On the convergence of decentralized gradient descent with diminishing stepsize, revisited

Abstract: Distributed optimization has received a lot of interest in recent years due to its wide applications in various fields. In this work, we revisit the convergence property of the decentralized gradient descent [A. Nedi{\'c}-A.Ozdaglar (2009)] on the whole space given by $$x_i(t+1) = \sum^{m_{j=1}w_{ij}x_j(t)} - \alpha(t) \nabla f_i(x_i(t)),$$ where the stepsize is given as $\alpha (t) = \frac{a}{(t+w)<sup>p}$ with $0&lt; p\leq 1$. Under the strongly convexity assumption on the total cost function $f$ with local cost functions $f_i$ not necessarily being convex, we show that the sequence converges to the optimizer with rate $O(t<sup>{-p})$ when the values of $a&gt;0$ and $w&gt;0$ are suitably chosen.