CCoin Theorems and the Fourier Expansion
Rohit Agrawal ∗ June 11, 2019
Abstract
In this note we compare two measures of the complexity of a class F of Boolean functions studied in(unconditional) pseudorandomness: F ’s ability to distinguish between biased and uniform coins (the coinproblem ), and the norms of the different levels of the Fourier expansion of functions in F (the Fouriergrowth ). We show that for coins with low bias ε = o (1 /n ), a function’s distinguishing advantage inthe coin problem is essentially equivalent to ε times the sum of its level 1 Fourier coefficients, whichin particular shows that known level 1 and total influence bounds for some classes of interest (such asconstant-width read-once branching programs) in fact follow as a black-box from the corresponding cointheorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it iswell-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, evenfor large ε , and we discuss here the possibility of a converse. A natural question one can ask when studying a limited model of computation is how well it can solve somebasic computational task, such as distinguishing between an unfair and a fair coin:
Definition 1.1 (Coin problem) . The advantage α of a Boolean function f : {− , } n → {− , } in distin-guishing ε -biased coins from uniform coins is defined to be α = (cid:12)(cid:12)(cid:12) E [ f ( X nε )] − E [ f ( X n )] (cid:12)(cid:12)(cid:12) , where X nδ are iid random variables over {− , } with expectation δ , so that X n are uniform random bits.A set F of Boolean functions is said to solve the ε -coin problem with advantage α for α = max f ∈F (cid:12)(cid:12)(cid:12) E [ f ( X nε )] − E [ f ( X n )] (cid:12)(cid:12)(cid:12) . One can see (e.g. via the equivalence of ‘ and total variation distance of probability distributions) thatthe unique Boolean function f achieving the greatest distinguishing advantage is of the form f ( x ) = 1 ifand only if the number of 1s in x is at least k , for the smallest k such that (1 + ε ) k (1 − ε ) n − k ≥
1. Thisis simply a (symmetric) linear threshold function, and so one sees that the coin problem for a class F ofBoolean functions is closely related to the ability of F to approximate threshold functions, and in particularthe majority function. The study of threshold functions in limited models is extensive, with early works dueto e.g. Ajtai [Ajt83] and Valiant [Val84], and early explicit consideration of the coin problem by Shaltiel andViola [SV10], Brody and Verbin (who introduced the name) [BV10], and Aaronson [Aar10]. These resultsare generally stated as a coin theorem , giving a statement of the form F n (parametrized by some parameter,e.g. the input length) cannot solve the ε -coin problem except with some advantage β ( n, ε ). Subsequent ∗ John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. Email: [email protected] . Supported by the Department of Defense (DoD) through the National Defense Science &Engineering Graduate Fellowship (NDSEG) Program. a r X i v : . [ c s . CC ] J un ork has given tight coin theorems on constant-width read-once branching programs (Steinberger [Ste13]), AC (Cohen, Ganor, and Raz [CGR14]), AC [ ⊕ ] (Limaye, Sreenivasaiah, Srinivasan, Tripathi, and Venkitesh[LSS + f , for which we follow the notation ofO’Donnell [O’D14]. Definition 1.2.
