Fractional Pseudorandom Generators from Any Fourier Level
Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, Abhishek Shetty
aa r X i v : . [ c s . CC ] A ug Fractional Pseudorandom Generators from Any Fourier Level
Eshan Chattopadhyay ∗ Cornell University [email protected]
Jason Gaitonde † Cornell University [email protected]
Chin Ho Lee ‡ Columbia University [email protected]
Shachar Lovett § University of California, San Diego [email protected]
Abhishek Shetty ¶ Cornell University [email protected]
August 11, 2020
Abstract
We prove new results on the polarizing random walk framework introduced in recent works ofChattopadhyay et al. [CHHL19, CHLT19] that exploit L Fourier tail bounds for classes of Booleanfunctions to construct pseudorandom generators (PRGs). We show that given a bound on the k -thlevel of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k . This interpolates previous works, which either require Fourier bounds on all levels [CHHL19], orhas polynomial dependence on the error parameter in the seed length [CHLT19], and thus answersan open question in [CHLT19]. As an example, we show that for polynomial error, Fourier boundson the first O (log n ) levels is sufficient to recover the seed length in [CHHL19], which requiresbounds on the entire tail.We obtain our results by an alternate analysis of fractional PRGs using Taylor’s theorem andbounding the degree- k Lagrange remainder term using multilinearity and random restrictions. In-terestingly, our analysis relies only on the level-k unsigned Fourier sum , which is potentially a muchsmaller quantity than the L notion in previous works. By generalizing a connection established in[CHH + F polynomials with seedlength close to the state-of-the-art construction due to Viola [Vio09], which was not known to bepossible using this framework. A central pursuit in complexity theory is to understand the need of randomness in efficient computa-tion. Indeed there are important conjectures (such as P = BPP ) in complexity theory which statethat one can completely remove the use of randomness without losing much in efficiency. While we arequite far from proving P = BPP , a rich line of work has focused on derandomizing simpler modelsof computation (see Vadhan [Vad12] for a survey of prior work on derandomization). A key tool forproving such derandomization results is through the notion of a pseudorandom generator defined asfollows. ∗ Supported by NSF grant CCF-1849899. † Supported by NSF grant CCF-1408673 and AFOSR grant F5684A1. ‡ Supported by a fellowship from the Croucher Foundation and by the Simons Collaboration on Algorithms andGeometry. § Supprted by NSF grants CCF-02006443 and DMS-1953928. ¶ Supported by a Cornell University Fellowship and a JP Morgan Chase Faculty Fellowship. efinition 1.1. Let F be a class of n -variate Boolean functions. Then a pseudorandom generator (PRG) for F with error ǫ > X ∈ {− , } n such that for all f ∈ F , | E X [ f ( X )] − E U n [ f ( U n )] | ≤ ǫ, where U n is the uniform distribution on {− , } n . If X = G ( U s ) for some explicit function G : {− , } s → {− , } n , then X has seed length s .There is a long line of research on explicit constructions of PRGs (for various classes of Booleanfunctions) in the literature and it is well beyond our scope to survey prior work here. Instead, wefocus on a recent line of work initiated by Chattopadhyay et al. [CHHL19, CHLT19] that providesa framework for constructing pseudorandom generators for any Boolean function class that exhibitFourier tail bounds (we discuss this in more details in the next subsection; see Section 2.1 for a briefintroduction to Fourier analysis of Boolean functions). This provides a unified PRG for several well-studied function classes such as small-depth circuits, low-sensitivity functions, and read-once branchingprograms that exhibit such Fourier tails. We discuss this new framework in Section 1.1, and presentour results in Section 1.2. We now briefly explain the polarizing random walk framework introduced by [CHHL19]. The authorsshow that for classes of n -variate Boolean functions that are closed under restrictions, one can quiteflexibly construct pseudorandom generators via a local-to-global principle that works as follows: it issufficient to construct fractional pseudorandom generators , a notion that generalizes a PRG to allowthe random variable X (in Definition 1.1) to be supported on the solid cube [ − , n so that it fools themultilinear expansion of each Boolean function in the class. Ideally, the variance of each coordinateof the random variable should be as large as possible while still provably fooling the class. Towardsthis, define a fractional PRG X to be p -noticeable if the variance in each of its coordinates is least p .(See Definition 2.5 for a formal definition of a fractional PRG.)To construct the full pseudorandom generator, the authors give a random walk gadget that com-poses together independent copies of such a fractional generator as steps in a random walk thatpolarizes quickly to the Boolean hypercube. The analysis for how the error accumulates in this pro-cess relies on interpreting the intermediate points of this pseudorandom walk as an average of randomrestrictions of the original function; because the fractional generator locally fools the class, this inter-pretation shows that it does not incur much error at each intermediate step, and the rapid polarizationshows that it does not take too many steps. Taken together, these two facts imply the resulting randomvariable successfully fools the class.The above framework shows that if one can construct non-Boolean random variables, with suf-ficiently large variance in each coordinate, that can locally fool any function in the class, then oneimmediately obtains a pseudorandom generator using their random walk gadget. As these genera-tors need not be Boolean, the construction of fractional pseudorandom generators is only easier thanconstructing pseudorandom generators. To that end, [CHHL19] further show how to construct suchfractional pseudorandom generators for any class of functions satisfying Fourier tail bounds . Namely,they show that if every function in the class is such that the L Fourier mass at each level 1 ≤ k ≤ n isat most b k for some fixed b ≥
