Fluctuations, stability and instability of a distributed particle filter with local exchange

Abstract

We study a distributed particle filter proposed by Bolić et al. (2005). This algorithm involves m groups of M particles, with interaction between groups occurring through a "local exchange" mechanism. We establish a central limit theorem in the regime where M is fixed and m → ∞. A formula we obtain for the asymptotic variance can be interpreted in terms of colliding Markov chains, enabling analytic and numerical evaluations of how the asymptotic variance behaves over time, with comparison to a benchmark algorithm consisting of m independent particle filters. We prove that subject to regularity conditions, when m is fixed both algorithms converge time-uniformly at rate M −1/2 . Through use of our asymptotic variance formula we give counter-examples satisfying the same regularity conditions to show that when M is fixed neither algorithm, in general, converges time-uniformly at rate m −1/2 .

Parallelizing particle filters with butterfly interactions

“…One of the most important earlier contributions to the design of parallel SMC algorithms is (Bolić et al, 2005) which introduced a modification of the BPF whereby the particle interactions are constrained by allowing the m PEs to exchange subsets of particles according specific local schemes. The theoretical properties of these popular local exchange particle filters (LEPF) was further investigated in (Míguez, 2007(Míguez, , 2014Míguez and Vázquez, 2015;Heine and Whiteley, 2016). The analysis of Míguez (2014); Míguez and Vázquez (2015) proved that under specific assumptions, the LEPF was uniformly convergent in time as m Ñ 8, but interestingly, in addition to the central limit theorem for the LEPF, it was shown in (Heine and Whiteley, 2016) that under some regularity assumptions, LEPF cannot be uniformly convergent in mean of order r ě 1 at rate m´1 {2 .…”

“…The theoretical properties of these popular local exchange particle filters (LEPF) was further investigated in (Míguez, 2007(Míguez, , 2014Míguez and Vázquez, 2015;Heine and Whiteley, 2016). The analysis of Míguez (2014); Míguez and Vázquez (2015) proved that under specific assumptions, the LEPF was uniformly convergent in time as m Ñ 8, but interestingly, in addition to the central limit theorem for the LEPF, it was shown in (Heine and Whiteley, 2016) that under some regularity assumptions, LEPF cannot be uniformly convergent in mean of order r ě 1 at rate m´1 {2 . Whether the time uniform convergence holds at any slower rate remains an open question.…”

On the performance of parallelisation schemes for particle filtering

“…The scheme termed distributed resampling with nonproportional allocation (DRNA) for the parallelisation of PFs was originally introduced in [2] (Section IV.A.3), but it has been only recently that a theoretical characterisation of its performance has been obtained [10,11,26]. The same as in Section 3.1, assume that we have a budget of K = MN particles, which are split into M subsets with N particles each.…”

“…Then, we describe and provide a basic convergence result for the standard PF and proceed to describe four different approaches to its parallelisation: the simple averaging of statistically independent (i.e. non interacting) low-complexity PFs, the method based on the distributed resampling with non-proportional allocation (DRNA) procedure of [2,10,11], the particle island model of [13,14] and the adaptive interaction scheme termed α-sequential Monte Carlo (α-SMC) in [12].…”

Convergence Properties of Weighted Particle Islands with Application to the Double Bootstrap Algorithm

“…Parallel implementation of SMC was first proposed in [2] in the form of an algorithm referred to as the local exchange particle filter (LEPF), in which groups of particles are spread across computational units. This algorithm was later improved in [1] (see also [21] where a detailed convergence analysis of the LEPF is carried out). As indicated by the almost 300 Google Scholar citations at the time of writing, the LEPF has triggered a substantial interest in parallelization of SMC.…”