Large-deviation theory and coverage in mobile phone networks

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics Large-deviation theory and coverage in mobile phone networks Paul Keeler, Weierstrass Institute for Applied Analysis and Stochastics, Berlin joint work with Christian Hirsch, Benedikt Jahnel, and Robert Patterson. Second half of talk written by Christian Hirsch. Melbourne, August 17th, 2015

2 1 Introduction to large deviations theory 2 SINR definition and model description 3 LDP for averages of connectable receivers in large domains 4 LDP for connectable receivers for small SINR threshold 5 Importance sampling for rare-event probability estimation LDT in phone networks August 17th, 2015 Page 2

3 Large deviations theory (LDT): overview Simple idea: for sample means, the probability distributions of large deviations away from the (true) mean often contain exponential terms LDT bridges the law of large numbers and central limit theorem, which is concerned with small deviations around the mean Statistical physics can often be reformulated as applications of large deviation theory acccording to Touchette (2009) so some ideas go back to Boltzmann. Cramér (1938) derived an early result of LDT known as Cramér s theorem. Connection between LDT and entropy by Sanov. Much theory developed (independently) by Donsker and Varadhan (in the US) and Freidlin and Wentzell (in the USSR) say Dembo and Zeitouni (2009) Mathematics of LDT soon becomes quite abstract and technical S. R. Srinivasa Varadhan, a key figure, won the 2007 Abel prize. Analysis and topology knowledge required for a rigorous treatment, leading to a jungle of deltas and epsilons to establish results Key ideas can be illustrated with well-behaved random variables LDT in phone networks August 17th, 2015 Page 2

4 Simple examples with random variables Collection of i.i.d. random variables {T i } and their sample or empirical mean S n = 1 n T i n If T n are Gaussian with mean µ and standard deviation σ, then the sample mean s probability density p(s) behaves asymptotically (of course, the S n is i=1 another Gaussian, but we neglect the subdominant n term here) p(s) e nj(s) (s µ)2, J(s) = 2σ 2, s R where we make the symbol more clear soon. If T n are exponential with mean 1/µ then the sample mean s probability density p(s) behaves asymptotically (S n is a gamma/erlang random variable) p(s) e nj(s), J(s) = s µ 1 log s µ, s > 0 Rate function of binomial variable with parameter p is the relative entropy of two coins with success probability s and p ie J(s) = s log( s (1 s) p ) + (1 s) log (1 p). J is called the rate function both positive and convex in these examples LDT in phone networks August 17th, 2015 Page 3

5 Large deviation principle (lite or diet version) To show that such exponential behaviour exists is to show a large deviation principle (LDP) exists A n is a random variable indexed by parameter n, assumed to be large P(A n B) is the probability that A n takes on a value in some set (Borel) B Definition (Large deviation principle greatly simplified) P(A n B) satisfies a large deviation principle with rate J B if the following limit exists 1 lim n n log P(A n B) = J B. This is what we mean by P(A n B) e nj B, or with small-o notation log P(A n B) = nj B + o(n) If the above limit doesn t exist, one can sometimes still derive bounds such as e nj B LDT in phone networks August 17th, 2015 Page 4 P(An B) e nj + B

6 Large deviation principle (more general) Definition (Rate function) For a metric space M, a function J : M [0, ] is called a rate function if it is lower semincontinuous, which means the levels sets {x M : J(x) a) are closed for any a 0. Moreover, a rate function is called a good rate function if the levels sets are compact for any a 0. Definition (Large deviation principle) A sequence of random variables T 1, T 2,... (with values in a metric space) satisfy a large deviation principle with speed a n and rate function J if, for all Borel sets B M, the following holds lim sup n lim inf n 1 log P(T n B) inf J(x) a n x clb (1) 1 log P(T n B) inf J(x) a n x intb (2) LDT in phone networks August 17th, 2015 Page 5

