Efficient Gradient Tracking Algorithms for Distributed Optimization Problems with Inexact Communication

Abstract: Distributed optimization problems usually face inexact communication issues induced by communication quantization, differential privacy protection, or channels noise. Most existing algorithms need 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 a convergence rate of $\mathcal{O}\left(k^{-1}\right)$ in the mean-square sense when the objective function is strongly convex and smooth. For distributed stochastic optimization problem,VRA-DSGT, where a hybrid variance reduction technique has been introduced in VRA-DGT, VRA-DGT,, maintains the convergence rate of $\mathcal{O}\left(k^{-1}\right)$ for strongly convex and smooth objective function. Simulated experiments on logistic regression problem with real-world data verify the effectiveness of the proposed algorithms.