Improving the Transient Times for Distributed Stochastic Gradient Methods

We consider the distributed optimization problem where $n$ agents, each possessing a local cost function, collaboratively minimize average of functions over connected network. Assuming stochastic gradient information is available, we study algorithm, called exact diffusion with adaptive stepsizes (EDAS) adapted from Exact Diffusion method [1] and NIDS [2] perform non-asymptotic convergence analysis. not only show that EDAS asymptotically achieves same network independent rate as centralized descent (SGD) for minimizing strongly convex smooth objective functions, but also characterize transient time needed algorithm to approach asymptotic rate, which behaves notation="LaTeX">$K_{T}=\mathcal {O}(\frac{n}{1-\lambda _{2}})$, notation="LaTeX">$1-\lambda _{2}$ stands spectral gap mixing matrix. To best our knowledge, shortest when function smooth. Numerical simulations further corroborate strengthen obtained theoretical results.