On the Convergence of Decentralized Gradient Descent

Consider the consensus problem of minimizing f(x) = ∑n i=1 fi(x) where each fi is only known to one individual agent i belonging to a connected network of n agents. All the agents shall collaboratively solve this problem and obtain the solution via data exchanges only between neighboring agents. Such algorithms avoid the need of a fusion center, offer better network load balance, and improve data privacy. We study the decentralized gradient descent method in which each agent i updates its variable x(i), which is a local approximate to the unknown variable x, by taking the average of its neighbors’ followed by making a local negative gradient step −α∇fi(x(i)). The iteration is x(i)(k + 1)← ∑ j wijx(j)(k)− α∇fi(x(i)(k)), for each agent i, where the coefficients wij form a symmetric doubly stochastic matrix W = [wij ] ∈ Rn×n. As agent i does not communicate to non-neighbors, wij 6= 0 only if i = j or j is a neighbor of i. We analyze the convergence of this iteration and derive its rate, assuming that each fi is proper closed convex and lower bounded, ∇fi is Lipschitz continuous with constant Lfi , and stepsize α is fixed. Provided that α < O(1/Lh) where Lh = maxi{Lfi}, the objective error at the averaged solution, f( 1 n ∑ i x(i)(k))− f ∗ where f∗ is the optimal objective value, reduces at a speed of O(1/k) until it reaches O(α). If fi are (restricted) strongly convex, then both 1 n ∑ i x(i)(k) and each x(i)(k) converge to the global minimizer x ∗ at a linear rate until reaching an O(α)-neighborhood of x∗. We also develop an iteration for decentralized basis pursuit and establish its linear convergence to an O(α)-neighborhood of the true sparse signal. This analysis reveals how convergence depends on the stepsize, function convexity, and network spectrum.