1, then one can construct a fractional pseudorandom generator for error ǫ with seed length O (log log n + log(1 /ǫ )) and variance Θ( b − ) in each coordinate. Combining thisfractional pseudorandom generator with their random walk gadget yields a pseudorandom generatorwith seed length b · polylog( n/ǫ ). As a result, if one can show that a function class admits nontrivialFourier tail bounds, then the [CHHL19] construction immediately implies a pseudorandom generator.Some examples of classes of Boolean functions that exhibit such tail bounds include AC circuits with2he parameter b = poly(log n ) [LMN89, Tal17], constant width read-once branching programs with b = poly(log n ) [CHRT18], and low-sensitivity functions with b = O ( s ) [GSW16, Tal17]. Using suchFourier bounds, [CHHL19] immediately gave a polylogarithmic seed length PRG for these functionclasses. It was also conjectured in [CHHL19] that the class of n -variate degree- d polynomials over F satisfy such tail bounds. We discuss this in more detail in Section 1.2.In the work by [CHLT19], the authors show how to construct fractional pseudorandom generatorsusing far fewer assumptions on the Fourier tails. Building on the analysis of the celebrated workof [RT19], which gave an oracle separation of BQP and PH (which itself relies on the [CHHL19]random walk framework), they show that one can use this same framework to obtain a pseudoran-dom generator with seed length depending only on bounds for the second Fourier level of the class.However, with these weaker assumptions, they require a different fractional PRG. To do this, theyessentially derandomize the result of [RT19], which shows that classes of multilinear functions withlow level-two Fourier mass cannot nontrivially distinguish between a suitable variant of the Forre-lation distribution and the uniform distribution. It turns out that this can be interpreted via Itˆo’sLemma, which shows that the local behavior of a smooth function of Brownian motion is essentiallydetermined by the first two derivatives [Wu20]. [CHLT19] show that one can derandomize this anal-ysis by efficiently constructing fractional PRGs that simulate Gaussian random variables with smallcovariance using the best-known constructions of error-correcting codes. However, this constructionincurs exponentially worse dependence on the error parameter in each fractional step to nearly samplesufficiently good Gaussian random variables. The final seed length that this framework obtains hasthe form O (( b /ǫ ) o (1) polylog( n )), where b is the level-two Fourier mass of the class. Comparedto [CHHL19], this yields exponentially worse dependence on the error, as well as quadratically worsedependence on the level-two mass (though [CHLT19] assume nothing about the rest of the Fourierlevels). Given these two works, a very natural question (explicitly asked in [CHLT19]) is whether it is possibleto interpolate between these constructions by assuming Fourier bounds on an intermediate level.Concretely, can this framework still succeed if one has Fourier control at just level- k ? If the classfurther has such Fourier bounds up to and including level- k , can one interpolate between the seedlengths of [CHHL19] and [CHLT19]? Given Fourier bounds from level-1 up to k , what range of error ǫ > /ǫ in the seedlength (or put contrapositively, given a desired error ǫ >
0, how many levels of Fourier bounds aresufficient to ensure that the seed length remains polylogarithmic in 1 /ǫ )?Moreover, the recent work by Chattopadhyay et al.[CHH +
20] shows that the problem of boundingthe level-two unsigned Fourier sum , defined by the absolute value of the sum of the Fourier coefficientsinstead of the sum of their absolute values (as in the definition of L Fourier mass) that is required in[CHHL19, CHLT19], corresponds to the problem of bounding the covariance of the function class andthe k - XOR of shifted majority functions. In particular, using this connection to this better-studiedobject, they explicitly ask whether it suffices to give bounds on the weaker Fourier quantity M ( F ) (ormore generally, M k ( F ), see Section 2 for the precise definition) to obtain pseudorandom generators.The reason for doing so is that for many classes of functions, we currently do not have strong enough L Fourier tail bounds, like the class of low-degree polynomials over F (though see Section 4). However,if one can show this, one now reduces the construction of pseudorandom generators to proving weakerFourier bounds, which are hopefully more amenable to known techniques and thereby making theproblem easier.In this work, we make progress on all of these questions, by providing a new analysis of the fractionalpseudorandom generator of [CHHL19]. Informally, we show that by only assuming a bound on some3evel- k Fourier quantity of the class, the [CHHL19] fractional pseudorandom generator will still bevalid. Moreover, our analysis actually shows that the error term that is induced by the high-ordercomponent can be improved from L ,k ( F ) (the level- k Fourier mass) to M k ( F ) (the level- k unsignedFourier sum , see Section 2), which is a priori significantly smaller. Here, L ,k ( f ) , X S ⊆ [ n ]: | S | = k | ˆ f ( S ) | and M k ( f ) , max x ∈ [ − , n (cid:12)(cid:12)(cid:12)(cid:12) X S : | S | = k ˆ f ( S ) x S (cid:12)(cid:12)(cid:12)(cid:12) = max x ∈{− , } n (cid:12)(cid:12)(cid:12)(cid:12) X S : | S | = k ˆ f ( S ) x S (cid:12)(cid:12)(cid:12)(cid:12) . and L ,k ( F ) and M k ( F ) refer to the maximum of L ,k and M k taken over functions in the class F .Our main result is the following analysis of a fractional pseudorandom generator: Theorem 1.1.
Let F be any class of n -variate Boolean functions that is closed under restrictions andnegations. Suppose that M k ( F ) ≤ b k for some b ≥ and k ≥ . Then for any ǫ > , there exists anexplicit Ω( ǫ /k /b ) -noticeable fractional PRG for F with error ǫ and seed length O ( k · log( n )) . Further, if it holds that L ,i ( F ) ≤ b i for all ≤ i < k , then the seed length can be improved to O (log log( n ) + log k + log(1 /ǫ )) . Note that for some Boolean classes of great interest such as the class of low-degree F polynomials,Fourier tail bounds as required by [CHHL19] are not yet known and thus Theorem 1.1 allows us toleverage potentially much weaker bounds proved in [CHHL19] to construct a PRG with polylogarithmicdependence on n/ǫ in the seed length (see Theorem 1.3), almost matching the best known PRG dueto Viola [Vio09]. As we discuss below, the results in [CHHL19, CHLT19] are not quite good enoughto use known Fourier tail bounds to obtain such a PRG for the class of F polynomials.Using the fractional pseudorandom generator from Theorem 1.1, we obtain the following conse-quences almost immediately from the random walk gadget of [CHHL19] (see Theorem 2.2):1. Pseudorandom Generators from Fourier Bounds at Level- k : From our fractional pseu-dorandom generator, we show that the random walk framework yields nontrivial pseudorandomgenerators assuming Fourier bounds just at level- k of the associated class, with improvements ifwe assume bounds from level-1 up to level- k . The informal statement is the following: Theorem 1.2.
