Robustesse des techniques de Monte Carlo dans l analyse d événements rares

Transcription

1 Institut National de Recherche en Informatique et Automatique Institut de Recherche en Informatique et Systèmes Aléatoires Robustesse des techniques de Monte Carlo dans l analyse d événements rares H. Cancela, Univ. de la Republica, Montevideo G. Rubino, B. Tuffin, IRISA-INRIA, Rennes btuffin@irisa.fr ACI Sure-Paths, Oct

2 Robustness of estimators in rare event simulation ACI Sure-Paths, Oct Outline 1 Generalities : ARMOR s work on Monte Carlo Standard Monte Carlo Rarity Bounded Relative Error (BRErr) Bounded Normal Approximation (BNA) Need for an extension of BRErr : BREff Illustrations Generalized Bounded Normal Approximation Conclusions /31

3 Generalities : ARMOR s work on Monte Carlo ACI Sure-Paths, Oct Generalities : ARMOR s work on Monte Carlo focus on the rare event case static and dynamic (Markov) models in the dynamic case, transient and stationary analysis algorithm design and generic analysis of Monte Carlo techniques 1/31

4 Generalities : ARMOR s work on Monte Carlo ACI Sure-Paths, Oct More precisely : (i) (static) reliability analysis (network reliability area), (ii) transient analysis of Markov models : estimation of the reliability at time t, (iii) transient analysis of Markov models : Importance Sampling applied to estimating the MTTF of a system, (iv) theoretical analysis of properties of Monte Carlo evaluation methods : illustration in the case of static models. and variations around (performability,... ). 2/31

5 Generalities : ARMOR s work on Monte Carlo ACI Sure-Paths, Oct Example of (i) : a communication network nodes are perfect, lines (edges) can fail lines are up or down, independently nodes 1 and 20 must talk to each other network reliability R is the probability that there is at least, a path up between 1 and /31

6 Generalities : ARMOR s work on Monte Carlo ACI Sure-Paths, Oct Example of (ii), (iii) : a multi-component system (say, a database), K types of components (disks, power units, servers,... ), type-k components fail with rate λ k, possible failure propagations reparations following an exponential distribution system is up iff at least n k type-k components are, we get a Markov model, homogeneous, finite, with states classed up or down. 4/31

7 Generalities : ARMOR s work on Monte Carlo ACI Sure-Paths, Oct An interesting metric is MTTF = mean time to system down. A more detailed metric is the reliability at time t, R(t) = Pr(system is up from time 0 to time t). 5/31

8 Standard Monte Carlo ACI Sure-Paths, Oct Standard Monte Carlo Consider a real random variable X and a real function ψ(). Let γ = E(ψ(X)) and σ 2 = Var(ψ(X)). The standard estimator of γ : build n independent copies X 1,, X n of X return γn STD = 1 n ψ(x i ). n E(γ STD n ) = γ, i=1 Var(γ STD n ) = σ 2 /n 6/31

9 Standard Monte Carlo ACI Sure-Paths, Oct A centered confidence interval for γ with (confidence) level δ is ( ) C STD n = γn STD ± z δ Var(γn STD ) the meaning is Pr(C STD n γ) = δ factor z δ is where N(x) = 1 2π z δ = N 1 ( 1 + δ 2 x 0 ) exp( u 2 /2)d u. (z δ is the 1 δ/2 quantile of the std. normal distr.)., 7/31

10 Rarity ACI Sure-Paths, Oct Rarity Rarity happens when γ 1. Examples : In the network reliability problem, X = 1 {chosen nodes can not communicate}, and ψ(x) = x. (That is, γ = 1 R). When network components are very reliable, γ 1. 8/31

11 Rarity ACI Sure-Paths, Oct When estimating MTTF in a Markov model, we first write MTTF = E(min{U, V }) Pr(U < V ), where U is the r.v. time to return to 0, V is the r.v. time to absorption, and 0 is the initial state of the chain, assumed to be an up state. Focus on estimating the denominator, the only difficult component of the formula : γ = Pr(U < V ) and we can put this into the general framework as before. If the system is highly reliable, again γ 1. 9/31

12 Rarity ACI Sure-Paths, Oct Rarity parameter We assume that X depends on some real parameter ε (and, thus, γ, σ are also functions of ε). We call ε a rarity parameter because it verifies lim γ = 0. ε 0 Example : in the static model, we can assume that the reliability of line i, r i, has the form r i = 1 a i ε b i. where a i and b i are positive constants. 10/31

13 Rarity ACI Sure-Paths, Oct In this case, we can prove that there exists some r > 0 such that γ = Θ(ε r ). If we look at the relative error when estimating γ using the standard estimator γ STD lim ε 0 z δ n, Var(γ STD n ) γ = z δ n lim ε 0 1 γ γ =. This is why, in general, we can not use γ STD n events. in case of rare 11/31