The
Fourier expansion of a function f : {− , } n → R is its unique representation as amultilinear polynomial f ( x ) = f ( x , . . . , x n ) = X S ⊆ [ n ] b f ( S ) Y i ∈ S x i , where [ n ] def = { , , . . . , n } and b f ( S ) is called the Fourier coefficient of f on S . The coefficient b f ( S ) is said tobe at level | S | , and can be expressed as b f ( S ) = E x " f ( x ) Y i ∈ S x i where the expectation is taken over the uniform distribution over {− , } n . In particular, b f ( ∅ ) = E x [ f ( x )].The total influence of f , denoted I [ f ], is defined as I [ f ] = P ni =1 P S i b f ( S ) = P S ⊆ [ n ] | S | b f ( S ) . If f : {− , } n → {− , } is monotone, then the total influence has the simpler expression I [ f ] = P ni =1 b f ( { i } ).The use of Fourier analysis in the study of Boolean functions was pioneered by the work of Kahn, Kalai,and Linial [KKL88], which kicked off a long line of work, including early results analyzing DNFs and CNFs byBrandman, Orlitsky, and Hennessy [BOH90], and AC by Linial, Mansour, and Nisan [LMN93]. More recently,Reingold, Steinke, and Vadhan [RSV13] introducted the concept of Fourier growth for use in constructions ofpseudorandom generators via the iterative random restriction framework of Ajtai and Wigderson [AW89],and particularly in the sense of the later work of Gopalan, Meka, Reingold, Trevisan, and Vadhan [GMR + Definition 1.3 ([RSV13]) . Given f : {− , } n → R , we define its level k ‘ norm to be L k ( f ) = X | S | = k (cid:12)(cid:12)(cid:12) b f ( S ) (cid:12)(cid:12)(cid:12) . For a class F of functions, we define the level k ‘ norm of F to be L k ( F ) = sup f ∈F L k ( f ). F is said tohave Fourier growth with base t if L k ( F ) ≤ t k for all k .Fourier growth bounds were introduced by [RSV13] to capture the fact that (roughly speaking) a randomrestriction that keeps input bits alive with probability p reduces the level k ‘ norm of a function by p k , andso a random restriction keeping an O (1 /t ) fraction of bits alive simplifies a function with Fourier growth withbase t into one with constant total Fourier ‘ norm, which is then easily fooled by a small-bias generator (Naorand Naor [NN93]). Fourier growth bounds have been studied for classes such as AC [LMN93, Hås01, Tal17],regular and constant-width read-once branching programs [RSV13, CHRT18], and product tests [Lee19].In this work, we study implications between bounds on the advantage of a class of Boolean functions inthe coin problem and bounds on the Fourier spectrum. Some connections along this line are well known, forexample it is folklore (see e.g. Tal [Tal17]) that Fourier growth bounds of the form L k ( f ) ≤ t k imply a strongcoin theorem with bound β ( n, ε ) = ε · O ( t ) for all ε = O (1 /t ). Recently, Tal [Tal19] (who graciously allowedthe author to include his result in this note) showed that if a class of functions F is closed under restriction,then for F to satisfy such a strong coin theorem it is enough for it to merely have L ( F ) ≤ t rather thanhave Fourier growth with base t .In this note, we give the first (to the best of our knowledge) results in the other direction, showing thatcoin theorems for small ε = o (1 /n ) are equivalent to bounds on the sum of the level 1 Fourier coefficients,and in particular to bounds on L ( F ) assuming that these coefficients can be taken to be non-negative (e.g.if F consists of monotone functions, or is closed under negating input variables). Thus, perhaps surprisingly,2ven coin theorems for only a small range of ε are sufficient to bound the level 1 spectrum of the class F ,even for F not closed under restriction. This allows us to give simple “black-box” proofs of existing level 1bounds in the literature, such as the bound for constant-width read-once branching programs due to Steinke,Vadhan, and Wan [SVW17] in 2014, which was conjectured by [RSV13].These results are summarized in Figure 1, showing the implications between bounds on coin problemsand Fourier growth for a class F , arranged from top to bottom in order of strength (under the assumptionthat F is monotone or closed under negation of input variables, but need not be closed under restriction). If F is additionally closed under restriction, then the bottom two layers collapse. L k ( F ) ≤ t k F has a coin theorem O ( t · ε ) for ε ≤ O (1 /t ) L ( F ) ≤ t F has a coin theorem O ( t · ε ) for ε ≤ /n ImmediateImmediate Folklore FolkloreLemma 3.2 [Tal19] Corollary 2.3 andLemma 3.2 [Tal19]Corollary 2.6Corollary 2.3Corollary 2.3 Any FF monotone or closed under negation of input variables F closed under restrictionNew to this workFigure 1: Implications studied in this work for a class F of Boolean functions, assuming t ≥ F closed under restrictionand negations which satisfies a coin theorem of the form ε · O (log n ) for all ε ≤ O (1 / log n ) (and therefore has L ( F ) ≤ O (log n )), but yet has L ( F ) = Ω(log n · n ). This result may be of interest because, intriguingly,many natural classes of Boolean functions F with known coin theorems or level-1 bounds are also known(or conjectured) to satisfy corresponding Fourier growth bounds (and in fact we are unaware of any naturalclass of Boolean functions satisfying these closure conditions and a L ( F ) ≤ t bound or coin theorem ε · O ( t )which does not at least conjecturally also have Fourier growth with base O ( t )). We hope that this resultmay help point the way to giving some additional constraints under which one could hope for a L or coinproblem bound to imply a general L k bound. The main technical result of this section is the following simple proposition.3 roposition 2.1.
