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AutoStep: Locally adaptive involutive MCMC

Tiange Liu, Nikola Surjanovic, Miguel Biron-Lattes, Alexandre Bouchard-Côté, Trevor Campbell

TL;DR

This work tackles the challenge of step-size tuning in involutive MCMC by introducing AutoStep MCMC, a locally adaptive scheme that operates on an augmented space to jointly learn a tuning parameter $\theta$ at each iteration. By employing a symmetric step-size selection mechanism and a round-based tuning strategy, AutoStep achieves $\,\pi$-invariance, irreducibility, and aperiodicity under mild conditions, while providing bounds on energy jump distance and iteration cost. Empirical results show AutoStep is robust to the initial $\theta_0$ and competitive with state-of-the-art adaptive samplers across synthetic and real targets, especially on multiscale/geometrically challenging distributions. Overall, AutoStep offers a principled, theoretically supported framework for locally adaptive step sizes in involutive MCMC with practical impact for Bayesian inference and complex target distributions.

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 $π$-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.

AutoStep: Locally adaptive involutive MCMC

TL;DR

This work tackles the challenge of step-size tuning in involutive MCMC by introducing AutoStep MCMC, a locally adaptive scheme that operates on an augmented space to jointly learn a tuning parameter at each iteration. By employing a symmetric step-size selection mechanism and a round-based tuning strategy, AutoStep achieves -invariance, irreducibility, and aperiodicity under mild conditions, while providing bounds on energy jump distance and iteration cost. Empirical results show AutoStep is robust to the initial and competitive with state-of-the-art adaptive samplers across synthetic and real targets, especially on multiscale/geometrically challenging distributions. Overall, AutoStep offers a principled, theoretically supported framework for locally adaptive step sizes in involutive MCMC with practical impact for Bayesian inference and complex target distributions.

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 -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.

Paper Structure

This paper contains 17 sections, 9 theorems, 62 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. AutoStep MCMC
  4. Step size selection
  5. Round-based tuning
  6. Theoretical Analysis
  7. Invariance
  8. Irreducibility and aperiodicity
  9. Step size selector termination
  10. Energy jump distance
  11. Experiments
  12. Symmetric step size selection criterion
  13. Tuning the initial step size $\theta_0$
  14. Robustness to initial step size $\theta_0$
  15. Comparison with other adaptive methods
  16. ...and 2 more sections

Key Result

Proposition 4.2

Under assum:diff_involution, AutoStep MCMC is $\sbpi$-invariant, and hence the $\scX$-marginal is $\pi$-invariant.

Figures (5)

  • Figure 1: Comparison of the symmetric (this work, blue) versus asymmetric (BironLattes24, orange) step size criteria in \ref{['alg:stepsize_selector']}, in terms of the move acceptance probability of AutoStep RWMH as a function of the state norm $\|x\|$. Note that the asymmetric criterion yields very low acceptance probabilities for states near the mode (left side of the plot) and in the tails (right side of the plot).
  • Figure 2: The effect of tuning $\theta_0$, showing traces of $\theta_0$ (\ref{['fig:stability']}) and cost per iteration (\ref{['fig:stabilitycost']}) versus tuning round when $\theta_0$ is initialized in $\{10^{-7}, \dots, 10^7\}$. Despite the wide range of initializations, the tuned values and cost per iteration stabilize quickly and reliably.
  • Figure 3: KSESS per unit cost for AutoStep (blue) and fixed-step (orange) RWMH (\ref{['fig:ksess_rwmh']}) and MALA (\ref{['fig:ksess_mala']}).
  • Figure 4: Energy jump distance per iteration (\ref{['fig:ejump_rwmh']}), acceptance probability (\ref{['fig:acc_rwmh']}), and cost per iteration (\ref{['fig:cost_rwmh']}) for AutoStep and fixed step RWMH versus initial step size $\theta_0$.
  • Figure 5: min KSESS per second for AutoStep and state-of-the-art samplers across five benchmarked models.

Theorems & Definitions (9)

  • Proposition 4.2
  • Proposition 4.5
  • Corollary 4.6
  • Proposition 4.8
  • Proposition 4.9
  • Corollary 4.10
  • Proposition 4.11
  • Lemma 1.1
  • Lemma 1.2