Penalty-Based Method for Decentralized Optimization over Time-Varying Graphs

Abstract

We consider a distributed stochastic optimization problem that is solved by a decentralized network of agents with only local communication between neighboring agents. The goal of the whole system is to minimize a global objective function given as a sum of local objectives held by each agent. Each local objective is defined as an expectation of a convex smooth random function and the agent is allowed to sample stochastic gradients for this function. For this setting we propose the first accelerated (in the sense of Nesterov's acceleration) method that simultaneously attains optimal up to a logarithmic factor communication and oracle complexity bounds for smooth strongly convex distributed stochastic optimization. We also consider the case when the communication graph is allowed to vary with time and obtain complexity bounds for our algorithm, which are the first upper complexity bounds for this setting in the literature.

Recent theoretical advances in decentralized distributed convex optimization

“…) which is optimal up to a logarithmic term. Similar results are attained in works which use penalty-based methods [75,101,39] (see Appendix B in [39]) for details.…”

“…find ∇W µ (p, q) satisfying (102) with δ = O(µε 2 ). Taking into account the relation (101) we get that it is needed to solve the problem (99) with accuracy δ = O(µε 2 ) in terms of the distance to the optimum. i.e.…”