Efficient Gradient Tracking Algorithms for Distributed Optimization Problems with Inexact Communication

TL;DR

This paper introduces two single-timescale distributed optimization algorithms, VRA-DGT and VRA-DSGT, that effectively handle inexact communication and achieve optimal convergence rates, validated by experiments on real data.

Contribution

The paper proposes novel single-timescale algorithms for distributed optimization with inexact communication, improving convergence rates over existing two-timescale methods.

Findings

VRA-DGT achieves $ ext{O}(k^{-1})$ convergence rate in mean square.

VRA-DSGT maintains $ ext{O}(k^{-1})$ convergence rate for stochastic problems.

Experiments on logistic regression validate the algorithms' effectiveness.

Abstract

Distributed optimization problems usually face inexact communication issues induced by channel noise, communication quantization or differential privacy protection. Most existing algorithms need a two-timescale setting of the stepsize of gradient descent and the parameter of noise suppression to ensure the convergence to the optimal solution. In this paper, we propose two single-timescale algorithms, VRA-DGT and VRA-DSGT, for distributed deterministic and stochastic optimization problems with inexact communication respectively. VRA-DGT integrates the Variance-Reduced Aggregation (VRA) mechanism with the distributed gradient tracking framework, which achieves the convergence rate of $\mathcal{O}\left(k^{-1}\right)$ in the mean square sense and $\mathcal{O}\left(\frac{\ln(k+1)}{k^b}\right)$, $\forall b\in(0.5,1)$ in the almost sure sense when the objective function is strongly convex and smooth. For stochastic optimization problems, VRA-DSGT, where a hybrid variance-reduced technique has been introduced in VRA-DGT, maintains the convergence rate of $\mathcal{O}\left(k^{-1}\right)$ in the mean square sense and $\mathcal{O}\left(\frac{\ln(k+1)}{k^b}\right)$, $\forall b\in(0.5,1)$ in the almost sure sense. Simulated experiments on a logistic regression problem with real-world data verify the effectiveness of the proposed algorithms.