Sparse Covers for Sums of Indicators

Sparse Co v ers for Sums of Indicators Constan tinos Dask alakis ∗ EECS and CSAIL, MIT costis@mit.edu Christos P apadimitriou † Computer Science, U.C. Berk eley c hristos@cs.berkele y .edu Octob er 3, 2014 Abstract F or all n, ǫ > 0, we show that the set of Poisson Binomial distributions on n v ariable s admits a prop er ǫ -co ver in total v ariation distance of size n 2 + n · (1 /ǫ ) O (log 2 (1 /ǫ )) , which can also be computed in po lynomial time. W e discuss the implications of o ur construction for appr o ximation algorithms and th e computation of appro ximate Nash eq uilibr ia in anonymous games. 1 In tro duction A P oisson Binomial Distribu tion of or der n is the d iscrete probability distribution of the su m of n ind ep enden t in d icato r rand om v ariables. The distribution is parameterized b y a v ector ( p i ) n i =1 ∈ [0 , 1] n of p robabilities, and is denoted PBD ( p 1 , . . . , p n ). In th is pap er w e establish that the set S n of all Poisson Binomial distrib utions of order n admits certain usefu l co v ers w ith resp ect to the total v ariation d istance d TV ( · , · ) b et w een distributions. Namely Theorem 1 (Main Th eorem) . F or al l n , ǫ > 0 , ther e e xi sts a set S n,ǫ ⊂ S n such that: 1. S n,ǫ is an ǫ -c over of S n in total variation distanc e; that is, for al l D ∈ S n , ther e exists some D ′ ∈ S n,ǫ such that d TV ( D , D ′ ) ≤ ǫ 2. |S n,ǫ | ≤ n 2 + n ·  1 ǫ  O (log 2 1 /ǫ ) 3. S n,ǫ c an b e c ompute d in time O ( n 2 log n ) + O ( n log n ) ·  1 ǫ  O (log 2 1 /ǫ ) . Mor e over, al l distributions PBD( p 1 , . . . , p n ) ∈ S n,ǫ in the c over satisfy at le ast one of the fol lowing pr op erties, for some p ositive inte ger k = k ( ǫ ) = O (1 /ǫ ) : • ( k -sp arse form) ther e is some ℓ ≤ k 3 such that, for al l i ≤ ℓ , p i ∈ n 1 k 2 , 2 k 2 , . . . , k 2 − 1 k 2 o and, for al l i > ℓ , p i ∈ { 0 , 1 } ; or • ( ( n, k ) -Binomial form) ther e i s some ℓ ∈ { 1 , . . . , n } and q ∈  1 n , 2 n , . . . , n n  such that, for al l i ≤ ℓ , p i = q and, for al l i > ℓ , p i = 0 ; mor e over, ℓ and q satisfy ℓq ≥ k 2 and ℓq (1 − q ) ≥ k 2 − k − 1 . ∗ Supp orted by a Sloan F oundation fello wship, a Microsoft Research faculty fello wship and NSF Award CCF- 0953960 (CAR EER ) and CCF-110149 1. † Supp orted by NSF grant CCF-0964033 and a Go ogle Un iv ersity Researc h Awa rd. 1 Co v ers such as the one pro vided b y Theorem 1 are of in terest in the d esign of algorithms, when one is searc hing a class of distributions C to iden tify an element of the class w ith some quan titativ e prop ert y , or in optimizing o v er a class with resp ect to some ob jective . If the metric used in the construction of the co v er is r elev an t for the problem at hand, and the co v er is d iscrete, relativ ely small and easy to construct, th en one can provide a useful appr o ximation to the sought distribution b y searc hing th e cov er, instead of searc hing all of C . F o r example, it is sho wn in [ DP07 , DP09 , DP13 ] th at Theorem 1 imp lies efficien t algorithms for computing appro ximate Nash equilibria in an imp ortant class of m ultipla y er games, called anon ymous [ Mil96 , Blo99 ]. W e p ro ceed with a fairly detailed sketc h of the pro of of our main co ve r theorem, T heorem 1 , stating t w o additional results, Theorems 2 and 3 . The complete p ro ofs of Theorems 1 , 2 and 3 are deferred to Sections 3 , 4 and 5 r esp ectiv ely . Secti on 1. 4 d iscusses r elat ed w ork, wh ile S ectio n 2 pro vides formal defin itions, as well as kn o wn appro ximations to the P oisson Binomial distribution b y s impler distribu tions, w hic h are used in the p r oof. 1.1 Pro of Outline and Additional Results A t a high leve l, th e pro of o f Theorem 1 is obtained in t w o steps. First, we establish the existence of an ǫ -co ver whose size is p olynomial in n and (1 /ǫ ) 1 /ǫ 2 , via Theorem 2 . W e then sho w that this co v er can b e prun ed to s ize p olynomial in n and (1 /ǫ ) log 2 (1 /ǫ ) using Theorem 3 , wh ic h pro vides a quan tification of how the total v ariation d istance b etw een Po isson Binomial distr ib utions dep ends on the n umber of their first moments that are equal. W e pr oceed to state the t w o ingredien ts of the p roof, Theorems 2 and 3 . W e start with Th eo- rem 2 wh ose d etail ed sk etc h is giv en in Section 1.2 , and co mplete pro of in Section 4 . Theorem 2. L et X 1 , . . . , X n b e arbitr ary mutual ly indep endent indic a tors, and k ∈ N . Then ther e exist mutual ly indep endent indic at ors Y 1 , . . . , Y n satisfying the fol lowing: 1. d TV ( P i X i , P i Y i ) ≤ 41 /k ; 2. at le ast one of the fol lowing is true: (a) ( k -sp arse form) ther e exists some ℓ ≤ k 3 such that, for al l i ≤ ℓ , E [ Y i ] ∈ n 1 k 2 , 2 k 2 , . . . , k 2 − 1 k 2 o and, for al l i > ℓ , E [ Y i ] ∈ { 0 , 1 } ; or (b) ( ( n, k ) -Binomial f orm) ther e is some ℓ ∈ { 1 , . . . , n } and q ∈  1 n , 2 n , . . . , n n  such that, for al l i ≤ ℓ , E [ Y i ] = q and, for al l i > ℓ , E [ Y i ] = 0; mor e over, ℓ and q satisfy ℓq ≥ k 2 and ℓq (1 − q ) ≥ k 2 − k − 1 . Theorem 2 implies the existence of an ǫ -co ve r o f S n whose size is n 2 + n · (1 /ǫ ) O (1 /ǫ 2 ) . This co v er can b e obtained by enumerating o v er all Poisson Binomial distributions of order n that are in k -sparse or ( n, k )-Binomial form as defined in the statemen t of the theorem, for k = ⌈ 41 /ǫ ⌉ . The next step is to sparsify this co ve r by remo ving elements to obtain Theorem 1 . Note that the term n · (1 /ǫ ) O (1 /ǫ 2 ) in the size of the co v er is due to the en umeration o ve r d istributions in sparse form. Using Theorem 3 b elo w, we argue that there is a lot of redun dancy in those distributions, and that it suffices to only include n · (1 /ǫ ) O (log 2 1 /ǫ ) of them in the co v er. In particular, Th eorem 3 establishes th at, if t wo Po isson Binomial distribu tions ha ve their fir st O (log 1 /ǫ ) momen ts equal, then their distance is at most ǫ . So w e only need to include at most one sparse form distribution with the same first O (log 1 /ǫ ) momen ts in our co ver. W e proceed to state Theorem 3 , p ostp onin g its p ro of to Section 5 . In Sectio n 1.3 we provide a sk etc h of the p ro of. 2 Theorem 3. L et P := ( p i ) n i =1 ∈ [0 , 1 / 2] n and Q := ( q i ) n i =1 ∈ [0 , 1 / 2] n b e two c o l le ctions of pr ob- ability values. L et also X := ( X i ) n i =1 and Y := ( Y i ) n i =1 b e two c ol le ctions of mutual ly indep endent indic at ors with E [ X i ] = p i and E [ Y i ] = q i , for al l i ∈ [ n ] . If for some d ∈ [ n ] the fol lowing c ond ition is satisfie d : ( C d ) : n X i =1 p ℓ i = n X i =1 q ℓ i , for al l ℓ = 1 , . . . , d, then d TV X i X i , X i Y i ! ≤ 13( d + 1) 1 / 4 2 − ( d +1) / 2 . (1) Remark 1. Condition ( C d ) in the statem ent of The or em 3 c onstr ains the first d p ower sums of the exp e ctations of th e c onst ituent indic at ors of two P oisson Binomial distributions. T o r elate these p ower sums to the moments of these distributions we c an u se th e the ory of symmetric p ol ynomials to arrive at the fol lowing e quivalent c o ndition to ( C d ) : ( V d ) : E   n X i =1 X i ! ℓ   = E   n X i =1 Y i ! ℓ   , for al l ℓ ∈ [ d ] . We pr ovide a pr o of that ( C d ) ⇔ ( V d ) in Pr op osition 2 of Se ction 6 . Remark 2. In view of R emark 1 , The or em 3 says the fol lowing: “If tw o sums of indep end en t indicators w ith exp ectatio ns in [0,1/2] ha v e equal fi rst d momen ts, then their to tal v ariation distance is 2 − Ω( d ) .” We note that the b ound ( 1 ) do es not dep end on the numb er of variables n , and in p a rticular do es not r ely on su mming a lar ge numb er of variables. W e al so note th at, sinc e we imp ose no c onstr aint on the exp e ctations of the indic ators, we also i mp ose no c onstr aint on th e varianc e of the r esulting Poisson Binomial distributions. Henc e we c anno t use Berry-Ess´ een typ e b ounds to b ound the total variation distanc e of the two Poisson Binomial distributions by appr oximating them with Normal distributions. Final ly, it is e asy to se e that The or em 3 ho lds if we r eplac e [0 , 1 / 2] with [1 / 2 , 1] . Se e Cor ol lary 1 in Se ction 6 . In Section 3 w e show how to use Theorems 2 and 3 to obtain Theorem 1 . W e contin ue with the outlines of the pro ofs of Theorems 2 and 3 , p ostp oning their co mplete pro ofs to Sectio ns 4 and 5 . 1.2 Outline of P ro of of Theorem 2 Giv en arbitrary indicators X 1 , . . . , X n w e obtai n indicators Y 1 , . . . , Y n , satisfying the r equ iremen ts of Theorem 2 , in tw o steps. W e first massage the give n v ariables X 1 , . . . , X n to obtain v ariables Z 1 , . . . , Z n suc h that d TV X i X i , X i Z i ! ≤ 7 /k ; (2) and E [ Z i ] / ∈  0 , 1 k  ∪  1 − 1 k , 1  ; that is, we eliminate from our collectio n v aria bles that hav e exp ectatio ns very close to 0 or 1, without trav eli ng to o m uc h distance from the starting P oisson Binomial distribu tion. 3 V ariables Z 1 , . . . , Z n do n ot necessarily satisfy P rop erties 2a or 2b in the statemen t of Theorem 2 , but allo w us to define v ariables Y 1 , . . . , Y n whic h do satisfy one of these p rop erties and, moreov er, d TV X i Z i , X i Y i ! ≤ 34 /k . (3) ( 2 ), ( 3 ) and the triangle inequalit y im p ly d TV ( P i X i , P i Y i ) ≤ 41 k , concluding the p r oof of T heo- rem 2 . Let us call Stage 1 the pro cess of d etermining the Z i ’s and Stage 2 the p rocess of determining the Y i ’s. The t wo stages are describ ed briefly b elow, and in detail in S ectio ns 4.1 and 4.2 resp ectiv ely . F or conv enience, w e use the follo wing notation: for i = 1 , . . . , n , p i = E [ X i ] will denote the exp ectatio n of the given indicator X i , p ′ i = E [ Z i ] the exp ectatio n of the int ermediate indicator Z i , and q i = E [ Y i ] the exp ecta tion of the final indicator Y i . Stage 1: Recall that our goal in this stage is to defi ne a Poisson Binomial distrib u tion P i Z i whose constituen t ind icat ors hav e no exp ectat ion in T k := (0 , 1 k ) ∪ (1 − 1 k , 1). The exp ectations ( p ′ i = E [ Z i ]) i are defined in terms of the corresp onding ( p i ) i as follo ws. F or all i , if p i / ∈ T k w e set p ′ i = p i . Th en, if L k is the set of indices i su c h that p i ∈ (0 , 1 /k ), w e c ho ose any collection ( p ′ i ) i ∈L k so as to satisfy | P i ∈L k p i − P i ∈L k p ′ i | ≤ 1 /k and p ′ i ∈ { 0 , 1 /k } , for all i ∈ L k . That is, we round all indicato rs’ exp ectat ions to 0 or 1 /k wh ile preserving the exp ectat ion of their sum, to within 1 /k . Using the Poisson appro ximation to the Po isson Binomial distribu tion, giv en as Th eorem 4 in Section 2.1 , w e can argue that P i ∈L k X i is within 1 /k of a P oisson distribu tion with the same mean. By the same tok en, P i ∈L k Z i is 1 /k -close to a Poisson d istribution w ith the same mean. An d the t w o resulting Po isson distribu tions h a v e means th at are within 1 /k , and are therefore 1 . 5 /k -close to eac h other (see Lemm a 3 ). Hence, by triangle inequalit y P i ∈L k X i is 3 . 5 /k -close to P i ∈L k Z i . A s imilar construction is used to defi n e the p ′ i ’s corresp ond in g to the p i ’s lying in (1 − 1 /k , 1). The details of this step can b e found in Section 4.1 . Stage 2: The definition of ( q i ) i dep ends on the num b er m of p ′ i ’s whic h are not 0 or 1. Th e case m ≤ k 3 corresp onds to Case 2a in the statemen t of Theorem 2 , while the case m > k 3 corresp onds to Case 2b . • Case m ≤ k 3 : First, we s et q i = p ′ i , if p ′ i ∈ { 0 , 1 } . W e then argue that eac h p ′ i , i ∈ M := { i p ′ i / ∈ { 0 , 1 }} , can b e rounded to some q i , whic h is an in teger multiple of 1 /k 2 , so that ( 3 ) holds. Notice that, if w e were allo we d to use multiples o f 1 /k 4 , this w ould b e immediate via an ap p licati on of Lemma 2 : d TV X i Z i , X i Y i ! ≤ X i ∈M | p ′ i − q i | . W e imp ro v e the required accuracy to 1 /k 2 via a series of Binomial appro ximations to the Poi s- son Binomial distribu tion, using Ehm’s b ound [ Ehm91 ] stated as Theorem 5 in S ectio n 2.1 . The details in v olv e partitioning the interv al [1 /k, 1 − 1 /k ] in to irregularly sized sub in terv al s, whose endp oints are in teger multiples of 1 /k 2 . W e then round all but one of the p ′ i ’s falling in eac h subinte rv al to th e endp oints of th e subinterv al so as to main tain th eir total exp ec- tation, and apply Eh m’s appro ximation to argue th at the distribution of their sum is not affected by more th an O (1 /k 2 ) in total v ariation distance. It is cr u cial that the total num b er of sub in terv a ls is O ( k ) to get a total hit of at most O (1 /k ) in v ariation distance in the o v erall distribution. The details are giv en in Section 4.2.1 . 