form of weighted average of the neighbors' local variables is utilized, while [33] uses the method of multipliers and [34] uses ADMM to design their algorithms. We also mention that, our Algorithm 2 also employs the gradient tracking technique, which, to our best knowledge, has not been discussed in existing literature on distributed zero-order optimization yet.…”