This paper considers distributed optimization algorithms, with application in binary classification via distributed support-vector-machines (D-SVM) over multi-agent networks subject to some link nonlinearities. The agents solve a consensus-constraint distributed optimization cooperatively via continuous-time dynamics, while the links are subject to strongly sign-preserving odd nonlinear conditions. Logarithmic quantization and clipping (saturation) are two examples of such nonlinearities. In contrast to existing literature that mostly considers ideal links and perfect information exchange over linear channels, we show how general sector-bounded models affect the convergence to the optimizer (i.e., the SVM classifier) over dynamic balanced directed networks. In general, any odd sector-bounded nonlinear mapping can be applied to our dynamics. The main challenge is to show that the proposed system dynamics always have one zero eigenvalue (associated with the consensus) and the other eigenvalues all have negative real parts. This is done by recalling arguments from matrix perturbation theory. Then, the solution is shown to converge to the agreement state under certain conditions. For example, the gradient tracking (GT) step size is tighter than the linear case by factors related to the upper/lower sector bounds. To the best of our knowledge, no existing work in distributed optimization and learning literature considers non-ideal link conditions.

翻译：本文研究分布式优化算法，并将其应用于多智能体网络在存在链路非线性条件下的二分类问题（通过分布式支持向量机D-SVM）。智能体通过连续时间动力学协同求解带有一致性约束的分布式优化问题，而链路受到强保号奇数非线性条件约束。对数量化和裁剪（饱和）是此类非线性的两个典型实例。与现有文献多假设理想链路和线性信道完美信息交换不同，本文展示了广义扇区有界模型如何影响动态平衡有向网络中优化器（即SVM分类器）的收敛性。通常，任何奇数扇区有界非线性映射均可应用于我们的动力学系统。主要挑战在于证明所提出的系统动力学始终存在一个零特征值（与一致性相关），而其他特征值均具有负实部。通过引入矩阵扰动理论的相关论证实现该目标，并证明解在特定条件下收敛至一致状态。例如，梯度跟踪步长需通过上下扇区边界因子比线性情况更严格。据我们所知，现有分布式优化与学习文献中尚未考虑非理想链路条件。