Original title: Geometric Ergodicity of Affine Invariant Ensemble Langevin and its Discrete Time Variants
Hong Ye Tan, Yifan Chen · September 2026
Scientists use sampling algorithms to explore complex probability targets. One popular method uses a team of interacting particles. These particles use their group spread, called empirical covarianceThe covariance matrix estimated from the current positions of all particles, used to adjust their movements., to guide their steps. In practice, this works well. However, researchers lacked a clear mathematical guarantee that it settles down quickly. A major challenge is that the particles can bunch up, making their spread matrix collapse toward zero. The authors resolved this. For certain well-behaved targets, they proved the process converges at a fast, steady geometric rate. Next, they showed that standard step-by-step computer simulations can fail and shoot off to infinity. To fix this, they introduced a time adjustment based on the spread's trace. This fix guarantees rapid convergence and matches the true target distribution.
A team of dancers needs to stay in formation while moving across a slippery floor. If they focus only on each other's positions without considering the floor, they can slip and fall. Here, the dancers are like particles in a sampling method, and their tendency to collapse the formation causes problems. The paper adds a gentle push to maintain space, similar to the regularization that prevents divergence.
Bayesian statisticians and scientists using these samplers need to know when they fail. This work shows a common method can diverge and provides a regularized version that works reliably.
The results are limited to potentials with bounded Hessian and strong convexity outside a ball. The regularization requires a sufficiently small step size, which might not be practical for all applications.
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Geometric Ergodicity of Affine Invariant Ensemble Langevin and its Discrete Time Variants — Hong Ye Tan, Yifan Chen
Abstract licence: CC0
Explanation generated by Synthesis from the abstract — it is not the original paper. This is a preprint, so it hasn't necessarily been peer reviewed.