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On randomized step sizes in Metropolis-Hastings algorithms

Sebastiano Grazzi, Samuel Livingstone, Lionel Riou-Durand

TL;DR

This work addresses the sensitivity of Metropolis–Hastings algorithms to step-size tuning by introducing randomized step-size schemes, namely an auxiliary-variable kernel $\overline{P}_h$ and a marginalized kernel $M_h$. The authors prove that, under mild conditions, these randomized kernels inherit the base algorithm's spectral-gap properties and maintain robustness to tuning, with $M_h$ offering smaller asymptotic variance when implementable. They establish scaling-limit results showing that high-dimensional performance is preserved and that step-size jitter can increase the optimal acceptance rate for Langevin and Hamiltonian samplers when $\mu$ is Uniform or Exponential; theoretical findings are complemented by extensive simulations on Neal's funnel, Rosenbrock, and Poisson-regression benchmarks. Overall, randomized step sizes provide a principled and practical route to more robust MCMC performance in high dimensions.

Abstract

The performance of Metropolis-Hastings algorithms is highly sensitive to the choice of step size, and miss-specification can lead to severe loss of efficiency. We study algorithms with randomized step sizes, considering both auxiliary-variable and marginalized constructions. We show that algorithms with a randomized step size inherit weak Poincaré inequalities/spectral gaps from their fixed-step-size counterparts under minimal conditions, and that the marginalized kernel should always be preferred in terms of asymptotic variance to the auxiliary-variable choice if it is implementable. In addition we show that both types of randomization make an algorithm robust to tuning, meaning that spectral gaps decay polynomially as the step size is increasingly poorly chosen. We further show that step-size randomization often preserves high-dimensional scaling limits and algorithmic complexity, while increasing the optimal acceptance rate for Langevin and Hamiltonian samplers when an Exponential or Uniform distribution is chosen to randomize the step size. Theoretical results are complemented with a numerical study on challenging benchmarks such as Poisson regression, Neal's funnel and the Rosenbrock (banana) distribution.

On randomized step sizes in Metropolis-Hastings algorithms

TL;DR

This work addresses the sensitivity of Metropolis–Hastings algorithms to step-size tuning by introducing randomized step-size schemes, namely an auxiliary-variable kernel and a marginalized kernel . The authors prove that, under mild conditions, these randomized kernels inherit the base algorithm's spectral-gap properties and maintain robustness to tuning, with offering smaller asymptotic variance when implementable. They establish scaling-limit results showing that high-dimensional performance is preserved and that step-size jitter can increase the optimal acceptance rate for Langevin and Hamiltonian samplers when is Uniform or Exponential; theoretical findings are complemented by extensive simulations on Neal's funnel, Rosenbrock, and Poisson-regression benchmarks. Overall, randomized step sizes provide a principled and practical route to more robust MCMC performance in high dimensions.

Abstract

The performance of Metropolis-Hastings algorithms is highly sensitive to the choice of step size, and miss-specification can lead to severe loss of efficiency. We study algorithms with randomized step sizes, considering both auxiliary-variable and marginalized constructions. We show that algorithms with a randomized step size inherit weak Poincaré inequalities/spectral gaps from their fixed-step-size counterparts under minimal conditions, and that the marginalized kernel should always be preferred in terms of asymptotic variance to the auxiliary-variable choice if it is implementable. In addition we show that both types of randomization make an algorithm robust to tuning, meaning that spectral gaps decay polynomially as the step size is increasingly poorly chosen. We further show that step-size randomization often preserves high-dimensional scaling limits and algorithmic complexity, while increasing the optimal acceptance rate for Langevin and Hamiltonian samplers when an Exponential or Uniform distribution is chosen to randomize the step size. Theoretical results are complemented with a numerical study on challenging benchmarks such as Poisson regression, Neal's funnel and the Rosenbrock (banana) distribution.
Paper Structure (18 sections, 8 theorems, 49 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 8 theorems, 49 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Related literature
  4. MCMC Algorithms with randomized step size
  5. Metropolis--Hastings algorithms
  6. Randomized step sizes via auxiliary-variable and marginalized kernels
  7. Examples of randomized kernels
  8. Spectral gap analysis
  9. Dirichlet forms, Poincaré inequalities & spectral gaps
  10. Orderings and Robustness to tuning
  11. Spectral gap results for kernels with randomized step size
  12. Scaling limits
  13. Numerical experiments
  14. ESJD
  15. Neal's funnel distribution
  16. ...and 3 more sections

Key Result

Proposition 1

For any $f \in L^2_0(\pi)$, any $h>0$ and any base kernel $P_h$ we have from which it follows that and that an ergodic average constructed using $M_h$ will always have smaller asymptotic variance than the same ergodic average using $\overline P_h$.

Figures (5)

  • Figure 1: ESJD estimates (log-scale) as a function of the step size ($x$-axis, log-scale) of standard MALA (blue), Auxiliary-variable MALA with Exponential (red) and Uniform (purple) randomized step sizes and Marginalized MALA with Exponential (green) and Uniform (yellow) randomized step sizes, for a one-dimensional standard Normal (left), Laplace (center), Student-$t$ with $5$ degrees of freedom (right).
  • Figure 2: Contour plots of a 2 dimensional Neal's Funnel target with $\sigma^2 = 9$ (left panel) and a 2 dimensional Rosenbrock's banana distribution with $a = 1/2$, $b = 50$ (right panel).
  • Figure 3: Left panel: histograms and Q-Q plots of the first components $x_1$ for MALA (blue) and Auxiliary-variable MALA with Uniform (green) and Exponential (red) randomized step sizes. The red solid line shows the true marginal density of $X_1$. Right panel: box plots for the estimation $\mathbb{P}(X_1 < \xi_{0.05})$ over $20$ independent simulations of each algorithm, as the number of iterations ($x$-axis) increases.
  • Figure 4: Left panel: same as in Figure \ref{['fig:FUNNEL']}. Right panel: box plots for the estimation $\mathbb{P}(|X_1| > \xi_{0.95})$ over $20$ independent simulations of each algorithm, as the number of iterations increases ($x$-axis).
  • Figure 5: Convergence of MALA (blue) and Auxiliary-variable MALA with Uniform (green) and Exponential (red) randomized step sizes initialized in the tails of the distribution. Left panel: trajectories of the first two coordinates. Right panel: traces of the first coordinate.

Theorems & Definitions (15)

  • Proposition 1
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1
  • Example 1
  • Proposition 3
  • proof
  • Theorem 4
  • ...and 5 more