# AutoStep: Locally adaptive involutive MCMC

arXiv: https://arxiv.org/abs/2410.18929 · HTML: https://mlsanity.com/paper/2410.18929/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 51 references that are in the map.
- First posted on arXiv 24 Oct 2024.

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

- [Involutive MCMC: a Unifying Framework](https://mlsanity.com/paper/2006.16653/) (2020)
- [Efficient and Generalizable Tuning Strategies for Stochastic Gradient MCMC](https://mlsanity.com/paper/2105.13059/) (2021)
- [A general perspective on the Metropolis-Hastings kernel](https://mlsanity.com/paper/2012.14881/) (2020)
- [GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2404.15253/) (2024)
- [Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions](https://mlsanity.com/paper/2110.00610/) (2021)
- [Incorporating Local Step-Size Adaptivity into the No-U-Turn Sampler using Gibbs Self Tuning](https://mlsanity.com/paper/2408.08259/) (2024)
- [Metropolis-Hastings view on variational inference and adversarial training](https://mlsanity.com/paper/1810.07151/) (2018)
- [Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance](https://mlsanity.com/paper/1507.05780/) (2015)
- [Air Markov Chain Monte Carlo](https://mlsanity.com/paper/1801.09309/) (2018)
- [Adaptive step size selection for Hessian-based manifold Langevin samplers](https://mlsanity.com/paper/1501.07454/) (2015)
- [Selecting the Metric in Hamiltonian Monte Carlo](https://mlsanity.com/paper/1905.11916/) (2019)
- [Markov Kernels Local Aggregation for Noise Vanishing Distribution Sampling](https://mlsanity.com/paper/1806.09000/) (2018)

## Built on by (most cited first, 4 of 4)

- [The Within-Orbit Adaptive Leapfrog No-U-Turn Sampler](https://mlsanity.com/paper/2506.18746/) (2025)
- [Harnessing the Power of Reinforcement Learning for Adaptive MCMC](https://mlsanity.com/paper/2507.00671/) (2025)
- [Stop the Sampler! Classifier-Based Adaptive Stopping for Sampling Kernels](https://mlsanity.com/paper/2606.16073/) (2026)
- [Optimal Slice-Adaptive Tuning of Hybrid Slice Sampling](https://mlsanity.com/paper/2609.08172/) (2026)

## Abstract (arXiv)

Many common Markov chain Monte Carlo (MCMC) kernels can be formulated using a deterministic involutive proposal with a step size parameter. Selecting an appropriate step size is often a challenging task in practice; and for complex multiscale targets, there may not be one choice of step size that works well globally. In this work, we address this problem with a novel class of involutive MCMC methods -- AutoStep MCMC -- that selects an appropriate step size at each iteration adapted to the local geometry of the target distribution. We prove that under mild conditions AutoStep MCMC is π-invariant, irreducible, and aperiodic, and obtain bounds on expected energy jump distance and cost per iteration. Empirical results examine the robustness and efficacy of our proposed step size selection procedure, and show that AutoStep MCMC is competitive with state-of-the-art methods in terms of effective sample size per unit cost on a range of challenging target distributions.