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.
This paper studies a variation of the classic network alignment problem, named diffusion-network alignment. The goal is to align the vertices of a rooted diffusion tree to the vertices of a network, where the diffusion tree could be from a communication trace or contact tracing, and the network could be an online or offline social network. Different from the classic network alignment where both networks are fully observed, this model captures the information asymmetry of two networks. To solve this problem, this paper presents an efficient algorithm based on tree correlation tests to extract alignment information from local neighborhoods. We analyze the performance of the algorithm in the sparse graph regime and show that with high probability, all matched pairs are correct. Furthermore, for each vertex on the diffusion tree, this paper establishes an explicit lower bound on the probability that the vertex is correctly matched. These lower bounds are depth-dependent and increase as vertices get closer to the root.