Distributed Nash Equilibrium Seeking With Limited Cost Function Knowledge via a Consensus-Based Gradient-Free Method

Pang, Yipeng; Hu, Guoqiang

doi:10.1109/tac.2020.2995666

Cited by 43 publications

(15 citation statements)

References 37 publications

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“…Specifically, randomized gradient-free oracles are built locally to estimate the partial gradient information, while gradient tracking is employed to trace the averaged gradient of the cluster-level cost function. For a review of the relevant methods, Gaussian smoothing techniques were firstly introduced in [36] for general convex optimization problem, and have received a renewed attention in both distributed optimization [37]- [42] and non-cooperative games [43]. One major advantage of such techniques is a free of the knowledge on the explicit form of the cost functions, which is essentially a nonmodel-based approach.…”

Section: A Literature Reviewmentioning

confidence: 99%

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

Pang¹,

Hu²

2020

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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 agents' local cost functions can be non-smooth, the explicit analytical expressions 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 to choose their own specific 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 heterogeneity of the step-size. Moreover, we provide both upper and lower bounds for the rate of convergence of the proposed algorithm. The performance of the proposed algorithm is validated by numerical simulations.

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Section: A Literature Reviewmentioning

confidence: 99%

“…It is assumed that (k, k) ∈ E i , ∀k ∈ V i . Moreover, we assume that the explicit analytical expressions of the agents' local cost functions are unknown, but the function values can be measured, similar to the settings in [35] and [43]. We aim to develop an NE seeking strategy to drive all agents' actions towards the NE.…”

Section: A Problem Formulationmentioning

confidence: 99%

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

Pang¹,

Hu²

2020

Preprint

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“…Remark 3: In contrast to existing works in [8]- [16], solving Problem 1 is much more challenging at least from the following aspects: (1) Prescribed-time convergence: different from NE seeking results that guarantee a semi-global exponential convergence in [8], [9], asymptotic or exponential convergence in [10]- [12], linear convergence in [13], and UUB convergence in [14], it is desirable to solve Problem 1 in the prescribed time (priori-given and user-defined) that is independent of any initial states, communication graphs, and control gains. (2) Player communication network: the topology is jointly switching rather than the static graphs in [8]- [16].…”

Section: B Main Objectivementioning

confidence: 99%

“…Then, it follows from the notations of R and F in (11) that F( ψ) = 0 n . Then, submitting it into (14) gives rise to…”

Section: A Prescribed-time Distributed Ne Seeking Designmentioning

confidence: 99%

Prescribed-Time Fully Distributed Nash Equilibrium Seeking in Noncooperative Games

Feng,

2020

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In this paper, we investigate a prescribed-time and fully distributed Nash Equilibrium (NE) seeking problem for continuoustime noncooperative games. By exploiting pseudo-gradient play and consensus-based schemes, various distributed NE seeking algorithms are presented over either fixed or switching communication topologies so that the convergence to the NE is reached in a prescribed time. In particular, a prescribed-time distributed NE seeking algorithm is firstly developed under a fixed graph to find the NE in a prior-given and user-defined time, provided that a static controller gain can be selected based on certain global information such as the algebraic connectivity of the communication graph and both the Lipschitz and monotone constants of the pseudo-gradient associated with players' objective functions. Secondly, a prescribed-time and fully distributed NE seeking algorithm is proposed to remove global information by designing heterogeneous dynamic gains that turn on-line the weights of the communication topology. Further, we extend this algorithm to accommodate jointly switching topologies. It is theoretically proved that the global convergence of those proposed algorithms to the NE is rigorously guaranteed in a prescribed time based on a time function transformation approach. In the last, numerical simulation results are presented to verify the effectiveness of the designs.The advantages of the proposed NE seeking algorithm include: 1) the convergence time is user-defined according to task requirements, which is neither dependent on any initial states or the parameters of the algorithm; 2) the proposed algorithm is fully distributed without requiring any global information on the graph's algebraic connectivity, the pseudo-gradient's Lipschitz and monotone constants, and the number of players; and 3) the communication graph is allowed to be jointly switching. The aforementioned requirements can improve the practical relevance of the problem to be addressed and meanwhile, it poses some technical challenges to the algorithm design and stability analysis, which makes that the NE seeking algorithms in the existing literature cannot be directly applied.

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“…Literature review: various distributed continuous-and discretetime algorithms are developed in [8]- [18] to solve distributed NE seeking problems in games, in which each player cannot observe all other players' actions, but can exchange information between neighbors over an undirected graph or weight-balanced digraph. Gradient-based NE seeking algorithms are popular techniques to find the NE of games with differentiable objective functions where each player modifies its current action based on the gradient with respect to its own action.…”

Section: Introductionmentioning

confidence: 99%

Attack-Resilient Distributed Algorithms for Exponential Nash Equilibrium Seeking

Feng,

2020

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This paper investigates a resilient distributed Nash equilibrium (NE) seeking problem on a directed communication network subject to malicious cyber-attacks. The considered attacks, named as Denial-of-Service (DoS) attacks, are allowed to occur aperiodically, which refers to interruptions of communication channels carried out by intelligent adversaries. In such an insecure network environment, the existence of cyber-attacks may result in undesirable performance degradations or even the failures of distributed algorithm to seek the NE of noncooperative games. Hence, the aforementioned setting can improve the practical relevance of the problem to be addressed and meanwhile, it poses some technical challenges to the distributed algorithm design and exponential convergence analysis. In contrast to the existing distributed NE seeking results over a prefect communication network, an attack-resilient distributed algorithm is presented such that the NE can be exactly reached with an exponential convergence rate in the presence of DoS attacks. Inspired by the previous works in [21]-[26], an explicit analysis of the attack frequency and duration is investigated to enable exponential NE seeking with resilience against attacks. Examples and numerical simulation results are given to show the effectiveness of the proposed design.

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Distributed Nash Equilibrium Seeking With Limited Cost Function Knowledge via a Consensus-Based Gradient-Free Method

Cited by 43 publications

References 37 publications

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

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

Prescribed-Time Fully Distributed Nash Equilibrium Seeking in Noncooperative Games

Attack-Resilient Distributed Algorithms for Exponential Nash Equilibrium Seeking

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