For every f : {− , } n → R , the rate of convergence of the limit lim ε → ε (cid:16) E [ f ( X nε )] − E [ f ( X n )] (cid:17) = n X i =1 b f ( { i } ) can be bounded for every | ε | ≤ / √ n as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε (cid:16) E [ f ( X nε )] − E [ f ( X n )] (cid:17) − n X i =1 b f ( { i } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ε | · n · q Var( f ( X n )) , where in particular if f : {− , } n → [ − , then the right-hand side is bounded by | ε | · n . In particular, bounds on the coin problem for f for small ε (e.g. ε = o (1 /n )) are equivalent to bounds onthe sum of the level 1 Fourier coefficients of f . Before giving the (straightforward) proof of Proposition 2.1(along with some slightly tighter but more technical bounds in terms of quantities smaller than Var( f ( X n ))),we first note some immediate corollaries, including some simple new proofs of known results in the literatureon Fourier growth.Most basically, if we can assume that the level 1 Fourier coefficients of f are non-negative, we get upperbounds on the ‘ norm of these coefficients: Corollary 2.2.
Let f : {− , } n → [ − , be such that b f ( { i } ) ≥ for every ≤ i ≤ n (e.g. f is monotone).Then L ( f ) = n X i =1 (cid:12)(cid:12)(cid:12) b f ( { i } ) (cid:12)(cid:12)(cid:12) ≤ inf | ε |≤ / √ n (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) ε (cid:16) E [ f ( X nε )] − E [ f ( X n )] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + | ε | n (cid:27) In particular, if f is monotone, then its total influence I [ f ] = L ( f ) satisfies the same bound. Beyond monotonicity, another way to establish such a bound is if f is part of a larger class of functionswith basic closure properties that all simultaneously satisfy a coin theorem. Corollary 2.3.
Let F be a class of functions f : {− , } n → [ − , such that1. F satisfies a coin theorem with bound β ( n, ε ) , meaning | E [ f ( X nε )] − E [ f ( X n )] | ≤ β ( n, ε ) for every f ∈ F .2. For every f ∈ F there exists g ∈ F such that b g ( { i } ) = (cid:12)(cid:12)(cid:12) b f ( { i } ) (cid:12)(cid:12)(cid:12) for every ≤ i ≤ n (e.g. F is closedunder negation of input variables, or consists only of monotone functions).Then L ( F ) = sup f ∈F n X i =1 (cid:12)(cid:12)(cid:12) b f ( { i } ) (cid:12)(cid:12)(cid:12) ≤ inf | ε |≤ / √ n (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) β ( n, ε ) ε (cid:12)(cid:12)(cid:12)(cid:12) + | ε | n (cid:27) and in particular, every monotone f ∈ F has I [ f ] = L ( f ) at most the above. Note that taking ε = 1 /n in Corollary 2.3 gives the simple upper bound of n · | β ( n, /n ) | + 1. To showthe applicability of this result, we show how it can be used to give simple proofs of several existing results inthe literature. As it is not relevant for us, we will not give formal definitions of the classes F involved anddefer to the original papers for such details.For constant-width read-once branching programs, Brody and Verbin [BV10] first claimed a coin theorem,which was improved by Steinberger [Ste13] to give an optimal bound. Interestingly, Steinberger also separatelyproved (using the same techniques) a total influence bound on monotone constant-width read-once branchingprograms, which was generalized (again using the same techniques) by Steinke, Vadhan, and Wan [SVW17]in 2014 to a corresponding level 1 Fourier bound for (non-necessarily monotone) constant-width read-oncebranching programs. Applying Corollary 2.3, we see that these latter results can in fact be derived (up toconstant factors) using Steinberger’s coin theorem as a black box.4 orollary 2.4 ([Ste13, SVW17]) . Let f : {− , } n → {− , } be computable by a width- w read-once branchingprogram. Then L ( f ) ≤ O w (log n ) w − + O (1) . In particular, if f is monotone then f has total influence I [ f ] = L ( f ) satisfying the same bound.Proof. Steinberger’s full coin theorem [Ste13, Full version, Corollary 1] states that for every integer r ≥ (cid:12)(cid:12)(cid:12) E [ f ( X nε )] − E [ f ( X n )] (cid:12)(cid:12)(cid:12) ≤ εr w − + (cid:0) n + r w − (cid:1) ( w − (cid:18) − ε (cid:19) r − so since width- w read-once branching programs are closed under negation if input variables, Corollary 2.3implies that for 0 < ε ≤ / √ n that L ( f ) ≤ r w − + 1 ε (cid:0) n + r w − (cid:1) ( w − (cid:18) − ε (cid:19) r − + εn so setting ε = 1 /n and taking r = d n e + 1 gives the result.One thing to note about the above proof is that the coin theorem we used of [Ste13] was suboptimal inthe range ε = n − ω (1) , but still implied the optimal level 1 bound of [SVW17]. Using Proposition 2.1, we canalso improve the [Ste13] coin theorem for small ε by setting ε = 1 /n in the following corollary: Corollary 2.5.
