# R'enyi Tracking Bounds for Langevin Dynamics with Moving Targets

arXiv: https://arxiv.org/abs/2609.17577 · HTML: https://mlsanity.com/paper/2609.17577/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 11 of 11)

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling](https://mlsanity.com/paper/2407.16936/) (2024)
- [Non-asymptotic Analysis of Diffusion Annealed Langevin Monte Carlo for Generative Modelling](https://mlsanity.com/paper/2502.09306/) (2025)
- [Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics](https://mlsanity.com/paper/2410.09697/) (2024)
- [Shifted Composition III: Local Error Framework for KL Divergence](https://mlsanity.com/paper/2412.17997/) (2024)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [Forward-KL Convergence of Time-Inhomogeneous Langevin Diffusions](https://mlsanity.com/paper/2601.22349/) (2026)

## Abstract (arXiv)

We study Langevin diffusion and Langevin Monte Carlo (LMC) when the target distribution changes over time. Under a log-Sobolev inequality (LSI), we derive non-asymptotic R'enyi-divergence guarantees for tracking the current target. The framework covers continuous-time Langevin diffusion and its discretizations. We then apply the results to nonsmooth sampling based on successive Moreau envelopes. For this scheme, we give explicit choices of the smoothing parameters and step sizes, together with corresponding complexity bounds. To our knowledge, these are the first non-asymptotic R'enyi-divergence tracking bounds for Langevin dynamics with discrete target updates.