Optimal design of the Barker proposal and other locally balanced Metropolis–Hastings algorithms

Abstract

SUMMARY We study the class of first-order locally-balanced Metropolis–Hastings algorithms introduced in Livingstone & Zanella (2022). To choose a specific algorithm within the class the user must select a balancing function g: ℝ + → ℝ + satisfying g(t) = tg(1/t), and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a general limiting optimal acceptance rate of 57% and scaling of n−1/3 as the dimension n tends to infinity among all members of the class under mild smoothness assumptions on g and when the target distribution for the algorithm is of the product form. In particular we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimize this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, optimal choice of balancing function under a Gaussian noise distribution, and optimal choice of first-order locally-balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings and in particular show that a bi-modal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.

Metropolis–Hastings reversiblizations of non-reversible Markov chains

“…Remark 3.1. We remark that the harmonic or P −1 -reversiblization is in fact the Barker proposal in the Markov chain Monte Carlo literature Livingstone and Zanella (2022); Vogrinc et al (2022);Zanella (2020). See also the discussion in Section 5 below.…”

“…where we take this choice of g as the balancing function in the context of locally-balanced Markov chains Livingstone and Zanella (2022); Vogrinc et al (2022); Zanella (2020). Now, we note that φ 1 , φ 2 are homogeneous with degree p − 1, q − 1 respectively, and so φ 1 /φ 2 is homogeneous with degree p − q.…”

“…According to Livingstone and Zanella (2022); Vogrinc et al (2022); Zanella (2020), it suffices to show that…”

“…More recently, higher order multiplicative reversiblizations are used to define a pseudo-spectral gap and develop concentration inequalities of non-reversible Markov chains (Paulin, 2015), while the second Metropolis-Hastings reversiblization is proposed and investigated in Choi (2020); Choi and Huang (2020). The geometric mean reversiblization is analyzed in Diaconis and Miclo (2009) in continuous-time and in Wolfer and Watanabe (2021) in discrete-time, while the Barker proposal (which is in fact a harmonic mean reversiblization, see Section 3.3 and Section 5 below) has recently enjoyed considerable interest in the Markov chain Monte Carlo literature in a series of papers Vogrinc et al, 2022;Zanella, 2020).…”

“…In the classical Metropolis-Hastings algorithm, one takes in a target distribution π and a proposal chain with generator L to transform L into a generator that is reversible with respect to the target π. Thus, in a broad sense, sampling from π amounts to reversiblizing a given proposal generator L, and being able to generate new reversiblizations may inspire design of improved stochastic algorithms, see for instance the second Metropolis-Hastings generator in Choi and Huang (2020) or the Barker proposal in Livingstone and ; Vogrinc et al (2022); Zanella (2020). Third, new reversiblizations also give rise to new symmetrizations of non-symmetric and non-negative matrices or in general non-self-adjoint kernel operators.…”