Nonasymptotic analysis of adaptive and annealed Feynman–Kac particle models

Abstract

Sequential and quantum Monte Carlo methods, as well as genetic type search algorithms can be interpreted as a mean field and interacting particle approximations of Feynman-Kac models in distribution spaces. The performance of these population Monte Carlo algorithms is strongly related to the stability properties of nonlinear Feynman-Kac semigroups. In this paper, we analyze these models in terms of Dobrushin ergodic coefficients of the reference Markov transitions and the oscillations of the potential functions. Sufficient conditions for uniform concentration inequalities w.r.t. time are expressed explicitly in terms of these two quantities. We provide an original perturbation analysis that applies to annealed and adaptive Feynman-Kac models, yielding what seems to be the first results of this kind for these types of models. Special attention is devoted to the particular case of Boltzmann-Gibbs measures' sampling. In this context, we design an explicit way of tuning the number of Markov chain Monte Carlo iterations with temperature schedule. We also design an alternative interacting particle method based on an adaptive strategy to define the temperature increments. The theoretical analysis of the performance of this adaptive model is much more involved as both the potential functions and the reference Markov transitions now depend on the random evolution on the particle model. The nonasymptotic analysis of these complex adaptive models is an open research problem. We initiate this study with the concentration analysis of a simplified adaptive models based on reference Markov transitions that coincide with the limiting quantities, as the number of particles tends to infinity.

Fluctuation analysis of adaptive multilevel splitting

“…, ⌈Nα⌉} are i.i.d. random vectors (see equation (5.1) page 22), and that with similar arguments they are independent of the subsample strictly below L N q , it is clear that (Z N j ) 0≤j≤(p+1)(⌈N α⌉+1)−1 is a triangular array of martingale increments adapted to the filtration J . It is then straightforward to check that…”

“…As a consequence, even if the global goal here is roughly the same as in [16,3], the techniques developed for establishing our convergence results are quite different. Note also that in the context of adaptive tempering (a context considered in [3]), Giraud and Del Moral give non-asymptotic bounds on the error in [22].…”

On the convergence of adaptive sequential Monte Carlo methods

“…To construct the particle approximation η N n , the practical SMC algorithm exploits summary statistics ξ n : E n−1 → R d by reweighing and propagating the particle approximation η N n−1 through the potential G n,η N n−1 (ξn) and the Markov kernel M n,η N n−1 (ξn) . This is a substitute for the perfect algorithm (as also used by [16] and which cannot be implemented) which employs the Markov kernel M n,ηn−1(ξn) and weight function G n,ηn−1(ξn) . We prove a WLLN and a CLT for both the approximation of the probability distribution η n and its normalising constant.…”

“…Some preliminary results can be found, under exceptionally strong conditions, in [8,17]. Proof sketches are given in [12] with some more realistic but limited analysis in [16]. We are not aware of any other asymptotic analysis of these particular class of algorithms in the literature.…”

Particle Filtering for Stochastic Navier--Stokes Signal Observed with Linear Additive Noise

“…In Algorithm 2 we use m iterations of (20) with Q n,l specified as above. The corresponding m-iterate of the MCMC transition kernel is denoted as K m n,l and is presented in Algorithm 3 in an algorithmic form.…”