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Distributed Saddle-Point Problems Under Data Similarity
Aleksandr Beznosikov · Gesualdo Scutari · Alexander Rogozin · Alexander Gasnikov

Thu Dec 09 08:30 AM -- 10:00 AM (PST) @ Virtual #None
We study solution methods for (strongly-)convex-(strongly)-concave Saddle-Point Problems (SPPs) over networks of two type--master/workers (thus centralized) architectures and mesh (thus decentralized) networks. The local functions at each node are assumed to be \textit{similar}, due to statistical data similarity or otherwise. We establish lower complexity bounds for a fairly general class of algorithms solving the SPP. We show that a given suboptimality $\epsilon>0$ is achieved over master/workers networks in $\Omega\big(\Delta\cdot \delta/\mu\cdot \log (1/\varepsilon)\big)$ rounds of communications, where $\delta>0$ measures the degree of similarity of the local functions, $\mu$ is their strong convexity constant, and $\Delta$ is the diameter of the network. The lower communication complexity bound over mesh networks reads $\Omega\big(1/{\sqrt{\rho}} \cdot {\delta}/{\mu}\cdot\log (1/\varepsilon)\big)$, where $\rho$ is the (normalized) eigengap of the gossip matrix used for the communication between neighbouring nodes. We then propose algorithms matching the lower bounds over either types of networks (up to log-factors). We assess the effectiveness of the proposed algorithms on a robust regression problem.

Author Information

Aleksandr Beznosikov (Moscow Institute of Physics and Technology)
Gesualdo Scutari (Purdue University)
Alexander Rogozin (Moscow Institute of Physics and Technology)
Alexander Gasnikov (Moscow Institute of Physics and Technology)

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