4 • Case m > k 3 : W e appro ximate P i Z i with a T rans lated Po isson distribu tion (defined for- mally in Section 2 ), usin g T h eorem 6 of Section 2.1 due to R¨ ollin [ R ¨ 07 ]. T he qualit y of the appro ximation is inv erse p rop ortional to the standard deviation of P i Z i , w hic h is at least k , b y the assumption m > k 3 . Hence, w e sho w that P i Z i is 3 /k -close to a T ran s late d Po isson distribution. W e th en argue that the latter is 6 /k -clo se to a Bi nomial distrib u tion B ( m ′ , q ), where m ′ ≤ n and q is an in teger multiple of 1 n . In particular, we sho w that an app ropriate c hoice of m ′ and q implies ( 3 ), if we set m ′ of the q i ’s equ al to q and the remaining equal to 0. T he details are in Section 4.2.2 . 1.3 Outline of P ro of of Theorem 3 Using Ro os’s exp ansion [ Ro o00 ], giv en as Theorem 7 of Section 2.1 , we express PBD( p 1 , . . . , p n ) as a w eigh ted sum of the Binomial distribu tion B ( n, p ) at p = ¯ p = P p i /n and its first n deriv ativ es with resp ect to p also at v alue p = ¯ p . (These deriv ativ es corresp ond to finite signed measur es.) W e notice that th e co efficien ts of the first d + 1 term s of this expansion are symmetric p olynomials in p 1 , . . . , p n of degree at most d . Hence, from the theory of symmetric p olynomials, eac h of these co efficien ts can b e written as a fu nction of the p o w er-sum symm etric p olynomials P i p ℓ i for ℓ = 1 , . . . , d . So, whenev er t wo Poisson Binomial distribu tions satisfy Condition ( C d ), the fi rst d + 1 terms of their expansions are exactly id entical , and the total v ariation distance of the d istributions dep en ds only on the other terms of the expansion (those corresp onding to h igher deriv ativ es of th e Binomial distribution). Th e pro of is concluded b y s ho wing that the joint cont ribu tion of these terms to the total v ariatio n distance can b e b oun ded by 2 − Ω( d ) , usin g Prop osition 1 of S ectio n 2.1 , whic h is also due to Roos [ Ro o00 ]. The details are pro vided in Section 5 . 1.4 Related W ork It is b eliev ed that P oisson [ P oi37 ] was the first to study the P oisson Binomial distribu tion, hence its name. Sometimes the d istribution is also referred to as “Poisso n’s Binomial Distribution.” PBDs h a v e many uses in research areas suc h as survey samp ling, case-con trol stud ies, and su rviv a l analysis; see e.g. [ CL97 ] for a survey of their uses. They are also very imp ortan t in the design of randomized algorithms [ MR95 ]. In Probability and S tatistic s th er e is a broad literature studying v arious prop erties of these distributions; see [ W an93 ] for an introd uction to some of this work. Many results provide approxi- mations to the Poisson Binomial distribution via simpler distribu tions. In a wel l-known r esult, Le Cam [ LC60 ] sho ws that, for an y v ector ( p i ) n i =1 ∈ [0 , 1] n , d TV PBD( p 1 , . . . , p n ) , Po isson n X i =1 p i !! ≤ n X i =1 p 2 i , where Po isson ( λ ) is the P oisson distribution with parameter λ . Sub sequen tly man y other p ro ofs of this b ound and imp ro v ed ones, suc h as Theorem 4 of Section 2.1 , were giv en, usin g a range of d ifferen t tec hn iques; [ HC60 , Che74 , BH84 , DP86 ] is a s amp ling of work along these lines, and Steele [ Ste94 ] give s an extensive list of relev an t references. Muc h work has also b een don e on appro ximating PBDs b y No rmal distributions (see e.g. [ Ber41 , Ess 42 , Mik93 , V ol95 , CGS10 ]) and b y Binomia l distributions; see e.g. Ehm’s resu lt [ Ehm91 ], giv en as T h eorem 5 of Section 2.1 , as w ell as So on’s result [ So o96 ] and Roos’s result [ Ro o00 ], give n as Th eorem 7 of Sectio n 2.1 . These r esults provide stru ctural inf ormation ab out PBDs that can b e well approximat ed by sim- pler distr ib utions, but fall short of our goal of appro ximating a PBD to within arbitr ary ac cur acy . Indeed, the app ro ximations obtained in the probability literature (suc h as the Poisson, Normal and 5 Binomial appr o ximatio ns) t ypically d ep end on the first f ew momen ts of the PBD b eing app ro xi- mated, while higher momen ts are crucial for arbitrary approximat ion [ Ro o00 ]. A t the same time, algorithmic app licati ons often r equire that the appr o ximating distribu tion is of the same kind as the distribution that is b eing appro ximated. E.g., in the anonymous game applicatio n ment ioned earlier, the p arameters of th e given PBD corresp ond to mixed strategies of p lay ers at Nash equilib- rium, and the parameters of the appro ximating PBD corresp ond to mixed s tr ateg ies at appro ximate Nash equ ilibr ium. Approxima ting the giv en PBD via a P oisson or a Normal distribu tion wo uld not ha v e an y meaning in the con text of a game. As outlined ab o v e, the p ro of of our main result, Theorem 1 , bu ilds on Theorems 2 and 3 . A w eak er form of these theorems w as announced in [ Das08 , DP09 ], wh ile a w eak er form of Theorem 1 w as announced in [ DDS12 ]. 2 Preliminaries F or a p ositiv e in teger ℓ , we denote by [ ℓ ] the set { 1 , . . . , ℓ } . F or a ran d om v ariable X , w e denote b y L ( X ) it s distribution. W e further n eed the follo wing definitions. T otal variation distanc e: F or t w o distributions P and Q sup p orted on a fi nite set A their total variation distanc e is d efined as d TV ( P , Q ) := 1 2 X α ∈ A | P ( α ) − Q ( α ) | . An equiv ale nt w a y to define d TV ( P , Q ) is to view P and Q as v ectors in R A , and d efine d TV ( P , Q ) = 1 2 k P − Q k 1 to equal half of their ℓ 1 distance. I f X and Y are r andom v aria bles ranging o v er a finite set, their total v ariation distance, denoted d TV ( X , Y ) , is defined to equ al d TV ( L ( X ) , L ( Y )). Covers: Let F b e a set of probabilit y d istributions. A s ubset G ⊆ F is ca lled a (pr op er) ǫ -c o ver o f F in total v ariation d istance if, for all D ∈ F , there exists some D ′ ∈ G suc h that d TV ( D , D ′ ) ≤ ǫ . Poisson Binomial Distribution: A Poisson Binomial distribution of or der n ∈ N is the discrete probabilit y distribution of the su m P n i =1 X i of n mutually in dep enden t Bernoulli random v ariables X 1 , . . . , X n . W e denote the set of all P oisson Binomial distributions of order n by S n . By d efinition, a P oisson Binomial distrib u tion D ∈ S n can b e r epresen ted b y a v ector ( p i ) n i =1 ∈ [0 , 1] n of p r obabilities as follo ws. W e map D ∈ S n to a v ector of p robabilities by finding a collect ion X 1 , . . . , X n of m utually ind ep enden t indicators such that P n i =1 X i is distr ibuted according to D , and setting p i = E [ X i ] for all i . Th e follo wing lemma implies that th e resu lting v ector of probabilities is unique up to a p erm utation, so that there is a one-to-one corresp ondence b et wee