Distributed Variable Sample-size Stochastic Optimization with Fixed Step-sizes

TL;DR

This paper introduces a distributed stochastic optimization algorithm with variance reduction that guarantees convergence under fixed step-sizes over randomly switching networks, with proven convergence rates and empirical validation.

Contribution

It develops a novel distributed variance-reduced stochastic gradient tracking method that achieves convergence with fixed step-sizes in dynamic network settings.

Findings

Almost sure convergence to the optimal solution.

Geometric convergence under strongly convex costs with increasing sample size.

Analyzed rate and complexity with constant and polynomially increasing sample sizes.

Abstract

The paper considers distributed stochastic optimization over randomly switching networks, where agents collaboratively minimize the average of all agents' local expectation-valued convex cost functions. Due to the stochasticity in gradient observations, distributedness of local functions, and randomness of communication topologies, distributed algorithms with a convergence guarantee under fixed step-sizes have not been achieved yet. This work incorporates variance reduction scheme into the distributed stochastic gradient tracking algorithm, where local gradients are estimated by averaging across a variable number of sampled gradients. With an identically and independently distributed (i.i.d.) random network, we show that all agents' iterates converge almost surely to the same optimal solution under fixed step-sizes. When the global cost function is strongly convex and the sample size increases at a geometric rate, we prove that the iterates geometrically converge to the unique optimal solution, and establish the iteration, oracle, and communication complexity. The algorithm performance including rate and complexity analysis are further investigated with constant step-sizes and a polynomially increasing sample size. Finally, the empirical algorithm performance are illustrated with numerical examples.