Let f : {− , } n → [ − , satisfy a coin theorem of | ε | · B for all | ε | ≥ ε , where B ≥ and ε ≤ / √ n . Then for all | ε | < ε , it holds that (cid:12)(cid:12)(cid:12) E [ f ( X nε )] − E [ f ( X n )] (cid:12)(cid:12)(cid:12) ≤ | ε | · (cid:0) B + n ( | ε | + ε ) (cid:1) ≤ | ε | · ( B + 2 n · ε ) . In particular, f satisfies a coin theorem with bound | ε | · ( B + 2 n · ε ) for all | ε | ≤ . The proof of Corollary 2.5 goes by using Proposition 2.1 applied with ε to derive a bound on the sum ofthe level 1 Fourier coefficients of f , then applying the following converse of Corollary 2.3 to derive a cointheorem for small ε from such a bound: Corollary 2.6.
Let f : {− , } n → [ − , have (cid:12)(cid:12)(cid:12)P ni =1 b f ( { i } ) (cid:12)(cid:12)(cid:12) ≤ t (e.g. L ( f ) ≤ t ). Then for all | ε | ≤ / √ n , (cid:12)(cid:12)(cid:12) E [ f ( X nε )] − E [ f ( X n )] (cid:12)(cid:12)(cid:12) ≤ | ε | · ( t + | ε | · n ) , so that in particular f satisfies a coin theorem with bound t · | ε | for all | ε | ≤ t/n .Proof. The triangle inequality and Proposition 2.1 imply that | E [ f ( X nε )] − E [ f ( X n )] | = | ε | · (cid:12)(cid:12)(cid:12)(cid:12) ε · (cid:16) E [ f ( X nε )] − E [ f ( X n )] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ε | · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε (cid:16) E [ f ( X nε )] − E [ f ( X n )] (cid:17) − n X i =1 b f ( { i } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 b f ( { i } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! ≤ | ε | · ( | ε | · n + t )as desired. Proof of Corollary 2.5.
By the triangle inequality and Proposition 2.1 we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 b f ( { i } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 b f ( { i } ) − ε (cid:16) E (cid:2) f (cid:0) X nε (cid:1)(cid:3) − E [ f ( X n )] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ε (cid:16) E (cid:2) f (cid:0) X nε (cid:1)(cid:3) − E [ f ( X n )] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ε | · n + B, so the result follows by applying Corollary 2.6 with t = B + | ε | · n .5t remains to prove Proposition 2.1, for which we will need just one simple fact about the Fourier expansion. Lemma 2.7 (Parseval’s identity) . The Fourier coefficients of f : {− , } n → R satisfy X S ⊆ [ n ] b f ( S ) = E h f ( X n ) i . In particular,
Var( f ( X n )) = P | S |≥ b f ( S ) .Proof of Proposition 2.1. Let f : {− , } n → R . Then by the multilinearity of the Fourier expansion, wehave E [ f ( X nε )] − E [ f ( X n )] = X S ⊆ [ n ] b f ( S ) E x ∼ X nε "Y i ∈ S x i − E x ∼ X n "Y i ∈ S x i = X ∅6 = S ⊆ [ n ] b f ( S ) · ε | S | ε (cid:16) E [ f ( X nε )] − E [ f ( X n )] (cid:17) − n X i =1 b f ( { i } ) = 1 ε n X k =2 ε k X | S | = k b f ( S )The limit claim follows since the right-hand side is a polynomial in ε with no constant term, and thus goes to0 with ε . For an effective bound for nonzero ε , we see (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε (cid:16) E [ f ( X nε )] − E [ f ( X n )] (cid:17) − n X i =1 b f ( { i } ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ε | n X k =2 | ε | k X | S | = k (cid:12)(cid:12)(cid:12) b f ( S ) (cid:12)(cid:12)(cid:12) ≤ | ε | n X k =2 | ε | k vuut(cid:18) nk (cid:19) X | S | = k b f ( S ) (by Cauchy-Schwarz)We show two different techniques to bound this sum, in terms of either Var( f ( X n )) = P | S |≥ b f ( S ) ormax k ≥ P | S | = k b f ( S ) (though the latter result is not needed to prove the claim, we present the simple proofhere since depending on the function f it may be significantly stronger). For the former, we have1 | ε | n X k =2 | ε | k vuut(cid:18) nk (cid:19) X | S | = k b f ( S ) ≤ | ε | vuut n X k =2 (cid:18) nk (cid:19) · ε k · s X | S |≥ b f ( S ) (by Cauchy-Schwarz)= 1 | ε | q (1 + ε ) n − (1 + nε ) · vuut Var( f ( X n )) − n X i =1 b f ( { i } ) Then (1 + ε ) n − (1 + nε ) ≤ e nε − (1 + nε ) ≤ ( nε ) for nε ≤