14 Bounded Relative Error (BRErr) ACI Sure-Paths, Oct Bounded Relative Error (BRErr) Consider now any estimator γ n of γ (for instance, an estimator built using some Importance Sampling method). Assume E(γ n ) = γ (that is, γ n is unbiased) and denote σn 2 = Var(γ n ). We can again build a centered confidence interval for γ from γ n, writing C n = (γ n ± z δ σ n ) The relative error is RErr = z δ σ n γ. 12/31

15 Bounded Relative Error (BRErr) ACI Sure-Paths, Oct We say that we have a bounded relative error (BRErr) if RErr remains bounded as ε 0. This desirable property states that the relative size of the confidence interval remains bounded as ε 0. In the case of the standard estimator, σ 2 n = γ(1 γ)/n, and σ n 1 γ RErr = z δ γ = z δ nγ when ε 0. 13/31

16 Bounded Relative Error (BRErr) ACI Sure-Paths, Oct Asymptotic Optimality Often used in queuing theory when γ n comes from the Importance Sampling method. γ = E f [g(x)] = E f [g(x)l(x)] where L(x) = f(x)/f (x). ˆγ IS n is called asymptotically optimal, if lim ε 0 lne f [g(x) 2 L(X) 2 ] lnγ = 2. Bounded Relative Error implies Asymptotic Optimality (recently proved by Sandmann), the reciprocal being false. Bounded Relative Error appears to be the right property. 14/31

17 Bounded Normal Approximation (BNA) ACI Sure-Paths, Oct Bounded Normal Approximation (BNA) Berry-Esseen : if F n () is the cdf of (γ n γ)/ˆσ n, where ˆσ 2 n is the standard estimator of σ 2, we have that for all real x, with ϱ = E( ψ(x) 3 ). F n (x) N(x) aϱ, σn 3 n The estimator γ n has Bounded Normal Approximation iff the ratio ϱ/σ 3 n remains bounded when ε 0 Proved that BNA implies BRErr, the reciprocal being false. This means again that BNA appears as the right property to look for. 15/31

18 Need for an extension of BRErr : BREff ACI Sure-Paths, Oct Need for an extension of BRErr : BREff What is important in simulation? the RErr for a given simulation time, not for a number n of replications. BRErr does not incorporate the second important characteristic of an estimator : simulation time (the computational cost). The average simulation time to get one replication can increase with ε, or decrease with ε. We will illustrate next this last case for a simulation method estimating the reliability of a static stochastic network. 16/31

19 Need for an extension of BRErr : BREff ACI Sure-Paths, Oct Illustration : simulation for static reliability analysis Consider the network reliability problem. Denote by G the undirected graph modelling the network, made of M links. Computing R, the probability that two fixed nodes s and t are connected, is an NP-hard problem. The method : Let P = {P 1,P 2,,P H } be a set of elementary (disjoint) paths connecting nodes s and t, Let π h be the event all links of path P h work and p h = Pr(π h ). Assume an infinite sequence of independent copies of G is built. Let F be the random variable first element in the sequence where every path in P has at least one link that does not work. 17/31

20 Need for an extension of BRErr : BREff ACI Sure-Paths, Oct Variable F is geometrically distributed with parameter q = H h=1 (1 p h) = Pr(no path in P works ) : and, in particular, Pr(F = f) = (1 q) f 1 q, E(F) = 1 q. Idea : sample first from the geometric distribution of F. The estimator is then built assuming that in the first F 1 copies, nodes s and t are connected. Assume F = f. To know if they are connected in the f th copy, we must sample the network conditioning on the fact that at least one edge in each path is down. 18/31

21 Need for an extension of BRErr : BREff ACI Sure-Paths, Oct Consider a path P h. Call W h the r.v. giving the first failed edge of P h. Write P h = (i h,1,i h,2,,i h,kh ). We have Pr(W h = w) = r i h,1 r ih,2 r ih,w 1 (1 r ih,w ) 1 r ih,1 r ih,2 r ih,kh. Assume W h = w. Edges i h,1,i h,2,,i h,w 1 are set to up, edge i h,w to down. Remaining edges in P h are sampled from their a priori Bernoulli distributions. We have built (virtually) f copies of G by sampling F once and the network configuration (the states of the M links) once. 19/31

22 Need for an extension of BRErr : BREff ACI Sure-Paths, Oct Observe that We have unbounded RErr (same variance than for crude Monte Carlo), We sample F and the network, on the average, nq (= n/(1/q)) times, and we test the connectivity between s and t also nq times on the average. The simulation time (or computing cost) is, on the average, O(nq(M + K K H )) (= O(nqM) if we wish). It decreases with ε if r i = 1 a i ε b i, since q decreases with ε. 20/31

