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(b) Assign the coordinates in [MATH] using a [MATH] -biased distribution [MATH] Equivalently, we prove that the following distribution [MATH] -fools ROBPs of width- [MATH] and length- [MATH] |
1. Pick [MATH] using a [MATH] -biased distribution with marginals [MATH] 2. Assign the coordinates in [MATH] uniformly at random. |
3. (a) Pick [MATH] using a [MATH] -biased distribution with marginals [MATH] (b) Assign the coordinates in [MATH] uniformly at random. |
(c) Assign the coordinates in [MATH] using a [MATH] -biased distribution [MATH] Let [MATH] . Let [MATH] be the event that [MATH] can be computed by a ROBP of the form [MATH] where [MATH] are defined over disjoint sets of at most [MATH] variables, and each [MATH] is a width- [MATH] ROBP with at most [MATH] vertices on t... |
[MATH] Assuming that [MATH] happened, then by Theorem 4.2 [MATH] can be written as [MATH] where [MATH] are subprograms of [MATH] and [MATH] For each [MATH] , using Theorem we have that |
[EQUATION] By linearity of expectation and the triangle inequality [EQUATION] [EQUATION] Overall, we get [EQUATION] [EQUATION] [EQUATION] |
where the last equality is due to the fact for any [MATH] the distribution of [MATH] is the uniform distribution over [MATH] We bound Expression ( ) by |
[EQUATION] where the second summand is bounded by [MATH] according to the above discussion using Theorem 4.2 and Theorem Pseudorandom restrictions for the XOR of short ROBPs |
In this section, we prove Theorem Let [MATH] be pairwise disjoint subsets of [MATH] , each of size at most [MATH] . For [MATH] let [MATH] be a width [MATH] ROBP. We construct a pseudorandom generator that [MATH] -fools [MATH] We recall the statement of Theorem and the construction. See |
Recall that the pseudorandom restriction assigns [MATH] fraction of the variables as follows: 1. Choose a set of coordinates [MATH] according to a [MATH] -biased distribution with marginals [MATH] , for [MATH] |
2. Assign the variables in [MATH] according to a [MATH] -biased distribution, for [MATH] Analysis. We shall assume without loss of generality that for all [MATH] it holds that [MATH] We shall also assume without loss of generality that for all [MATH] it holds that [MATH] (i.e., that the functions are non-constant). Sin... |
We partition the functions into [MATH] buckets according to their variance. Let [MATH] for every [MATH] , let [MATH] and [MATH] Let [MATH] be a sufficiently large constant. We consider two cases in our analysis: |
Low-Variance Case: For every [MATH] we have [EQUATION] High-Variance Case: There exists a [MATH] with [EQUATION] Setting Up Parameters: |
Let [MATH] be a sufficiently large constant. Set [EQUATION] 5.1 Low-Variance Case For [MATH] let [MATH] Thus, [MATH] Let [MATH] be any [MATH] -biased distribution with marginals [MATH] For [MATH] , we shall show that with probability at least [MATH] over the choice of [MATH] , it holds that |
[EQUATION] for any [MATH] -biased distribution [MATH] over [MATH] Thus, by union bound Eq. ( ) holds for all [MATH] simultaneously with probability at least [MATH] over [MATH] Using the following XOR lemma for small-biased distributions from GMR 12 we get that any |
[MATH] -biased distribution, fools [MATH] with error at most [MATH] (using [MATH] ). Lemma 5.1 GMR 12 , Thm. 4.1] , restated) Let [MATH] Let [MATH] be functions on disjoint input variables such that each [MATH] is [MATH] -fooled by any [MATH] -biased distribution. Let [MATH] be a multilinear function in its inputs. The... |
In Appendix A.2 , we show how to derive Lemma 5.1 from GMR 12 , Thm. 4.1] In the remainder of this section, we focus on fooling a single [MATH] , that is, fooling the product (i.e., XOR) of functions [MATH] for which [MATH] We note that since we are in the “Low-Variance Case”, then |
[EQUATION] We handle two cases depending on whether [MATH] is big or not. The case of [MATH] In this case there are at most [MATH] functions in [MATH] , each computed by a width- [MATH] ROBP on at most [MATH] bits. Thus, [MATH] can be computed by a ROBP of length at most [MATH] and width at most [MATH] Using Theorem 3.... |
The case of [MATH] In this case all variances in [MATH] are certainly smaller than [MATH] , and hence for all [MATH] , we have [MATH] Let |
[EQUATION] Then, [EQUATION] We have [MATH] and [MATH] We will show that with high probability over [MATH] , any [MATH] -biased distribution fools [MATH] |