Let F be any class of n -variate Boolean that is closed under restrictionsand negations. Suppose that F satisfies M k ( F ) ≤ b k for some b ≥ and k > .Then there exists an explicit pseudorandom generator for F for error ǫ with seed length k · b / ( k − polylog( n/ǫ ) /ǫ / ( k − . The seed length can be improved if L ,i ( F ) ≤ b i for all levels i ≤ k . See Theorem 3.7 for the precise statement. One immediate consequence is that if one has com-parable Fourier bounds at level k = 3, one recovers the seed length in [CHLT19]. Further, fromsuch a Fourier bound for just the level k = 4, one obtains quadratically better dependence on theerror in the seed length (as well as polylogarithmic factors in n/ǫ ) compared to [CHLT19]. In par-ticular, given an appropriate Fourier bound of b k on just some level k ≤ polylog( n ), one obtainsa pseudorandom generator with error ǫ with seed length O ( b / ( k − polylog( n/ǫ ) /ǫ / ( k − ). We remark that at this level of generality, this linear dependence on k is essentially necessary. Indeed, any Booleanfunction on n -variables has L level- n mass at most 1, but one cannot hope to generically fool all Boolean functionssimultaneously without using n bits.
4e note that the fractional PRG from Theorem 1.1 cannot be converted into a PRG for k = 1 , k >
2. This leads toa non-trivial PRG when k >
2. See Remark 3.8 for more discussion.2.
Pseudorandom Generators with Polylogarithmic Error Dependence from up toLevel- k Bounds : A simple corollary of our fractional pseudorandom generator is that onecan recover the polylogarithmic dependence on 1 /ǫ from [CHHL19] if ε ≥ b · log n · − O ( k ) andwe have Fourier bounds up to level- k . Corollary 1.1.
Let F be any class of n -variate Boolean functions that is closed under restrictionsand negations. Suppose that for some level k > and b ≥ , we have M k ( F ) ≤ b k and L ,i ( F ) ≤ b i for i < k . Then, for any ǫ ≥ b · log n · − O ( k ) , there exists an explicit pseudorandom generatorfor F for error ǫ with seed length O ( b polylog( n/ǫ )) . This actually covers the analysis of [CHHL19] without requiring anything on the full Fouriertail, and addresses an open question of [CHLT19] asking how many levels of Fourier bounds oneneeds control of to regain polylogarithmic dependence on ε . In particular, if one requires error ǫ = 1 / poly( n ), then it suffices to have Fourier bounds up to level Θ(log( n )) to get the samedependence.We view this work as a proof-of-concept that it is indeed possible to interpolate between the twoextremes of [CHHL19, CHLT19] in the polarizing random walk framework and obtain better resultsusing weakened Fourier assumptions. We prove Theorem 1.1 in Section 3, from which Theorem 1.2 andCorollary 1.1 follow without much difficulty using the existing random walk technique of [CHHL19].As a concrete possible application of this approach which would provably improve on these works,both [CHHL19] and [CHLT19] conjecture Fourier bounds on the L mass of the class of F polynomialsof degree at most d . The former conjectures that this class satisfies a tail bound of the form c kd forsome constant c d at all levels 1 ≤ k ≤ n (so as to apply their approach), while the latter conjecturesjust that the level-two L mass is O ( d ). While neither conjecture seems close yet to being resolved,one can imagine that should the latter be proved, it may be feasible to extend the analysis to achievea bound of (poly( d )) k for k = Ω(1), or even more optimistically, k = Ω(log n ). The pseudorandomgenerator implied by the analysis here would thus apply and yield significantly improved seed lengthcompared to [CHLT19], though we note that our generator does not actually apply if such a boundonly holds at level k = 2. Such a result would imply a pseudorandom generator with improved seedlength compared to [CHLT19] for AC [ ⊕ ] (see the discussion in [CHLT19], using results of Razborov[Raz87] and Smolensky [Smo87, Smo93]). Finally note that we just need to bound the quantity M k as opposed to L ,k required in prior works.Nonetheless, our analysis here, coupled with weaker Fourier tail bounds obtained in [CHHL19],immediately implies the following: Theorem 1.3.
Let F be the class of degree- d polynomials over F on n variables. Then there existsan explicit pseudorandom generator for F with error ǫ and seed length O ( d ) polylog( n/ǫ ) . See Section 4 for the precise dependences in the seed length. While this result does not quitematch the current state-of-the-art PRG for this class due to Viola [Vio09] (and similarly, fails to giveanything nontrivial for d = Ω(log n )), we view this as a conceptual contribution that the random walkframework can yield an explicit pseudorandom generator with error dependence that is polylogarithmic5n n/ǫ , which was not previously known from [CHHL19] or [CHLT19]. The Fourier tail bounds thatare sufficient for our analysis are too weak to be employed in [CHHL19], and leads to a quadratic errordependence in the level-two fractional pseudorandom generator by [CHLT19]. We present the proofof Theorem 1.3 in Section 4.Moreover, as stated before, recent work [CHH +
20] has shown that it is possible to deduce boundson M ( F ) using covariance bounds with the XOR of certain resilient functions. As we are able toshow that bounds on such quantities imply pseudorandom generators, we give an analogous argumentfor an appropriate generalization of this result to M k ( F ) in Section 5, thus reducing the problem ofconstructing PRGs in this framework to proving correlation bounds. To prove our results, we rely on an alternate, simple analysis of the fractional pseudorandom generatorconsidered by [CHHL19], where they assume control on the entire Fourier tail, and then use theirgadget construction to obtain the full pseudorandom generator. As this latter part can be done in anentirely black-box fashion, we need only focus on the former. For this first part, their approach is thefollowing: consider a random variable X . By writing out f in the multilinear (Fourier) expansion, onehas | E X [ f ( X )] − E U n [ f ( U