Linear Convergence Rate of a Class of Distributed Augmented Lagrangian Algorithms

Abstract

Abstract-We study distributed optimization where nodes cooperatively minimize the sum of their individual, locally known, convex costs fi(x)'s, x ∈ R d is global. Distributed augmented Lagrangian (AL) methods have good empirical performance on several signal processing and learning applications, but there is limited understanding of their convergence rates and how it depends on the underlying network. This paper establishes globally linear (geometric) convergence rates of a class of deterministic and randomized distributed AL methods, when the fi's are twice continuously differentiable and have a bounded Hessian. We give explicit dependence of the convergence rates on the underlying network parameters. Simulations illustrate our analytical findings.

Tracking-ADMM for distributed constraint-coupled optimization

“…If we ignore the cross term 2c[z k+1 − A d x ] e λ k+1 , then (14) tells us that as long asλ k changes from one iteration to the next one, the network of agents is, on average, getting closer to the optimal solution of D. The additional cross term depends on the consensus error e λ k+1 = λ k+1 −λ k+1 and is clearly a distinctive feature of Tracking-ADMM (as opposed to the parallel ADMM) arising from its distributed nature. In the parallel ADMM, a condition similar to (14) can be derived and leveraged to prove the algorithm convergence. However, as expected, this is not the case here: (14) is not sufficient to prove convergence of the proposed scheme and we are required to study the interplay between the minimization step in (6a) and the two consensus steps in (6b) and (6c).…”

“…In the parallel ADMM, a condition similar to (14) can be derived and leveraged to prove the algorithm convergence. However, as expected, this is not the case here: (14) is not sufficient to prove convergence of the proposed scheme and we are required to study the interplay between the minimization step in (6a) and the two consensus steps in (6b) and (6c).…”

“…However, the consensus system is not autonomous, thus the inequality describing the variation of such Lyapunov function across iterations contains terms that are not defined in sign, so that it is no longer necessarily decreasing. The idea is to properly combine such inequality with (14) of Proposition 1 so as to "balance out" the terms that are not defined in sign. In this way, we obtain an aggregate descent condition that allows us to prove the asymptotic stability of the overall (nonlinear) dynamical system modeling our Tracking-ADMM algorithm.…”

“…Even though constraint-coupled problems arise more naturally in practical applications the main effort in distributed optimization, until recently, has been devoted to solve optimization problems in which there is no coupling constraint but the agents are required to agree on a common decision. For this set-up, consensus methods based on (sub)gradient iterations and proximal operators (see [1][2][3][4][5][6][7][8]), distributed algorithms based on duality (see [9,10]), and distributed approaches based on the Alternating Direction Method Multipliers (ADMM) (see [11][12][13][14][15][16]) have been proposed. We refer the reader to [17,18] for an indepth discussion on ADMM.…”

Distributed Proximal Algorithms for Multiagent Optimization With Coupled Inequality Constraints

“…Generally speaking, a consensus-based algorithm is to directly integrate consensus theory into an optimization algorithm which only involves primal decision variables, and the distributed algorithms along this line subsume distributed subgradient [2], distributed primal-dual subgradient algorithms [3], distributed quasi-monotone subgradient algorithm [4], asynchronous distributed gradient [5], Newton-Raphson consensus [6], dual averaging [7], diffusion adaptation strategy [8], fast distributed gradient [9], and stochastic mirror descent [10]. On the other hand, the dual-decomposition-based algorithms aim at handling the alignment of all local decision variables by equality constraints, through introducing corresponding dual variables, and typical algorithms include alternating direction method of multipliers (ADMM) [11], augmented Lagrangian method [12], distributed dual proximal gradient [13], EXTRA [14], and distributed forwardbackward Bregman splitting [15].…”