For ease of notation, in this case we think of [MATH] as [MATH] and denote by [MATH] The proof strategy for this part follows the work of Gopalan and Yehudayoff GY14 We note that |
[EQUATION] where [MATH] is the [MATH] -symmetric polynomial given by [MATH] We show that [MATH] and [MATH] fool the low-degree symmetric polynomials. Then, the following theorem by Gopalan and Yehudayoff GY14 bootstraps this to show that [MATH] and [MATH] also fool the sum of all high-degree symmetric polynomials. |
Theorem 5.2 (Gopalan-Yehudayoff Tail Inequalities GY14 Let [MATH] Suppose [MATH] and [MATH] for some [MATH] and [MATH] Then, for every |
[MATH] it holds that [MATH] Furthermore, if [MATH] , then [EQUATION] Analyzing the Symmetric Polynomials From Eq. ( ) and our assumption that |
[MATH] we get that [MATH] Recall that [MATH] is a sufficiently large constant and recall the definition of [MATH] from Eqs. ( ), ( ) and ( ). We set |
[EQUATION] In the following, we shall use the facts that [MATH] and [MATH] Claim 5.3 Let [MATH] Let [MATH] be a set of size at most [MATH] Then, with probability at least [MATH] over the choice of [MATH] |
[MATH] has spectral-norm at most [MATH] Proof. Note that [MATH] can be computed by a ROBP with length [MATH] and width [MATH] (as in the case where [MATH] is big). Apply Theorem 3.10 to [MATH] |
We say that [MATH] is a good set if for all sets [MATH] of size at most [MATH] the spectral-norm of [MATH] is at most [MATH] We observe that by Claim 5.3 , the probability that [MATH] is good is at least |
[MATH] (using Eq. ( ) and ( )). Claim 5.4 If [MATH] is good, then for any [MATH] [MATH] has spectral-norm at most [MATH] Proof. We expand the [MATH] -symmetric polynomial: |
[MATH] Since [MATH] is good, each summand has spectral-norm [EQUATION] (using [MATH] ). Summing over all [MATH] summands completes the proof. |
We wish to show that with high probability the total variance under restrictions [MATH] is small. Towards this goal, we prove a bound on the [MATH] -th moment of the total variance. |
Claim 5.5 [MATH] Proof. Fix [MATH] , not necessarily distinct indices. By Lemma 3.14 [EQUATION] from which we deduce [EQUATION] We are left to bound [MATH] By Fact 3.5 , for any [MATH] , the random variable [MATH] (whose value depends on the choice of [MATH] ) is bounded in [MATH] By Theorem 3.13 , its expected value i... |
we have [MATH] In such a case [MATH] We get [MATH] , which gives [EQUATION] We say that a set [MATH] is excellent if [MATH] is good and [MATH] |
Claim 5.6 [MATH] Proof. Note that [MATH] and apply Markov’s inequality on [MATH] using Claim 5.5 Claim 5.7 Let [MATH] be an excellent set. Let [MATH] be any [MATH] -biased distributions. Then, for [MATH] we have |
[EQUATION] and [EQUATION] Proof. Recall that [MATH] and [MATH] . The first claim relies on the following: 1. [MATH] has small spectral-norm (using Claim 5.4 , since [MATH] is good) and hence is fooled by [MATH] . In details, its spectral-norm is at most [MATH] and [MATH] is [MATH] -biased. Thus |
[EQUATION] 2. The expectation of [MATH] on a uniformly chosen [MATH] is at most [EQUATION] The second claim relies on the following: |
1. [MATH] has small spectral-norm (using Claim 5.4 , since [MATH] is good) and hence is fooled by [MATH] . In details, its spectral-norm is at most [MATH] and [MATH] is [MATH] -biased. Thus |
[EQUATION] 2. The expectation of [MATH] on a uniformly chosen [MATH] is [MATH] . ∎ The next lemma combined with Claim 5.7 concludes the low-variance case, since it shows that with high probability, [MATH] is excellent, and then [MATH] is an [MATH] -PRG for [MATH] (for a sufficiently large choice of [MATH] ). |
Lemma 5.8 If [MATH] is excellent, then [MATH] Proof. Let [MATH] and let [MATH] be the event that [MATH] and [MATH] Picking [MATH] , and using Claim 5.7 the event [MATH] happens with probability at least [MATH] Assuming [MATH] occurs, Theorem 5.2 gives |
[EQUATION] Furthermore, for sets of smaller cardinality, i.e., for [MATH] , Claim 5.7 gives [EQUATION] We would like to bound [MATH] for [MATH] . Towards this end, we consider the expectation of [MATH] by partitioning into the two cases depending on whether the event [MATH] occurred or not. |
[EQUATION] Thus, [MATH] and we get [EQUATION] Since the [MATH] ’s and [MATH] ’s are bounded in [MATH] , we get [EQUATION] 5.2 High-Variance Case |
In the high-variance case, there exists a [MATH] and an interval [MATH] (the constant [MATH] handles the case [MATH] ) satisfying: |