n )] | = (cid:12)(cid:12) E X (cid:2) X ∅6 = S ⊆ [ n ] ˆ f ( S ) X S (cid:3)(cid:12)(cid:12) (1) ≤ k − X i =1 X S ⊆ [ n ]: | S | = i | ˆ f ( S ) || E X [ X S ] | | {z } low-order terms + n X i = k X S ⊆ [ n ]: | S | = i | ˆ f ( S ) || E X [ X S ] | | {z } high-order terms , (2)where we used the triangle inequality in (2). [CHHL19] use two different methods to control eachcomponent in (2); the first part is handled by sufficiently strong independence in the coordinates, whilethe latter is handled by physical smallness of the random variable. To control the low-degree terms,they require X to be O ( ǫ )-almost k -wise independent. By definition, this forces the biases of all theFourier characters to be small in the low-order component. To deal with the higher-order component,they use the fact that X need not be Boolean ; that is, they can scale any such distribution down.Because they assume that the Fourier mass on each level- k is bounded by b k , it suffices to scale downa {± } n valued random variable by Θ(1 /b ), forcing the higher-order error terms to form a geometricseries. One can show that to balance these terms, one may take the threshold at k = Θ(log(1 /ǫ )) toensure that this yields ǫ error and seed length O (log log n +log(1 /ǫ )) using known explicit constructionsof almost k -wise independent distributions.Our analysis of this fractional PRG is instead driven by exploiting Taylor’s theorem, multilinearity,and the technique of random restrictions that is used critically in [CHHL19]. The general approachof using Taylor’s theorem in the construction of PRGs has been quite fruitful. The typical way inwhich it is used is to write some smooth function to be fooled as a low-degree polynomial, which isrelatively easy to fool, plus an error term bounded by the next derivatives of the function which ideallyis negligible; this approach is often tied to invariance principles . We do the same decomposition, whichinitially agrees with [CHHL19]: | E X [ f ( X )] − E U [ f ( U )] | ≤ k − X i =1 X S ⊆ [ n ]: | S | = i | ˆ f ( S ) || E X [ X S ] | | {z } low-order terms + | E X [ R k ( X )] | | {z } high-order term , (3)6here our higher order term arises from Taylor’s theorem. Na¨ıvely, if one wanted to analyze thequality of a fractional PRG for the multilinear expansion of f using Taylor’s theorem, this might doprecisely nothing; indeed, the multilinear expansion of f is by definition a polynomial, so is equalto its Taylor series. Therefore, “using Taylor’s theorem” would give exactly nothing but the original[CHHL19] approach! To get anything different than [CHHL19], we must truncate the Taylor series upto some level k − k − k -th order derivatives of the Fourier expansion. Thehope in doing this is that implicitly, Taylor’s Theorem collapses the higher-order terms into level- k derivatives. One might then hope that there are significant sign cancellations that thus avoid the useof the complete Fourier tail assumption and triangle inequality as in (2). In particular, we only paythe triangle inequality at level- k by doing so, though we now must study these new error terms.If these derivatives in the error term were evaluated at the origin ∈ R n , then we would immedi-ately be able to bound the entire error term by any level- k bounds on our class F . This is because thesederivatives at are precisely the level- k Fourier coefficients. However, the remainder term is evaluatedat some intermediate point on the line between and the realization of the supposed fractional PRG X . It turns out, though, that using the closure of the class under restrictions and negations, one canobtain an upper bound on this quantity using just the weaker Fourier quantity M k ( F ).To obtain pseudorandom generators, we then need only apply the random walk gadget of [CHHL19].We refer the reader to Section 3 for formal proofs of the ideas sketched in this section. Remark 1.2.
A previous version of this paper took an alternate approach that works as follows: bymultilinearity, one can evaluate the error term in the Taylor expansion at the corner of some cube.By leveraging known connections between these derivatives and p -biased Fourier coefficients (see, forinstance, Chapter 8 of [O’D14]), one can then apply a result of Keller [Kel12] that reduces bounds onthe resulting L mass of these biased coefficients at level- k to the unbiased L Fourier mass of some newfunction defined on n variables that roughly simulates biased bits. For many classes, this reductioncomes at little loss in known Fourier bounds, but requires awkward bounds on an associated classof functions . While this result still manages to partially interpolate between the previous approachesin [CHHL19, CHLT19], as well as give a new PRG for the class of low-degree F polynomials, ournew approach removes this artificial caveat, as well as allows for the weaker Fourier requirements via M k ( F ) as opposed to L ,k ( F ) . As in [CHHL19] and [CHLT19], we study PRGs for classes F of n -variate Boolean functions that areclosed under restriction and negation (that is, fixing any subset and flipping any subset of the bits ofthe variables yields a function that remains in the class). We briefly recall basic Fourier analysis: any Boolean function f : {− , } n → {− , } admits a uniquemultilinear expansion, also known as the Fourier expansion , given by f ( x ) = X S ⊆ [ n ] ˆ f ( S ) x S , (4)where we write x S , Q i ∈ S x i . The Fourier coefficient ˆ f ( S ) is given byˆ f ( S ) = E X ∼{− , } n [ f ( X ) X S ] . f to [ − , n , where f ( x ) for arbitrary x is evaluated according to theexpression in (4). Note that in this case, f ( ) = ˆ f ( ∅ ) = E U n [ f ( U n )]. One of the main parameters ofinterest from the Fourier expansion for this framework is the following: Definition 2.1.
The level- k mass of a Boolean function f is L ,k ( f ) , X S ⊆ [ n ]: | S | = k | ˆ f ( S ) | , and the level- k mass of a class F is L ,k ( F ) , max f ∈F L ,k ( f ).In this work, we will show how to construct PRGs whose seed length depends on the following,smaller quantity: Definition 2.2.