n P oisson Binomial distributions and v ecto rs ( p i ) n i =1 ∈ [0 , 1] n suc h that 0 ≤ p 1 ≤ p 2 ≤ . . . ≤ p n ≤ 1. Th e pro of of this lemma can b e fou n d in Section 6 . Lemma 1. L et X 1 , . . . , X n b e mutual ly indep endent indic ator s with exp e ctations p 1 ≤ p 2 ≤ . . . ≤ p n r esp e ctively. Similarly let Y 1 , . . . , Y n b e mutual ly indep endent indic ator s with exp e ctations q 1 ≤ . . . ≤ q n r esp e ctively. The distributions of P i X i and P i Y i ar e differ ent if and only if ( p 1 , . . . , p n ) 6 = ( q 1 , . . . , q n ) . W e will b e d enoting a P oisson Binomial distribu tion D ∈ S n b y PBD( p 1 , . . . , p n ) w hen it is the distribution of th e s um P n i =1 X i of mutually ind ep enden t indicators X 1 , . . . , X n with exp ectations p i = E [ X i ], for all i . Giv en the ab o v e d iscussion, the repr esen tation is unique up to a p erm utation of the p i ’s. 6 T r ans late d Poisson D istribution: W e sa y that an in teger r andom v aria ble Y has a tr ansla te d Poisson distribution with parameters µ and σ 2 and wr ite L ( Y ) = T P ( µ, σ 2 ) iff L ( Y − ⌊ µ − σ 2 ⌋ ) = P oisson( σ 2 + { µ − σ 2 } ) , where { µ − σ 2 } rep r esen ts the fr acti onal part of µ − σ 2 . Or der Notation: Let f ( x ) and g ( x ) b e t w o p ositiv e fun ctions defined on some infinite subset of R + . One writes f ( x ) = O ( g ( x )) if and only if, f or sufficien tly large v alues of x , f ( x ) is at most a constan t times g ( x ). That is, f ( x ) = O ( g ( x )) if and only if th ere exist p ositiv e real num b ers M and x 0 suc h that f ( x ) ≤ M g ( x ) , for a ll x > x 0 . Similarly , we write f ( x ) = Ω( g ( x )) if and only if there exist p ositive reals M and x 0 suc h that f ( x ) ≥ M g ( x ) , for a ll x > x 0 . W e are casual in our use of the order notati on O ( · ) and Ω( · ) th r oughout the paper . Whenev er w e wr ite O ( f ( n )) or Ω( f ( n )) in some b ound where n ranges o v er the intege rs, w e mean that there exists a constant c > 0 suc h that the b ound holds true for s ufficien tly large n if we replace the O ( f ( n )) or Ω( f ( n )) in th e b ound by c · f ( n ). On the other hand, whenev er w e write O ( f (1 /ǫ )) or Ω( f (1 /ǫ )) in s ome b ound where ǫ ranges o ve r the p ositiv e reals, w e mean that there exists a constan t c > 0 suc h that the b ound holds true for su ffi cien tly smal l ǫ if we replace the O ( f (1 /ǫ )) or Ω( f (1 /ǫ )) in the b oun d with c · f (1 /ǫ ). W e conclude with an easy but u seful lemma whose p ro of w e defer to Section 6 . Lemma 2. L et X 1 , . . . , X n b e mutual ly indep endent r and om variables, and let Y 1 , . . . , Y n b e mutu - al ly indep endent r andom variables. Then d TV n X i =1 X i , n X i =1 Y i ! ≤ n X i =1 d TV ( X i , Y i ) . 2.1 Appro ximations to the Poisson Binomial Distribution W e presen t a collection of kno wn appro ximations to the P oisson Binomia l distribution via s impler distributions. The qualit y of these appro ximations can b e quant ified in term s of the fi rst few momen ts of the P oisson Binomial distribu tion that is b eing approxima ted. W e will make u se of these b ound s to approxima te P oisson Binomial distrib utions in different regimes of their momen ts. Theorems 4 — 6 are obtained via the S tein-Chen metho d. Theorem 4 (Po isson Approximati on [ BH84 , BHJ92 ]) . L et J 1 , . . . , J n b e mutual ly indep endent indic at ors with E [ J i ] = t i . Then d TV n X i =1 J i , Poisso n n X i =1 t i !! ≤ P n i =1 t 2 i P n i =1 t i . Theorem 5 (Binomial Appr o ximatio n [ Ehm91 ]) . L et J 1 , . . . , J n b e mutual ly indep endent indic ators with E [ J i ] = t i , and ¯ t = P i t i n . Then d TV n X i =1 J i , B ( n, ¯ t ) ! ≤ P n i =1 ( t i − ¯ t ) 2 ( n + 1) ¯ t (1 − ¯ t ) , wher e B ( n, ¯ t ) is the Binomial distribution with p ar ameters n and ¯ t . 7 Theorem 6 (T ranslated Poi sson Approxima tion[ R ¨ 07 ]) . L et J 1 , . . . , J n b e mutual ly i ndep endent indic at ors with E [ J i ] = t i . Then d TV n X i =1 J i , T P ( µ, σ 2 ) ! ≤ q P n i =1 t 3 i (1 − t i ) + 2 P n i =1 t i (1 − t i ) , wher e µ = P n i =1 t i and σ 2 = P n i =1 t i (1 − t i ) . The app ro ximation theorems stated ab o v e d o not alw a ys p r o vide tigh t enough ap p ro ximations. When these fail, we emplo y the follo wing theorem of Ro os [ Ro o00 ], which pro vides an expans ion of the Poisson Binomial d istribution as a w eigh ted sum of a fin ite num b er of signed measures: the Binomial distribution B ( n , p ) (for an arbitrary v alue of p ) and its fi rst n d eriv ativ es with resp ect to the parameter p , at the c hosen v al ue of p . F or the pu rp oses of the follo wing statemen t w e denote b y B n,p ( m ) the probabilit y assigned b y th e Binomial distribu tion B ( n, p ) to in teger m . Theorem 7 ([ Ro o00 ]) . L et P := ( p i ) n i =1 ∈ [0 , 1] n , X 1 , . . . , X n b e mutual ly indep endent indic ators with exp e ctations p 1 , . . . , p n , and X = P i X i . Then, for al l m ∈ { 0 , . . . , n } and p ∈ [0 , 1] , P r [ X = m ] = n X ℓ =0 α ℓ ( P , p ) · δ ℓ B n,p ( m ) , (4) wher e for the purp oses of the ab o ve expr ession: • α 0 ( P , p ) := 1 and for ℓ ∈ [ n ] : α ℓ ( P , p ) := X 1 ≤ k (1) <... ℓ th at ha v e exp ectation equal to 1. 1 Notice that en umerating o v er the ab o v e distributions tak es time O ( n 2 log n ) + O ( n log n ) ·  1 ǫ  O (1 /ǫ 2 ) , as a num b er in { 0 , . . . , n } and a pr ob ab ility in  1 n , 2 n , . . . , n n  can b e r epresen ted using O (log n ) bits, wh ile a n umb er in { 0 , . . . , k 3 } and a probabilit y in n 1 k 2 , 2 k 2 , . . . , k 2 − 1 k 2 o can b e represented u sing O (log k ) = O (log 1 /ǫ ) bits. W e next sh ow that we can remov e from S ′ n,ǫ a large n um b er of the sparse-form distributions it con tains to obtain a 2 ǫ -co v er of S n . In particular, we shall only k eep n ·  1 ǫ  O (log 2 1 /ǫ ) sparse-form distributions by app ealing to Theorem 3 . T o explai n the p runing we in tro duce some notation. F or a collec tion P = ( p i ) i ∈ [ n ] ∈ [0 , 1] n of probabilit y v al ues we denote b y L P = { i | p i ∈ (0 , 1 / 2] } and b y R P = { i | p i ∈ (1 / 2 , 1) } . Th eorem 3 , Corollary 1 , Lemma 2 and Lemma 1 imply that if t w o collect ions P = ( p i ) i ∈ [ n ] and Q = ( q i ) i ∈ [ n ] of p robabilit y v alues satisfy X i ∈L P p t i = X i ∈L Q q t i , for all t = 1 , . . . , d ; X i ∈R P p t i = X i ∈R Q q t i , for al l t = 1 , . . . , d ; and ( p i ) [ n ] \ ( L P ∪R P ) and ( q i ) [ n ] \ ( L Q ∪R Q ) are equal up to a p erm utation; then