23 Need for an extension of BRErr : BREff ACI Sure-Paths, Oct Bounded relative Efficiency It basically addresses the (relative) variance of an estimator obtained during a given simulation time. Consider an estimator γ n of γ, with variance σn 2, built from n replications (possibly dependent), and denote by t n the average simulation time needed to get these n replications. The relative efficiency of γ n is REff = γ2 σ 2 nt n. The estimator γ n has bounded relative efficiency (BREff) if there exists d > 0 such that REff is minored by d for all ε. 21/31

24 Need for an extension of BRErr : BREff ACI Sure-Paths, Oct Sufficient condition on the static reliability example Let r > 0 be the real such that γ = Θ(ε r ). P h P, let P h = (i h,1,, i h,kh ) and b h = min 1 k Kh b ih,k (b h is the exponent of ε in the most reliable edge of P h (as ε 0)). Theorem : the estimator of the static unreliability described in previous section verifies Bounded Relative Efficiency if H h=1 b h r. 22/31

25 Illustrations ACI Sure-Paths, Oct Illustrations Illustration on a (very) small problem s 1 ε P2 v P1 1 ε 1 ε t γ = 1 R = ε 3 + 2ε 2 (1 ε) 2ε 2 σ 2 = γ(1 γ) 2ε 2 σ/γ 1/( 2ε) : no BRErr p 1 = 1 ε, p 2 = (1 ε) 2 q = (1 p 1 )(1 p 2 ) 2ε 2 ; t n proportional to q REff γ2 σ 2 q bounded : BREff. 23/31

26 Illustrations ACI Sure-Paths, Oct Results on the simple topology, with a number of replications fixed to n = 10 4 : r i i Est. Conf. interval RErr KS stat e-01 ( e-01, e-01) e e e-02 ( e-02, e-02) e e e-04 ( e-05, e-04) e e (0, 0) (0, 0) 1 Results for a simulation time T = 10 seconds : r i i Est. Conf. interval RErr e-01 ( e-01, e-01) e e-02 ( e-02, e-02) e e e-04, e-04) e e-06 ( e-06, e-06) e e e-08, e-08) e-03 24/31

27 Illustrations ACI Sure-Paths, Oct Illustration : efficiency on the dodecahedron topology with s = 1 and t = 20 r i i Relative eff. w.r.t. crude MC /31

28 Illustrations ACI Sure-Paths, Oct r i i Est. Conf. interval RErr KS stat e-01 ( e-01, e-01) e e e-03 ( e-03, e-03) e e (0, 0) e (0, 0) 1 Results for a simulation time T = 5 seconds : r i i Est. Conf. interval RErr KS stat e-01 (7.0918e-01, e-01) 3.98e e e-03 (2.7776e-03, e-03) 3.87e e e-06 (1.9211e-06, e-06) 6.41e e e-09 (1.8705e-09, e-09) 6.69e e e-12 (1.8474e-12, e-12) 6.75e e-02 26/31

29 Generalized Bounded Normal Approximation ACI Sure-Paths, Oct Generalized Bounded Normal Approximation Again, in BNA, time (cost) not taken into account. Fix time (computational budget) T and let n = n(t) be the corresponding average number of of iterations. The estimator γ n(t) verifies GBNA if the ratio ϱ n(t) /(σ 3 n(t) n(t)) remains bounded when ε 0. It can be proved now that GNBA implies BREff and that in the case of the static reliability problem, both properties are equivalent. 27/31

30 Generalized Bounded Normal Approximation ACI Sure-Paths, Oct Coverage function Confidence interval R(η, X) for γ, at confidence level η with (random) data X. We should have Pr[γ R(η, X)] = η. If η = inf{η [0, 1] : γ R(η, X)}, then, η should be uniformly distributed : F η (η) = Pr[η η] = η. F η (η) is the actual coverage level : empirical estimation. 28/31

31 Generalized Bounded Normal Approximation ACI Sure-Paths, Oct Illustration on the simple topology 1 with eps=0.5 with eps=0.9 with eps=0.99 with eps= with eps=0.5 with eps=0.9 with eps=0.99 with eps=0.999 with eps= Obtained confidence level Obtained confidence level Desired confidence Level Desired confidence Level 29/31

32 Generalized Bounded Normal Approximation ACI Sure-Paths, Oct Illustration on the Dodecahedron topology 1 with eps=0.5 with eps=0.9 with eps= with eps=0.5 with eps=0.9 with eps=0.99 with eps=0.999 with eps= Obtained confidence level Obtained confidence level Desired confidence Level Desired confidence Level 30/31

33 Conclusions ACI Sure-Paths, Oct Conclusions Rare event simulation requires sophisticated techniques with robustness properties. Bounded Relative Error does not take into account the estimator s full information. We have defined Bounded Relative Efficiency to cope with this problem defined Generalized Bounded Normal Approximation (and previously BNA) Illustrated the validity of the approach on a reliability analysis problem. deep investigation of the coverage of the confidence interval 31/31