For any multilinear polynomial f : R n → R given by f ( x ) = P S ⊆ [ n ] ˆ f ( S ) x S , definethe level- k part by f k ( x ) , X S ⊆ [ n ]: | S | = k ˆ f ( S ) x S , (5)and further define f Definition 2.3. Let F be a class of n -variate multilinear polynomials that is closed under restrictionsand negations. Then define conv( F ) as the convex closure of F i.e.conv( F ) , (cid:26) X f ∈F λ f f (cid:12)(cid:12)(cid:12)(cid:12) X f ∈F λ f = 1 , λ f ≥ ∀ f ∈ F (cid:27) . (9)We briefly note the following two elementary facts: first, by the assumption that F is closed underrestrictions and negations, the same is true of conv( F ). Moreover, we have M k ( F ) = M k (conv( F ))by the triangle inequality. 8 .2 (Fractional) Pseudorandom Generators We now recall the (well-known) definition of a pseudorandom generator, as well as the generalizationof a fractional pseudorandom generator as introduced by [CHHL19]: Definition 2.4. Let F be a class of n -variate Boolean functions. Then a pseudorandom generator (PRG) for F with error ǫ > X ∈ {− , } n such that for all f ∈ F , | E X [ f ( X )] − E U n [ f ( U n )] | ≤ ǫ, where U n is the uniform distribution on {− , } n . If X = G ( U s ) for some explicit function G : {− , } s → {− , } n , then X has seed length s . Definition 2.5. A fractional pseudorandom generator (fractional PRG) for F with error ǫ > X ∈ [ − , n such that for all f ∈ F (identifying f with its multilinear expansion) | E X [ f ( X )] − f ( ) | ≤ ǫ, where the definition of seed length is the same. A fractional PRG is p - noticeable if for each i ∈ [ n ], E [ X i ] ≥ p .We now state the main results of [CHHL19] and [CHLT19] that show how to construct PRGs fromsuitably combining noticeable fractional PRGs. This is done by the following amplification theorem ,which roughly composes fractional random variables into a random walk inside the Boolean hypercube: Theorem 2.2. Suppose F is class of n -variate Boolean functions that is closed under restrictions, andthat X is a p -noticeable fractional PRG with error ǫ and seed length s . Then there exists an explicitPRG for F with seed length O ( s log( n/ǫ ) /p ) and error O ( ǫ log( n/ǫ ) /p ) . Using this result, [CHHL19] proved the following theorem that exploits strong L control of eachFourier level: Theorem 2.3. Let F be any class of n -variate Boolean functions that is closed under restrictions.Suppose that L ,k ( F ) ≤ b k for some b ≥ and all ≤ k ≤ n . Then for any ǫ > , there exists anexplicit PRG for F with error ǫ and seed length b · polylog( n/ǫ ) . This is achieved by constructing a fractional PRG that is a scaled version of a nearly log(1 /ǫ )-wiseindependent distribution. As we will be analyzing a similar fractional PRG, we defer the details tonext section.To lessen the requisite assumptions on the Fourier spectrum, [CHLT19] derandomize a construc-tion of [RT19] to prove the following result that requires only level-two control, albeit at a cost ofexponentially worse dependence on the error ε , and quadratically worse dependence on the level-twomass: Theorem 2.4. Let F be any class of n -variate Boolean functions that is closed under restrictions.Suppose that L , ( F ) ≤ b for some b ≥ . Then for any ǫ > , there exists an explicit PRG for F with error ǫ and seed length O (( b /ε ) o (1) polylog( n )) . k -th Fourier Level We now turn to the proof of our main result yielding a fractional pseudorandom generator from level- k bounds, Theorem 1.1. Throughout the remainder of this section, we assume that F is closed underrestrictions and negations of variables. We restate the result here:9 heorem 3.1 (Theorem 1.1, restated) . Let F be any class of n -variate Boolean functions that isclosed under restrictions and negations. Suppose that M k ( F ) ≤ b k for some b ≥ and k > . Thenfor any ǫ > , there exists a Ω( ǫ /k /b ) -noticeable explicit fractional PRG for F with error ǫ and seedlength O ( k · log( n )) .If it further holds that L ,i ( F ) ≤ b i for all ≤ i < k , then the seed length can be improved to O (log log( n ) + log k + log(1 /ǫ )) . To set up the proof of this theorem, we require some auxilliary results. The main technical claimwe need is the following, which bounds the error term we will encounter using the quantity M k ( F ) wedefined before (Definition 2.2). Lemma 3.2. Let f ∈ F . Then for all c ∈ (0 , , we have max x ∈ [ − c,c ] n | f ≥ k ( x ) | ≤ (cid:18) c − c (cid:19) k M k ( F ) . (10)To prove this lemma, we require the following simple results. The first simply shows that wemay always bound the contribution of the level- k part of any function in F by simply rescaling theargument: Lemma 3.3. Let f ∈ conv( F ) . Then, for all c ∈ (0 , and x ∈ [ − c, c ] n , we have | f k ( x ) | ≤ c k M k ( F ) . (11) Proof. Simply observe that c − x ∈ [ − , n by assumption, and by homogeneity of f k as a polynomial,we have | f k ( x ) | = c k | f k ( c − x ) | ≤ c k M k (conv( F )) = c k M k ( F ) . (12)The next simple, but powerful, claim simply shows that one can “recenter” functions in F andremain in conv( F ) (and therefore, enjoy the same Fourier bounds). This random restriction techniqueis a key tool in [CHHL19]: Lemma 3.4. Let f ∈ conv( F ) and a , b ∈ [ − , n such that | a i | + | b i | ≤ for all i ∈ [ n ] . Define ˜ f by ˜ f ( x ) = f ( a + b ◦ x ) , where ◦ denotes componentwise multiplication. Then, ˜ f ∈ conv( F ) .Proof. Given a , b , define a distribution D i on Z i = {− , , x i , − x i } where x i is treated as formalvariable, such that E y i ∼ D i [ y i ] = a i + b i x i ; note that this is possibly by the assumption that | a i | + | b i | ≤ D = Q i D i be the product distribution of the D i . For any z ∈ Q i Z i , define f z ( x ) as thefunction obtained by setting x i = z i for each i ; in particular, each variable gets set to ± F , we clearlyhave f z ∈ F for any z . By multilinearity and independence of the product distribution, we have f ( a + b ◦ x ) = E z ∼ D [ f z ( x )]. Thus ˜ f ∈ conv( F ).As mentioned before, our fundamental approach will be to bound the higher-order terms of theFourier expansion at the fractional points of the fractional generator via the error term that arises inTaylor’s theorem. Denote