Tractability Using Periodized Generalized Faure Sequences Christiane Lemieux Department of Statistics and Actuarial Science University of Waterloo, Canada Information-Based Complexity and Stochastic Computation Workshop ICERM, September 15, 2014 Christiane Lemieux University of Waterloo Tractability using PGFS 1 / 26
Motivation Faure sequences and their generalizations are often used because they achieve the optimal value of 0 for the quality parameter t . Problem is they are not extensible in the dimension, since the base b must be at least as large as the dimension s to achieve t = 0 Can we make them extensible in the dimension, and if so, for what kind of problems would that work? Based on * C. Lemieux, H. Faure, New perspectives on (0 , s ) -sequences, in: P. L’Ecuyer, A. Owen (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2008, Springer-Verlag, 2009, pp. 113–130. * C. Lemieux, Tractability using periodized generalized Faure sequences, to appear in Journal of Complexity, 2014. Christiane Lemieux University of Waterloo Tractability using PGFS 2 / 26
Outline 1 Problem Setup and Review 2 Periodized Generalized Faure Sequences 3 Tractability Results 4 Numerical Results Christiane Lemieux University of Waterloo Tractability using PGFS 3 / 26
1 – Problem Setup and Review Goal is to estimate � I s ( f ) = [0 , 1) s f ( x ) d x where f : I s = [0 , 1) s → R is a real-valued function. Monte Carlo or (randomized) quasi-Monte Carlo estimate I s ( f ) by N Q N , s ( f , P N ) = 1 � f ( x i ) N i =1 where P N = { x 1 , . . . , x N } ⊆ [0 , 1) s . With (r)QMC, P N is a low-discrepancy point set: N s s E ( P N , z ) = 1 � � � 1 x i , j ≤ z j − z j , N i =1 j =1 j =1 then low (star-)discrepancy means z ∈ I s | E ( P N , z ) | ∈ O ( N − 1 log s N ) . D ∗ ( P N ) = sup Christiane Lemieux University of Waterloo Tractability using PGFS 4 / 26
Example: want to estimate the exp. nb of clients waiting more than 5 minutes in a bank over fixed horizon T ; fcts g and h resp. generate inter-arrival and service times A i , S i , so f ( x ) = φ ( Y ) = nb of clients who waited more than 5 minutes Y = ( A 1 = g ( x 1 ) , S 1 = h ( x 2 ) , . . . , A N = g ( x 2 N− 1 ) , S N = h ( x 2 N )) N = number of clients observed over horizon [0 , T ] Here s = 2 N is not known ahead of time. In such cases, we need P N to be extensible in the dimension, i.e., coordinates for each x i can be added “on the fly”. Christiane Lemieux University of Waterloo Tractability using PGFS 5 / 26
Review of Faure sequences Elementary intervals in base b: subsets of I s of the form � l j s � b r j , l j + 1 � (volume is b − M where M = r 1 + . . . + r s ) b r j j =1 where 0 ≤ l j < b r j and r j ≥ 0, j = 1 , . . . , s . M = 5 A (0 , m , s )-net in base b is a point set P N with N = b m points s.t. any elementary interval of volume b − M contains b m − M pts from P N , when M ≤ m . A (0 , s )-sequence in base b is a sequence of points x 1 , x 2 , . . . such that { x kb m +1 , . . . , x ( k +1) b m } is a (0 , m , s )-net for all m ≥ 0 and all k ≥ 0. Christiane Lemieux University of Waterloo Tractability using PGFS 6 / 26
Linearly scrambled van der Corput sequence in base b: For a prime base b , it is obtained by choosing a matrix C = ( c r , k ) r , k ≥ 1 with elements in Z b and an infinite number of rows and columns, and then defining the n th term of this sequence as ∞ ∞ y n , r � � S C c r +1 , k +1 · a k ( n ) , b ( n ) := in which y n , r = b r +1 r =0 k =0 where digits a k ( n ) come from n = � k ≥ 0 a k ( n ) b k . Sequence of points in I s can be constructed using ( S C 1 b , . . . , S C s b ), where C 1 , . . . , C s are generating matrices . Original Faure sequences (1982): C j = P j − 1 , where P b is the (upper b triangular) Pascal matrix P b in Z b ⇒ (0 , s )-sequence if b ≥ s . Generalized by Tezuka (1994) to C j = A j P j − 1 , where each A j is an b (NLT) matrix: also a (0 , s )-sequence if b ≥ s . Christiane Lemieux University of Waterloo Tractability using PGFS 7 / 26
Insight from Korobov Lattices Korobov lattices are extensible in the dimension, but coordinates start repeating if s ≥ N : � i − 1 � (1 , a , a 2 mod N , . . . , a s − 1 mod N ) mod 1 , i = 1 , . . . , N P N = . N Can take N prime and a primitive element modulo N to get cycle of maximal period N − 1. With s ≥ N means projections over indices with lag of N − 1 are bad, but N large can mitigate this issue since corresponding projections of f are often not important in that case. Bank example where s is not fixed but on average equal to 2000; use Korobov lattice with N = 1021 and a = 76 MC Kor Sobol’ Q N , s ( f , P N ) 543 544 544 HW 2.21 0.93 0.81 Christiane Lemieux University of Waterloo Tractability using PGFS 8 / 26
