Avoiding unsafe sets when training with Langevin Dynamics

Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics on the loss landscape, and a natural safety question is to bound the probability $ν_t(\mathcal{A}_H) = \mathbb{P}(Q_t \in \mathcal{A}_H)$ that the trajectory lies in a designated failure region $\mathcal{A}_H$. We study this for a smooth, strongly convex loss in $d$ dimensions and a failure region separated from the minimizer by an energy gap. Three bounds emerge. At the end of training, the equilibrium mass $π(\mathcal{A}_H)$ is exponentially small in $d$, with a complementary energy-barrier rate when the noise is small. Along the trajectory, a shape-free bound $ν_t(\mathcal{A}_H) \le π(\mathcal{A}_H)(1 + \sqrt{χ_0^2/π(\mathcal{A}_H)}\,e^{-mt})$ shows that the in-set probability relaxes to (twice) the static value after a burn-in time of order $d$, using only the global spectral gap $m$ of the loss. A worked Ornstein-Uhlenbeck example shows this burn-in is necessary: an angular slice of the equilibrium shell can transiently swell by a factor exponential in $d$, even though its equilibrium mass is tiny. To rule such swelling out we introduce a local relaxation rate attached to the failure region, defined through the spectral measure of its centered indicator rather than a Dirichlet-form Rayleigh quotient. For geometrically isolated regions this rate exceeds the global one, shrinking the burn-in proportionally, and combined with a maximum-principle ceiling it caps the trajectory probability uniformly in time. The picture is that strong convexity sets how fast training relaxes, but the shape of the unsafe set decides whether the trajectory bulges through it on the way home.

When Stopping Fails: Rethinking Minimal Risk Conditions through Human-Interactive Autonomous Driving for Safe Transportation Systems

Autonomous vehicles (AVs) are increasingly deployed in urban environments, yet their safety frameworks remain primarily designed around collision avoidance and minimal risk condition (MRC) behaviors such as slowing or stopping when uncertainty arises. Although effective in reducing immediate crash risk, real-world deployments indicate that stopping alone does not guarantee safe integration into human-governed roadway systems. Incidents reported by municipalities and public records show that AV fallback behaviors can obstruct traffic, interfere with emergency response operations, and create accessibility challenges for passengers and pedestrians. This paper presents an analysis of publicly documented incidents involving AV stopping behavior and human-AV interaction failures. We categorize these incidents according to limitations in perception, planning, and control within current AV architectures. Using this taxonomy, we identify key gaps in existing safety paradigms, particularly the lack of mechanisms for interpreting human authority, responding to multimodal instructions, and adapting to dynamic, socially regulated traffic conditions. We then review emerging research directions that support human-interactive perception, language-grounded and accessibility-aware planning, and assisted control through remote guidance and teleoperation. The analysis highlights the need to augment current AV safety frameworks with capabilities that enable cooperative interaction with human agents and infrastructure. These findings suggest that reliable urban deployment of AVs requires moving beyond passive fallback strategies toward human-interactive autonomy.