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

Abstract

We study the class of first-order locally-balanced Metropolis-Hastings algorithms introduced in [9]. 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 universal 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 optimise 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.

Particle algorithms for maximum likelihood training of latent variable models

“…Furthermore, much like in [20], we circumvent the degeneracy with latent-variable dimension that plagues common MCMC methods (e.g. see [5,65,39,40] and references therein) by avoiding accept-reject steps and employing ULA kernels (known to have favourable properties [18,25,26]). Lastly, computations within our algorithms are easily vectorized across particles: a significant boon in practice (Sect.…”

“…The methods have one practical downside that limits their scalability: similarly as with standard Metropolis-Hastings algorithms (e.g. see [5,65,39,40] and references therein), the acceptance probability degenerates with increasing latent variable dimensions D x and the particle numbers N . This, in turn, forces us to choose small step sizes h for large D x and N , which leads to slow convergence.…”

“…The mean-field (N → ∞) limit of PGA ( 9) is (65,66), that of PQN (9RHS,13) is (66,67), and that of PMGA ( 15) is (68):…”

Metropolis Adjusted Langevin Trajectories: a robust alternative to Hamiltonian Monte Carlo

“…However their lack of robustness to tuning is often a major issue for their practical implementations. The challenge of developing gradient-based samplers that combine high dimensional efficiency and robustness to tuning has received a particular interest lately; see [11,38,45,75]. The focus of our work is motivated by this main challenge, and falls under the continuity of these recent studies.…”

“…Note that since the function t → 1 ∧ e t is 1-Lipschitz, Proposition 9 guaranteesP U is between 1 ∧ e − d j=1 ∆ h,j and 1 ∧ e − d j=2 ∆ h,j ≤ E 1 ∧ e − d j=1 ∆ h,j − 1 ∧ e − d j=2 ∆ h,j ≤ E [|∆ h |] d→∞ −−−→ 0 , Since f is bounded this implies E f (x L (1))1 [f (x L (1))] P U ≤ 1 ∧ e − d j=2 ∆ h,j + lim d→∞ E [f (x 0 (1))] P U > 1 ∧ e − d j=2 ∆ h,j = E [f (X T )] a( ) + E [f (X 0 )] (1 − a( )) = E [f (X T )] .The first equality holds by design of MALT accept-reject mechanism, the third by independence of the coordinates of MALT trajectory and the fourth by Proposition 3 and point (i).Proof of Proposition 7. Mimicking the argument of[75, Theorem 3] we will first establish thatE f (X n+1 (1)) − f (X n (1)) E (f (x L (1)) − f (x 0 (1))) 2 1 ∧ e − d j=1 ∆ h,j≤ 2E (f (x L (1)) − f (x 0 (1))) e − d j=2 ∆ h,j (56). …”