Stability of Feynman-Kac Semigroup and SMC Parameters' Tuning
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1 Stability of Feynman-Kac Semigroup and SMC Parameters' Tuning François GIRAUD PhD student, CEA-CESTA, INRIA Bordeaux Tuesday, February 14th 2012
2 1 Recalls and context of the analysis Feynman-Kac semigroup SMC algorithm Quantities of interest 2 Parameters' tuning Estimation of q p,n and b p,n Examples of tuning and consequences 3 The particular case of Gibbs measures Associated FK structure Dobrushin analysis Parameters β p and m p Concentration inequality
3 Feynman-Kac semigroup Feynman-Kac formulae Let us x (E, E) a measurable state space. A sequence of measures (η n) on E admits a FK structure associated with potentials g n : E R + and Markov kernels M n : E E [0, 1] if n, η n = φ n(η n 1) with, for all measure µ on E and bounded function f : E R, or in a equivalent way : φ n(µ)(f ) = µ(gn Mn.f ) µ(g n) φ n(µ) = ψ gn (µ).m n FK semigroup so that φ p,n(η p) = η n. φ p,n = φ n... φ p+1
4 Feynman-Kac semigroup Structure of φ p,n The composed function φ p,n admits a structure similar to that of each function φ p, involving some selecting function g p,n and some Markov kernel M p,n, so that : that is to say : φ p,n(µ)(f ) = given by the backward formulae : µ(gp,n Mp,n.f ) µ(g p,n) φ p,n(µ) = ψ gp,n (µ).m p,n g n,n = 1 g p 1,n = g p.m p.g p,n M n,n = Id M p 1,n.f = Mp.(gp,n.Mp,n.f ) M p.g p,n
5 SMC algorithm Sequential Monte Carlo (SMC) SMC algorithm consists in approximating a theoretical FK sequence (η n) by a large cloud of random samples termed particles (ζ k n ) 1 k N E N dening at each generation n the occupation distribution : η N n = 1 N NX k=1 δ ζ n k We run from generation (ζ k n 1) to generation (ζ k n ) through a selection step using positive function g n on E, and a mutation step, using Markov kernel M n. ζ 1 ṇ. ζ i ṇ. ζ N n 2 3 bζ n 1 S n,η N n. bζ n i bζ n N M n+1 ζ 1 n+1. ζ i n+1. ζ N n+1 with the selection Markov transition : S η N (x, dy) = εg n 1 n(x).δ x(dy) + (1 εg(x))ψ gn η N n 1 (dy) 3 7 5
6 SMC algorithm Natural decomposition of the error nx ηn N η n = φ p,n(ηp N ) φ p,n φ p(ηp 1) N p=0 No explosion in time n under stability conditions on φ p,n.
7 Quantities of interest Dobrushin coecient To each Markov kernel K on E, is associated its Dobrushin coecient β(k) [0, 1] dened by : or in an equivalent way : β(k) = Sup{K(x, A) K(y, A); x, y E, A E} β(k) = Sup{ µ.k ν.k tv µ ν tv ; µ, ν P(E), µ ν} β satises β(k 1.K 2) β(k 1).β(K 2), for any kernels K 1, K 2, and by denition, for any measures µ, ν on E and any Markov kernel K, one have : µ.k ν.k tv β(k). µ ν tv.
8 Quantities of interest "calculable" quantities of interest We denotes : b p = β(m p) q p = Sup x,y E g p(x) g p(y) "theoretical" quantities of interest b p,n = β(m p,n) q p,n = Sup x,y E g p,n(x) g p,n(y) a total variation stability result For any measures µ, ν on E, and any integers p n : φ p,n(µ) φ p,n(ν) tv q p,n.b p,n µ ν tv This simple result already highlights the product q p,n.b p,n as being a quantity representing the semigroup's stability between transformations p and n.
9 Quantities of interest L 2 mean error For any function f, uniformly bounded by 1 on E : A concentration inequality ηn N (f ) η n(f ) 2 1 N For any bounded by 1 function f and any ε > 0 : nx q k,n b k,n 1» «1 N log P ηn N (f ) η n(f ) rn N + ε ε2 bnβ 2rn 2 n + + ε 2r n + b n N 3 where r n, β n and b n are constants so that : 8 >< k=0 r n P n p=0 4q3 p,nb p,n β n 2 P n p=0 4q2 p,nb 2 p,n >: bn sup 2q p,nb p,n 0 p n
10 Estimation of qp,n and bp,n Estimation of q p,n For any integers p n : nx q p,n 1 (q k 1)b p+1...b k 1 k=p+1 Estimation of b p,n For any integers p n, ny b p,n b k q k,n k=p+1
11 Examples of tuning and consequences Theorem If q p are bounded by constant M, then b p a (for any a (0, 1)) ensures M+a L p mean error bound : η N n (f ) η n(f ) p Bp N 1 1 a and concentration inequality : P ηn N (f ) η n(f ) r «1 N + r 2 y e y N 2
12 Examples of tuning and consequences Theorem If q p tends to 1 (decreasingly), then if b p satises p 1, (with α = b p b p qp α 1 qp α+1 a 1 p + a qp α+1 p + a a 1 a ), we still have the following Lp mean error bound : η N n (f ) η n(f ) p Bp N 1 1 a and the concentration inequality : P ηn N (f ) η n(f ) r «3 N + r 4 y e y N 2
13 Associated FK structure Particular case of Gibbs measures Let us x a potential V : E R and a strictly increasing sequence of "temperatures" β n so that β n +. n + η n(dx) = µ βn (dx) = 1 Z βn e βnv (x) m(dx) with m a reference measure. (Gibbs measures) g p(x) = e (βp β p 1)V (x) and then q p = e osc(v ) p, with p = β p β p 1 M n satisfying η n.m n = η n (for example the Simulated-annealing kernel) SMC = Interacting simulated annealing.
14 Dobrushin analysis Simulated annealing kernel K β, Dobrushin estimation temperature β ; proposition kernel K(x, dy) In general, this kernel doesn't have any mixing property in the sense of Dobrushin. However, under the assumption K k 0 (x,.) δν(.) for some integer k 0,, some measure ν and some δ > 0, we can show that : β(k k 0 β ) 1 δe β V (k 0) To obtain suitable mixing properties, we choose M p = K k 0.m p β p.
15 Parameters βp and mp Tuning of β p and m p For all b ]0, 1[, condition b p b is then turned into 1 δe βp V (k 0) mp b, which can also be written : m p log( 1 b )e V (k 0).β p δ
16 Concentration inequality Theoretical Gibbs measure's concentration on V 's global minimizer ε > 0, ε > 0 so that 0 < ε < ε, η n (V V min + ε) e ε.βn m ε SMC's concentration on V 's global minimizer Taking f = 1 {V Vmin +ε} in the general concentration inequality and noticing that η N n.f then designates the proportion p N n (ε) of particules (ζ i n) satisfying V (ζ i n) V min + ε, we obtain : P pn N (ε) e ε.βn + r i N + r m ε N 2 j y! e y.
17 Concentration inequality Thank you for your attention!
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