# Avoiding unsafe sets when training with Langevin Dynamics

arXiv: https://arxiv.org/abs/2607.07538 · HTML: https://mlsanity.com/paper/2607.07538/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 30 references that are in the map.
- First posted on arXiv 8 Jul 2026.

## Builds on (most cited first, 12 of 12)

- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Risks from Learned Optimization in Advanced Machine Learning Systems](https://mlsanity.com/paper/1906.01820/) (2019)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Defining and Characterizing Reward Hacking](https://mlsanity.com/paper/2209.13085/) (2022)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [Implicit Gradient Regularization](https://mlsanity.com/paper/2009.11162/) (2020)
- [Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations](https://mlsanity.com/paper/1811.01558/) (2018)
- [A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1702.05575/) (2017)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [On Large Batch Training and Sharp Minima: A Fokker-Planck Perspective](https://mlsanity.com/paper/2112.00987/) (2021)
- [Safety from Honesty in a Disinterested AI Predictor](https://mlsanity.com/paper/2606.29657/) (2026)

## Abstract (arXiv)

Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics, and a natural safety question is to bound the probability nut(AH) = mathbbP(Qt ∈ AH) that the trajectory lies in a designated failure region AH. We study this for a smooth, strongly convex loss in d dimensions, with AH separated from the minimizer by an energy gap. At the end of training, the equilibrium mass π(AH) is exponentially small in d, with a complementary energy-barrier rate when the noise is small. Along the trajectory, a shape-free bound nut(AH) ≤ π(AH)(1 + √chi02/π(AH),e-mt) shows the in-set probability relaxes to (twice) the static value after a burn-in of order d, using only the global spectral gap m. 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, though its equilibrium mass is tiny. To rule this out we introduce a local relaxation rate, defined through the spectral measure of the region's centered indicator rather than a Dirichlet-form Rayleigh quotient. For geometrically isolated regions this rate exceeds the global one, shrinking the burn-in, and with a maximum-principle ceiling it caps the trajectory probability uniformly in time. Strong convexity sets how fast training relaxes, but the shape of the unsafe set decides whether the trajectory bulges through it on the way to equilibrium.