7 Cramér s theorem For a random variable T, define the cumulant generating function Theorem (Cramér s theorem) φ(λ) := log Ee λt <, λ R (3) Let T 1, T 2,... be i.i.d. random variables with mean µ satisfying (3) and S n be their sample mean. Then for any x > µ, we have 1 lim n n log P(S n > x) = φ (x), where φ (x) is given by the Legendre(-Fenchel) transform of φ, namely φ (x) := sup[λx φ(λ)] λ R An important generalization is the Gärtner-Ellis theorem: if φ(λ) exists and is differentiable for all λ R, then a LDP exists with rate function J(x) = φ (x) LDT in phone networks August 17th, 2015 Page 6

8 Empirical measure and Sanov s theorem Large deviation principles can be established for more descriptive objects than sample means Assume T 1, T 2,... are i.i.d random variables taking values in a finite set T, and define the empirical measure of a sample vector T = (T 1,..., T n ) by L T n (t) := 1 n 1[T i = t] = 1 n δ Ti (t), t T n n i=1 (interpreted as a random element of the space of probability measures on T ) Theorem (Sanov s theorem) Assume T 1, T 2,... as above and denote their distribution by µ. Then the empirical measures L T n satisfy a large deviation principle with speed n and a good rate function J given by the relative entropy (or Kullback-Leibler divergence) J(ν) = h(ν µ) := t T i=1 ν(t) log ν(t) µ(t) LDT in phone networks August 17th, 2015 Page 7

9 Empirical processes and LDP levels Large deviation principles can be applied to a generalization of an empirical measure: the empirical pair measure L T n,2(t, v) := 1 n 1[T i 1 = t, T i = v], (t, v) T T n i=1 ( For symmetry, on sets T 0 := T n ) Infinite-dimensional version of the empirical pair measure is called an empirical process, introduced by Donsker and Varadhan Levels of large deviation principles: Level 1 for sample/empirical means, Level 2 for empirical measures, and Level 3 for empirical processes Large deviation frameworks have been adapted for various random objects eg for marked point processes with Level-3 results by Georgii and Zessin (1993) LDT in phone networks August 17th, 2015 Page 8

10 Applications and importance sampling Applications: Statistical physics (eg Ising model, Curie-Weiss-Potts model) as the number of particles n approaches infinity (ie thermodynamic limit), with a focus on Gibbsian systems Used to study Markov processes and stochastic differential equations Risk and finance (remember Cramér worked as an actuary for many years) Information theory (which can considered as a generalization of certain parts of statistical physics) Speed up stochastic or Monte Carlo simulations by importance sampling Importance sampling: Large deviation principles gives a more formed way to perform stochastic or Monte Carlo simulations Studying a rare event by simulation takes (by definition) a long time. Change the probability measure so the events occurs more often (compare to change of variables for an integral) LDT in phone networks August 17th, 2015 Page 9

11 Signal-to-noise-plus-interference-ratio (SINR) USEFUL SIGNAL RECEIVED POWER SINR = ALL OTHER (ie INTERFERING) SIGNALS RECEIVED POWER + NOISE Network transmitters located at {X i } in R 2 and a user at the origin o. R i is the received power of a signal from a transmitter at X i R 2. SINR in relation to a transmitter at X i SINR(X i, o) = R i. (4) R j j i SINR is motivated by information theory; eg maximize SINR(X i, o) Propagation model: R i = F i,j l( X i X j ), l(r) is a decreasing function eg l(r) = min[1, r α ] where α > 2, Assume F i,j are i.i.d. random variables representing propagation effects such as multi-path fading and/or shadowing ie signals colliding with obstacles. LDT in phone networks August 17th, 2015 Page 10

12 SINR-based network model Homogeneous Poisson point process of transmitters Homogeneous Poisson point process of receivers Connectable receivers associated with each transmitter Goal: exponential decay of probability of unlikely configurations eg large proportion of isolated transmitters LDT in phone networks August 17th, 2015 Page 11

13 SINR-based network model Homogeneous Poisson point process of transmitters Homogeneous Poisson point process of receivers Connectable receivers associated with each transmitter Goal: exponential decay of probability of unlikely configurations eg large proportion of isolated transmitters LDT in phone networks August 17th, 2015 Page 11

