We present stochastic consensus and convergence of the discrete consensus-based optimization (CBO) algorithm with random batch interactions and heterogeneous external noises. Despite the wide applications and successful performance in many practical simulations, the convergence of the discrete CBO algorithm was not rigorously investigated in such a generality. In this work, we introduce a generalized discrete CBO algorithm with a weighted representative point and random batch interactions, and show that the proposed discrete CBO algorithm exhibits stochastic consensus and convergence toward the common equilibrium state exponentially fast under suitable assumptions on system parameters. For this, we recast the given CBO algorithm with random batch interactions as a discrete consensus model with a random switching network topology, and then we use the mixing property of interactions over sufficiently long time interval to derive stochastic consensus and convergence estimates in mean square and almost sure senses. Our proposed analysis significantly improves earlier works on the convergence analysis of CBO models with full batch interactions and homogeneous external noises.

我们提出了具有随机批次交互和异构外部噪声的离散基于一致性优化 (CBO) 算法的随机一致性和收敛性. 尽管在许多实际模拟中得到了广泛的应用和成功的表现，但离散 CBO 算法的收敛性并没有得到如此普遍的严格研究。在这项工作中，我们介绍了一种具有加权代表点和随机批次交互的广义离散 CBO 算法，并表明所提出的离散 CBO 算法在系统参数的适当假设下表现出随机一致性和以指数速度快速收敛到共同平衡状态。为此，我们将具有随机批量交互的给定 CBO 算法重铸为具有随机切换网络拓扑的离散共识模型，然后我们使用在足够长的时间间隔内交互的混合特性来推导出均方的随机共识和收敛估计，几乎可以肯定的感觉。