A Primal-Dual Framework for Decentralized Stochastic Optimization

We consider the decentralized convex optimization problem, where multiple agents must cooperatively minimize a cumulative objective function, with each local function expressible as an empirical average of data-dependent losses. State-of-the-art approaches for decentralized optimization rely on gradient tracking, where consensus is enforced via a doubly stochastic mixing matrix. Construction of such mixing matrices is not straightforward and requires coordination even prior to the start of the optimization algorithm. This paper puts forth a primal-dual framework for decentralized stochastic optimization that obviates the need for such doubly stochastic matrices. Instead, dual variables are maintained to track the disagreement between neighbors. The proposed framework is flexible and is used to develop decentralized variants of SAGA, L-SVRG, SVRG++, and SEGA algorithms. Using a unified proof, we establish that the oracle complexity of these decentralized variants is $O(1/\epsilon)$, matching the complexity bounds obtained for the centralized variants. Additionally, we also present a decentralized primal-dual accelerated SVRG algorithm achieving $O(1/\sqrt{\epsilon})$ oracle complexity, again matching the bound for the centralized accelerated SVRG. Numerical tests on the algorithms establish their superior performance as compared to the variance-reduced gradient tracking algorithms.