[EQUATION] In this case, the expected value of [MATH] under the uniform distribution is rather small: [EQUATION] Recall that the pseudorandom restriction samples a set [MATH] according to some [MATH] -biased distribution [MATH] with marginals [MATH] , and a partial assignment to the bits in [MATH] according to some [MA... |
[EQUATION] Thus, it suffices to show that for most [MATH] [MATH] we have [MATH] Theorem 5.9 (Theorem - High Variance Case) With probability [MATH] over [MATH] and [MATH] , it holds that [MATH] |
Proof. Denote by [MATH] By our assumption, [MATH] Since all functions in [MATH] have variance at least [MATH] we have [EQUATION] |
We remark that in this case, unlike the low-variance case, we do not know how to handle large [MATH] easily, so for the rest of the proof [MATH] can be anything between [MATH] and [MATH] |
Fix [MATH] and [MATH] . We expand [MATH] [EQUATION] For any fixed [MATH] , using [MATH] gives [EQUATION] Claim 5.10 (Most [MATH] ’s preserve variance in expectation) |
With probability at least [MATH] over the choice of [MATH] , it holds that [MATH] Proof. Since [MATH] , it suffices to show that with probability [MATH] over the choice of [MATH] we have [MATH] To show that |
[MATH] is well-concentrated we analyze its [MATH] -th moment for [MATH] where [MATH] is a sufficiently large constant. [EQUATION] |
Fix [MATH] , (not necessarily distinct), then by Lemma 3.14 [EQUATION] Overall, we get [EQUATION] To bound [MATH] we use the fact that by Theorem 3.13 |
[EQUATION] and then by Chernoff’s bound [MATH] with probability at least [MATH] Since [MATH] is always upper bounded by [MATH] , the [MATH] -moment of the sum is at most |
[EQUATION] We get that [MATH] Since [MATH] this is at most [MATH] Thus, using Markov’s inequality, the probability that [MATH] is at most [MATH] which completes the proof. |
Let [EQUATION] where [MATH] is a sufficiently large constant declared before Eq. ( ). Assume that [MATH] is an even integer. Recall that [MATH] We again define [MATH] to be a good set if [MATH] has spectral-norm at most [MATH] for all sets [MATH] of size at most [MATH] . As in Claim 5.3 the probability that [MATH] is g... |
Claim 5.11 If [MATH] is a good set, then at most [MATH] of the [MATH] ’s have [MATH] Proof. If [MATH] have [MATH] , then their product has spectral-norm at least [MATH] , since [MATH] for functions defined on disjoint variables. |
Fix an excellent set [MATH] . Let [MATH] be the of indices [MATH] with [MATH] . We show that with high probability over [MATH] [MATH] . We denote by |
[EQUATION] Obviously [MATH] and [MATH] is bounded in [MATH] Furthermore, we have that [EQUATION] Thus, the error term have small spectral-norm since [MATH] . We use this fact to bound |
[MATH] . (recall that [MATH] is an even integer.) Claim 5.12 [EQUATION] Proof. The spectral-norm of [MATH] is at most [MATH] . Thus, any [MATH] -biased distribution fools [MATH] with error at most [MATH] and we get |
[EQUATION] To bound [MATH] we use Lemma 3.8 We observe that [MATH] are independent random variables, where each [MATH] is bounded in [MATH] with mean zero, and hence [MATH] (See Lemma 3.12 ). Applying Lemma 3.8 gives |
[EQUATION] Since [MATH] , the upper bound on [MATH] is at most [MATH] Finally, the upper bound with respect to [MATH] is at most |
[EQUATION] Using Markov’s Inequality and Claim 5.12 gives [EQUATION] using [MATH] and [MATH] in the last inequality. Furthermore, using Eqs. ( ) and ( 10 ): [MATH] In the complement event, |
[EQUATION] Since [MATH] , we get that with probability at least [MATH] over [MATH] and [MATH] [MATH] . (End of Proof of Theorem 5.9 |
Assigning all the variables: a pseudorandom generator for the XOR of short ROBPs In Theorem , we proved that we can pseudorandomly assign [MATH] -fraction of the coordinates of [MATH] , while maintaining its acceptance probability up to an additive error of [MATH] , using [MATH] random bits. In this section, we will co... |
[MATH] times, combined with Lovett’s Lov08 or Viola’s Vio08 pseudorandom generator for low-degree polynomials, and CHRT’s pseudorandom generator for constant-width ROBPs CHRT17 . Our main result is: See |
Assigning [MATH] -fraction of the variables. The first step is rather standard. By making [MATH] recursive calls to Theorem we can assign all but [MATH] fraction of the coordinates while maintaining the acceptance probability. Note that we rely on the fact that under restrictions, the restricted function is still of th... |