by h ( k ) the k -th derivative of any C k function h : R → R . We then havethe following claim: Lemma 3.5. Let f : R n → R be multilinear and let x ∈ R n . Define g : R → R by g ( t ) = f ( t x ) . Then, g ( k ) (0) = k ! · f k ( x ) . (13)10 roof. From the definition, it follows that g ( t ) = X S ⊆ [ n ] t | S | ˆ f ( S ) x S . Differentiating with respect to t , we get g ( k ) ( t ) = X S : | S |≥ k (cid:18) k − Y i =0 ( | S | − i ) (cid:19) t | S |− k ˆ f ( S ) x S . Setting t = 0 eliminates all of the monomials with | S | > k , giving us the required bound.Finally, we connect the function defined in the previous part with our assumed Fourier bounds: Lemma 3.6. Let f ∈ conv( F ) , c ∈ (0 , and x ∈ [ − c, c ] n . Define g as in Lemma 3.5. Then, max s ∈ [0 , | g ( k ) ( s ) | ≤ (cid:18) c − c (cid:19) k · k ! · M k ( F ) Proof. Fix s ∈ [0 , 1] and let λ = 1 − c . Define the auxiliary function ˜ f ( y ) = f ( s x + λ y ). Writing a = s x and b = ( λ, . . . , λ ), we clearly have s | x i | + | λ | ≤ 1, so we may apply Lemma 3.4 to see that˜ f ∈ conv( F ). Now writing ˜ g ( t ) = ˜ f ( t x ) = f ( s x + λt x ), we also have ˜ g ( t ) = g ( s + tλ ). By the chainrule, differentiating both sides k times and then setting t = 0, λ k g ( k ) ( s ) = ˜ g ( k ) (0) . (14)On the other hand, by Lemma 3.5, we have ˜ g ( k ) (0) = k ! · ˜ f k ( x ), and as ˜ f ∈ conv( F ) by Lemma 3.4,we conclude using Lemma 3.3 that g ( k ) ( s ) = ˜ g ( k ) (0) λ k ≤ (cid:18) c − c (cid:19) k · k ! · M k ( F ) , (15)as desired.With these in order, we can finally return to the proof of Lemma 3.2: Proof of Lemma 3.2. Let f ∈ F , x ∈ [ − c, c ] n and define g ( t ) = f ( t x ). Then, by Taylor expandingabout t = 0 and evaluating at t = 1, we have g (1) = X i Fix f ∈ F , and let X be an arbitrary random variable such that | X i | = x ≤ / i for some x > | E X [ f ( X )] − f ( ) | = (cid:12)(cid:12)(cid:12)(cid:12) E X (cid:20) X S ⊆ [ n ]:1 ≤| S |≤ k − ˆ f ( S ) X S (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) + | E X [ f ≥ k ( X )] | . To deal with the second term, by Lemma 3.2, we have | E X [ f ≥ k ( X )] | ≤ (cid:18) x − x (cid:19) k M k ( F ) . (19)By assumption, M k ( F ) ≤ b k for some b ≥ 1; therefore, by taking x = Θ( ǫ /k /b ), this term is at most ǫ/ j mass for j < k , then one may take X = x · Y , where Y ∈ {− , } n is ( k − O ( k log( n )) [Vad12]and gives no additional error. Note that then X is Θ( ǫ /k /b )-noticeable.If one further assumes that L ,i ( F ) ≤ b i for all i < k , then one may apply their same analysisand instead let X = x · Y ′ , where Y ′ is an ( ǫ/ k − (cid:12)(cid:12)(cid:12)(cid:12) E X (cid:20) X S ⊆ [ n ]:1 ≤| S |≤ k − ˆ f ( S ) X S (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k − X i =1 x i X S : | S | = i | ˆ f ( S ) || E [ Y ′ S ] | ≤ ( ǫ/ k − X i =1 ( bx ) i ≤ ǫ/ , (20)as we choose x such that ( bx ) ≤ / 2. By standard constructions, such a random variable can beefficiently sampled with seed length O (log log( n ) + log k + log(1 /ǫ )) [NN93]. Combining these twoerrors yields the claim. Again, X remains Θ( ǫ /k /b )-noticeable. Using Theorem 1.1 and Theorem 2.2, it is fairly immediate to obtain PRGs that rely only on level- k bounds. Similarly, bounds on levels up to k can be leveraged to get an improved seed length. Theorem 3.7 (Theorem 1.2, restated) . Let F be any class of n -variate Boolean functions that isunder restrictions and negations. Suppose that M k ( F ) ≤ b k for some b ≥ and k > . Then for any ǫ > , there exists an explicit PRG for F with error ǫ with seed length s = O (cid:18) b k − · k · log( n ) log k − ( n/ǫ ) ǫ k − (cid:19) . (21) If it further holds that L ,i ( F ) ≤ b i for all ≤ i < k , then the seed length can be improved to s = O (cid:18) b k − · (log log n + log k + log( b/ǫ )) · log k − ( n/ǫ ) ǫ k − (cid:19) . (22) Proof. By Theorem 2.2, given an explicit p -noticeable fractional PRG for F with error δ and seedlength s , one immediately obtains an explicit PRG for F with error O ( δ log( n/δ ) /p ) and seed length O ( s log( n/δ ) /p ). 12or the first statement, by our assumption and using the fractional PRG guaranteed byTheorem 1.1, for any δ > 0, we immediately obtain an explicit PRG for F with error O ( b δ − /k log( n/δ )) and seed length O ( b k log( n ) log( n/δ ) /δ /k ). To get the error below ǫ , we set δ = Θ (cid:18)(cid:18) ǫb log( n/ǫ ) (cid:19) kk − (cid:19) (23)(the astute reader may notice we implicitly use b ≤ n here). This yields a PRG with error ǫ and seedlength s = O (cid:18) b k − · k · log( n ) log k − ( n/ǫ ) ǫ k − (cid:19) . (24)The second statement follows in an identical manner from using the improved seed length fromthe second part of Theorem 1.1 in the case that one has control on the L Fourier mass on the lowerlevels.Corollary 1.1 is now an immediate consequence of Theorem 3.7; for any desired ǫ > b · log( n ) · − O ( k ) ,one can simply apply Theorem 3.7 using level k = Θ(log( b log( n ) /ǫ )) to obtain a PRG for F with errorat most ǫ with seed length s = O ( b · log( b log( n ) /ǫ ) · log( n/ǫ )) . (25)Note that if ǫ = 1 / poly( n ), then this means that one needs bounds only up to level Θ(log( n )) (again,using the fact that b ≤ n ). This also partially answers an open question of [CHLT19], which asks howmany levels of Fourier bounds suffice to recover polylogarithmic dependence in 1 /ε . Remark 3.8. Note that this Taylor approach does not yield anything nontrivial given just level-two bounds, unlike the fractional generator in [CHLT19]. This is actually a necessary byproduct ofcombining this approach with the random walk gadget of [CHHL19]. Given only level-two bounds, thisapproach attempts to use j -wise independence for j < k = 2 and smallness to deal with errors on thehigh degree terms ( k ≥ ). However, the trivial random variable that is ± with equal probability istrivially -wise independent, as each component is a uniform random bit, albeit trivially correlated.No matter how we scale them, one can show that composing arbitrarily many independent copies ofthis random variable via the random walk gadget will necessarily polarize to ± at termination, whichclearly cannot fool nontrivial functions. F