Section: New Results

Optimal algorithms for smooth and strongly convex distributed optimization in networks

In [35], we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings: centralized and decentralized communications over a network. For centralized (i.e. master/slave) algorithms, we show that distributing Nesterov's accelerated gradient descent is optimal and achieves a precision $\epsilon >0$ in time $O(\sqrt{{\kappa }_g}(1+\Delta \tau )ln(1/\epsilon ))$, where ${\kappa }_g$ is the condition number of the (global) function to optimize, $\Delta$ is the diameter of the network, and $\tau$ (resp. 1) is the time needed to communicate values between two neighbors (resp. perform local computations). For decentralized algorithms based on gossip, we provide the first optimal algorithm, called the multi-step dual accelerated (MSDA) method, that achieves a precision $\epsilon >0$ in time $O(\sqrt{{\kappa }_l}(1+\frac{\tau }{\sqrt{\gamma }})ln(1/\epsilon ))$, where ${\kappa }_l$ is the condition number of the local functions and $\gamma$ is the (normalized) eigengap of the gossip matrix used for communication between nodes. We then verify the efficiency of MSDA against state-of-the-art methods for two problems: least-squares regression and classification by logistic regression.