Claim 6.1 Let [MATH] , with block-length [MATH] Then, there is a pseudorandom restriction [MATH] using at most [MATH] random bits, and changing the acceptance probability by at most [MATH] Furthermore, [MATH] is [MATH] biased with marginals [MATH] , and |
[MATH] is [MATH] biased. Proof Sketch. Apply Theorem with error [MATH] for [MATH] times recursively, with independent random bits per each iteration. Denoting by [MATH] , this generates [MATH] -pseudorandom restrictions |
[MATH] where [MATH] and [MATH] for all [MATH] To be more precise, for each [MATH] [MATH] is the [MATH] -biased subset with marginals [MATH] in the description of the generator in Theorem and [MATH] is its assignment sampled from a [MATH] -biased distribution. We take [MATH] and [MATH] to be the concatenation of [MATH] ... |
[EQUATION] The amount of random bits used to sample the restriction is [EQUATION] Next, we claim that [MATH] is a [MATH] -biased with marginals [MATH] Recall that [MATH] are [MATH] -biased with marginals [MATH] , and that [MATH] By Claim 3.7 , for any subset [MATH] of size at most [MATH] we have |
[EQUATION] which implies by monotonicity that for larger subsets [MATH] , we have [MATH] Finally, we claim that conditioned on [MATH] and in fact for any choice of [MATH] [MATH] is [MATH] -biased. This is due to the fact that [MATH] is the concatenation of [MATH] where each [MATH] is [MATH] -biased. |
We would like to claim that [MATH] simplifies after assigning [MATH] of the coordinates. For a particular function [MATH] , with high probability, at least [MATH] , the block length decreases under a random restriction by a factor of [MATH] This is due to the fact that on expectation at most [MATH] of the variables wil... |
We run into trouble if at some iteration we have more than [MATH] functions of block-length [MATH] The first observation is that in this case the total variance of the functions is extremely high, exponential in [MATH] . Recall that the expected value of the product is exponentially small in the total variance. This me... |
The nice thing about these “aggressive pseudorandom restrictions” is that they keep variables alive with such small probability that with high probability each function [MATH] will depend on at most [MATH] variables after the restriction, except for a small number of functions covering at most [MATH] “bad variables”. T... |
6.1 PRG for the XOR of many functions with block-length [MATH] Let [MATH] be the class of functions of the form [MATH] where [MATH] are Boolean functions on disjoint sets of variables, [MATH] (the ‘junta’) depends on at most [MATH] variables, [MATH] are non-constant and depend on at most [MATH] variables and |
[MATH] Lemma 6.2 There exists a constant [MATH] such that the following holds. For all [MATH] such that [MATH] , there exists a log-space explicit pseudorandom generator [MATH] that [MATH] -fools [MATH] |
Algorithm 1 The Pseudorandom Generator [MATH] 1: A block-length [MATH] , the output length [MATH] , a junta-size [MATH] , an error parameter [MATH] |
2: Set [MATH] 3: Pick [MATH] using a [MATH] -biased distribution with marginals [MATH] 4: Assign coordinates of [MATH] in [MATH] using a [MATH] -biased distribution. |
5: Assign coordinates of [MATH] in [MATH] using Viola’s generator with error [MATH] and degree [MATH] 6: return [MATH] Lemma 6.3 |
Let [MATH] be a sufficiently large constant. Let [MATH] be some integer. Let [MATH] be non-constant Boolean functions that depend on disjoint sets of at most [MATH] variables each. Assume [MATH] Suppose [MATH] is [MATH] -biased distribution with marginals [MATH] Suppose [MATH] is sampled from a [MATH] -biased distribut... |
Proof. Without loss of generality [MATH] Let [MATH] , and note that [MATH] since [MATH] for a sufficiently large constant [MATH] |
Let [MATH] be the disjoint sets of variables on which [MATH] depend respectively. For any function [MATH] , there exists a sensitive pair of inputs [MATH] such that [MATH] and [MATH] differ in exactly one coordinate [MATH] and such that [MATH] We say that the sensitive pair “survives” the random restriction defined by ... |
[EQUATION] We upper bound the first and second summands separately. By Lemma 3.8 and Lemma 3.12 , the first summand is upper bounded by [MATH] where |
[EQUATION] Since [MATH] , the first summand is upper bounded by [MATH] Next, we upper bound the second summand. By Vazirani’s XOR lemma, since [MATH] is [MATH] -biased, we have that the marginal distribution of any set of at most [MATH] bits in [MATH] is [MATH] -close to uniform in statistical distance. Since [MATH] is... |
