Stochastic Subgradient Algorithms for Strongly Convex Optimization over Distributed Networks

TL;DR

This paper introduces a stochastic gradient descent-based algorithm for distributed convex optimization over networks, achieving optimal convergence rates with minimal communication and computational complexity, suitable for large-scale data.

Contribution

The paper presents a novel distributed SGD algorithm with a carefully designed averaging scheme, achieving optimal convergence rates and efficiency in both computation and communication.

Findings

Achieves convergence rate of O(N√N/T) after T updates.

Attains mean square deviation of O(√N/T) after T oracle calls.

Matches performance lower bounds, demonstrating optimality.

Abstract

We study diffusion and consensus based optimization of a sum of unknown convex objective functions over distributed networks. The only access to these functions is through stochastic gradient oracles, each of which is only available at a different node, and a limited number of gradient oracle calls is allowed at each node. In this framework, we introduce a convex optimization algorithm based on the stochastic gradient descent (SGD) updates. Particularly, we use a carefully designed time-dependent weighted averaging of the SGD iterates, which yields a convergence rate of $O\left(\frac{N\sqrt{N}}{T}\right)$ after $T$ gradient updates for each node on a network of $N$ nodes. We then show that after $T$ gradient oracle calls, the average SGD iterate achieves a mean square deviation (MSD) of $O\left(\frac{\sqrt{N}}{T}\right)$. This rate of convergence is optimal as it matches the performance lower bound up to constant terms. Similar to the SGD algorithm, the computational complexity of the proposed algorithm also scales linearly with the dimensionality of the data. Furthermore, the communication load of the proposed method is the same as the communication load of the SGD algorithm. Thus, the proposed algorithm is highly efficient in terms of complexity and communication load. We illustrate the merits of the algorithm with respect to the state-of-art methods over benchmark real life data sets and widely studied network topologies.