A Two-timescale Primal-dual Algorithm for Decentralized Optimization with Compression

This paper proposes a two-timescale compressed primal-dual (TiCoPD) algorithm for decentralized optimization with improved communication efficiency over prior works on primal-dual decentralized optimization. The algorithm is built upon the primal-dual optimization framework and utilizes a majorization-minimization procedure. The latter naturally suggests the agents to share a compressed difference term during the iteration. Furthermore, the TiCoPD algorithm incorporates a fast timescale mirror sequence for agent consensus on nonlinearly compressed terms, together with a slow timescale primal-dual recursion for optimizing the objective function. We show that the TiCoPD algorithm converges with a constant step size. It also finds an $\mathcal{O}(1/T)$ stationary solution after T iterations. Numerical experiments on decentralized training of a neural network validate the efficacy of TiCoPD algorithm.