1, which gives the claim. For the other6ound, we have1 | ε | n X k =2 | ε | k vuut(cid:18) nk (cid:19) X | S | = k b f ( S ) ≤ | ε | · s max k ≥ X | S | = k b f ( S ) · n X k =2 (cid:0) | ε |√ n (cid:1) k (since (cid:18) nk (cid:19) ≤ n k )= s max k ≥ X | S | = k b f ( S ) · | ε | · | ε | n − | ε | n +1 √ n n +1 − | ε |√ n = s max k ≥ X | S | = k b f ( S ) · | ε | n − | ε | n √ n n +1 − | ε |√ n ≤ s max k ≥ X | S | = k b f ( S ) · · | ε | · n (if | ε | ≤ / (2 √ n ))as desired. The previous section demonstrated that bounds on the level 1 Fourier coefficients are essentially equivalent tocoin theorems for inverse-polynomially small error ε = o (1 /n ). This raises two natural questions: can we sayanything about either coin theorems for larger ε , or about bounds on the Fourier spectrum beyond level 1?These questions are of interest because, perhaps surprisingly, many natural classes of Boolean functions F for which we know level-1 bounds are also known (or conjectured) to satisfy corresponding Fourier growthbounds L k ( F ) ≤ O (cid:0) L ( F ) (cid:1) k for all k . For example, AC (Tal [Tal17]) and the class of product tests (Lee[Lee19]) are known to have this property, and constant-width read-once branching programs (cwROBPs) arebelieved to (Chattopadhyay, Hatami, Reingold, and Tal [CHRT18] explicitly make this conjecture and provealmost this optimal result). Furthermore, these classes all have known corresponding coin theorems which arenot only capable of proving the known L bound via Corollary 2.3, but are also valid for all ε = O (1 / L ( F ))(Cohen, Ganor, and Raz [CGR14] for AC , Lee and Viola [LV18] for product tests, and Steinberger [Ste13]for cwROBPs). One might therefore hope that there is a stronger relationship between Fourier growth andcoin theorems.One direction, that Fourier growth bounds of this form imply coin theorems, is well-known (see e.g. [Tal17],this can also be seen in the proof of Proposition 2.1 by replacing the first Cauchy-Schwarz step), so the goal ofthis section is to explore the possibility of a converse. Generally, allowing for both additive and multiplicativelosses, one might ask something like the following: Question . Is there a “natural” set of conditions C such that the following is true: Let F = ( F n ) n ∈ N be aclass of Boolean functions f n : {− , } n → {− , } satisfying C . Then if F satisfies a coin theorem with bound | ε | · B ( n ), meaning for all n ∈ N , f n ∈ F n and | ε | ≤ | E [ f n ( X nε )] − E [ f n ( X n )] | ≤ | ε | · B ( n ),then there exists a constant c F such that L k ( F ) ≤ ( c F · (1 + B ( n ))) k for all k .Perhaps the most natural condition to impose, beyond closure under negations as considered in theprevious section, is closure under restriction , that is, fixing parts of the input, since intuitively this has theproperty of “reducing the level” of any Fourier coefficient containing one of the fixed inputs. Furthermore,all the classes of Boolean functions mentioned earlier in this section are closed under restriction, and