A Decentralized Proximal-Gradient Method With Network Independent Step-Sizes and Separated Convergence Rates

Abstract

This paper considers the problem of decentralized optimization with a composite objective containing smooth and non-smooth terms. To solve the problem, a proximal-gradient scheme is studied. Specifically, the smooth and nonsmooth terms are dealt with by gradient update and proximal update, respectively. The studied algorithm is closely related to a previous decentralized optimization algorithm, PG-EXTRA [37], but has a few advantages. First of all, in our new scheme, agents use uncoordinated step-sizes and the stable upper bounds on step-sizes are independent from network topologies. The step-sizes depend on local objective functions, and they can be as large as that of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of our new scheme is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We also provide some numerical experiments to demonstrate the efficacy of the introduced algorithms and validate our theoretical discoveries. 1 We believe that agent i's instantaneous estimation on the optimal solution is not a piece of sensitive information but the functions s i and r i are.

Analysis and Design of First-Order Distributed Optimization Algorithms Over Time-Varying Graphs

“…Although the update (3b) has two gradient evaluations, they are evaluated at successive iterates so EXTRA can easily be implemented with one gradient evaluation per iteration by storing the previous gradient in memory. Several additional linear-rate algorithms have since been proposed [9,13,16,21,36,38,39]. Each of these methods have updates similar to (3) in that they require agents to store the previous iterate and/or gradient in memory.…”

“…Other distributed optimization algorithms solve (1) but lie outside the scope of the present work. This includes algorithms involving dual decomposition [4,24,25], inexact dual methods [7], proximal algorithms [27], asynchronous algorithms [15,37], weakly convex cases [13,20,26], accelerated methods [20,33,34]. Although linear convergence rates were obtained for many of the algorithms cited above, each algorithm differs in the nature and strength of its convergence analysis guarantees.…”

“…Although linear convergence rates were obtained for many of the algorithms cited above, each algorithm differs in the nature and strength of its convergence analysis guarantees. For example, some works show (non-constructively) the existence of a linear rate [35] whereas others provide specific tuning recommendations with associated analytic rate bounds [13,26]. Numerical simulations are another means for comparing performance [33], but they run the risk of being misleading because algorithm performance often depends on the graph topology, choice of functions, hyperparameter tuning, or state initializations.…”

A Unified Contraction Analysis of a Class of Distributed Algorithms for Composite Optimization

“…Our focus is on the design of distributed algorithms for Problem (P) that provably converge at a linear rate. When G = 0, several distributed schemes have been proposed in the literature enjoying such a property; examples include EXTRA [1], AugDGM [2], NEXT [3], SONATA [4], [5], DIGing [6], NIDS [7], Exact Diffusion [8], MSDA [9], and the distributed algorithms in [10], [11], and [12]. When G = 0 results are scarce; to our knowledge, the only two schemes available in the literature achieving linear rate for (P) are SONATA [5] and the distributed proximal gradient algorithm [13].…”

“…Because of that, in general, they cannot achieve the rate of the centralized gradient algorithm (addressing thus Q2). Works partially addressing Q2 are the following: MSDA [9] uses multiple communication steps to achieve the lower complexity bound of (P) when G = 0; and the algorithms in [16] and [7] achieve linear rate and can adjust the number of communications performed at each iteration to match the rate of the centralized gradient descent. However it is not clear how to extend (if possible) these methods and their convergence analysis to the more general composite (i.e., G = 0) setting (P).…”