Distributed Stochastic Optimization with Gradient Tracking over Time- Varying Directed Networks

We study a distributed method called SAB-TV, which employs gradient tracking to collaboratively minimize the strongly-convex sum of smooth local cost functions for networked agents communicating over a time-varying directed graph. Each agent, assumed to have access to a stochastic first- order oracle for obtaining an unbiased estimate of the gradient of its local cost function, maintains an auxiliary variable to asymptotically track the stochastic gradient of the global cost. The optimal decision and gradient tracking are updated over time through limited information exchange with local neighbors using row- and column-stochastic weights, guaranteeing both consensus and optimality. With a sufficiently small constant step-size, we demonstrate that, in expectation, SAB- TV converges linearly to a neighborhood of the optimal solution. Numerical simulations illustrate the effectiveness of the proposed algorithm.