We develop policy gradient algorithms with global optimality and convergence guarantees for reinforcement learning (RL) with proportional-integral-derivative (PID) parameterized control policies. RL enables learning control policies through direct interaction with a system, without explicit model knowledge that is typically assumed in classical control. The PID policy architecture offers built-in structural advantages, such as superior tracking performance, elimination of steady-state errors, and robustness to model error that have made it a widely adopted paradigm in practice. Despite these advantages, the PID parameterization has received limited attention in the RL literature, and PID control designs continue to rely on heuristic tuning rules without theoretical guarantees. We address this gap by rigorously integrating PID control with RL, offering theoretical guarantees while maintaining the practical advantages that have made PID control ubiquitous in practice. Specifically, we first formulate PID control design as an optimization problem with a control policy that is parameterized by proportional, integral, and derivative components. We derive exact expressions for policy gradients in these parameters, and leverage them to develop both model-based and model-free policy gradient algorithms for PID policies. We then establish gradient dominance properties of the PID policy optimization problem, and provide theoretical guarantees on convergence and global optimality in this setting. Finally, we benchmark the performance of our algorithms on the controlgym suite of environments.

The Lipschitz constant plays a crucial role in certifying the robustness of neural networks to input perturbations. Since calculating the exact Lipschitz constant is NP-hard, efforts have been made to obtain tight upper bounds on the Lipschitz constant. Typically, this involves solving a large matrix verification problem, the computational cost of which grows significantly for both deeper and wider networks. In this paper, we provide a compositional approach to estimate Lipschitz constants for deep feed-forward neural networks. We first obtain an exact decomposition of the large matrix verification problem into smaller sub-problems. Then, leveraging the underlying cascade structure of the network, we develop two algorithms. The first algorithm explores the geometric features of the problem and enables us to provide Lipschitz estimates that are comparable to existing methods by solving small semidefinite programs (SDPs) that are only as large as the size of each layer. The second algorithm relaxes these sub-problems and provides a closed-form solution to each sub-problem for extremely fast estimation, altogether eliminating the need to solve SDPs. The two algorithms represent different levels of trade-offs between efficiency and accuracy. Finally, we demonstrate that our approach provides a steep reduction in computation time (as much as several thousand times faster, depending on the algorithm for deeper networks) while yielding Lipschitz bounds that are very close to or even better than those achieved by state-of-the-art approaches in a broad range of experiments. In summary, our approach considerably advances the scalability and efficiency of certifying neural network robustness, making it particularly attractive for online learning tasks.