weare not aware of any natural class of Boolean functions with these properties which does not (at leastconjecturally) satisfy a corresponding L k bound. However, Tal [Tal19] recently gave evidence suggesting thatthis is not enough, showing that any F closed under restriction satisfies a coin theorem of ε · O ( L ( F )) for7 = O (1 / L ( F )), so that for such F there is essentially no difference between coin theorems for ε = o (1 /n )and ε = O (1 / L ( F )). Lemma 3.2 ([Tal19] ) . Let F be a class of Boolean functions which is closed under restriction and satisfies (cid:12)(cid:12)(cid:12)P ni =1 b f ( { i } ) (cid:12)(cid:12)(cid:12) ≤ t for every f ∈ F (e.g. L ( F ) ≤ t ). Then for all | ε | < it holds that (cid:12)(cid:12)(cid:12) E [ f ( X nε )] − E [ f ( X n )] (cid:12)(cid:12)(cid:12) ≤ ln (cid:18) − ε (cid:19) · t. In particular, F satisfies a coin theorem of | ε | · O ( t ) for all | ε | = min(1 /t, . . As an example of the power of this result, note that it implies the coin theorem for AC of Cohen, Ganor,and Raz [CGR14] as a corollary of Boppana’s [Bop97] bound on the total influence of that class. Corollary 3.3 ([CGR14]) . Let f : {− , } n → {− , } be computable by a size s , depth d Boolean circuitand | ε | ≤ . Then | E [ f ( X nε )] − E [ f ( X n )] | ≤ | ε | · O d (log d − ( s )) . We include the proof of Lemma 3.2 at the end of this section, but we will first use Tal’s result to providea simple proof that there exist classes of Boolean functions F which are closed under restriction and negationsof input variables and satisfy a coin theorem and level 1 bound of B ( n ), but have L ( F ) ≥ Ω( n · B ( n )),thereby showing that these properties themselves are not enough to give a positive answer to Question 3.1. Proposition 3.4.
For every function √ log n + O (1) < B ( n ) < √ n and sufficiently large odd integer n ,there is a class F B of Boolean functions on at most n bits that is closed under restriction, negation of inputvariables, and negation of outputs such that L ( F ) ≤ B and F satisfies a coin theorem of | ε | · O ( B ) for all ε = O (1 /B ) , but L ( F ) = Ω( B · n ) . The idea is that it is easy to construct such a family F if we consider functions f : {− , } n → [ − , f takes on the values {± B ( n ) / √ n } , then f will satisfy a coin theorem of bound | ε | · B ( n ) and a corresponding L bound, butneed not satisfy higher L k bounds. By randomly rounding such a function f to have Boolean outputs, theresulting function (and its closure under restriction and negation of input variables) will with high probabilitystill satisfy a coin theorem but will keep its large L k mass. Thus, this suggests that any positive answer toQuestion 3.1 will require a condition which is in some sense “not linear” and can detect such bad examples.Formally, Proposition 3.4 follows from the following lemma which shows that the above process indeedpreserves the sums of Fourier coefficients with high probability. Lemma 3.5.
Let m be a positive integer and g : {− , } m → [ − , be a function. Then defining ˜ g ˜ g ˜ g : {− , } m → {− , } as the random function which for each x ∈ {− , } m independently sets ˜ g ˜ g ˜ g ( x ) ∈ {− , } to have expectation g ( x ) , it holds for every collection T ⊆ [ m ] of subsets of [ m ] that Pr "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ∈T b ˜ g ˜ g ˜ g ( S ) − X S ∈T b g ( S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ε ≤ (cid:0) − m − ε / |T | (cid:1) Proof.
