S-ADDOPT: Decentralized stochastic first-order optimization over directed graphs
This addresses the problem of efficient optimization in distributed networks for applications like machine learning, though it appears incremental as it builds on existing gradient tracking methods.
The paper tackles decentralized stochastic optimization over directed graphs by proposing the S-ADDOPT algorithm, which uses gradient tracking to achieve linear convergence to an error ball with constant step-sizes and sublinear convergence to exact solutions with decaying step-sizes, comparable to centralized stochastic gradient descent.
In this report, we study decentralized stochastic optimization to minimize a sum of smooth and strongly convex cost functions when the functions are distributed over a directed network of nodes. In contrast to the existing work, we use gradient tracking to improve certain aspects of the resulting algorithm. In particular, we propose the~\textbf{\texttt{S-ADDOPT}} algorithm that assumes a stochastic first-order oracle at each node and show that for a constant step-size~$α$, each node converges linearly inside an error ball around the optimal solution, the size of which is controlled by~$α$. For decaying step-sizes~$\mathcal{O}(1/k)$, we show that~\textbf{\texttt{S-ADDOPT}} reaches the exact solution sublinearly at~$\mathcal{O}(1/k)$ and its convergence is asymptotically network-independent. Thus the asymptotic behavior of~\textbf{\texttt{S-ADDOPT}} is comparable to the centralized stochastic gradient descent. Numerical experiments over both strongly convex and non-convex problems illustrate the convergence behavior and the performance comparison of the proposed algorithm.