Concentration inequalities for occupancy models with log-concave - PowerPoint PPT Presentation

Concentration inequalities for occupancy models with log-concave marginals Jay Bartroff, Larry Goldstein, and Ümit I¸ slak International Colloquium on Stein’s Method, Concentration Inequalities, and Malliavin Calculus 2014 Thanks to Janna Beling and the Borchard Foundation Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 1 / 28

Concentration and Coupling Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 2 / 28

Concentration inequalities for occupancy models with log-concave marginals Main idea: How to get bounded, size biased couplings for certain multivariate occupancy models, then use existing methods to get concentration inequalities Outline 1. The models 2. Some methods for concentration inequalities 3. Our main result 4. Applications ◮ Erdös-Rényi random graph ◮ Germ-grain models ◮ Multinomial counts ◮ Multivariate hypergeometric sampling 5. Comparisons ◮ McDiarmid’s Inequality ◮ Negative association ◮ Certifiable functions Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 3 / 28

Setup Occupancy model M = ( M α ) M α may be ◮ degree count of vertex α in an Erdös-Rényi random graph ◮ # of grains containing point α in a germ-grain model ◮ # of balls in box α in multinomial model ◮ # balls of color α in sample from urn of colored balls We consider statistics like m � � Y ge = 1 { M α ≥ d } , Y eq = 1 { M ( x ) = d } dx α = 1 m � � Y ge = w α 1 { M α ≥ d α } , Y eq = w ( x ) 1 { M ( x ) = d ( x ) } dx α = 1 Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 4 / 28

Some Methods for Concentration Inequalities McDiarmid’s (Bounded Difference) Inequality If X 1 , . . . , X n independent Y = f ( X 1 , . . . , X n ) , f measurable there are c i such that sup � f ( x 1 , . . . , x i , . . . , x n ) − f ( x 1 , . . . , x ′ � ≤ c i , � � i , . . . , x n ) x i , x ′ i then � � t 2 P ( Y − µ ≥ t ) ≤ exp for all t > 0 , − 2 � n i = 1 c 2 i and a similar left tail bound. Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 5 / 28

Some Methods for Concentration Inequalities Negative Association (NA) X 1 , X 2 , ..., X m are NA if E ( f ( X i ; i ∈ A 1 ) g ( X j ; j ∈ A 2 )) ≤ E ( f ( X i ; i ∈ A 1 )) E ( g ( X j ; j ∈ A 2 )) for any A 1 , A 2 ⊂ [ m ] disjoint, f , g coordinate-wise nondecreasing. Dubashi & Ranjan 98 If X 1 , X 2 , ..., X m are NA indicators, then Y = � m i = 1 X i satisfies � t + µ � µ e t for all t > 0 P ( Y − µ ≥ t ) ≤ µ + t = O ( exp ( − t log t )) as t → ∞ . Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 6 / 28

Some Methods for Concentration Inequalities Certifiable Functions McDiarmid & Reed 06 If X 1 , X 2 , ..., X n independent and Y = f ( X 1 , X 2 , ..., X n ) where f is certifiable: There is c such that changing any coordinate x j changes the value of f ( x ) by at most c , If f ( x ) = s then there is C ⊂ [ n ] with | C | ≤ as + b such that that y i = c i ∀ i ∈ C implies f ( y ) ≥ s , Then t 2 � � P ( Y − µ ≤ − t ) ≤ exp for all t > 0, − 2 c 2 ( a µ + b + t / 3 c ) = O ( exp ( − t )) as t → ∞ . A similar right tail bound. Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 7 / 28

Some Methods for Concentration Inequalities Bounded Size Bias Couplings If there is a coupling Y s of Y with the Y -size bias distribution and Y s ≤ Y + c for some c > 0 with probability one, then max { P ( Y − µ ≥ t ) , P ( Y − µ ≤ − t ) } ≤ b µ, c ( t ) . Ghosh & Goldstein 11: For all t > 0, − t 2 t 2 � � � � P ( Y − µ ≤ − t ) ≤ exp P ( Y − µ ≥ t ) ≤ exp − . 2 c µ 2 c µ + ct b exponential as t → ∞ . Arratia & Baxendale 13: � t � �� − µ b µ, c ( t ) = exp h ( x ) = ( 1 + x ) log ( 1 + x ) − x . c h , µ b Poisson as t → ∞ . Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 8 / 28

Main Result M α lattice log-concave M = ( M α ) α ∈ [ m ] , � � Y ge = w α 1 { M α ≥ d α } , Y ne = w α 1 { M α � = d α } . α ∈ [ m ] α ∈ [ m ] Main Result (in words) 1. If M is bounded from below and can be closely coupled to a version M ′ having the same distribution conditional on M ′ α = M α + 1, then there is a bounded size biased coupling Y s ge ≤ Y ge + C and the above concentration inequalities hold. 2. If M is non-degenerate at ( d α ) and can be closely coupled to a version M ′ having the same distribution conditional on M ′ α � = d α , ne ≤ Y ne + C ′ and then there is a bounded size biased coupling Y s the above concentration inequalities hold. Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 9 / 28