14 SINR-based network model Homogeneous Poisson point process of transmitters Homogeneous Poisson point process of receivers Connectable receivers associated with each transmitter Goal: exponential decay of probability of unlikely configurations eg large proportion of isolated transmitters LDT in phone networks August 17th, 2015 Page 11

15 Network model Network participants X = homogeneous Poisson point process of transmitters in R d ; intensity λ T Y = homogeneous Poisson point process of receivers in R d ; intensity λ R SINR model: for X i X and Y j Y, define the SINR as SINR(X i, Y j ) = SINR(X i, Y j, X) = F Xi,Y j l( X i Y j ) w + k i F X k,y j l( X k Y j ) w = constant thermal noise l(r) = min{1, r α } path-loss function with path-loss exponent α > d Compact support {F x,y } x X, y Y = i.i.d. fading variables Compact support Distribution q of Fx,y 1 is globally Lipschitz LDT in phone networks August 17th, 2015 Page 12

16 1 Introduction to large deviations theory 2 SINR definition and model description 3 LDP for averages of connectable receivers in large domains 4 LDP for connectable receivers for small SINR threshold 5 Importance sampling for rare-event probability estimation LDT in phone networks August 17th, 2015 Page 13

17 Reachable receivers For τ > 0, define connectable receivers Y (i) associated with transmitter X i Y (i) = {Y j Y : SINR(X i, Y j ) τ} Possible applications Device-to-device (D2D) networks ie mobile phones communicate directly to each other First step in a multi-hop network Goal: exponential decay of probability that an atypically large proportion of transmitters suffers from some frustration event Y (i) = Isolation #Y (i) 1 At most 1 connectable receiver Y (i) B ε (X i ) Impossibility of long hops LDT in phone networks August 17th, 2015 Page 14

18 Empirical measure of connectable receivers Empirical measure of connectable receivers L n = 1 Λ n X i Λ n δ Y (i) X i, Λ n = [ n/2, n/2] d For example, L n ({ }) = 1 Λ n #{X i Λ n : X i isolated} For Q P θ any jointly stationary point process of transmitters and receivers, let Q ( ) = E#{X i Λ 1 : Y (i) X i } be the Palm mark measure of {(X i, Y (i) X i )} i 1 For a marked point process on Λ n = [ n/2, n/2] d, the specific relative entropy / Kullback-Leibler divergence is 1 ( h(q P) = lim n Λ n E dq Λn P log dq ) Λ n. dp Λn dp Λn LDT in phone networks August 17th, 2015 Page 15

19 Large deviation principle for the empirical measure Theorem (Level-2 result) The random measures {L n } n 1 satisfy an LDP in the τ -topology with speed Λ n and good rate function Loosely speaking, I(Q) = inf h(q P). Q P θ Q =Q P(L n F ) exp( Λ n inf Q F I(Q)). τ -topology (finer than the usual weak topologies) is generated by maps B µ(b), where B is a bounded Borel set. LDPs/importance sampling for interference in (Ganesh and Torrisi, 2008) or (Torrisi and Leonardi, 2013) Proof based on Level-3 LDP for Poisson point processes (Georgii and Zessin, 1993) LDT in phone networks August 17th, 2015 Page 16

20 1 Introduction to large deviations theory 2 SINR definition and model description 3 LDP for averages of connectable receivers in large domains 4 LDP for connectable receivers for small SINR threshold 5 Importance sampling for rare-event probability estimation LDT in phone networks August 17th, 2015 Page 17

21 SINR-based network model Y τ = connectable receivers for a (typical) transmitter located at the origin o Y τ = {Y j Y : SINR(o, Y j ) τ} Asymptotic behavior as technically-dependent parameter τ 0 LDT in phone networks August 17th, 2015 Page 18

22 SINR-based network model Y τ = connectable receivers for a (typical) transmitter located at the origin o Y τ = {Y j Y : SINR(o, Y j ) τ} Asymptotic behavior as technically-dependent parameter τ 0 LDT in phone networks August 17th, 2015 Page 18