2 – Periodized Generalized Faure Sequences (PGFS) Fix the base b , and for any s ≥ 1, use sequence defined over Z b based on generating matrices C j = A j P j − 1 , j = 1 , . . . , s (can be ≥ b ) . b Scrambling matrices A j fixed as follows: choose a period p ≈ b / 2 and then let A j = f j mod p I (diagonal) where the multipliers f j ∈ Z b are ordered according to the quality of the one-dimensional van der Corput sequence they generate (similar idea used to define generalized Faure sequences in LF09). Christiane Lemieux University of Waterloo Tractability using PGFS 9 / 26
(49th,50th) coordinates of 1000 first points of Faure (left) and PGFS with b = 97 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Christiane Lemieux University of Waterloo Tractability using PGFS 10 / 26
Of course, if s ≥ b , t > 0 for a PGFS, but we have t b = 0, where: Definition For sequence { x 1 , x 2 , . . . } over Z b , t k =0 if for each u = { j 1 , . . . , j r } satisfying 1 ≤ j 1 < · · · < j r ≤ s , 1 ≤ | u | = r ≤ k , and r ( u ) := j r − j 1 + 1 ≤ k , the corresponding projection { ( x n , j 1 , x n , j 2 , . . . , x n , j r ) , n ≥ 1 } is a (0 , r )-sequence over Z b . Result: P N the first N points of a digital sequence over Z b with t b = 0 ; P u N the projection of P N over u ⊆ { 1 , . . . , s } with r ( u ) ≤ b, then N ) ≤ 1 b + 1 D ∗ ( P u 2 b ( b log b ( bN )) | u | . N Christiane Lemieux University of Waterloo Tractability using PGFS 11 / 26
3 – Tractability Results Look at worst-case error over Hilbert space H s with norm � · � H s : e ( P N ; H s ) = sup {| I s ( f ) − Q N , s ( f ; P N ) | : f ∈ H s , � f � H s ≤ 1 } , and compare it with the initial error, defined as e (0; H s ) = sup {| I s ( f ) | : f ∈ H s , � f � H s ≤ 1 } . We then define n ( ǫ, H s ) as the smallest n for which there exists P n such that e ( P n ; H s ) ≤ ǫ · e (0; H s ), where ǫ is in (0 , 1). Integration over H s is said to be QMC-tractable if there exist non-negative numbers C , p , and q such that n ( ǫ, H s ) ≤ C ǫ − p s q for all ǫ ∈ (0 , 1) and all s ≥ 1 . (1) If q = 0 in (1), then integration over H s is QMC-strongly tractable , and the infimum of the numbers p satisfying (1) with q = 0 is called the ǫ − exponent of QMC-strong tractability . Christiane Lemieux University of Waterloo Tractability using PGFS 12 / 26
Can adapt the results and proofs from SWW04 to our settings. As in SWW04, we assume H s is a reproducing kernel Hilbert space with a reproducing kernel of the form � � K s ( x , y ) = γ s , u η j ( x j , y j ) , (2) u ⊆{ 1 ,..., s } j ∈ u and the weights γ s , u are arbitrary non-negative numbers. Hence any f ∈ H s satisfies f ( · ) = � f ( x ) , K s ( x , · ) � . Case A: anchored Sobolev space H ( K s , A ): take η j , A ( x , y ) = min( | x j − a j | , | y − a j | ) if ( x − a j )( y − a j ) > 0 and 0 otherwise. The point ( a 1 , . . . , a s ) is called the anchor . Case B: unanchored Sobolev space H ( K s , B ): take � � � � η j , B ( x , y ) = 1 x − 1 y − 1 2 B 2 ( | x − y | ) + , where B 2 ( · ) is the 2 2 Bernoulli polynomial of degree 2, i.e., B 2 ( x ) = x 2 − x + 1 6 Christiane Lemieux University of Waterloo Tractability using PGFS 13 / 26
Choice of weights Tractability over weighted spaces usually occurs by choosing weights of the form γ s , u = � j ∈ u γ j for some 0 ≤ γ j ≤ 1 , j = 1 , . . . , s , or when there exists an integer r such that γ s , u = 0 for all u with | u | > r (finite order) . Here, we propose a special case of the latter, which makes use of the notion of range r ( u ) defined earlier. Definition A set of weights { γ s , u } u ⊆{ 1 ,..., s } is said to be of finite-range if there exists an integer R ∈ { 0 , . . . , s − 1 } (called the range) such that γ s , u = 0 if r ( u ) = max j { j ∈ u } − min j { j ∈ u } + 1 > R . Christiane Lemieux University of Waterloo Tractability using PGFS 14 / 26
Theorem: Let H ( K s ) be the anchored Sobolev space H ( K s , A ) with an arbitrary anchor a , or the unanchored Sobolev space H ( K s , B ) , and assume we have finite-range weights { γ s , u } u ⊆{ 1 ,..., s } of range R ≥ 1 . Let P N be the first N points of a digital sequence over Z b such that t R = 0 , where R ≤ b. Then � b + 1 � 2 e 2 ( P N ; H ( K s )) ≤ 1 � (2 b log b bN ) 2 | u | . γ s , u (3) N 2 2 b ∅� = u ⊆{ 1 ,..., s } r ( u ) ≤ R Christiane Lemieux University of Waterloo Tractability using PGFS 15 / 26
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