神经网络反馈系统的可扩展增量鲁棒性分析

中文摘要 AI 辅助

包含深度神经网络（NN）的反馈系统的半定规划（SDP）证书通常随神经元总数扩展，而小增益测试虽可扩展但可能非常保守。本文开发了一个统一且可扩展的框架，用于涉及高维神经网络和未建模动态的反馈互连的增量鲁棒稳定性与性能分析。通过将全阶SDP条件的结构化分解与可扩展的Lipschitz常数估计算法相结合，我们推导出简化的验证条件，以证明增量收敛性和增量$\ell_2$-增益界。所得控制分析线性矩阵不等式（LMI）的维数仅取决于最后两个网络层的宽度，并且与网络深度无关。该框架保留了被控对象与神经网络之间的耦合，并将增量小增益条件作为特例恢复。为进一步降低保守性，我们开发了一种多轮交替更新方案，在保持可扩展性的同时迭代细化耦合变量。数值实验表明，所提出的框架在大规模神经网络反馈系统中实现了最先进的增量$\ell_2$-增益界。

英文摘要

Semidefinite programming (SDP) certificates for feedback systems containing deep neural networks (NNs) typically scale with the total number of neurons, whereas small-gain tests are scalable but can be highly conservative. This paper develops a unified and scalable framework for incremental robust stability and performance analysis of feedback interconnections involving high-dimensional NNs and unmodeled dynamics. By combining a structured decomposition of the full-order SDP condition with scalable Lipschitz constant estimation algorithms, we derive reduced verification conditions that certify incremental convergence and incremental $\ell_2$-gain bounds. The dimensions of the resulting control-analysis linear matrix inequalities (LMIs) depend only on the widths of the last two network layers and are \textit{independent of network depth}. The framework preserves the coupling between the plant and the NN, with the incremental small-gain condition recovered as a special case. To further reduce conservatism, we develop a multi-round alternating update scheme that iteratively refines the coupling variables while preserving scalability. Numerical experiments show that the proposed framework achieves state-of-the-art incremental $\ell_2$-gain bounds for large-scale NN feedback systems.

发表机构

• University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）
• University of Michigan（密歇根大学）
• University of Minnesota（明尼苏达大学）

机构由 AI 辅助整理，请以论文原文为准。