Avoiding unsafe sets when training with Langevin Dynamics
Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics, 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, with $\mathcal{A}_H$ separated from the minimizer by an energy gap. 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)}\,...
| Authors | Adam M. Oberman |
|---|---|
| Primary category | cs.LG |
| Published | 8 July 2026 |
| Last updated at source | 15 July 2026 |
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