Randomness Buys Depth for Approximate Counting

Viola, Emanuele

doi:10.1007/s00037-013-0076-6

Cited by 9 publications

(6 citation statements)

References 26 publications

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“…Lower bounds for the closely related promise and approximate majorities were proved by Viola [Vio14] and O'Donnell and Wimmer [OW07] respectively. Viola [Vio14] shows that any poly(n)sized depth-d AC 0 circuit cannot compute a promise majority for δ = o(1/(log n) d−2 ).…”

Section: Known Upper Boundsmentioning

confidence: 99%

A fixed-depth size-hierarchy theorem for AC ⁰ [⊕] via the coin problem

Limaye

Sreenivasaiah

Srinivasan

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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In this work we prove the first Fixed-depth Size-Hierarchy Theorem for uniform AC 0 [⊕]. In particular, we show that for any fixed d, the class C d,k of functions that have uniform AC 0 [⊕] formulas of depth d and size n k form an infinite hierarchy. We show this by exibiting the first class of explicit functions where we have nearly (up to a polynomial factor) matching upper and lower bounds for the class of AC 0 [⊕] formulas.The explicit functions are derived from the δ-Coin Problem, which is the computational problem of distinguishing between coins that are heads with probability (1 + δ)/2 or (1 − δ)/2, where δ is a parameter that is going to 0. We study the complexity of this problem and make progress on both upper bound and lower bound fronts.• Upper bounds. For any constant d ≥ 2, we show that there are explicit monotone AC 0 formulas (i.e. made up of AND and OR gates only) solving the δ-coin problem that have depth d, size exp(O(d(1/δ) 1/(d−1) )), and sample complexity (i.e. number of inputs) poly(1/δ). This matches previous upper bounds of O'Donnell and Wimmer (ICALP 2007) and Amano (ICALP 2009) in terms of size (which is optimal) and improves the sample complexity from exp(O(d(1/δ) 1/(d−1) )) to poly(1/δ). • Lower bounds. We show that the above upper bounds are nearly tight (in terms of size) even for the significantly stronger model of AC 0 [⊕] formulas (which are also allowed NOT and Parity gates): formally, we show that any AC 0 [⊕] formula solving the δ-coin problem must have size exp(Ω(d(1/δ) 1/(d−1) )). This strengthens a result of Shaltiel and Viola (SICOMP 2010), who prove a exp(Ω((1/δ) 1/(d+2) )) lower bound for AC 0 [⊕], and a result of Cohen, Ganor and Raz (APPROX-RANDOM 2014), who show a exp(Ω((1/δ) 1/(d−1) )) lower bound for the smaller class AC 0 .The upper bound is a derandomization involving a use of Janson's inequality (from probabilistic combinatorics) and classical combinatorial designs; as far as we know, this is the first such use of Janson's inequality. For the lower bound, we prove an optimal (up to a constant factor) degree lower bound for multivariate polynomials over F 2 solving the δ-coin problem, which may be of independent interest.

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Section: Known Upper Boundsmentioning

confidence: 99%

A fixed-depth size-hierarchy theorem for AC ⁰ [⊕] via the coin problem

Limaye

Sreenivasaiah

Srinivasan

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…We consider tests on m bits that can be written as the product of k bounded functions on disjoint inputs of bits each. Such tests generalize the well-studied combinatorial rectangles [1,31,32,22,12,3,26,40,16,19] as well as other classes of tests, see [15]. They were introduced in [15] by Gopalan, Kane, and Meka who call them Fourier shapes.…”

Section: Fooling Productsmentioning

confidence: 97%

Untitled

Lee¹,

Viola²

2020

Theory of Comput.

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We construct pseudorandom generators with improved seed length for several classes of tests. First, we consider the class of read-once polynomials over GF(2) in m variables. For error ε we obtain seed lengthÕ(log(m/ε) log(1/ε)). This is optimal up to a factor of log(1/ε) • poly log log(m/ε). The previous best seed length was polylogarithmic in m and 1/ε. Second, we consider product tests f : {0, 1} m → C ≤1. These tests are the product of k functions f i : {0, 1} → C ≤1 , where the inputs of the f i are disjoint subsets of the m variables and C ≤1 is the complex unit disk. Here we obtain seed length • polylog(m/ε). This implies better generators for other classes of tests. If moreover the f i have output range {−1, 0, 1} then we obtain seed lengthÕ((log(k/ε) +)(log(1/ε) + log log m)). This is again optimal up to a factor of log(1/ε) • polylog(, log k, log m, log(1/ε)), while the previous best seed length was ≥ √ k. A main component of our proofs is showing that these classes of tests are fooled by almost d-wise independent distributions perturbed with noise.

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“…A leading goal is to prove RL=L by constructing generators with logarithmic seed length that fool one-way, space-bounded algorithms, but here the seminal papers [42,37,43] remain the state of the art and have larger seed lengths. However, somewhat better generators have been obtained for several special cases, including for example combinatorial rectangles [2,42,43,37,25,8,40,52,28,31], combinatorial shapes [29,21,27], and product tests [27]. In particular, for combinatorial rectangles f : ({0, 1} n ) k → {0, 1} two incomparable results are known.…”

Section: Application: Pseudorandomnessmentioning

confidence: 99%

Bounded Independence Plus Noise Fools Products

Haramaty¹,

Lee²,

Viola³

2018

SIAM J. Comput.

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Let D be a b-wise independent distribution over {0, 1} m. Let E be the "noise" distribution over {0, 1} m where the bits are independent and each bit is 1 with probability η/2. We study which tests f : {0, 1} m → [−1, 1] are ε-fooled by D + E, i.e., | E[f (D + E)] − E[f (U)]| ≤ ε where U is the uniform distribution. We show that D + E ε-fools product tests f : ({0, 1} n) k → [−1, 1] given by the product of k bounded functions on disjoint n-bit inputs with error ε = k(1 − η) Ω(b 2 /m) , where m = nk and b ≥ n. This bound is tight when b = Ω(m) and η ≥ (log k)/m. For b ≥ m 2/3 log m and any constant η the distribution D + E also 0.1-fools log-space algorithms. We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed-Solomon codes of dimension m/k where k = O(1), communication Ω(ηm) − O(log m) is required to decode one message symbol from a codeword with ηm errors, and communication O(ηm log m) suffices. Second, we obtain pseudorandom generators. We can ε-fool product tests f : ({0, 1} n) k → [−1, 1] under any permutation of the bits with seed lengths 2n +Õ(k 2 log(1/ε)) and O(n) + O(nk log 1/ε). Previous generators have seed lengths ≥ nk/2 or ≥ n √ nk. For the special case where the k bounded functions have range {0, 1} the previous generators have seed length ≥ (n + log k) log(1/ε).

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Randomness Buys Depth for Approximate Counting

Cited by 9 publications

References 26 publications

A fixed-depth size-hierarchy theorem for AC ⁰ [⊕] via the coin problem

A fixed-depth size-hierarchy theorem for AC ⁰ [⊕] via the coin problem

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Bounded Independence Plus Noise Fools Products

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Randomness Buys Depth for Approximate Counting

Cited by 9 publications

References 26 publications

A fixed-depth size-hierarchy theorem for AC 0 [⊕] via the coin problem

A fixed-depth size-hierarchy theorem for AC 0 [⊕] via the coin problem

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Bounded Independence Plus Noise Fools Products

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A fixed-depth size-hierarchy theorem for AC ⁰ [⊕] via the coin problem

A fixed-depth size-hierarchy theorem for AC ⁰ [⊕] via the coin problem