Our analysis recovers all the existing applications of [CHHL19] (among them, AC circuits, low-sensitivity functions, and read-once branching programs); indeed, all the classes considered theresatisfy L Fourier bounds on the entire tail. To our knowledge, our new analysis does not immediatelyimprove the seed lengths obtained there, though it shows (i) the seed lengths there can potentiallybe improved using stronger bounds on M k , and (ii) the analysis would still have been valid had theseFourier bounds been known only up to some level k .However, the generality afforded to us by this new analysis allows us to obtain a new PRG forlow-degree polynomials over F , which addresses an open question of [CHHL19] by showing that thisframework can handle this class. Indeed, let F be the set of n -variate, degree- d polynomials over F .As a preliminary step towards deriving Fourier tail bounds that would imply a nontrivial PRG forthis class using their framework, [CHHL19] prove the following Fourier bounds: Proposition 4.1 (Theorem 6.1 of [CHHL19]) . Let p : F n → F be a degree- d polynomial, and let f ( x ) = ( − p ( x ) . Then L ,k ( f ) ≤ ( k · d ) k . k = Ω( √ n ) and any d . While Theorem 2.4 can yielda nontrivial PRG by just applying the level-two bound, the dependence on 1 /ǫ is at least quadratic. However, using our new, more flexible analysis, one can obtain a nontrivial PRG with polylogarithmicdependence on the error parameter. Our formal result is the following: Theorem 4.2. Let F be the class of degree- d polynomials over F on n variables. Then there existsan explicit pseudorandom generator for F with error ǫ and seed length s = O (2 O ( d ) · log (log( n ) /ε ) · log( n/ε )) . (26) Proof. Fix ε > k = Θ(log(log( n ) /ε )). By Proposition 4.1, we have that for all j ≤ k , L ,j ( F ) ≤ (Θ(log(log( n ) /ε ) · d )) j . (27)By setting b = Θ(log(log( n ) /ε ) · d ), we may apply Theorem 3.7 for F and error ε . Note that ε − Θ(1 / log(1 /ε )) = O (1), so plugging in this value of b , we immediately obtain the desired pseudorandomgenerator.For comparison, the best known construction by Viola [Vio09], obtained by summing d independentcopies of a sufficiently good small-biased space, attains seed length d · log( n ) + O ( d · d log(1 /ε )), whichfor constant ε and d is within an additive constant of the optimal possible seed. The generatorimplied by our analysis recovers this polylogarithmic dependence in n/ε , although with slightly worsedependence on log n and polynomially worse dependence in log(1 /ε ). Neither generator can handlesuperlogarithmic degree. While this result clearly falls short of the state-of-the-art, we emphasize thatthis generator is conceptually distinct from the existing constructions, and yet belongs to this genericrandom walk framework.Our analysis allows us to exploit known Fourier bounds that are too weak for the existing analysesto obtain polylogarithmic error dependence. In particular, to get a nontrivial pseudorandom gen-erator for polynomials of superlogarithmic degree with nontrivial seed length, our work shows thatthe following weaker conjecture would suffice to break the logarithmic degree barrier and still achievepolylogarithmic (in n ) seed length for ǫ = 1 / poly( n ): Conjecture 4.3. Let F be the class of degree- d polynomials over F on n variables. Then M k ( F ) ≤ (poly( k, log n ) · o ( d ) ) k (28) for k ≤ O (log n ) . In fact, we observe that to break the logarithmic degree barrier, it actually suffices that this holdsjust at k = 3, though with poor dependence on ε . Note that this is a significantly weaker conjecturethan positing that the same bounds hold for L ,k ( F ). Moreover, as we explain in the next section, M k ( F ) can be controlled using correlation bounds, which are much better studied than L Fourierbounds. By applying this Fourier bound at level-two, one can use the fractional PRG of [CHLT19] to obtain seed length2 O ( d ) polylog( n ) /ǫ o (1) using the random walks framework. This gives exponentially worse error dependence comparedto our approach. Bounds on M k ( F ) via Correlation with Shifted Majorities As we have seen, this analysis allows for the construction of PRGs from the weaker quantity M k ( F ).In this section, we repeat the argument of [CHH + 20] to show how bounds on M k ( F ) follow fromcovariance bounds with certain resilient functions (in particular, shifted majorities). In their paper,they deal with the case of k = 2; we rather straightforwardly generalize this argument, but stressthat the approach is the same as in Section 6 of their paper. To that end, for convenience andconsistency with their argument, we adopt their conventions and requisite definitions just for thissection. We will now consider Boolean functions written as f : { , } n → { , } . Translating to thisnotation, for any such Boolean function, let e ( f )( x ) , ( − f ( x ) . Then, letting F = e ( f ), we now haveˆ F ( S ) = E x [ F ( x ) e ( P i ∈ S x i )]. Definition 5.1. The covariance between f and g , where f, g are Boolean iscov( f, g ) , | E [ e ( f ( x )) e ( g ( x ))] − E [ e ( f ( x ))] E [ e ( g ( x ))] | , (29)while the covariance between a function f and a class G is defined as cov( f, G ) , max g ∈G cov( f, g ).For any x ∈ { , } n , we write | x | for the Hamming weight, i.e. P ni =1 x i . For any a ∈ { , , . . . , n } ,[CHH + 20] define Maj a by Maj a ( x ) , ( | x | > a , (30)as well as the following associated functions for any θ ∈ [ n/ θ ( x ) , ( ( − Maj n/ ( x ) if (cid:12)(cid:12) | x | − n/ (cid:12)(cid:12) > θ M k ( F ) with covariance bounds against k - XOR s of thesefunctions: Lemma 5.1 (Lemma 6.1 of [CHH + . Let F be any family of ( kn ) -variate Boolean func-tions that is closed under relabeling variables. Further, suppose that for any a , . . . , a k such that | a i − n/ | = O ( √ kn log n ) for all i ∈ [ k ] , and all f ∈ F , we have for some t ≥ f ( x , . . . , x k ) , ⊕ ki =1 Maj a i ) ≤ (cid:18)r tn (cid:19) k , (32) where x i ∈ { , } n and ⊕ denotes the XOR function.Then, it holds that M k ( F ) ≤ O ( p tk log n ) k . (33)To prove this lemma, [CHH + 20] use the following sequence of claims. Fact 5.1 (Claim 6 . + . For