Distributed accelerated Nash equilibrium learning for two-subnetwork zero-sum game with bilinear coupling

summary:This paper proposes a distributed accelerated first-order continuous-time algorithm for $O({1}/{t^2})$ convergence to Nash equilibria in a class of two-subnetwork zero-sum games with bilinear couplings. First-order methods, which only use subgradients of functions, are frequently used in distributed/parallel algorithms for solving large-scale and big-data problems due to their simple structures. However, in the worst cases, first-order methods for two-subnetwork zero-sum games often have an asymptotic or $O(1/t)$ convergence. In contrast to existing time-invariant first-order methods, this paper designs a distributed accelerated algorithm by combining saddle-point dynamics and time-varying derivative feedback techniques. If the parameters of the proposed algorithm are suitable, the algorithm owns $O(1/t^2)$ convergence in terms of the duality gap function without any uniform or strong convexity requirement. Numerical simulations show the efficacy of the algorithm

Distributed Algorithm for Continuous-type Bayesian Nash Equilibrium in Subnetwork Zero-sum Games

In this paper, we consider a continuous-type Bayesian Nash equilibrium (BNE) seeking problem in subnetwork zero-sum games, which is a generalization of deterministic subnetwork zero-sum games and discrete-type Bayesian zero-sum games. In this continuous-type model, because the feasible strategy set is composed of infinite-dimensional functions and is not compact, it is hard to seek a BNE in a non-compact set and convey such complex strategies in network communication. To this end, we design two steps to overcome the above bottleneck. One is a discretization step, where we discretize continuous types and prove that the BNE of the discretized model is an approximate BNE of the continuous model with an explicit error bound. The other one is a communication step, where we adopt a novel compression scheme with a designed sparsification rule and prove that agents can obtain unbiased estimations through compressed communication. Based on the above two steps, we propose a distributed communication-efficient algorithm to practicably seek an approximate BNE, and further provide an explicit error bound and an $O(\ln T/\sqrt{T})$ convergence rate.Comment: Submitted to SIAM Journal on Control and Optimizatio

Fully Distributed Nash Equilibrium Seeking in N-Cluster Games

Distributed optimization and Nash equilibrium (NE) seeking problems have drawn much attention in the control community recently. This paper studies a class of non-cooperative games, known as $N$-cluster game, which subsumes both cooperative and non-cooperative nature among multiple agents in the two problems: solving distributed optimization problem within the cluster, while playing a non-cooperative game across the clusters. Moreover, we consider a partial-decision information game setup, i.e., the agents do not have direct access to other agents' decisions, and hence need to communicate with each other through a directed graph whose associated adjacency matrix is assumed to be non-doubly stochastic. To solve the $N$-cluster game problem, we propose a fully distributed NE seeking algorithm by a synthesis of leader-following consensus and gradient tracking, where the leader-following consensus protocol is adopted to estimate the other agents' decisions and the gradient tracking method is employed to trace some weighted average of the gradient. Furthermore, the algorithm is equipped with uncoordinated constant step-sizes, which allows the agents to choose their own preferred step-sizes, instead of a uniform coordinated step-size. We prove that all agents' decisions converge linearly to their corresponding NE so long as the largest step-size and the heterogeneity of the step-size are small. We verify the derived results through a numerical example in a Cournot competition game

Gradient-Free Nash Equilibrium Seeking in N-Cluster Games with Uncoordinated Constant Step-Sizes

In this paper, we consider a problem of simultaneous global cost minimization and Nash equilibrium seeking, which commonly exists in $N$-cluster non-cooperative games. Specifically, the agents in the same cluster collaborate to minimize a global cost function, being a summation of their individual cost functions, and jointly play a non-cooperative game with other clusters as players. For the problem settings, we suppose that the explicit analytical expressions of the agents' local cost functions are unknown, but the function values can be measured. We propose a gradient-free Nash equilibrium seeking algorithm by a synthesis of Gaussian smoothing techniques and gradient tracking. Furthermore, instead of using the uniform coordinated step-size, we allow the agents across different clusters to choose different constant step-sizes. When the largest step-size is sufficiently small, we prove a linear convergence of the agents' actions to a neighborhood of the unique Nash equilibrium under a strongly monotone game mapping condition, with the error gap being propotional to the largest step-size and the smoothing parameter. The performance of the proposed algorithm is validated by numerical simulations