Gradient-Type Methods For Decentralized Optimization Problems With Polyak-{\L}ojasiewicz Condition Over Time-Varying Networks

TL;DR

This paper introduces gradient-type methods for decentralized optimization problems with Polyak-Łojasiewicz conditions over dynamic networks, addressing both minimization and saddle-point problems with theoretical guarantees and numerical validation.

Contribution

It proposes novel gradient descent and ascent algorithms with consensus and inexact gradients for decentralized problems under PL conditions, extending existing methods to time-varying networks.

Findings

Algorithms converge under PL conditions on dynamic networks.

Numerical experiments demonstrate efficiency on robust least squares.

Methods outperform existing approaches in convergence speed.

Abstract

This paper focuses on the decentralized optimization (minimization and saddle point) problems with objective functions that satisfy Polyak-{\L}ojasiewicz condition (PL-condition). The first part of the paper is devoted to the minimization problem of the sum-type cost functions. In order to solve a such class of problems, we propose a gradient descent type method with a consensus projection procedure and the inexact gradient of the objectives. Next, in the second part, we study the saddle-point problem (SPP) with a structure of the sum, with objectives satisfying the two-sided PL-condition. To solve such SPP, we propose a generalization of the Multi-step Gradient Descent Ascent method with a consensus procedure, and inexact gradients of the objective function with respect to both variables. Finally, we present some of the numerical experiments, to show the efficiency of the proposed algorithm for the robust least squares problem.