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Published byΞΟ Ξ»Ξ¬Ξ»ΞΉΞΏΟ ΞΞΏΟ ΞΌΟΞΉΟΟΞ·Ο Modified over 6 years ago
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Tight Fourier Tails for AC0 Circuits
Avishay Tal (IAS) CCC β2017
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Bounded Depth Circuits
A C 0 (π,π): π variables π gates (size of the circuit) depth π alternating gates A C 0 βA C 0 ππππ¦ π ,π 1
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Brief History Parity π₯ 1 , β¦, π₯ π = π₯ 1 + π₯ 2 +β¦+ π₯ π (πππ 2)
[Ajtaiβ83, Furst-Saxe-Sipserβ84, Yaoβ85]: Parity is not in AC0 [HΓ₯stad β86]: any depth-π circuit computing parity is of size at least exp π 1/(πβ1) . Result is tight: there exists a circuit of size exp π 1/(πβ1) and depth π computing Parity Challenge: Give an explicit function with better lower bounds. Really good lower bounds will imply lower bounds for NC1 & log-space.
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Brief History [Linial-Mansour-Nisanβ89]: Bounded depth circuits are well-approximated in L2 by low degree polynomials. Theorem: Let πβA C 0 (π,π). Then, βπ of deg p =π log π/π π s.t. π π₯ π π₯ βπ π₯ 2 β€π [HΓ₯stad β12]: any πβA C 0 (π,π) may agree with Parity on at most expβ‘(βπ/ log (π) πβ1 ) of the inputs. [Imagaliazzo-Matthews-Paturiβ12]: β¦ 1 2 +expβ‘(βπ/ log (π/π) πβ1 ) [HΓ₯stad β12] and [IMPβ12] results are tight!
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Discrete Fourier Analysis 101
For functions π,π: β1,1 π ββ define inner-product as π,π = π¬ π₯ [π π₯ β
π(π₯)] The characters π π π₯ = πβπ π₯ π for πβ[π] form an orthonormal basis. Hence, any function π: β1,1 π ββ has a unique expansion π(π₯) = πβ[π] π π β
πβπ π₯ π called the Fourier expansion. The Fourier coefficients π (π) are real numbers given by π π = π, π π = π π₯ π π₯ β
πβπ π₯ π Plancherelβs Identity: π π₯ π π₯ β
π(π₯) = π,π = π π π β
π (π) Parsevalβs Identity: π π₯ π π₯ 2 = π,π = π π π 2 If π is Boolean, i.e., π: β1,1 π β{β1,1}, then π π π 2 =1 Example: Majority MAJ(x_1, x_2, x_3) = Β½ x1 + Β½ x2 + Β½ x3 β Β½ x1x2x3
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Discrete Fourier Analysis 101
The Fourier transform of a Boolean function π naturally defines a distribution π· π over sets πβ[π]: Denote by π π π = ππ« πβΌ π· π [|π|=π] = π:|π|=π π π 2 Denote by π β₯π π = ππ« πβΌ π· π [ π β₯π] = π: π β₯π π π 2 The probability to sample π from π· π equals π π 2 .
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Tails and Low-Degree Approximation Equivalence
Let π: β1,1 π ββ. The truncated Fourier expansion of π at level π is a degree π polynomial defined by π β€π π₯ = π: π β€π π π β
πβπ π₯ π By Parseval: π π₯ π π₯ β π β€π π₯ 2 = πΎ >π [π]. By Parseval: this is the best L2-approx. of π among degree π polys. π has a degree-π L2-approximation with error π iff πΎ >π π β€π
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π π πππππ‘π¦ π π π
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Comparison of Results in Fourier language
W π π LMNβ89 exp β π 1/π decay Boppanaβ97 Our Result 1/π decay HΓ₯stadβ01 Lower Bound exp βπ decay HΓ₯stadβ12 IMPβ12 π log π πβ1 log π π π
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Comparison of Results in Polynomial Language
If π can be computed by a circuit with size π and depth π, then π can be π-approximated in L2 by polynomials of degree: LMNβ89 π(log π/π π ) Boppanaβ97 π(log π πβ1 /π) HΓ₯stadβ01 π(log π/π πβ2 β
log (π) β
log (1/π) ) This Work π(log π πβ1 β
log (1/π) )
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Main Theorem A significant improvement for πβͺ 1 poly(π) .