Main Result A few more details on Part 1 M = f ( U ) where U is some collection of random variables f is measurable Closely coupled means given U k ∼ L ( V k ) := L ( U| M α ≥ k ) , there is coupling U + k and constant B such that L ( U + k |U k ) = L ( V k | M + Y + k ,α = M k ,α + 1 ) and k , ge , � = α ≤ Y k , ge , � = α + B , where Y k , ge , � = α = � β � = α 1 ( M k ,β ≥ d β ) . The constant is C = | w | ( B | d | + 1 ) where | w | = max w α , | d | = max d α . Part 2 is similar. Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 10 / 28

Main Result Main Ingredients in Proof Incrementing Lemma If M is lattice log-concave then there is π ( x , d ) ∈ [ 0 , 1 ] such that if M ′ ∼ L ( M | M ≥ d ) B | M ′ ∼ Bern ( π ( M ′ , d )) , and then M ′ + B ∼ L ( M | M ≥ d + 1 ) . Extension of Goldstein & Penrose 10 for M Binomial, d = 0 Analogous versions for → L ( M | M ≤ d − 1 ) L ( M | M ≤ d ) ֒ L ( M ) ֒ → L ( M | M � = d ) where ֒ → means “coupled to” Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 11 / 28

Main Result Main Ingredients in Proof Mixing Lemma (Goldstein & Rinnott 96) A nonnegative linear combination of Bernoullis with positive mean can be size biased by 1. choosing a summand with probability proportional to its mean, 2. replacing chosen summand by 1, and 3. modifying other summands to have correct conditional distribution. Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 12 / 28

Main Result Main Steps in Proof of Part 1 1. Induction on k : Given U k , U + k , let � U + with probability π ( M k ,α , k ) k U k + 1 = otherwise. U k U k + 1 has correct distribution by Incrementing Lemma . 2. Using k = d α case of induction and Mixing Lemma , mixing Y α ge = f ( U d α ) with probabilities ∝ w α P ( M α ≥ d α ) yields size biased Y s ge ≤ Y ge + | w | ( B | d | + 1 ) . Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 13 / 28

Application 1: Erdös-Rényi random graph m vertices Independent edges with probability p α,β = p β,α ∈ [ 0 , 1 ) . Constructing U + k from U k : 1. Selection non-neighbor β of α with probability p α,β ∝ 1 − p α,β 2. Add edge connecting β to α This affects at most 1 other vertex so B = 1 and Y s ge ≤ Y ge + | w | ( | d | + 1 ) . Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 14 / 28

Application 1: Erdös-Rényi random graph Applying this to Y is = m − Y ge with d α = 1: − t 2 � � P ( Y is − µ is ≤ − t ) = P ( Y ge − µ ge ≥ t ) ≤ exp 4 ( m − µ is + t / 3 ) Ghosh, Goldstein, & Raiˇ c 11 studied Y is using an unbounded size biased coupling � − t 2 � P ( Y is − µ is ≤ − t ) ≤ exp 4 µ is New bound ◮ an improvement for t ≤ 6 µ is − 3 m ◮ applicable for all d α Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 15 / 28

Application 2: Germ-Grain Models Used in forestry, wireless sensor networks, material science, . . . Germs U α ∼ f α strictly positive on [ 0 , r ) p Grains B α = closed ball of radius ρ α centered at U α d : [ 0 , r ) p → { 0 , 1 , . . . , m } = # of intersections we’re interested in at x ∈ [ 0 , r ) p Choice of r relative to p , ρ α guarantees nontrivial distribution of M ( x ) = # of grains containing at point x ∈ [ 0 , r ) p � = 1 { x ∈ B α } α ∈ [ m ] � Y ge = [ 0 , r ) p w ( x ) 1 { M ( x ) ≥ d ( x ) } dx = (weighted) volume of d -way intersections of grains Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 16 / 28

Application 2: Germ-Grain Models Main ideas in proof Different approach: 1. Generate U 0 independent of U 1 , . . . , U m 2. Compute U 0 , . . . , U d ( U 0 ) and set Y s ge = Y ge ( M d ( U 0 ) ) 3. Y s ge has size bias distribution by Conditional Lemma with A = { M ( U 0 ) ≥ d ( U 0 ) } : Conditional Lemma (Goldstein & Penrose 10) If P ( A ) ∈ ( 0 , 1 ) < 1 and Y = P ( A |F ) , then Y s has the Y -size bias distribution if L ( Y s ) = L ( Y | A ) . Jay Bartroff (USC) Concentration for occupancy models 1.Jul.14 17 / 28