Bounds on Deep Neural Network Partial Derivatives with Respect to Parameters
For researchers in safety-critical control systems, this work strengthens the mathematical foundation by providing explicit, computable bounds that were previously assumed, addressing a known gap in Lyapunov-based DNN applications.
This paper provides rigorous polynomial bounds on first and second partial derivatives of DNNs with respect to parameters, enabling precise stability guarantees for Lyapunov-based DNNs and convergence analysis in gradient-based learning.
Deep neural networks (DNNs) have emerged as a powerful tool with a growing body of literature exploring Lyapunov-based approaches for real-time system identification and control. These methods depend on establishing bounds for the second partial derivatives of DNNs with respect to their parameters, a requirement often assumed but rarely addressed explicitly. This paper provides rigorous mathematical formulations of polynomial bounds on both the first and second partial derivatives of DNNs with respect to their parameters. We present lemmas that characterize these bounds for fully-connected DNNs, while accommodating various classes of activation function including sigmoidal and ReLU-like functions. Our analysis yields closed-form expressions that enable precise stability guarantees for Lyapunov-based deep neural networks (Lb-DNNs). Furthermore, we extend our results to bound the higher-order terms in first-order Taylor approximations of DNNs, providing important tools for convergence analysis in gradient-based learning algorithms. The developed theoretical framework develops explicit, computable expressions, for previously assumed bounds, thereby strengthening the mathematical foundation of neural network applications in safety-critical control systems.