The application of reinforcement learning to safety-critical systems is limited by the lack of formal methods for verifying the robustness and safety of learned policies. This paper introduces a novel framework that addresses this gap by analyzing the combination of an RL agent and its environment as a discrete-time autonomous dynamical system. By leveraging tools from dynamical systems theory, specifically the Finite-Time Lyapunov Exponent (FTLE), we identify and visualize Lagrangian Coherent Structures (LCS) that act as the hidden "skeleton" governing the system's behavior. We demonstrate that repelling LCS function as safety barriers around unsafe regions, while attracting LCS reveal the system's convergence properties and potential failure modes, such as unintended "trap" states. To move beyond qualitative visualization, we introduce a suite of quantitative metrics, Mean Boundary Repulsion (MBR), Aggregated Spurious Attractor Strength (ASAS), and Temporally-Aware Spurious Attractor Strength (TASAS), to formally measure a policy's safety margin and robustness. We further provide a method for deriving local stability guarantees and extend the analysis to handle model uncertainty. Through experiments in both discrete and continuous control environments, we show that this framework provides a comprehensive and interpretable assessment of policy behavior, successfully identifying critical flaws in policies that appear successful based on reward alone.

The empirical success of deep learning is often attributed to SGD's mysterious ability to avoid sharp local minima in the loss landscape, which is well known to lead to poor generalization. Recently, empirical evidence of heavy-tailed gradient noise was reported in many deep learning tasks; under the presence of such heavy-tailed noise, it can be shown that SGD can escape sharp local minima, providing a partial solution to the mystery. In this work, we analyze a popular variant of SGD where gradients are truncated above a fixed threshold. We show that it achieves a stronger notion of avoiding sharp minima; it can effectively eliminate sharp local minima entirely from its training trajectory. We characterize the dynamics of truncated SGD driven by heavy-tailed noises. First, we show that the truncation threshold and width of the attraction field dictate the order of the first exit time from the associated local minimum. Moreover, when the objective function satisfies appropriate structural conditions, we prove that as the learning rate decreases the dynamics of the heavy-tailed SGD closely resemble that of a special continuous-time Markov chain which never visits any sharp minima. We verify our theoretical results with numerical experiments and discuss the implications on the generalizability of SGD in deep learning.