Distributed Stochastic Optimization under Heavy-Tailed Noises

TL;DR

This paper introduces a distributed optimization algorithm robust to heavy-tailed gradient noises without requiring a centralized server, ensuring convergence under general noise assumptions and validated by simulations.

Contribution

It proposes a novel distributed stochastic optimization method combining gradient clipping and projection that converges under heavy-tailed noise without centralized coordination.

Findings

Convergence to the optimal solution with probability 1.

Effective handling of heavy-tailed noise distributions like Pareto.

Validation through simulation confirms theoretical results.

Abstract

This paper studies the distributed optimization problem under the influence of heavy-tailed gradient noises. Here, a heavy-tailed noise means that the noise does not necessarily satisfy the bounded variance assumption. Instead, it satisfies a more general assumption. The commonly-used bounded variance assumption is a special case of the considered noise assumption. A typical example of this kind of noise is a Pareto distribution noise with tail index within (1,2], which has infinite variance. Despite that there has been several distributed optimization algorithms proposed for the heavy-tailed noise scenario, these algorithms need a centralized server in the network which collects the information of all clients. Different from these algorithms, this paper considers that there is no centralized server and the agents can only exchange information with neighbors in a communication graph. A distributed method combining gradient clipping and distributed stochastic subgradient projection is proposed. It is proven that when the gradient descent step-size and the gradient clipping step-size meet certain conditions, the state of each agent converges to the optimal solution of the distributed optimization problem with probability 1. The simulation results validate the algorithm.