A Communication-Efficient Stochastic Gradient Descent Algorithm for Distributed Nonconvex Optimization

This paper studies distributed nonconvex optimization problems with stochastic gradients for a multi-agent system, in which each agent aims to minimize the sum of all agents' cost functions by using local compressed information exchange. We propose a distributed stochastic gradient descent algorithm, suitable for a general class of compressors. We show that the proposed algorithm achieves the linear speedup convergence rate
$\mathcal{O}(-1/\sqrt{nT})$
for smooth nonconvex functions, where
$T$
and
$n$
are the number of iterations and agents, respectively. If the global cost function additionally satisfies the Polyak-Lojasiewicz condition, the proposed algorithm can linearly converge to a neighborhood of the global optimum, regardless of whether the stochastic gradient is unbiased or not. Numerical experiments are carried out to verify the efficiency of our algorithm.