d TV (PBD( P ) , PBD ( Q )) ≤ 2 · 13( d + 1) 1 / 4 2 − ( d +1) / 2 . In particular, for some d ( ǫ ) = O (log 1 /ǫ ), this b oun d b ecomes at most ǫ . F or a collection P = ( p i ) i ∈ [ n ] ∈ [0 , 1] n , we d efine its moment pr ofile m P to b e the (2 d ( ǫ ) + 1)- dimensional v ecto r m P =   X i ∈L P p i , X i ∈L P p 2 i , . . . , X i ∈L P p d ( ǫ ) i ; X i ∈R P p i , . . . , X i ∈R P p d ( ǫ ) i ; |{ i | p i = 1 }|   . By the p revious discussion, for t wo collecti ons P , Q , if m P = m Q then d TV (PBD( P ) , PBD( Q )) ≤ ǫ . Giv en the ab o v e we sparsify S ′ n,ǫ as follo ws: for ev ery p ossible momen t profile that can arise from a Po isson Binomial distribution in k -sparse form, we k eep in our cov er a single Poi sson Binomial distribution with s u c h momen t profile. T he co ver resu lting from this sparsification is a 2 ǫ -co v er, since th e sparsification loses us an additional ǫ in tot al v a riation distance, as argu ed ab o v e. W e now b ound the cardinalit y of the sparsified co v er. The total num b er of momen t profiles of k -sparse P oisson Binomial d istributions is k O ( d ( ǫ ) 2 ) · ( n + 1). In deed, consider a P oisson Binomial distribution PBD( P = ( p i ) i ∈ [ n ] ) in k -sparse form. There are at most k 3 + 1 choice s for |L P | , at 1 Note that imp osing the condition p 1 ≤ . . . ≤ p ℓ w on’t lose us any P oisson Binomial distribu t ion in k -sparse form giv en Lemma 1 . 9 most k 3 + 1 choic es for |R P | , and at most ( n + 1) choice s for |{ i | p i = 1 }| . W e also claim that the total num b er of p ossib le vecto rs   X i ∈L P p i , X i ∈L P p 2 i , . . . , X i ∈L P p d ( ǫ ) i   is k O ( d ( ǫ ) 2 ) . Indeed, if |L P | = 0 there is j u st one suc h vec tor, n amely the all-zero vecto r. If |L P | > 0, then, for all t = 1 , . . . , d ( ǫ ), P i ∈L P p t i ∈ (0 , |L P | ] and it must b e an intege r m ultiple of 1 /k 2 t . So the total n umber of p ossible v alues of P i ∈L P p t i is at most k 2 t |L P | ≤ k 2 t k 3 , and the tota l num b er of p ossible vecto rs   X i ∈L P p i , X i ∈L P p 2 i , . . . , X i ∈L P p d ( ǫ ) i   is at most d ( ǫ ) Y t =1 k 2 t k 3 ≤ k O ( d ( ǫ ) 2 ) . The same upp er b ound applies to the total n umber of p ossible v ectors   X i ∈R P p i , X i ∈R P p 2 i , . . . , X i ∈R P p d ( ǫ ) i   . The moment profiles we enumerated o v er are a su p erset of the moment profiles of k -spars e P oisson Binomial distr ib utions. W e call them c omp atible moment profiles. W e argued that there are at most k O ( d ( ǫ ) 2 ) · ( n + 1) compatible moment pr ofiles, so th e total num b er of Poisson Binomial distribu tions in k -sparse form that we keep in the co ver is at most k O ( d ( ǫ ) 2 ) · ( n + 1) = n ·  1 ǫ  O (log 2 1 /ǫ ) . The n umb er of Po isson Binomial distribu tions in ( n, k )-Binomial form is the s ame as b efore, i.e. at m ost n 2 , as w e d id not eliminate any of them. So the size of the spars ifi ed co v er is n 2 + n ·  1 ǫ  O (log 2 1 /ǫ ) . T o finish the pro of it remains to argue that we don’t actually need to fi rst compute S ′ n,ǫ and then sparsify it to obtain our cov er, but can p ro duce it directly in time O ( n 2 log n ) + O ( n log n ) ·  1 ǫ  O (log 2 1 /ǫ ) . W e claim th at, give n a moment profile m that is compatible with a k -sparse Poisson Binomial distribu tion, we can co mpu te some PBD( P = ( p i ) i ) in k -sparse form suc h that m P = m , if suc h a d istribution exists, in time O (log n )  1 ǫ  O (log 2 1 /ǫ ) . This follo ws from Claim 1 of Section 6 . 2 So our algorithm enumerate s o v er all moment p r ofiles that are compati ble with a k -sparse Poisson Binomial d istribution an d for ea c h profile in v ok es Claim 1 to find a P oisson Binomial d istribution with suc h moment profi le, if such distr ibution exists, addin g it to the co v er if it do es exist. It then en umerates ov er all Poisson Binomial distrib u tions in ( n, k )-Binomial form and adds them to the co v er as well . The o v erall running time is as promised. 2 A naive app licatio n of Claim 1 results in running time O ( n 3 log n ) ·  1 ǫ  O (log 2 1 /ǫ ) . W e can improv e this to th e claimed running time as follo ws: for all possible va lues |L P | , |R P | suc h that |L P | + |R P | ≤ min( k 3 , n − m 2 d ( ǫ )+1 ), we inv oke Claim 1 with ˜ n = |L P | + |R P | , δ = d ( ǫ ), B = k 3 , n 0 = n 1 = 0, n s = |L P | , n b = |R P | , and moments µ ℓ = m ℓ , for ℓ = 1 , . . . , d ( ǫ ) , and µ ′ ℓ = m d ( ǫ )+ ℓ , for ℓ = 1 , . . . , d ( ǫ ). If for some p air |L P | , |R P | the algorithm succeeds in findin g probabilities matching th e pro vided moments, w e set m 2 d ( ǫ )+1 of t h e remaining probabilities equ al to 1 and the rest to 0. Otherwise, we outpu t “fail.” 10 4 Pro of of Th eorem 2 W e organize the pro of according to the stru cture and notation of our outline in Sectio n 1.2 . In particular, w e pro ceed to provide the details of S tage s 1 and 2, describ ed in the outline. The reader should refer to Section 1.2 f or notation. 4.1 Details of Stage 1 Define L k := { i i ∈ [ n ] ∧ p i ∈ (0 , 1 /k ) } and H k := { i i ∈ [ n ] ∧ p i ∈ (1 − 1 /k , 1) } . W e define the exp ectatio ns ( p ′ i ) i of the in termediate ind icato rs ( Z i ) i as follo ws. First, we set p ′ i = p i , for all i ∈ [ n ] \ L k ∪ H k . It follo ws that d TV   X i ∈ [ n ] \L k ∪H k X i , X i ∈ [ n ] \L k ∪H k Z i   = 0 . (5) Next, w e d efine the probabilities p ′ i , i ∈ L k , using the follo wing pro cedure: 1. S et r =  P i ∈L k p i 1 /k  ; and let L ′ k ⊆ L k b e an arb itrary subset of cardinalit y |L ′ k | = r . 2. S et p ′ i = 1 k , for all i ∈ L ′ k , and p ′ i = 0, for all i ∈ L k \ L ′ k . W e b ound the total v ariatio n distance d TV  P i ∈L k X i , P i ∈L k Z i  using the P oisson app ro ximation to the P oisson Binomial distribution. In particular, Th eorem 4 imp lies d TV   X i ∈L k X i , Poisso n   X i ∈L k p i     ≤ P i ∈L k p 2 i P i ∈L k p i ≤ 1 k P i ∈L k p i P i ∈L k p i = 1 /k . Similarly , d TV  P i ∈L k Z i , Poisso n  P i ∈L k p ′ i  ≤ 1 /k . Finally , we use Lemm a 3 (giv en b elo w and pro ve d in Section 6 ) to b ound the distance d TV   P oisson   X i ∈L k p i   , Poisso n   X i ∈L k p ′ i     ≤ 1 2  e 1 k − e − 1 k  ≤ 1 . 5 k , where we used that | P i ∈L k p i − P i ∈L k p ′ i | ≤ 1 /k . Using the triangle inequalit y the a b ov e imply d TV   X i ∈L k X i , X i ∈L k Z i   ≤ 3 . 5 k . (6) Lemma 3 (V ariation Distance of P oisson Di stribu tions) . L e t λ 1 , λ 2 > 0 . Then d TV (P oisson( λ 1 ) , Po isson ( λ 2 )) ≤ 1 2  e | λ 1 − λ 2 | − e −| λ 1 − λ 2 |  . 