Convergence of a Multi-Agent Projected Stochastic Gradient Algorithm for Non-Convex Optimization

Abstract

We introduce a new framework for the convergence analysis of a class of distributed constrained non-convex optimization algorithms in multi-agent systems. The aim is to search for local minimizers of a non-convex objective function which is supposed to be a sum of local utility functions of the agents. The algorithm under study consists of two steps: a local stochastic gradient descent at each agent and a gossip step that drives the network of agents to a consensus. Under the assumption of decreasing stepsize, it is proved that consensus is asymptotically achieved in the network and that the algorithm converges to the set of Karush-Kuhn-Tucker points. As an important feature, the algorithm does not require the double-stochasticity of the gossip matrices. It is in particular suitable for use in a natural broadcast scenario for which no feedback messages between agents are required. It is proved that our results also holds if the number of communications in the network per unit of time vanishes at moderate speed as time increases, allowing potential savings of the network's energy. Applications to power allocation in wireless ad-hoc networks are discussed. Finally, we provide numerical results which sustain our claims.

Distributed black-box optimization of nonconvex functions

“…First, using earlier results on distributed stochastic approximation due to [22], we show that the sequence obtained from (19) is related to a mean ordinary differential equation (ODE). Then, we show that this ODE is similar to the one analyzed by [2] for the single-agent CE algorithm.…”

“…Nevertheless, there are much fewer studies on nonconvex optimization. Although there are some useful theoretical works [22,23] analyzing the convergence of diffusion and consensus first-order methods over nonconvex cost functions (indeed, we use results from [22] in this paper), first-order methods are not effective for optimizing blackbox multidimensional multi-extrema objectives. where a; is a vector of length M and X C 5ft is a non-empty compact set of solutions.…”

Distributed Adaptive Convex Optimization on Directed Graphs via Continuous-Time Algorithms

“…Most of existing algorithms are based on discrete-time dynamics (see e.g., [5]- [10], [21]). By designing the consensusbased dynamics, these discrete-time algorithms can find the solution of the optimization problem.…”

Event-Triggered Discrete-Time Distributed Consensus Optimization over Time-Varying Graphs

“…Convergence to the same optimal solution is proved for the cases when the weights were constant and equal and when the weights were uniform but all agents had the same constraint set. Further distributed algorithm for set constrained optimization was investigated in Bianchi and Jakubowicz [34] and Lou et al [35]. To work out the distributed optimization problems with 2 Complexity asynchronous step-sizes or inequality-equality constraints, distributed Lagrangian and penalty primal-dual subgradient algorithms were developed in Zhu and Martinez [36] and Towfic and Sayed [37].…”