Hit-and-run for numerical integration Daniel Rudolf University Jena February 2012, Sydney supported by Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 1 / 12
The problem Let D ⊂ R d . Compute � � E π ̺ f = f ( x ) π ̺ ( d x ) = f ( x ) · c ̺ ( x ) d x , D D where c ̺ > 0 is a density. Function evaluations of f and ̺ are possible. The normalizing constant 1 � ̺ ( x ) d x c = D is not known and hard to compute. Equivalent formulation: Compute � D f ( x ) ̺ ( x ) d x S ( f , ̺ ) = ( f , ̺ ) ∈ F ( D ) . � for D ̺ ( x ) d x Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 2 / 12
Markov chain Monte Carlo Idea: Run a suitable Markov chain ( X n ) n ∈ N with limit distribution � A ̺ ( x ) d x π ̺ ( A ) = � D ̺ ( x ) d x and compute n � S n , n 0 ( f , ̺ ) = 1 f ( X j + n 0 ) . n j = 1 Error: e ( S n , n 0 , ( f , ̺ )) = ( E | S ( f , ̺ ) − S n , n 0 ( f , ̺ ) | 2 ) 1 / 2 . Error bounds? Polynomially in d ? ( D ⊂ R d ) Goal: Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 3 / 12
Markov chains Suppose ν is an initial distribution and K is a Markov kernel. • For f ∈ L 1 ( π ̺ ) the Markov operator is defined by � � Pf ( x ) = f ( y ) K ( x , d y ) S ( f , ̺ ) = f ( x ) π ̺ ( d x ) . and D D • There exists an L 2 -spectral gap if β := � P − S � L 2 ( π ̺ ) → L 2 ( π ̺ ) < 1 . • For suitable burn-in n 0 one has 4 � f � 2 e ( S n , n 0 , ( f , ̺ )) 2 ≤ 4 n ( 1 − β ) . (see Rudolf 2011). Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 4 / 12
Hit-and-run algorithm Compute � D f ( x ) ̺ ( x ) d x S ( f , ̺ ) = � . D ̺ ( x ) d x Let B ( 0 , R ) be the ball with radius R around 0. Hit-and-run algorithm: assume x 1 , . . . , x k are already computed; • choose direction u k uniformly distributed on ∂ B ( 0 , 1 ) ; • return x k + 1 ∈ A k = { α ∈ R | x k + α u k ∈ D } distributed with density • ̺ ( x k + u k α ) � ℓ k ( α ) = A k ̺ ( x k + u k t ) d t . Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 5 / 12
Simulation for d = 2 Hit-and-run in the unit ball and the cube, time steps 10 4 and α = 20. 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 � d j = 1 xj Figure: On the left ̺ 1 ( x ) = e α x 1 and on the right ̺ 2 ( x ) = e α . d √ Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 6 / 12
Densities with certain structure The class R d , R : B ( 0 , 1 ) ⊂ D ⊂ B ( 0 , R ) ; • ̺ is log-concave, for x , y ∈ D and 0 < λ < 1 • ̺ ( λ x + ( 1 − λ ) y ) ≥ ̺ ( x ) λ · ̺ ( y ) 1 − λ . � � x ∈ R d | ̺ ( x ) ≥ c For c > 0 let L ( c ) = • . Assume that π ̺ ( L ( c )) ≥ 1 x ∈ D B (˜ x , 1 ) ⊂ L ( c ) . = ⇒ ∃ ˜ 8 For u ∈ ∂ B ( 0 , 1 ) and x ∈ D one can sample w.r.t. the density • ̺ ( x + u α ) A = { α ∈ R | x + u α ∈ D } . � ℓ ( α ) = where A ̺ ( x + ut ) d t Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 7 / 12
Hit-and-run algorithm Theorem (Lovász and Vempala 06) Assumptions: • let ̺ ∈ R d , R ; • initial distribution ν , with bounded d ν d π ̺ ; • let ν P n 0 be the distribution after n 0 steps of hit-and-run. If �� � � d ν dR � � � ν P n 0 − π ̺ � tv ≤ 2 ε. n 0 ≻ d 2 R 2 log 3 � � then , � d π ̺ � ε ∞ • Proof is based on an estimate of the s -conductance. Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 8 / 12
Question Assume that there exist numbers α < 1, k ∈ N and M < ∞ , such that � � � � � d ν 2 < ∞ we have for all ν with � d π ̺ √ n 0 . � ν P n 0 − π ̺ � tv ≤ M α k Question: • Does it imply the existence of an L 2 -spectral gap? • Additional assumptions? • Estimate of the L 2 -spectral gap? Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 9 / 12
Different approximation scheme Theorem (similar to Belloni and Chernozhukov 07) Assumptions: • n independent runs of a suitable Markov chain, n 0 , . . . , X n after n 0 steps one has X 1 n 0 . • let f be a bounded function. • compute n � S n , n 0 ( f ) = 1 f ( X i � n 0 ) . n i = 1 Then S n , n 0 , ( f , ̺ )) 2 ≤ 1 n � f � ∞ + 2 � f � ∞ � ν P n 0 − π ̺ � tv . e ( � Number of steps n · n 0 for � S n , n 0 , while n + n 0 for S n , n 0 . • Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 10 / 12
Error bound Theorem Let � � ( f , ̺ ) | sup ̺ inf ̺ ≤ C , ̺ ∈ R d , R , � f � ∞ ≤ 1 � F d , R , C = . Then � � n 1 / 3 F d , R , C ) 2 ≤ 1 n + 4 dR d + 1 C exp e ( � S har n , n 0 , � − 3 · 10 − 10 0 . ( dR ) 2 / 3 Consequently F d , R , C ) ≺ d 5 ε − 2 [ log ε − 2 ] 3 [ log C ] 3 R 2 [ log R ] 3 . comp ( ε, d , � Initial state chosen with respect to uniform distribution in B ( 0 , 1 ) . • • Constants are very large, of the magnitude of 10 30 . Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 11 / 12
References • Belloni and Chernozhukov 2009: On the computational complexity of MCMC-based estimators in large samples, Ann. Statist. 4, 2011-2055. • Lovász and Vempala 2006: Fast algorithms for logconcave functions: Sampling, rounding, integration and optimization, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), 57–68, 2006. • Mathé and Novak 2007: Simple Monte Carlo and the Metropolis algorithm, J. Complexity 23, 673-696. • Rudolf 2011: Explicit error bounds of Markov chain Monte Carlo, to appear in Dissertationes Math. Daniel Rudolf (University Jena) Hit-and-run for numerical integration February 2012, Sydney 12 / 12
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