11 W e follo w a s im ilar rounding sc heme to define ( p ′ i ) i ∈H k from ( p i ) i ∈H k . Th at is, w e r ound some of the p i ’s to 1 − 1 /k and some of them to 1 so that | P i ∈H k p i − P i ∈H k p ′ i | ≤ 1 /k . As a resu lt, we get (to see this, r ep eat the argument employ ed ab ov e to the v aria bles 1 − X i and 1 − Z i , i ∈ H k ) d TV   X i ∈H k X i , X i ∈H k Z i   ≤ 3 . 5 k . (7) Using ( 5 ), ( 6 ), ( 7 ) and Lemm a 2 we get ( 2 ). 4.2 Details of Stage 2 Recall that M := { i | p ′ i / ∈ { 0 , 1 }} and m := |M| . Dep ending on on wh ether m ≤ k 3 or m > k 3 w e follo w differen t strategies to define the exp ectatio ns ( q i ) i of indicators ( Y i ) i . 4.2.1 The Case m ≤ k 3 First we set q i = p ′ i , for all i ∈ [ n ] \ M . It follo ws that d TV   X i ∈ [ n ] \M Z i , X i ∈ [ n ] \M Y i   = 0 . (8) F or the definition of ( q i ) i ∈M , we make use of Eh m ’s Binomial approximat ion to the P oisson Binomial distribution, stat ed as Th eorem 5 in Section 2.1 . W e start by partitioning M as M = M l ⊔ M h , where M l = { i ∈ M | p ′ i ≤ 1 / 2 } , and describ e b elo w a pro cedure for defining ( q i ) i ∈M l so that the follo wing hold: 1. d TV  P i ∈M l Z i , P i ∈M l Y i  ≤ 17 /k ; 2. f or all i ∈ M l , q i is an in teger multiple of 1 /k 2 . T o define ( q i ) i ∈M h , we apply the same pro cedure to (1 − p ′ i ) i ∈M h to obtain (1 − q i ) i ∈M h . Assum ing the correctness of our procedu re for probabilities ≤ 1 / 2 the follo wing should also hold: 1. d TV  P i ∈M h Z i , P i ∈M h Y i  ≤ 17 /k ; 2. f or all i ∈ M h , q i is an integer multiple of 1 / k 2 . Using Lemm a 2 , the ab o v e b oun ds imply d TV X i ∈M Z i , X i ∈M Y i ! ≤ d TV   X i ∈M l Z i , X i ∈M l Y i   + d TV   X i ∈M h Z i , X i ∈M h Y i   ≤ 34 /k . (9) No w that w e ha v e ( 9 ), u sing ( 8 ) and Lemma 2 w e ge t ( 3 ). So it suffices to defin e the ( q i ) i ∈M l prop erly . T o do this, w e define the partition M l = M l, 1 ⊔ M l, 2 ⊔ . . . ⊔ M l,k − 1 where for all j : M l,j =  i    p ′ i ∈  1 k + ( j − 1) j 2 1 k 2 , 1 k + ( j + 1) j 2 1 k 2  . (Notice that the length of in terv al used in the definition of M l,j is j k 2 .) No w, for eac h j = 1 , . . . , k − 1 suc h that M l,j 6 = ∅ , w e defin e ( q i ) i ∈M l,j via the follo wing procedu re: 12 1. S et p j, min := 1 k + ( j − 1) j 2 1 k 2 , p j, max := 1 k + ( j +1) j 2 1 k 2 , n j = |M l,j | , ¯ p j = P i ∈M l,j p ′ i n j . 2. S et r = j n j ( ¯ p j − p j, min ) j /k 2 k ; let M ′ l,j ⊆ M l,j b e an arb itrary subset of cardin alit y r . 3. S et q i = p j, max , for all i ∈ M ′ l,j ; 4. f or an a rb itrary index i ∗ j ∈ M l,j \ M ′ l,j , set q i ∗ j = n j ¯ p j − ( r p j, max + ( n j − r − 1) p j, min ); 5. fi nally , set q i = p j, min , for all i ∈ M l,j \ M ′ l,j \ { i ∗ j } . It is easy to see that 1. P i ∈M l,j p ′ i = P i ∈M l,j q i ≡ n j ¯ p j ; 2. f or all i ∈ M l,j \ { i ∗ j } , q i is an in teger multiple of 1 /k 2 . Moreo v er Theorem 5 implies: d TV   X i ∈M l,j Z i , B ( n j , ¯ p j )   ≤ P i ∈M l,j ( p ′ i − ¯ p j ) 2 ( n j + 1) ¯ p j (1 − ¯ p j ) ≤    n j ( j 1 k 2 ) 2 ( n j +1) p j, min (1 − p j, min ) , w hen j < k − 1 n j ( j 1 k 2 ) 2 ( n j +1) p j, max (1 − p j, max ) , when j = k − 1 ≤ 8 k 2 . A similar deriv ation giv es d TV  P i ∈M l,j Y i , B ( n j , ¯ p j )  ≤ 8 k 2 . So by the triangle inequalit y: d TV   X i ∈M l,j Z i , X i ∈M l,j Y i   ≤ 16 k 2 . (10) As Eq ( 10 ) h olds for all j = 1 , . . . , k − 1, an applicatio n of Lemm a 2 giv es: d TV   X i ∈M l Z i , X i ∈M l Y i   ≤ k − 1 X j =1 d TV   X i ∈M l,j Z i , X i ∈M l,j Y i   ≤ 16 k . Moreo v er, the q i ’s defined ab o v e are inte ger m ultiples of 1 /k 2 , except ma yb e for q i ∗ 1 , . . . , q i ∗ k − 1 . But w e can roun d these to their closest multiple of 1 /k 2 , in creasing d TV  P i ∈M l Z i , P i ∈M l Y i  b y at most 1 /k . 4.2.2 The Case m > k 3 Let t = |{ i | p ′ i = 1 }| . W e show that the r andom v ariable P i Z i is within tota l v ariation distance 9 /k from the Binomial distribu tion B ( m ′ , q ) where m ′ := &  P i ∈M p ′ i + t  2 P i ∈M p ′ 2 i + t ' and q := ℓ ∗ n , where ℓ ∗ satisfies P i ∈M p ′ i + t m ′ ∈ [ ℓ ∗ − 1 n , ℓ ∗ n ]. Notic e that: 13 •  P i ∈M p ′ i + t  2 ≤ ( P i ∈M p ′ 2 i + t )( m + t ), by the Cauc h y-Sch w arz inequalit y; and • P i ∈M p ′ i + t m ′ ≤ P i ∈M p ′ i + t ( P i ∈M p ′ i + t ) 2 P i ∈M p ′ 2 i + t = P i ∈M p ′ 2 i + t P i ∈M p ′ i + t ≤ 1. So m ′ ≤ m + t ≤ n , and th ere exists some ℓ ∗ ∈ { 1 , . . . , n } so that P i ∈M p ′ i + t m ′ ∈ [ ℓ ∗ − 1 n , ℓ ∗ n ]. F or fi xed m ′ and q , we set q i = q , f or all i ≤ m ′ , an d q i = 0, for all i > m ′ , an d compare the distributions of P i ∈M Z i and P i ∈M Y i . F or con v enience w e defin e µ := E " X i ∈M Z i # and µ ′ := E " X i ∈M Y i # , σ 2 := V ar " X i ∈M Z i # and σ ′ 2 := V ar " X i ∈M Y i # . The follo wing lemma compares the v alues µ , µ ′ , σ , σ ′ . Lemma 4. The fol lowing hold µ ≤ µ ′ ≤ µ + 1 , (11) σ 2 − 1 ≤ σ ′ 2 ≤ σ 2 + 2 , (12) µ ≥ k 2 , (13 ) σ 2 ≥ k 2  1 − 1 k  . (14 ) The pro of of Lemma 4 is give n in S ecti on 6 . T o compare P i ∈M Z i and P i ∈M Y i w e approximat e b oth by T r anslated P oisson distributions. T heorem 6 implies that d TV X i Z i , T P ( µ, σ 2 ) ! ≤ q P i p ′ 3 i (1 − p ′ i ) + 2 P i p ′ i (1 − p ′ i ) ≤ p P i p ′ i (1 − p ′ i ) + 2 P i p ′ i (1 − p ′ i ) ≤ 1 p P i p ′ i (1 − p ′ i ) + 2 P i p ′ i (1 − p ′ i ) = 1 σ + 2 σ 2 ≤ 1 k p 1 − 1 /k + 2 k 2  1 − 1 k  (using ( 14 )) ≤ 3 k , where for the last inequalit y we assumed k ≥ 3, bu t the b oun d of 3 /k clearly also holds for k = 1 , 2. Similarly , d TV X i Y i , T P ( µ ′ , σ ′ 2 ) ! ≤ 1 σ ′ + 2 σ ′ 2 ≤ 1 k q 1 − 1 k − 1 k 2 + 2 k 2  1 − 1 k − 1 k 2  (using ( 12 ),( 14 )) ≤ 3 k , 14 where for the last inequalit y we assumed k ≥ 3, bu t the b oun d of 3 /k clearly also holds for k = 1 , 2. By the triangle inequalit y w e then ha v e that d TV X i Z i , X i Y i ! ≤ d TV X i Z i , T P ( µ, σ 2 ) ! + d TV X i Y i , T P ( µ ′ , σ ′ 2 ) ! + d TV  T P ( µ, σ 2 ) , T P ( µ ′ , σ ′ 2 )  = 6 /k + d TV  T P ( µ, σ 2 ) , T P ( µ ′ , σ ′ 2 )  . (15) It remains to b ound the total v ariatio n distance b et wee n the tw o T ran s late d P oisson distrib u tions. W e mak e use of the follo wing lemma. Lemma 5 ([ BL06 ]) . L et µ 1 , µ 2 ∈ R and σ 2 1 , σ 2 2 ∈ R + \ { 0 } b e such that ⌊ µ 1 − σ 2 1 ⌋ ≤ ⌊ µ 2 − σ 2 2 ⌋ . Then d TV  T P ( µ 1 , σ 2 1 ) , T P ( µ 2 , σ 2 2 )  ≤ | µ 1 − µ 2 | σ 1 + | σ 2 1 − σ 2 2 | + 1 σ 2 1 . Lemma 5 imp lies d TV  T P ( µ, σ 2 ) , T P ( µ ′ , σ ′ 2 )  ≤ | µ − µ ′ | min( σ , σ ′ ) + | σ 2 − σ ′ 2 | + 1 min( σ 2 , σ ′ 2 ) ≤ 1 k q 1 − 1 k − 1 k 2 + 3 k 2  1 − 1 k − 1 k 2  (using Lemma 4 ) ≤ 3 /k , (16) where for the last inequalit y w e assumed k > 3, bu t the b oun d clearly also holds for k = 1 , 2 , 3. Using ( 15 ) and ( 16 ) w e get d TV X i Z i , X i Y i ! ≤ 9 /k , (17) whic h implies ( 3 ). 