[MATH] Combining the bounds on both summands we get [EQUATION] Using [MATH] we get [EQUATION] Using [MATH] and [MATH] we get [MATH] In the complement event, at least [MATH] of the functions [MATH] are non-constant. |
Lemma 6.4 Let [MATH] Suppose [MATH] where [MATH] is a [MATH] -junta and [MATH] is a polynomial of degree- [MATH] over [MATH] If [MATH] fools degree- [MATH] polynomials over [MATH] with error [MATH] , then [MATH] fools [MATH] with error [MATH] |
Proof. Let [MATH] be the set of variables on which [MATH] depends. Using the Fourier transform of [MATH] [MATH] we write [MATH] as |
[MATH] Note that [MATH] is a polynomial of degree- [MATH] over [MATH] as well, thus we get [EQUATION] Proof of Lemma 6.2 First note that [MATH] has very small expectation under the uniform distribution |
[EQUATION] using the assumption [MATH] Thus, we need to maintain low-expectancy under the pseudorandom assignment. By Lemma 6.3 , with probability at least [MATH] after the aggressive random restriction at least [MATH] of the functions [MATH] remain non-constant. Since [MATH] we maintained the low-expectancy under aggr... |
Furthermore, we wish to show that with high probability, except for a set of at most [MATH] “bad variables” all functions have block-length at most [MATH] Recall that there are at most [MATH] functions. The probability that any particular [MATH] variables survive is at most [MATH] Pick [MATH] The probability that at le... |
[EQUATION] If [MATH] , then this probability is at most [MATH] This means that, with high probability, there are less than [MATH] variables from all functions with more than [MATH] effective variables remaining. Otherwise, there would have been [MATH] functions accountable to a total number of more than [MATH] variable... |
Overall, with probability at least [MATH] we are left with the XOR of a small-junta, on at most [MATH] variables, and an XOR of at least [MATH] non-constant functions on at most [MATH] variables (i.e., a degree [MATH] polynomial). Moreover, the restricted function has expected value at most [MATH] in absolute value und... |
6.2 A thought experiment We are ready to describe the pseudo-random restriction process in full detail. We start by describing a process that iteratively “looks” at the restricted functions in order to decide which pseudorandom restriction to apply next: the one described in Lemma 6.1 or the one from Lemma 6.3 This ult... |
We start with [MATH] blocks of length [MATH] We assume that [MATH] (if not set [MATH] ). Algorithm 2 an “adaptive pseudorandom generator” |
1: for [MATH] do 2: Let [MATH] 3: if [MATH] then apply CHRT’s PRG on the remaining coordinates, and Halt! 4: if more than [MATH] of the restricted functions are non-constant and depend on at most [MATH] variables then |
apply [MATH] from Lemma 6.2 on the remaining variables, and Halt! 5: else apply the pseudorandom restriction from Lemma 6.1 on the remaining variables. |
Next, we show that the process yields a pseudorandom string fooling [MATH] First, note that the process either stops at Step 3 or at Step 4. In both cases we assign all the variables according to some pseudorandom generator, hence all the variables will be assigned by the end of the process. |
For [MATH] Let [MATH] be the set of coordinates that remain alive at the beginning of the [MATH] -th iteration. Denote by [MATH] the [MATH] -th function under the restriction at the beginning of the [MATH] -th iteration. Define [MATH] to be the set of variables that affect the output of [MATH] For example if [MATH] is ... |
Let [MATH] be the set of functions that depend on some but not more than [MATH] variables, [MATH] be the set of functions that depend on more than [MATH] variables and [MATH] |
Claim 6.5 Let [MATH] Suppose [MATH] and [MATH] Then, with probability at least [MATH] we have [MATH] Proof. Under the assumptions we reach Step 5 in Algorithm . We show that: |
1. With probability at least [MATH] , at most [MATH] of the variables in [MATH] remain alive in Step 5. 2. With probability at least [MATH] , at most [MATH] new variables are added to [MATH] |
Both claims rely on the fact that any set of [MATH] variables remain alive under the pseudorandom restriction in Lemma 6.1 with probability at most [MATH] |
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