A Randomized Incremental Subgradient Method for Distributed Optimization in Networked Systems

Abstract

We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedić and Bertsekas [SIAM J. Optim., 12 (2001), pp. 109-138]. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e.g., parameter estimation in networks of tiny wireless sensors. The stochastic component in the algorithm is described by a Markov chain, which can be constructed in a distributed fashion using only local information. We provide a detailed convergence analysis of the proposed algorithm and compare it with existing, both deterministic and randomized, incremental subgradient methods.

Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs

“…Note that the equilibria of Ψ α-dis-opt are precisely the set of saddle points of F in (5). Let x * = 1 n ⊗x * be a solution of (4).…”

“…• Remark 5.7 (Discrete-time counterpart of (6) and (11)): It is worth noticing that the discretization of (6) for undirected graphs (performed in [12] for the case of continuously differentiable, strictly convex functions) and (11) for weight-balanced digraphs gives rise to different discretetime optimization algorithms from the ones considered in [1], [2], [3], [4], [5], [6].…”

Tradeoffs for task parallelization in distributed optimization

“…For this reason, we consider two classes of decentralized optimization methods which can naturally be implemented and analyzed in an asynchronous framework. The two concrete algorithms we consider are DDA [6] and MIGD [10].…”

“…Nedic et al [8] and Bert-sekas [9] analyze the situation where every component f i (·) has to be visited once in every cycle, essentially assuming that the nodes are connected as a complete graph. Johansson et al [10] generalize this approach for any connected graph in the Markov incremental gradient descent (MIGD) algorithm by having the task take a random walk on the graph. In contrast to consensus methods, all J tasks can be solved concurrently using MIGD by having each task take a random walk.…”

“…Incremental methods [8,9,10] are an alternative approach in which only a single node performs an update at any given time, and one single copy of the optimization variables is passed around the network. This reduces the communication cost by having the optimization method "visit" the function components f i one at a time.…”