[Paper Review] Exponential Convergence for Distributed Smooth Optimization Under the Restricted Secant Inequality Condition

Xinlei Yi, Shengjun Zhang+3 more|arXiv (Cornell University)|Sep 7, 2019

Distributed Control Multi-Agent Systems45 references4 citations

TL;DR

This paper establishes exponential convergence for a continuous-time distributed primal-dual gradient algorithm and linear convergence for its discrete-time Euler discretization under the restricted secant inequality condition—a weaker assumption than strong convexity that allows non-convex cost functions and non-unique global minimizers. The results extend convergence guarantees to broader classes of distributed smooth optimization problems without requiring convexity or uniqueness of solutions.

ABSTRACT

This paper considers the distributed smooth optimization problem in which the objective is to minimize a global cost function formed by a sum of local smooth cost functions, by using local information exchange. The standard assumption for proving exponential/linear convergence of first-order methods is the strong convexity of the cost functions, which does not hold for many practical applications. In this paper, we first show that the continuous-time distributed primal-dual gradient algorithm converges to one global minimizer exponentially under the assumption that the global cost function satisfies the restricted secant inequality condition. This condition is weaker than the strong convexity condition since it does not require convexity and the global minimizers are not necessary to be unique. We then show that the discrete-time distributed primal-dual algorithm constructed by using the Euler's approximation method converges to one global minimizer linearly under the same condition. The theoretical results are illustrated by numerical simulations.

Motivation & Objective

• To address the limitation of existing distributed optimization algorithms that rely on strong convexity, which does not hold in many practical applications such as least squares and logistic regression.
• To develop convergence guarantees for distributed primal-dual gradient algorithms under a weaker condition than strong convexity, enabling application to non-convex and non-unique minimizer scenarios.
• To establish both exponential convergence in continuous time and linear convergence in discrete time using the same relaxed condition.
• To provide theoretical justification for the use of first-order methods in distributed settings where cost functions are smooth but not strongly convex.

Proposed method

• Proposes a continuous-time distributed primal-dual gradient algorithm that uses local information exchange to minimize a global cost function composed of local smooth functions.
• Introduces the restricted secant inequality condition as a weaker alternative to strong convexity, which does not require convexity or uniqueness of global minimizers.
• Applies Lyapunov stability theory to prove exponential convergence of the continuous-time algorithm to a global minimizer under the restricted secant inequality condition.
• Constructs a discrete-time version of the algorithm using Euler’s approximation method, ensuring linear convergence under the same condition.
• Derives explicit stepsize constraints for linear convergence in the discrete-time case, involving parameters such as the network Laplacian, Lipschitz constant, and restricted secant inequality constant.
• Validates theoretical results via numerical simulations on a 10-agent ring network with non-convex local cost functions satisfying the restricted secant inequality.

Experimental results

Research questions

• RQ1 Can exponential convergence be achieved for distributed smooth optimization under a condition weaker than strong convexity?
• RQ2 Does the restricted secant inequality condition allow convergence guarantees when the global cost function is non-convex or has multiple global minimizers?
• RQ3 Can the discrete-time counterpart of the continuous-time distributed primal-dual algorithm achieve linear convergence under the same relaxed condition?
• RQ4 How do the stepsize constraints in the discrete-time algorithm relate to the system parameters and convergence rate?
• RQ5 Is the restricted secant inequality condition strictly weaker than metric subregularity and strong convexity in the context of distributed optimization?

Key findings

• The continuous-time distributed primal-dual gradient algorithm converges exponentially to a global minimizer when the global cost function satisfies the restricted secant inequality condition.
• The discrete-time distributed primal-dual algorithm, obtained via Euler’s method, converges linearly to a global minimizer under the same condition, without requiring convexity or uniqueness of the solution set.
• The restricted secant inequality condition is strictly weaker than strong convexity and does not require the global minimizers to be unique or the cost functions to be convex.
• The linear convergence rate of the discrete-time algorithm is bounded below by $1 - rac{h(2 heta_2 heta_4 - h heta_1 heta_3 heta_5)}{4 heta_3 heta_4}$, with explicit constraints on the stepsize $h$.
• Numerical simulations on a 10-agent ring network with non-convex local cost functions confirm linear convergence of the discrete-time algorithm, with all primal variables converging to zero (a global minimizer) and dual variables converging to zero.
• The simulation results validate the theoretical convergence rate and demonstrate the effectiveness of the proposed method in a non-convex, non-unique minimizer setting.