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Efficient Gradient Tracking Algorithms for Distributed Optimization Problems with Inexact Communication

Shengchao Zhao, Yongchao Liu

TL;DR

The paper tackles distributed optimization when communication is inexact due to quantization, privacy, or noise. It introduces two single-timescale algorithms, VRA-DGT for deterministic problems and VRA-DSGT for stochastic problems, by integrating Variance-Reduced Aggregation with gradient tracking and, for the stochastic case, coupling with a hybrid variance reduction. Both methods achieve a mean-square convergence rate of $O\left(\frac{1}{k}\right)$ under strongly convex and smooth objectives, without the two-timescale constraint traditional in prior work. Empirical tests on logistic regression with real data demonstrate robustness to channel noise and quantization and show improved performance over existing robust DGT schemes.

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.

Efficient Gradient Tracking Algorithms for Distributed Optimization Problems with Inexact Communication

TL;DR

The paper tackles distributed optimization when communication is inexact due to quantization, privacy, or noise. It introduces two single-timescale algorithms, VRA-DGT for deterministic problems and VRA-DSGT for stochastic problems, by integrating Variance-Reduced Aggregation with gradient tracking and, for the stochastic case, coupling with a hybrid variance reduction. Both methods achieve a mean-square convergence rate of under strongly convex and smooth objectives, without the two-timescale constraint traditional in prior work. Empirical tests on logistic regression with real data demonstrate robustness to channel noise and quantization and show improved performance over existing robust DGT schemes.

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 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 for strongly convex and smooth objective function. Simulated experiments on logistic regression problem with real-world data verify the effectiveness of the proposed algorithms.
Paper Structure (6 sections, 10 theorems, 91 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 6 sections, 10 theorems, 91 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. VRA-DGT and VRA-DSGT Algorithms
  3. Convergence Analysis
  4. Convergence Rate of VRA-DGT
  5. Convergence Rate of VRA-DSGT
  6. Experimental Results

Key Result

Lemma 1

Let $\beta_k=\frac{a}{k^{\beta}+b}, b\ge 0,a>0,\beta\in (1/2,1),$ or $\beta_k=\frac{a}{k+b}, b\ge 0, a\in(1,b+1)$. Then under Assumption , there exist constants $c_x$ and $c_s$ such that

Figures (2)

  • Figure 1: The value of $\frac{\|\bar{x}_k-x^*\|^2}{\|\bar{x}_1-x^*\|^2}$ w.r.t to the number of iterations (VRA-DGT).
  • Figure 2: The value of $\frac{\|\bar{x}_k-x^*\|^2}{\|\bar{x}_1-x^*\|^2}$ w.r.t to the number of iterations (VRA-DSGT).

Theorems & Definitions (19)

  • Lemma 1: The effectiveness of VRA zhao2023VRA
  • Lemma 2: gradient tracking error
  • proof
  • Lemma 3: agreement error
  • proof
  • Lemma 4: optimality gap
  • proof
  • Theorem 1
  • proof
  • Lemma 5: effectiveness of hybrid variance reduction
  • ...and 9 more