Closed-Form Robustness Bounds for Second-Order Pruning of Neural Controller Policies

Deep neural policies have unlocked agile flight for quadcopters, adaptive grasping for manipulators, and reliable navigation for ground robots, yet their millions of weights conflict with the tight memory and real-time constraints of embedded microcontrollers. Second-order pruning methods -- \emph{Optimal Brain Damage} (OBD) and its variants, including \emph{Optimal Brain Surgeon} (OBS) and the recent \textsc{SparseGPT} -- compress networks in a single pass by leveraging the local Hessian, achieving far higher sparsity than magnitude thresholding. Despite their success in vision and language, the consequences of such weight removal on \emph{closed-loop} stability, tracking accuracy, and safety have remained unclear. We present the first mathematically rigorous robustness analysis of second-order pruning in nonlinear discrete-time control. The system evolves under a continuous transition map \(f:X\times U\!\to\!X\), while the controller is an \(L\)-layer multilayer perceptron with ReLU-type activations that are \emph{globally} \(1\)-Lipschitz. Pruning the weight matrix of layer \(k\) replaces \(W_k\) with \(W_k+\delta W_k\), producing the perturbed parameter vector \(\widehat{\Theta}=\Theta+\delta\Theta\) and the pruned policy \(\pi(\cdot;\widehat{\Theta})\). For every input state \(s\in X\) we derive the closed-form inequality \( \|\pi(s;\Theta)-\pi(s;\widehat{\Theta})\|_2 \le C_k(s)\,\|\delta W_k\|_2, \) where the constant \(C_k(s)\) depends \emph{only} on unpruned spectral norms and biases, and can be evaluated in closed form from a single forward pass. When several layers \(S\subseteq\{1,\dots,L\}\) are pruned we show the deviations add \emph{linearly}, \( \textstyle \sum_{k\in S} C_k(s)\|\delta W_k\|_2, \) yielding an explicit state-dependent or worst-case robustness estimation. All constants are computed \emph{offline}. Therefore, environment roll-outs, back-propagation, or hyper-parameter tuning are \emph{not} necessary, what makes the framework directly usable inside compression pipelines. The derived bounds specify, prior to field deployment, the maximal admissible pruning magnitude compatible with a prescribed control-error threshold. By linking second-order network compression with closed-loop performance guarantees, our work narrows a crucial gap between modern deep-learning tooling and the robustness demands of safety-critical autonomous systems.