A Metropolis-class sampler for targets with non-convex support

Abstract

We aim to improve upon the exploration of the general-purpose random walk Metropolis algorithm when the target has non-convex support $$A\subset {\mathbb {R}}^d$$ A ⊂ R d , by reusing proposals in $$A^c$$ A c which would otherwise be rejected. The algorithm is Metropolis-class and under standard conditions the chain satisfies a strong law of large numbers and central limit theorem. Theoretical and numerical evidence of improved performance relative to random walk Metropolis are provided. Issues of implementation are discussed and numerical examples, including applications to global optimisation and rare event sampling, are presented.

Hopping between distant basins

“…In all simulations we set the perturbation q to be a spherically symmetric and Gaussian with standard deviation σ , although other choices are possible (see the discussion in Sect. 4.1 of [24]). In the next section we explore for which types of energy function f BH-S offers an advantage over BH, and also discuss the choice of the halting index.…”

“…Note that although in [24] the halting index K can be randomised, in the present setting with a known bounded domain D it is sufficient to consider only fixed halting indices.…”

“…In contrast, since non-local perturbation steps offer the possibility of direct moves between distant low-energy basins, they may possibly be effective on such surfaces within a monotonic optimisation algorithm. In this paper we explore the use of a particular non-local perturbation, the 'skipping perturbation' of [24].…”

“…Like a flat stone skimming across water, this involves repeated perturbations in a straight line until either a point of lower energy is found, or the skipping process is halted. The BH-S algorithm, which was first outlined in [24], thus provides a direct mechanism to escape local minima which contrasts with the indirect approach taken by BH. Another perspective is that BH-S alters the balance between the computational effort expended on local optimisation versus the effort spent on perturbation, typically increasing the latter while decreasing the former (cf.…”