5 Pro of of Th eorem 3 Let X and Y b e t wo collections of indicators as in the statemen t of Theorem 3 . F or α ℓ ( · , · ) defi ned as in the statemen t of Theorem 7 , we claim the follo wing. Lemma 6. If P , Q ∈ [0 , 1] n satisfy pr op ert y ( C d ) in the statement of The or em 3 , then for al l p , ℓ ∈ { 0 , . . . , d } : α ℓ ( P , p ) = α ℓ ( Q , p ) . Pro of of lemma 6 : Firs t α 0 ( P , p ) = 1 = α 0 ( Q , p ) by definition. No w fi x ℓ ∈ { 1 , . . . , d } and consider th e fu nction f ( ~ x ) := α ℓ (( x 1 , . . . , x n ) , p ) in the v ariables x 1 , . . . , x n ∈ R . Observe that f is a symmetric p olynomial of d egree ℓ on x 1 , . . . , x n . Hence, from the theory of symmetric 15 p olynomials, it follo ws that f can b e wr itten as a p olynomial fun ction of the p o wer-sum s ymmetric p olynomials π 1 , . . . , π ℓ , where π j ( x 1 , . . . , x n ) := n X i =1 x j i , for all j ∈ [ ℓ ] , as the elemen tary symmetric p olynomial of degree j ∈ [ n ] can b e written as a p olynomial function of the p o w er-sum symmetric p olynomials π 1 , . . . , π j (e.g. [ Zol87 ]). No w ( C d ) imp lies that π j ( P ) = π j ( Q ), f or all j ≤ ℓ . So f ( P ) = f ( Q ), i.e . α ℓ ( P , p ) = α ℓ ( Q , p ).  F or all p ∈ [0 , 1], by combining Theorem 7 and Lemma 6 and w e get that P r [ X = m ] − P r [ Y = m ] = n X ℓ = d +1 ( α ℓ ( P , p ) − α ℓ ( Q , p )) · δ ℓ B n,p ( m ) , for all m ∈ { 0 , . . . , n } . Hence, for all p : d TV ( X, Y ) = 1 2 n X m =0 | P r [ X = m ] − P r [ Y = m ] | ≤ 1 2 n X ℓ = d +1 | α ℓ ( P , p ) − α ℓ ( Q , p ) | · k δ ℓ B n,p ( · ) k 1 ≤ 1 2 n X ℓ = d +1 ( | α ℓ ( P , p ) | + | α ℓ ( Q , p ) | ) · k δ ℓ B n,p ( · ) k 1 . (18) Plugging p = ¯ p := 1 n P i p i in to P rop osition 1 , we get θ ( P , ¯ p ) = P n i =1 ( p i − ¯ p ) 2 n ¯ p (1 − ¯ p ) ≤    max i { p i } − min i { p i }    ≤ 1 2 (see [ Ro o00 ]) and then 1 2 n X ℓ = d +1 | α ℓ ( P , ¯ p ) | · k δ ℓ B n, ¯ p ( · ) k 1 ≤ √ e ( d + 1) 1 / 4 2 − ( d +1) / 2 1 − 1 √ 2 d d +1 ( √ 2 − 1) 2 ≤ 6 . 5( d + 1) 1 / 4 2 − ( d +1) / 2 . But ( C d ) imp lies that P i q i = P i p i = ¯ p . So we get in a s im ilar fashion 1 2 n X ℓ = d +1 | α ℓ ( Q , ¯ p ) | · k δ ℓ B n, ¯ p ( · ) k 1 ≤ 6 . 5( d + 1) 1 / 4 2 − ( d +1) / 2 . Plugging these b ounds in to ( 18 ) we get d TV ( X, Y ) ≤ 13( d + 1) 1 / 4 2 − ( d +1) / 2 . 16 6 Deferred Pro ofs Pro of of Lemma 1 : Let X = P i X i and Y = P i Y i . It is ob vious th at, if ( p 1 , . . . , p n ) = ( q 1 , . . . , q n ), then the distribu tions of X and Y are the same. In the other directio n, w e show that, if X and Y h av e th e s ame distr ibution, then ( p 1 , . . . , p n ) = ( q 1 , . . . , q n ). C onsider the p olynomials: g X ( s ) = E  (1 + s ) X  = n Y i =1 E  (1 + s ) X i  = n Y i =1 (1 + p i s ); g Y ( s ) = E  (1 + s ) Y  = n Y i =1 E  (1 + s ) Y i  = n Y i =1 (1 + q i s ) . Since X and Y h av e the same d istribution, g X and g Y are equal, so they h av e the same d egree and ro ots. Notice that g X has degree n − |{ i | p i = 0 }| and roots {− 1 p i | p i 6 = 0 } . S imilarly , g Y has degree n − |{ i | q i = 0 }| and ro ots {− 1 q i | q i 6 = 0 } . Hence, ( p 1 , . . . , p n ) = ( q 1 , . . . , q n ).  Pro of of Lemma 2 : It follo ws from the coupling lemma th at for an y coup lin g of the v ariables X 1 , . . . , X n , Y 1 , . . . , Y n : d TV n X i =1 X i , n X i =1 Y i ! ≤ Pr " n X i =1 X i 6 = n X i =1 Y i # ≤ n X i =1 Pr[ X i 6 = Y i ] . (19) W e pr oceed to fi x a sp ecific coup lin g. F or all i , it follo ws from the optimal coup ling theorem that there exists a coupling of X i and Y i suc h that Pr[ X i 6 = Y i ] = d TV ( X i , Y i ). Using these in dividual couplings for eac h i we define a grand coupling of the v ariables X 1 , . . . , X n , Y 1 , . . . , Y n suc h that Pr[ X i 6 = Y i ] = d TV ( X i , Y i ), for all i . This coupling is f aithful b ecause X 1 , . . . , X n are m utually indep endent and Y 1 , . . . , Y n are mutually indep end en t. Under this coup lin g Eq ( 19 ) imp lies: d TV n X i =1 X i , n X i =1 Y i ! ≤ n X i =1 Pr[ X i 6 = Y i ] ≡ n X i =1 d TV ( X i , Y i ) . (20)  Claim 1. Fix inte gers ˜ n, δ, B , k ∈ N + , ˜ n, k ≥ 2 . Given a set of values µ 1 , . . . , µ δ , µ ′ 1 , . . . , µ ′ δ , wher e, for al l ℓ = 1 , . . . , δ , µ ℓ , µ ′ ℓ ∈ ( 0 ,  1 k 2  ℓ , 2  1 k 2  ℓ , . . . , B ) , discr e te sets T 1 , . . . , T ˜ n ⊆  0 , 1 k 2 , 2 k 2 , . . . , 1  , and four inte gers n 0 , n 1 ≤ ˜ n , n s , n b ≤ B , it is p o ssible to solve the system of e quations: (Σ) : X p i ∈ (0 , 1 / 2] p ℓ i = µ ℓ , for al l ℓ = 1 , . . . , δ , X p i ∈ (1 / 2 , 1) p ℓ i = µ ′ ℓ , for al l ℓ = 1 , . . . , δ , |{ i | p i = 0 }| = n 0 |{ i | p i = 1 }| = n 1 |{ i | p i ∈ (0 , 1 / 2] }| = n s |{ i | p i ∈ (1 / 2 , 1) }| = n b 17 with r esp e ct to the variables p 1 ∈ T 1 , . . . , p ˜ n ∈ T ˜ n , or to determine that no solution exists, in time O ( ˜ n 3 log 2 ˜ n ) B O ( δ ) k O ( δ 2 ) . Pro of of C laim 1 : W e use dynamic programming. Let us consider the f ollo wing tensor of dimen- sion 2 δ + 5: A ( i, z 0 , z 1 , z s , z b ; ν 1 , . . . , ν δ ; ν ′ 1 , . . . , ν ′ δ ) , where i ∈ [ ˜ n ], z 0 , z 1 ∈ { 0 , . . . , ˜ n } , z s , z b ∈ { 0 , . . . , B } and ν ℓ , ν ′ ℓ ∈ ( 0 ,  1 k 2  ℓ , 2  1 k 2  ℓ , . . . , B ) , for ℓ = 1 , . . . , δ. The tot al n umber o f cells in A is ˜ n · ( ˜ n + 1) 2 · ( B + 1) 2 · δ Y ℓ =1 ( B k 2 ℓ + 1) ! 2 ≤ O ( ˜ n 3 ) B O ( δ ) k O ( δ 2 ) . Ev ery cell of A is assigned v alue 0 or 1, as follo ws: A ( i, z 0 , z 1 , z s , z b ; ν 1 , . . . , ν δ , ν ′ 1 , . . . , ν ′ δ ) = 1 ⇐ ⇒             There exist p 1 ∈ T 1 , . . . , p i ∈ T i suc h that |{ j ≤ i | p j = 0 }| = z 0 , |{ j ≤ i | p j = 1 }| = z 1 , |{ j ≤ i | p j ∈ (0 , 1 / 2] }| = z s , |{ j ≤ i | p j ∈ (1 / 2 , 1) }| = z b , P j ≤ i : p j ∈ (0 , 1 / 2] p ℓ j = ν ℓ , for all ℓ = 1 , . . . , δ , P j ≤ i : p j ∈ (1 / 2 , 1) p ℓ j = ν ′ ℓ , for all ℓ = 1 , . . . , δ .             . Notice that w e n eed O ( ˜ n 3 ) B O ( δ ) k O ( δ 2 ) bits to store A and O (log ˜ n + δ log B + δ 2 log k ) bits to address cells of A . T o complete A w e can w ork in la y ers of increasing i . W e in itialize all en tries to v al ue 0. Th en, the fir st la y er A (1 , · , · ; · , . . . , · ) can b e completed ea sily as follo ws: A (1 , 1 , 0 , 0 , 0; 0 , 0 , . . . , 0; 0 , 0 , . . . , 0) = 1 ⇔ 0 ∈ T 1 ; A (1 , 0 , 1 , 0 , 0; 0 , 0 , . . . , 0; 0 , 0 . . . , 0) = 1 ⇔ 1 ∈ T 1 ; A (1 , 0 , 0 , 1 , 0; p, p 2 , . . . , p δ ; 0 , . . . , 0) = 1 ⇔ p ∈ T 1 ∩ (0 , 1 / 2]; A (1 , 0 , 0 , 0 , 1; 0 , . . . , 0; p, p 2 , . . . , p δ ) = 1 ⇔ p ∈ T 1 ∩ (1 / 2 , 1) . Inductive ly , to complete la yer i + 1, we consider all the non-zero entries of lay er i and f or ev ery suc h non-zero en try and for every v i +1 ∈ T i +1 , we fin d whic h en try of la y er i + 1 w e w ould transition to if w e chose p i +1 = v i +1 . W e set that entry equ al to 1 and we also s a v e a p ointer to this en try f r om the corresp ondin g ent ry of la y er i , lab eling th at p ointer w ith the v alue v i +1 . The bit op erations required to complete la y er i + 1 are b oun ded b y |T i +1 | ( ˜ n + 1) 2 B O ( δ ) k O ( δ 2 ) O (log ˜ n + δ log B + δ 2 log k ) ≤ O ( ˜ n 2 log ˜ n ) B O ( δ ) k O ( δ 2 ) . Therefore, the o v erall time needed to complete A is O ( ˜ n 3 log ˜ n ) B O ( δ ) k O ( δ 2 ) . 18 Ha ving completed A , it is easy to chec k if there is a solution to (Σ). A solution exists if and only if A ( ˜ n, n 0 , n 1 , n s , n b ; µ 1 , . . . , µ δ ; µ ′ 1 , . . . , µ ′ δ ) = 1 , and can b e found by tracing th e p oin ters from this cell of A back to lev el 1. The o v erall r u nning time is dominated by the time needed to complete A .  Pro of of lemma 3 : Without loss of generalit y assum e th at 0 < λ 1 ≤ λ 2 and denote δ = λ 2 − λ 1 . F or all i ∈ { 0 , 1 , . . . } , denote p i = e − λ 1 λ i 1 i ! and q i = e − λ 2 λ i 2 i ! . Finally , define I ∗ = { i : p i ≥ q i } . W e ha ve X i ∈I ∗ | p i − q i | = X i ∈I ∗ ( p i − q i ) ≤ X i ∈I ∗ 1 i ! ( e − λ 1 λ i 1 − e − λ 1 − δ λ i 1 ) = X i ∈I ∗ 1 i ! e − λ 1 λ i 1 (1 − e − δ ) ≤ (1 − e − δ ) + ∞ X i =0 1 i ! e − λ 1 λ i 1 = 1 − e − δ . On th e other hand X i / ∈I ∗ | p i − q i | = X i / ∈I ∗ ( q i − p i ) ≤ X i / ∈I ∗ 1 i ! ( e − λ 1 ( λ 1 + δ ) i − e − λ 1 λ i 1 ) = X i / ∈I ∗ 1 i ! e − λ 1 (( λ 1 + δ ) i − λ i 1 ) ≤ + ∞ X i =0 1 i ! e − λ 1 (( λ 1 + δ ) i − λ i 1 ) = e δ + ∞ X i =0 1 i ! e − ( λ 1 + δ ) ( λ 1 + δ ) i − + ∞ X i =0 1 i ! e − λ 1 λ i 1 = e δ − 1 . Com bining the ab o v e w e get the result.  Pro of of lemma 4 : W e ha v e µ m ′ = P i ∈M p ′ i + t m ′ ≤ q = ℓ ∗ n ≤ P i ∈M p ′ i + t m ′ + 1 n = µ m ′ + 1 n . Multiplying by m ′ w e get: µ ≤ m ′ q ≤ µ + m ′ n . As µ ′ = m ′ q and m ′ ≤ n , w e get µ ≤ µ ′ ≤ µ + 1 . Moreo ver, since m ≥ k 3 , µ ≥ X i ∈M p ′ i ≥ m 1 k ≥ k 2 . 19 F or the v ariances we ha ve : σ ′ 2 = m ′ q (1 − q ) = m ′ · ℓ ∗ n ·  1 − ℓ ∗ − 1 n − 1 n  ≥ X i ∈M p ′ i + t ! ·  1 − 1 n − P i ∈M p ′ i + t m ′  = (1 − 1 /n ) X i ∈M p ′ i + t ! − ( P i ∈M p ′ i + t ) 2 m ′ ≥ (1 − 1 /n ) X i ∈M p ′ i + t ! − ( P i ∈M p ′ i + t ) 2 ( P i ∈M p ′ i + t ) 2 P i ∈M p ′ 2 i + t = X i ∈M p ′ i (1 − p ′ i ) − 1 n X i ∈M p ′ i + t ! = σ 2 − 1 n X i ∈M p ′ i + t ! ≥ σ 2 − 1 . (21) In the other direction: σ ′ 2 = m ′ q (1 − q ) = m ′ ·  ℓ ∗ − 1 n + 1 n  ·  1 − ℓ ∗ n  ≤ m ′ ·  ℓ ∗ − 1 n  ·  1 − ℓ ∗ n  + m ′ n ≤ X i ∈M p ′ i + t ! ·  1 − P i ∈M p ′ i + t m ′  + 1 = X i ∈M p ′ i + t ! −  P i ∈M p ′ i + t  2 m ′ + 1 ≤ X i ∈M p ′ i + t ! −  P i ∈M p ′ i + t  2 ( P i ∈M p ′ i + t ) 2 P i ∈M p ′ 2 i + t + 1 + 1 = X i ∈M p ′ i + t ! − X i ∈M p ′ 2 i + t !  P i ∈M p ′ i + t  2  P i ∈M p ′ i + t  2 + P i ∈M p ′ 2 i + t + 1 = X i ∈M p ′ i + t ! − X i ∈M p ′ 2 i + t ! 1 − P i ∈M p ′ 2 i + t  P i ∈M p ′ i + t  2 + P i ∈M p ′ 2 i + t ! + 1 = X i ∈M p ′ i (1 − p ′ i ) +  P i ∈M p ′ 2 i + t  2  P i ∈M p ′ i + t  2 + P i ∈M p ′ 2 i + t + 1 = σ 2 +  P i ∈M p ′ 2 i + t  2  P i ∈M p ′ i + t  2 + P i ∈M p ′ 2 i + t + 1 ≤ σ 2 + 2 . (22) Finally , σ 2 = X i ∈M p ′ i (1 − p ′ i ) ≥ m 1 k  1 − 1 k  ≥ k 2  1 − 1 k  . 20  Prop osition 2. F or al l d ∈ [ n ] , Condition ( C d ) in the statement of The or em 3 is e quivalent to the fol lowing c ondition: ( V d ) : E   n X i =1 X i ! ℓ   = E   n X i =1 Y i ! ℓ   , for al l ℓ ∈ [ d ] . Pro of of Prop osition 2 : ( V d ) ⇒ ( C d ): First n otic e that, for all ℓ ∈ [ n ], E h ( P n i =1 X i ) ℓ i can b e written as a weig hted sum of the elemen tary symmetric p olynomials ψ 1 ( P ), ψ 2 ( P ),..., ψ ℓ ( P ), wh ere, for all t ∈ [ n ], ψ t ( P ) is defined as ψ t ( P ) := ( − 1) t X S ⊆ [ n ] | S | = t Y i ∈ S p i . ( V d ) imp lies th en b y ind uction ψ ℓ ( P ) = ψ ℓ ( Q ) , for all ℓ = 1 , . . . , d . (23) Next, for all t ∈ [ n ], defin e π t ( P ) to b e the p o w er su m sy m metric p olynomial of degree t π t ( P ) := n X i =1 p t i . No w, fi x an y ℓ ≤ d . Sin ce π ℓ ( P ) is a symmetric p olynomial of d egree ℓ on the v aria bles p 1 , . . . , p n , it can b e expressed as a fu nction o f the elementa ry symmetric p olynomials ψ 1 ( P ) , . . . , ψ ℓ ( P ). S o, b y ( 23 ), π ℓ ( P ) = π ℓ ( Q ). Since this holds for any ℓ ≤ d , ( C d ) is satisfied. The imp lication ( C d ) ⇒ ( V d ) is established in a similar fashion. ( C d ) sa ys that π ℓ ( P ) = π ℓ ( Q ) , for all ℓ = 1 , . . . , d. (24) Fix some ℓ ≤ d . E h ( P n i =1 X i ) ℓ i can b e w ritten as a w eigh ted sum of the elementa ry symm etric p olynomials ψ 1 ( P ), ψ 2 ( P ),..., ψ ℓ ( P ). Also, for all t ∈ [ ℓ ], ψ t ( P ) can b e written as a p olynomial function of π 1 ( P ) , . . . , π t ( P ) (see, e.g., [ Zol87 ]). S o from ( 24 ) it follo ws that E h ( P n i =1 X i ) ℓ i = E h ( P n i =1 Y i ) ℓ i . Sin ce this h olds for an y ℓ ≤ d , ( V d ) is satisfied.  Corollary 1. L et P := ( p i ) n i =1 ∈ [1 / 2 , 1] n and Q := ( q i ) n i =1 ∈ [1 / 2 , 1] n b e two c ol le ctions of pr ob ability values in [1 / 2 , 1] . L et also X := ( X i ) n i =1 and Y := ( Y i ) n i =1 b e two c ol le ctions of mutual ly indep endent indic ators with E [ X i ] = p i and E [ Y i ] = q i , for al l i ∈ [ n ] . If for some d ∈ [ n ] Condition ( C d ) in the statement of The or em 3 is satisfie d, then d TV X i X i , X i Y i ! ≤ 13( d + 1) 1 / 4 2 − ( d +1) / 2 . 21 Pro of of C orollary 1 : Defin e X ′ i = 1 − X i and Y ′ i = 1 − Y i , for all i . Also, denote p ′ i = E [ X ′ i ] = 1 − p i and q ′ i = E [ Y ′ i ] = 1 − q i , for all i . By assumption: n X i =1  1 − p ′ i  ℓ = n X i =1  1 − q ′ i  ℓ , for all ℓ = 1 , . . . , d. (25) Using th e Binomial theorem and in duction, we see that ( 25 ) imp lies: n X i =1 p ′ ℓ i = n X i =1 q ′ ℓ i , for al l ℓ = 1 , . . . , d. Hence we can app ly Theorem 3 to deduce d TV X i X ′ i , X i Y ′ i ! ≤ 13( d + 1) 1 / 4 2 − ( d +1) / 2 . 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