In this paper, we perform Lyapunov based analysis of the loss function to derive an a priori upper bound on the settling time of deep neural networks. While previous studies have attempted to understand deep learning using control theory framework, there is limited work on a priori finite time convergence analysis. Drawing from the advances in analysis of finite-time control of non-linear systems, we provide a priori guarantees of finite-time convergence in a deterministic control theoretic setting. We formulate the supervised learning framework as a control problem where weights of the network are control inputs and learning translates into a tracking problem. An analytical formula for finite-time upper bound on settling time is computed a priori under the assumptions of boundedness of input. Finally, we prove the robustness and sensitivity of the loss function against input perturbations.

中文翻译：

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深度神经网络有限时间收敛的先验保证

在本文中，我们对损失函数进行基于李雅普诺夫的分析，以推导出深度神经网络稳定时间的先验上限。虽然之前的研究试图使用控制理论框架来理解深度学习，但在先验有限时间收敛分析方面的工作有限。借鉴非线性系统有限时间控制分析的进展，我们提供了确定性控制理论设置中有限时间收敛的先验保证。我们将监督学习框架制定为一个控制问题，其中网络的权重是控制输入，学习转化为跟踪问题。在输入有界的假设下，先验地计算了建立时间的有限时间上限的解析公式。最后，