any f ∈ F , let F ( x , . . . , x k ) = e ( f ( x , . . . , x k )) . Under thehypotheses of Lemma 5.1, for any ≤ a , . . . , a k ≤ O ( √ kn log n ) , | E x ,..., x k [( F ( x , . . . , x k ) − E [ F ]) k Y i =1 Thr a i ( x i )] | ≤ (cid:18)r tn (cid:19) k . (34) . Fact 5.2 (Claim 6.3 of [CHH + . For any x ∈ { , } n , P ni =1 e ( x i ) = 2 P ≤ a ≤ n/ Thr a ( x ) . act 5.3 (Claim 6.4 of [CHH + . For any Boolean function f : { , } kn → { , } , thereexists a k -equipartition of [ kn ] into disjoint sets S , . . . , S k such that (cid:12)(cid:12)(cid:12)(cid:12) X S ⊆ [ kn ]: | S | = k ˆ f ( S ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ S j ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) (35) for some absolute constant C > . As this fact is not quite identical to that in [CHH + Proof. We (effectively) use the probabilistic method: let P be the set of k -equipartitions of [ kn ]. Let T ⊆ [ kn ] of size k be arbitrary; without loss of generality, suppose T = [ k ]. Consider a uniformlyrandom k -equipartition P = S ⊔ . . . ⊔ S k ∈ P . The probability that each i ∈ T belong to distinct S j is easily seen to be k − Y i =1 ( k − i ) · nkn − i ≥ ( k − n k − ( kn ) k − = ( k − k k − = e − O ( k ) , (36)where the last line uses Stirling’s approximation. By symmetry, let α ∈ N be the number of k -equipartitions that any arbitrary subset T is in. Then we have α (cid:12)(cid:12)(cid:12)(cid:12) X S ⊆ [ kn ]: | S | = k ˆ f ( S ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) X P ∈P X i j ∈ S j ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) (37) ≤ X P ∈P (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ S j ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) (38) ≤ |P| max P ∈P (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ S j ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) . (39)The first line follows from simple counting, while the second is the triangle inequality. Rearranging,we deduce that (writing T as a generic subset of size k ) (cid:12)(cid:12)(cid:12)(cid:12) X S ⊆ [ kn ]: | S | = k ˆ f ( S ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ |P| α max P ∈P (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ S j ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) (40)= Pr P ∼P ( T ∈ P ) − max P ∈P (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ S j ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) (41) ≤ e O ( k ) max P ∈P (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ S j ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) . (42)The last fact that is needed can be deduced from the Chernoff bound: Fact 5.4 (Claim 6.5 of [CHH + . For any a ≥ Ω( √ kn log n ) , E [ | Thr a | ] ≤ O (1 /n k ) . With these facts, we can now prove Lemma 5.1 in an entirely analogous fashion to [CHH + Proof of Lemma 5.1. Fix f ∈ F , and again write F ( x , . . . , x k ) = e ( f ( x , . . . , x k )). Let F ′ = F − E [ F ].Let U j = { i : ( j − n + 1 ≤ i ≤ jn } . Then, possibly after relabelling variables, we have by Fact 5.3that M k ( f ) = (cid:12)(cid:12)(cid:12)(cid:12) X S ⊆ [ kn ]: | S | = k ˆ f ( S ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ U j , ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) , (43)16o we may turn to bounding this latter term. We have (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ U j , ∀ j ∈ [ k ] ˆ f ( { i , . . . , i k } ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) X i j ∈ U j , ∀ j ∈ [ k ] E [ F ′ ( x , . . . , x k ) k Y j =1 e (( x j ) i j )] (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) F ′ ( x , . . . , x k ) k Y j =1 (cid:18) X i j ∈ U j e (( x j ) i j ) (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k X ≤ a i ≤ n/ , ∀ i ∈ [ k ] (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) F ′ ( x , . . . , x k ) k Y i =1 Thr a i ( x i ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k (cid:18) X ≤ a i ≤ O ( √ kn log n ) , ∀ i ∈ [ k ] (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) F ′ ( x , . . . , x k ) k Y i =1 Thr a i ( x i ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) + O (1) (cid:19) ≤ k · O (cid:16)p kn log n (cid:17) k · (cid:18)r tn (cid:19) k = O (cid:16)p tk log n (cid:17) k . The first inequality follows from Fact 5.2, the second from Fact 5.4, and the last by Fact 5.1. As f ∈ F was arbitrary, and by absorbing the constant C from above into the implicit constant here inthis bound, we obtain the desired claim. In this work, we have given a nearly complete interpolation between the PRGs obtained in the polariz-ing random walks framework by exploiting level- k bounds on the class of functions, thus answering anopen question from [CHLT19]. We do so by exploiting an alternate Fourier analysis via Taylor’s the-orem and utilizing multilinearity and random restrictions. This new analysis enables us to constructPRGs from bounds on the potentially much smaller and better-understood, Fourier quantity M k ( F ),for any k ≥ 3. By generalizing the connection established in [CHH + L ,i , for all i ≤ k , where k ≥ 3. A natural openquestion along these lines is to obtain such an improved seed length using bounds on M i (instead of L ,i ) for all i ≤ k . Another natural question is to construct a PRG using bounds on just M (recallthat [CHLT19] gives such a construction using bounds on L , and our analysis only gives a non-trivialPRG from bounds on M k when k ≥ k bounds for F polynomials, our approach shows that the randomwalks framework can yield pseudorandom generators for the class of F polynomials that is competitivewith, though falls short of, the state-of-the-art. As mentioned, we hope this paper both gives evidencethat stronger Fourier control (perhaps via proving the required correlation bounds) can give betterPRGs using this framework, and can also handle classes that were previously not known to be possible.In particular, we emphasize that proving Conjecture 4.3 even for the case of k = 3 will lead to PRGsfor F polynomials with degree ω (log n ), a longstanding problem in complexity theory. References [CHH + 20] Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, and David Zucker-man. 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