AutoStep: Locally adaptive involutive MCMC (2410.18929v3)
Abstract: 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 $\pi$-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.
- A general perspective on the Metropolis–Hastings kernel. arXiv:2012.14881.
- A tutorial on adaptive MCMC. Statistics and Computing, 18(4):343–373.
- Atchadé, Y. F. (2006). An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift. Methodology and Computing in Applied Probability, 8(2):235–254.
- Comprehensive benchmarking of Markov chain Monte Carlo methods for dynamical systems. BMC Systems Biology.
- Hit-and-run algorithms for generating multivariate distributions. Mathematics of Operations Research, 18(2):255–266.
- autoMALA: Locally adaptive Metropolis-adjusted Langevin algorithm. In International Conference on Artificial Intelligence and Statistics, volume 238, pages 4600–4608.
- Incorporating local step-size adaptivity into the No-U-Turn Sampler using Gibbs self tuning. arXiv: 2408.08259.
- GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo. arXiv:2404.15253.
- Handling sparsity via the horseshoe. In International Conference on Artificial Intelligence and Statistics, volume 5, pages 73–80.
- Air Markov chain Monte Carlo. arXiv:1801.09309.
- Efficient and generalizable tuning strategies for stochastic gradient MCMC. Statistics and Computing, 33(3):66.
- Hybrid Monte Carlo. Physics Letters B, 195(2):216–222.
- Markov chain Monte Carlo: Can we trust the third significant figure? Statistical Science, 23(2).
- Fixed-width output analysis for Markov chain Monte Carlo. Journal of the American Statistical Association, 101(476):1537–1547.
- Turing: A language for flexible probabilistic inference. In International Conference on Artificial Intelligence and Statistics, pages 1682–1690.
- Bayesian Data Analysis. CRC Press, 3rd edition.
- Riemann manifold Langevin and Hamiltonian Monte Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(2):123–214.
- Delayed rejection in reversible jump Metropolis–Hastings. Biometrika, 88(4):1035–1053.
- An adaptive Metropolis algorithm. Bernoulli, 7(2):223–242.
- Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1):97–109.
- The No-U-Turn Sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. The Journal of Machine Learning Research, 15(1):1593–1623.
- Kleppe, T. S. (2016). Adaptive step size selection for Hessian-based manifold Langevin samplers. Scandinavian Journal of Statistics, 43(3):788–805.
- Livingstone, S. (2021). Geometric ergodicity of the random walk Metropolis with position-dependent proposal covariance. Mathematics, 9(4).
- Markov kernels local aggregation for noise vanishing distribution sampling. SIAM Journal on Mathematics of Data Science, 4(4):1293–1319.
- An adaptive approach to Langevin MCMC. Statistics and Computing, 22.
- Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6):1087–1092.
- Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions. Bayesian Analysis, pages 1–28.
- Neal, R. M. (1996). Bayesian Learning for Neural Networks, volume 118 of Lecture Notes in Statistics. Springer New York, New York, NY, 1 edition.
- Neal, R. M. (2003). Slice sampling. The Annals of Statistics, 31(3):705–767.
- Metropolis-Hastings view on variational inference and adversarial training. arXiv:1810.07151.
- Involutive MCMC: A unifying framework. In International Conference on Machine Learning.
- Variable length trajectory compressible hybrid Monte Carlo. arXiv:1604.00889.
- On the half-Cauchy prior for a global scale parameter. Bayesian Analysis, 7(4):887–902.
- Weak convergence and optimal scaling of random walk Metropolis algorithms. Annals of Applied Probability, 7(1):110–120.
- General state space Markov chains and MCMC algorithms. Probability Surveys, 1:20–71.
- Optimal scaling of discrete approximations to Langevin diffusions. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 60(1):255–268.
- Brownian dynamics as smart Monte Carlo simulation. The Journal of Chemical Physics, 69(10):4628–4633.
- Connectionist Bench (Sonar, Mines vs. Rocks). UCI Machine Learning Repository.
- Pigeons.jl: Distributed sampling from intractable distributions. arXiv:2308.09769.
- Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state spaces. The Annals of Statistics, 8(1):1–9.
- Some adaptive Monte Carlo methods for Bayesian inference. Statistics in Medicine, 18:2507–2515.
- Sampling from multiscale densities with delayed rejection generalized Hamiltonian Monte Carlo. arXiv:2406.02741.