23 Large deviation principle for the empirical measure Theorem (Level-1 result) As τ 0 the random measure { τ d α Y τ (τ 1 α B)} B B(Λ1) satisfies an LDP in the weak topology with rate τ d α and good rate function given by { Λ I(µ) = 1 I y (f(y))dy if f = dµ/dx exists, otherwise, where I y (s) = inf Q P θ ( h(q P) + h ( Pois(s) Pois(λR E Q [P(SINR(o, y) 1 X)]) )). Idea: Y τ is Cox point process with random intensity measure B λ R P(SINR(o, y) τ X)dy + (Georgii & Zessin, 1993) + (Dawson & B Gärtner) LDT in phone networks August 17th, 2015 Page 19

24 Exponential decay of isolation probability Corollary ( lim τ d α log P(Y τ = ) = lim τ d α log Eexp τ 0 τ 0 = Λ 1 λ R Λ τ 1/α ) P(SINR(o, y) τ X)dy ( inf h(q P) + λr E Q [P(SINR(o, y) 1 X)] ) dy. Q P θ Variational characterization of rate function intractable analytically and numerically Minimization problem can be solved for class of Poisson point processes importance-sampling algorithms LDT in phone networks August 17th, 2015 Page 20

25 1 Introduction to large deviations theory 2 SINR definition and model description 3 LDP for averages of connectable receivers in large domains 4 LDP for connectable receivers for small SINR threshold 5 Importance sampling for rare-event probability estimation LDT in phone networks August 17th, 2015 Page 21

26 Importance sampling for rare-event probability estimation Problem: estimate isolation probability p τ for small τ by Monte Carlo method Accurate estimation of p τ by naïve MC requires large number of simulation runs Idea: perform simulation under different probability measure Q that makes rare event more likely ( dp ) P(o is isolated) = Q dq (ω)1 o is isolated(ω). Heuristic: optimal Q conditional distribution under rare event Connection to LDPs: use minimizers in rate function α = 4, w = 1, constant fading, constant transmission powers LDT in phone networks August 17th, 2015 Page 22

27 Importance sampling for rare-event probability estimation Problem: estimate isolation probability p τ for small τ by Monte Carlo method Accurate estimation of p τ by naïve MC requires large number of simulation runs Idea: perform simulation under different probability measure Q that makes rare event more likely ( dp ) P(o is isolated) = Q dq (ω)1 o is isolated(ω). Heuristic: optimal Q conditional distribution under rare event Connection to LDPs: use minimizers in rate function α = 4, w = 1, constant fading, constant transmission powers lim τ 1/2 ( log p τ = inf h(q P) + λr Q(SINR(o, y) 1) ) dy. τ 0 Q P θ Λ 1 Optimization only over inhomogeneous Poisson point processes LDT in phone networks August 17th, 2015 Page 22

28 Importance sampling for isolation probability continuous family of 1D-optimization problems for optimal intensity λ opt (r) N = 1, 000, 000 simulation runs Variance reduction by 78% Isolation probability Variance λ( ) λ( ) λ opt ( ) LDT in phone networks August 17th, 2015 Page 23

29 Literature [1] F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks: Volume 1: Theory. Now Publishers Inc, [2] A. Dembo and O. Zeitouni. Large deviations techniques and applications, volume 38. Springer Science & Business Media, [3] A. J. Ganesh and G. L. Torrisi. Large deviations of the interference in a wireless communication model. IEEE Trans. Inform. Theory, 54(8): , [4] H.-O. Georgii and H. Zessin. Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Related Fields, 96(2): , [5] C. Hirsch, B. Jahnel, P. Keeler, and R. I. Patterson. Large-deviation principles for connectable receivers in wireless networks. arxiv preprint arxiv: , [6] G. L. Torrisi and E. Leonardi. Simulating the tail of the interference in a Poisson network model. IEEE Trans. Inform. Theory, 59(3): , [7] H. Touchette. The large deviation approach to statistical mechanics. Physics Reports, 478(1):1 69, LDT in phone networks August 17th, 2015 Page 24

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