MUSIC: Accelerated Convergence for Distributed Optimization With Inexact and Exact Methods

TL;DR

This paper tackles distributed optimization over networks by addressing slow convergence and high communication costs of gradient-based methods. It introduces MUSIC, a Multi-Updates Single-Combination framework, enabling each agent to perform multiple local updates ($E$) before a single consensus step, and develops two variants: inexact MUSIC, which preserves efficiency while delivering linear convergence to a neighborhood, and exact MUSIC, which employs bias-correction to achieve convergence to the exact solution with superior communication efficiency. The authors provide rigorous convergence analyses for both variants, showing linear (R-linear) rates and explicit steady-state error bounds that depend on $E$, the step size $\alpha$, the gradient bound $G_{max}$, the strong convexity parameter $\mu$, and the smoothness $L$, along with the necessity of local correction in exact MUSIC. Numerical experiments on synthetic least-squares problems and real data (including LIBSVM Letter and logistic regression) demonstrate that exact MUSIC achieves fast linear convergence to the exact solution with significantly reduced communication rounds (scaling as $\mathcal{O}(\frac{1}{E})$) compared with traditional diffusion methods, while inexact MUSIC confirms the rate–accuracy tradeoff controlled by $E$ and step-size scheduling. The framework thus offers a practical, topology-agnostic approach to accelerate distributed optimization with tunable communication efficiency and provable guarantees.

Abstract

Gradient-type distributed optimization methods have blossomed into one of the most important tools for solving a minimization learning task over a networked agent system. However, only one gradient update per iteration is difficult to achieve a substantive acceleration of convergence. In this paper, we propose an accelerated framework named as MUSIC allowing each agent to perform multiple local updates and a single combination in each iteration. More importantly, we equip inexact and exact distributed optimization methods into this framework, thereby developing two new algorithms that exhibit accelerated linear convergence and high communication efficiency. Our rigorous convergence analysis reveals the sources of steady-state errors arising from inexact policies and offers effective solutions. Numerical results based on synthetic and real datasets demonstrate both our theoretical motivations and analysis, as well as performance advantages.

Figures (7)

• Figure 1: Illustration of workflow in the inexact MUSIC. Note that a temporary variable $s$ is used to control the quantity of local updates.
• Figure 2: Performance of inexact MUSIC measured in terms of relative error with respect to communication rounds or iterations on the distributed quadratic problem (a) impact of $E$ under $\alpha=0.0001$ (b) impact of different fixed step sizes under $E=3$ (c) impact of $E$ under a diminishing step size $\alpha=\alpha^{0}/t^{\frac{1}{2}}$ with $\alpha^{0}=0.001$ (d) impact of diminishing step sizes under $E=3$.
• Figure 3: Illustration of workflow in the exact MUSIC.
• Figure 4: Performance comparisons measured in terms of relative error with respect to iterations on the least squares problem based on synthesized data.
• Figure 5: Performance comparisons measured in terms of relative error with respect to iterations on the least squares problem based on real dataset.

Theorems & Definitions (19)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Remark 1
• Lemma 4