arXiv subjects

Qiuchen Tian

Publications and source records attributed to Qiuchen Tian.

3 recordsLinked to original sources

arXiv2026

Optimal Design of Distributed Inexact Gradient Tracking: Exact Worst-case Convergence Rate and Explicit Parameters

This paper addresses the problem of designing distributed optimization algorithms. Different to existing methods that usually give sufficient conditions and conservative convergence rates, we propose a framework for determining the exact worst-case convergence rate. The worst-case convergence is defined over the set of $μ$-strongly convex and $L$-Lipschitz objective functions and the set of connected graphs with given algebraic connectivity. Focusing on the Distributed Inexact Gradient Tracking (DIGing)--an algorithm widely deployed in applications and extensively studied in recent years, we develop a systematic design methodology by integrating techniques from graph signal processing and robust control theory. First, we present a novel decomposition structure that enables partial subsystem decoupling. Then we derive explicit formulas for the exact worst-case convergence rate and the corresponding parameters. Numerical experiments validate the correctness and effectiveness of our theoretical results.

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arXiv2026

Optimal Parameter Design for DIGing on Minimizing Unweighted Sum of Squares

There is no general method for designing proper parameters to achieve faster convergence in distributed optimization algorithms. In this paper, we consider the distributed inexact gradient tracking (DIGing) algorithm with the objective function being the unweighted sum of squares. By representing the iteration algorithm as a dynamical linear system, we decompose it into different graph frequencies and obtain a set of decoupled subsystems, on which we can easily analyze the convergence rate. By using Routh stability criterion from control theory, we derive the explicit formula of the optimal worst-case convergence rate and the corresponding parameters. We can see that the convergence rate of DIGing is slow even for the simplest objective functions, thus acceleration is necessary for general application. The proposed method can be viewed as the first step toward optimal parameter design of DIGing algorithm in solving general objective functions.

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arXiv2026

Exact Worst-case Convergence Rates of Distributed Gradient Tracking Methods

Different from most existing literature in the analysis of distributed optimization algorithms that reports sufficient convergence conditions leading to a conservative convergence rate, this work provides the exact worst-case convergence rates for two typical gradient tracking algorithms, DIGing and AugDGM. By eigen-decomposition, we show that two algorithms share the same average-state dynamics, while they differ from each other in the gradient tracking subsystems, which entirely govern algorithm convergence. Exploiting the diagonal structure of this decomposition, we reduce the stability analysis of MIMO systems to that of a set of parameter-varying SISO systems, from which explicit formulas for the exact worst-case convergence rate can be derived. These formulas clearly show that the optimal worst-case convergence rate of DIGing is larger than that of AugDGM under the same assumptions on objective functions and communication networks. Furthermore, we find that there is an inflection point in the graph connectivity, which is $σ=1/3$. For graphs with connectivity better than this inflection point, the optimal convergence rate of centralized gradient descent can be achieved by AugDGM provided the condition number of objective functions is worse enough. On the other hand, for graphs with connectivity worse than this inflection point, the centralized optimal rate can never be achieved. Numerical experiments validate the theoretical results.

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