We first write X S ∈T b ˜ g ˜ g ˜ g ( S ) − X S ∈T b g ( S ) = 2 − m X x ∈{− , } m (˜ g ˜ g ˜ g ( x ) − g ( x )) · X S ∈T Y i ∈ S x i ! . by definition of Fourier coefficients. Then by definition of ˜ g ˜ g ˜ g , the right hand side is a sum of 2 m independentmean-zero random variables, one for each x ∈ {− , } n , bounded in a range of size 2 − m · · (cid:12)(cid:12)P S ∈T Q i ∈ S x i (cid:12)(cid:12) , We thank Avishay Tal for telling us about this result and allowing us to include it and its proof in this note. x . Thus, Hoeffding’s inequality [Hoe63] implies the concentration boundPr "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X S ∈T b ˜ g ˜ g ˜ g ( S ) − X S ∈T b g ( S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ε ≤ − · ε P x ∈{− , } m (cid:0) · − m · (cid:12)(cid:12)P S ∈T Q i ∈ S x i (cid:12)(cid:12)(cid:1) = 2 exp − m − · ε − m P x ∈{− , } m (cid:0)P S ∈T Q i ∈ S x i (cid:1) It remains to show that the denominator is equal to |T | : first note that this sum can be written as E x ∼ X m h(cid:0)P S ∈T Q i ∈ S x i (cid:1) i where the x i are distributed as iid random signs. Note that for S = ∅ we havethat Q i ∈ S x i has mean zero and is marginally distributed as a random sign, and for S = ∅ we have that Q i ∈ S x i = 1, so that if ∅ ∈ T we can write E x ∼ X m h(cid:0)P S ∈T Q i ∈ S x i (cid:1) i = 1 + E x ∼ X m (cid:20)(cid:16)P S ∈T \{∅} Q i ∈ S x i (cid:17) (cid:21) with |T \ {∅}| = |T | −
1, and thus it suffices to consider the case that T does not contain the empty set.In this case, we have that E x ∼ X n h(cid:0)P S ∈T Q i ∈ S x i (cid:1) i is the variance of a sum of |T | random variableseach marginally distributed as a random sign. Since all the terms are distinct, given S = S ∈ T we knowthere exists some j in the symmetric difference of S and S (without loss of generality in S ), and thus thecovariance of Q i ∈ S x i and Q i ∈ S x i is zero, as x j has mean zero even conditioned on the value of Q i ∈ S x i .Hence, these variables are all uncorrelated, and so the variance of their sum is simply the sum of the variances,which is |T | · |T | as desired.Applying Lemma 3.5 to the majority function proves Proposition 3.4. Proof of Proposition 3.4.
Let f : {− , } n → {− , } be the majority function on n bits for n odd, and˜ f ˜ f ˜ f : {− , } n → {− , } be the random function which independently for each x ∈ {− , } n sets ˜ f ˜ f ˜ f ( x )with expectation B/ √ n · f ( x ). Then define ˜ F ˜ F ˜ F to be the set of all restrictions of all functions of the form z τ · ˜ f ˜ f ˜ f ( z σ , . . . , z n σ n ) for τ ∈ {− , } and σ ∈ {− , } n .We claim that with positive probability ˜ F ˜ F ˜ F has the desired properties. Note that ˜ F ˜ F ˜ F is closed underrestriction, negation of input variables, and negation of the output by definition. To prove the coin theoremclaim, by Tal’s result (Lemma 3.2) it is enough to prove the level 1 bound L ( ˜ F ˜ F ˜ F ) ≤ B . Thus, we need toshow that with positive probability L ( ˜ F ˜ F ˜ F ) ≤ B and L ( ˜ F ˜ F ˜ F ) ≥ Ω( B · n ) (we will in fact show an upper boundof B + 1, which is equivalent by shifting).Since L ( f ) = Θ( n / ) (see e.g. [O’D14, Problem 5.26]), we have L ( B/ √ n · f ) = Θ( B · n ), so applyingLemma 3.5 to B/ √ n · f and T = { S ⊆ [ n ] | | S | = 3 } implies thatPr h L ( ˜ F ˜ F ˜ F ) ≥ L ( ˜ f ˜ f ˜ f ) ≥ L ( B/ √ n · f ) − n = Ω( B · n ) i ≥ − exp( − Ω(2 n /n )) . For the level one bound, since ˜ F ˜ F ˜ F is closed under negation of input variables, it suffices to bound the sum ofthe level 1 Fourier coefficients of each member of ˜ F ˜ F ˜ F . For this, consider a fixed sign τ ∈ {− , } , sign pattern σ ∈ {− , } n , and restriction ρ of z τ · ˜ f ˜ f ˜ f ( z σ , . . . , z n σ n ) keeping m variables alive, which we denote ˜ f ˜ f ˜ f | σ,τρ .By Parseval and Cauchy-Schwarz, the sum of the level 1 Fourier coefficients of any Boolean function on m variables is at most √ m , so if √ m ≤ B then L ( ˜ f ˜ f ˜ f | σ,τρ ) ≤ B with probability 1. Thus, assume without loss ofgenerality that m > B . Define g : {− , } m → {± B/ √ n } be given by g ( z , . . . , z m ) = B/ √ n · τ · f ( x , . . . , x n )where x i is equal to σ i · ρ ( i ) if i is fixed by R , and equal to σ j · z j for j the index of i in the free coordinatesotherwise. Then the sum of the singleton Fourier coefficients of the restriction ˜ f ˜ f ˜ f | σ,τρ is distributed exactly as thesum of the Fourier coefficients ˜ g ˜ g ˜ g : {− , } m → {− , } where ˜ g ˜ g ˜ g ( x ) is set independently for each x ∈ {− , } m with expectation g ( x ). Since the sum of g ’s singleton Fourier coefficients is at most ( B/ √ n ) · √ m ≤ B , byLemma 3.5 we have Pr h L ( ˜ f ˜ f ˜ f | σ,τρ ) ≥ B + 1 i ≤ (cid:0) − m − /m (cid:1) . m ≥ B ≥ log n + O ( √ log n ), so since m m /m is increasing in m for m ≥ / ln 2,this probability is at most exp( − ω ( n )). Thus, since there are 2 O ( n ) functions ˜ f ˜ f ˜ f | σ,τρ , we conclude by a unionbound.It remains to give the proof of Lemma 3.2. Proof of Lemma 3.2 [Tal19].
By Proposition 2.1, the sum of the partial derivatives of f at zero can bebounded in terms of L ( f ). Tal [Tal19] observed that by a technique of Chattopadhyay, Hatami, Hosseini,and Lovett [CHHL18], the partial derivatives at any other point can be bounded in terms of the Fouriercoefficients of appropriate restrictions of f . Formally, for ε ∈ [ − ,
1] we have that there exists a distribution D ( n ) ( ε ) over restrictions ρ such that for ε ∈ [ | ε | − , − | ε | ], E (cid:2) f (cid:0) X nε + ε (cid:1)(cid:3) = E ρ ∼D ( n ) ( ε ) h f | ρ (cid:16) X nε/ (1 −| ε | ) (cid:17)i . Therefore, letting g ( ε ) = E [ f ( X nε )], we have for | ε | < g ( ε ) = lim ε → g ( ε + ε ) − g ( ε ) ε = lim ε → E (cid:2) f (cid:0) X nε + ε (cid:1)(cid:3) − E (cid:2) f (cid:0) X nε +0 (cid:1)(cid:3) ε = lim ε → E ρ ∼D ( n ) ( ε ) E h f | ρ (cid:16) X nε/ (1 −| ε | ) (cid:17)i − E h f | ρ (cid:16) X n / (1 −| ε | ) (cid:17)i ε = E ρ ∼D ( n ) ( ε ) (cid:20) − | ε | · lim δ → E [ f | ρ ( X nδ )] − E [ f | ρ ( X n )] δ (cid:21) (setting δ = ε/ (1 − | ε | ))= 11 − | ε | · E ρ ∼D ( n ) ( ε ) " n X i =1 c f | ρ ( { i } ) . (by Proposition 2.1)Since F is closed under restriction, we have for every ρ that (cid:12)(cid:12)(cid:12)P ni =1 c f | ρ ( { i } ) (cid:12)(cid:12)(cid:12) ≤ t , and so we can bound | g ( x ) | ≤ t/ (1 − | x | ). The result follows since E [ f ( X nε )] − E [ f ( X n )] = g ( ε ) − g (0) = R ε g ( x ) dx . We thank Salil Vadhan for helpful conversations and his feedback on this note, and Avishay Tal for hisfeedback on this note, telling us about Lemma 3.2, and allowing us to include it and its proof in this note.
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