Anchored Langevin Algorithms
This addresses a problem in machine learning for sampling from complex distributions, offering a method that handles non-differentiable and heavy-tailed cases, though it appears incremental as it builds on existing Langevin frameworks.
The paper tackles the limitations of standard Langevin algorithms, which require differentiable log-densities and fail on heavy-tailed targets, by proposing anchored Langevin dynamics that accommodates non-differentiable and heavy-tailed distributions, with non-asymptotic guarantees in the 2-Wasserstein distance.
Standard first-order Langevin algorithms such as the unadjusted Langevin algorithm (ULA) are obtained by discretizing the Langevin diffusion and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they face two key limitations: (i) they require differentiable log-densities, excluding targets with non-differentiable components; and (ii) they generally fail to sample heavy-tailed targets. We propose anchored Langevin dynamics, a unified approach that accommodates non-differentiable targets and certain classes of heavy-tailed distributions. The method replaces the original potential with a smooth reference potential and modifies the Langevin diffusion via multiplicative scaling. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and provide an equivalent formulation derived via a random time change of the Langevin diffusion. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.