Optimal algorithms for smooth and strongly convex distributed optimization in networks

Kevin Scaman (1) , Francis Bach (2) , Sébastien Bubeck (3) , Yin Tat Lee (3) , Laurent Massoulié (1, 4)

1 MSR - INRIA - Microsoft Research - Inria Joint Centre
2 SIERRA - Statistical Machine Learning and Parsimony
3 Microsoft Corporation [Redmond]
4 DYOGENE - Dynamics of Geometric Networks

In this paper, 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 $\varepsilon > 0$ in time $O(\sqrt{\kappa_g}(1+\Delta\tau)\ln(1/\varepsilon))$, 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 $\varepsilon > 0$ in time $O(\sqrt{\kappa_l}(1+\frac{\tau}{\sqrt{\gamma}})\ln(1/\varepsilon))$, 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.

Mots clés: accelerated gradient descent; linear convergence; gossip; decentralized algorithms; distributed optimization

Domaines: Optimisation et contrôle [math.OC]; Machine Learning [stat.ML]

Kevin Scaman, Francis Bach, Sébastien Bubeck, Yin Tat Lee, Laurent Massoulié. Optimal algorithms for smooth and strongly convex distributed optimization in networks. 2017. ⟨hal-01478317⟩