If π can be computed by a circuit of size π and depth π, then βπ: πΎ β₯π π β€ exp βπ/ log (π) πβ1 . Alternatively, π can be π-approximated in L2 by a polynomial of degree π log π πβ1 β
log 1/π . πΎ π π A significant improvement for πβͺ 1 poly(π) . Tight (for any πβ«π)
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Applications to Pseudo-randomness
F PRG A distribution π· over Β±1 π is pseudorandom for crkts of class πΆ if βπβπΆ: π π₯~π· π π₯ β π π π₯βΌπ [π π₯ ] A pseudo-random generator (PRG) for πΆ is a function PRG: β1,1 π β β1,1 π such that PRG( π π ) is pseudorandom for πΆ.
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Summary of Applications
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Why should we care? Why are we not satisfied by exp β π 1/π decay in tails and want exp βπ decay? Motivating question: give a Fourier analytical proof that Majority cannot be approximated by AC0 circuits. (Other proofs: [Smolenskyβ93, OβDonnell-Wimmerβ07]) πβA C 0 π π π π π MAJ polylog(π)
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Different Notions of Fourier Concentration
Let π be a Boolean function and π‘ a parameter. TFAE: for all k: π β₯π π β€ πβ
π βπ/π(π‘) for all k: π πβΌ π· π |π| π β€π π‘ π for all p, k: ππ« πβΌ π
p β‘ deg π π β₯π β€π ππ‘ π . and they imply Exp. Small Fourier Tails Fourier Moments βSwitching Lemmaβ π: π =π | π π | =π π‘ π
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Majority is not approximated by AC0
Problem: both MAJ and AC0 are concentrated on lower levels of the Fourier spectrum. Idea: Recall πβπ π π ο¨ π =π π π β€polylog π π . ο¨ on the kβth level, πβs Fourier mass is concentrated on only polylog π π coefs out of all the π π coefs. Since MAJ is symmetric, it spreads its Fourier weight equally within each layer: every coefficient in the kβth level is at most 1 π π .
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Majority is not approximated by AC0
Using Plancherel: π π₯ π π₯ β
MAJ π₯ = π π π β
MAJ π β€ π=1 π π =π π π β
MAJ π For 1β€π< π 0.1 : π =π π π β
MAJ π β€ polylog π π π π For πβ₯ π 0.1 : π β₯ π π π β
MAJ π β€ πβ₯ π π π 2 β
πβ₯ π MAJ π 2 = π β₯ π π β
π β₯ π MAJ β€ exp (β π 0.1 /polylog π ) βͺ 1 π ο¨ π π₯ π π₯ β
MAJ π₯ β€ polylog π π
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Open Question Which distributions fool AC0? [Aaronsonβ10, Fefferman-Shaltiel-Umans-Violaβ12] Can you find a distribution which is pseudorandom for AC0 but not pseudorandom for log-time quantum algorithms? F ο¨ an oracle separation between BQP from PH
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Exponentially Small Fourier Tails
Definition: π has ESFT(t) if for all π: π β₯π π β€ πβ
π βπ/π‘ Several interesting classes of functions have ESFT(t) CNFs/DNFs of width-π€ [HΓ₯stadβ86, LMNβ89] π‘ = π(π€) Formulas of size π [Reichardtβ11] π‘ = π π Read-Once Formulas [Impagliazzo-Kabanetsβ14] π‘ = π π 1/3.27 Circuits of size π and depth π π‘ = π( log π πβ1 ) Functions with max-sensitivity π [Gopalan-Servedio-T-Wigdersonβ16]: π‘ = π(π )
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Thank You!
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