On the linear convergence of distributed Nash equilibrium seeking for multi-cluster games under partial-decision information

This paper considers the distributed strategy design for Nash equilibrium (NE) seeking in multi-cluster games under a partial-decision information scenario. In the considered game, there are multiple clusters and each cluster consists of a group of agents. A cluster is viewed as a virtual noncooperative player that aims to minimize its local payoff function and the agents in a cluster are the actual players that cooperate within the cluster to optimize the payoff function of the cluster through communication via a connected graph. In our setting, agents have only partial-decision information, that is, they only know local information and cannot have full access to opponents' decisions. To solve the NE seeking problem of this formulated game, a discrete-time distributed algorithm, called distributed gradient tracking algorithm (DGT), is devised based on the inter- and intra-communication of clusters. In the designed algorithm, each agent is equipped with strategy variables including its own strategy and estimates of other clusters' strategies. With the help of a weighted Fronbenius norm and a weighted Euclidean norm, theoretical analysis is presented to rigorously show the linear convergence of the algorithm. Finally, a numerical example is given to illustrate the proposed algorithm

Distributed Algorithms for Computing a Fixed Point of Multi-Agent Nonexpansive Operators

This paper investigates the problem of finding a fixed point for a global nonexpansive operator under time-varying communication graphs in real Hilbert spaces, where the global operator is separable and composed of an aggregate sum of local nonexpansive operators. Each local operator is only privately accessible to each agent, and all agents constitute a network. To seek a fixed point of the global operator, it is indispensable for agents to exchange local information and update their solution cooperatively. To solve the problem, two algorithms are developed, called distributed Krasnosel'ski\u{\i}-Mann (D-KM) and distributed block-coordinate Krasnosel'ski\u{\i}-Mann (D-BKM) iterations, for which the D-BKM iteration is a block-coordinate version of the D-KM iteration in the sense of randomly choosing and computing only one block-coordinate of local operators at each time for each agent. It is shown that the proposed two algorithms can both converge weakly to a fixed point of the global operator. Meanwhile, the designed algorithms are applied to recover the classical distributed gradient descent (DGD) algorithm, devise a new block-coordinate DGD algorithm, handle a distributed shortest distance problem in the Hilbert space for the first time, and solve linear algebraic equations in a novel distributed approach. Finally, the theoretical results are corroborated by a few numerical examples

Decentralized Non-Convex Learning with Linearly Coupled Constraints

Motivated by the need for decentralized learning, this paper aims at designing a distributed algorithm for solving nonconvex problems with general linear constraints over a multi-agent network. In the considered problem, each agent owns some local information and a local variable for jointly minimizing a cost function, but local variables are coupled by linear constraints. Most of the existing methods for such problems are only applicable for convex problems or problems with specific linear constraints. There still lacks a distributed algorithm for such problems with general linear constraints and under nonconvex setting. In this paper, to tackle this problem, we propose a new algorithm, called "proximal dual consensus" (PDC) algorithm, which combines a proximal technique and a dual consensus method. We build the theoretical convergence conditions and show that the proposed PDC algorithm can converge to an $\epsilon$-Karush-Kuhn-Tucker solution within $\mathcal{O}(1/\epsilon)$ iterations. For computation reduction, the PDC algorithm can choose to perform cheap gradient descent per iteration while preserving the same order of $\mathcal{O}(1/\epsilon)$ iteration complexity. Numerical results are presented to demonstrate the good performance of the proposed algorithms for solving a regression problem and a classification problem over a network where agents have only partial observations of data features