Quantified derandomization of linear threshold circuits

Tell, Roei

doi:10.1145/3188745.3188822

Cited by 15 publications

(5 citation statements)

References 72 publications

(61 reference statements)

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“…exceptional inputs for any ϕ < 1.62, then a full derandomization of TC 0 follows. They also provide a corresponding QD algorithm for n 1+1/c d -wires TC 0 d circuits for some constant c ≈ 30 [61]. However, in their case, the gap is between two constants 1.62 and 30 in the exponent; in our case, the gap in the exponent can be made arbitrarily small.…”

Section: Sharp Threshold For Qd Of Generalized Probabilistic Formulasmentioning

confidence: 89%

“…Comparison With Prior Work. Several papers [21,28,61,62] prove obtaining quantified derandomization "thresholds" for circuit classes such as AC 0 , TC 0 and AC 0 [⊕]. Most related to ours is [21,61], showing if there is a QD algorithm for…”

Section: Sharp Threshold For Qd Of Generalized Probabilistic Formulasmentioning

confidence: 92%

“…For a Boolean circuit class C and a function B : N → N, the QD problem for C with B exceptional inputs is the following: Given an n-input circuit C ∈ C that evaluates to 0 on at most B(n) inputs, deterministically output an n-bit string on which C evaluates to 1. 5 Note the standard derandomization problem has B(n) = 2 n /3, but studying the case of B(n) = 2 n α for α < 1 turns out to also be a very interesting and subtle problem [21,28,61,62], with implications for general derandomization.…”

Section: Sharp Threshold For Quantified Derandomizationmentioning

confidence: 99%

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Sharp threshold results for computational complexity

Chen

Jin

Williams

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We establish several "sharp threshold" results for computational complexity. For certain tasks, we can prove a resource lower bound of n c for c ≥ 1 (or obtain an efficient circuit-analysis algorithm for n c size), there is strong intuition that a similar result can be proved for larger functions of n, yet we can also prove that replacing "n c " with "n c+ε " in our results, for any ε > 0, would imply a breakthrough n ω(1) lower bound. We first establish such a result for Hardness Magnification. We prove (among other results) that for some c, the Minimum Circuit Size Problem for (log n) c-size circuits on length-n truth tables (MCSP[(log n) c ]) does not have n 2−o(1)-size probabilistic formulas. We also prove that an n 2+ε lower bound for MCSP[(log n) c ] (for any ε > 0 and c ≥ 1) would imply major lower bound results, such as NP does not have n k-size formulas for all k, and #SAT does not have log-depth circuits. Similar results hold for time-bounded Kolmogorov complexity. Note that cubic size lower bounds are known for probabilistic De Morgan formulas (for other functions). Next we show a sharp threshold for Quantified Derandomization (QD) of probabilistic formulas: (a) For all α, ε > 0, there is a deterministic polynomial-time algorithm that finds satisfying assignments to every probabilistic formula of n 2−2α −ε size with at most 2 n α falsifying assignments. (b) If for some α, ε > 0, there is such an algorithm for probabilistic formulas of n 2−α +ε-size and 2 n α unsatisfying assignments, then a full derandomization of NC 1 follows: a deterministic poly-time algorithm additively approximating the acceptance probability of any polynomial-size formula. Consequently, NP does not have n k-size formulas, for all k. Finally we show a sharp threshold result for Explicit Obstructions, inspired by Mulmuley's notion of explicit obstructions from GCT. An explicit obstruction against S(n)-size formulas is a poly-time algorithm A such that A(1 n) outputs a list {(x i , f (x i))} i ∈[poly(n)] ⊆ {0, 1} n ×{0, 1}, and every S(n)-size formula F is inconsistent with the (partially defined) function f. We prove that for all ε > 0, there is an explicit obstruction against n 2−ε-size formulas, and prove that there is an explicit obstruction against n 2+ε-size formulas for some ε > 0 if and only if there is an explicit obstruction against all * L.C. and R.W. are supported by NSF CCF-1741615 and a Google Faculty Research Award. Portions of this work were completed while L.C. and R.W. were visiting the Simons Institute at UC Berkeley.

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Section: Sharp Threshold For Qd Of Generalized Probabilistic Formulasmentioning

confidence: 89%

Section: Sharp Threshold For Qd Of Generalized Probabilistic Formulasmentioning

confidence: 92%

Section: Sharp Threshold For Quantified Derandomizationmentioning

confidence: 99%

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Sharp threshold results for computational complexity

Chen

Jin

Williams

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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“…Their argument requires the error correcting codes to have an efficient decoder. As it is unclear how to decode the codes constructed in [CT19], one can only use a construction in [Tel18] with a worse dependence on depth in their argument (they achieve O(1/ε) instead of O(log ε −1 ) in Theorem 1.6). Our argument does not need an efficient decoder at all, so we can apply the better construction from [CT19] to achieve a better magnification.…”

Section: Intuitionmentioning

confidence: 99%

Hardness Magnification for all Sparse NP Languages

Chen

Jin

Williams

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In the Minimum Circuit Size Problem (MCSP[s(m)]), we ask if there is a circuit of size s(m) computing a given truth-table of length n = 2 m . Recently, a surprising phenomenon termed as hardness magnification by [Oliveira and Santhanam, FOCS 2018] was discovered for MCSP[s(m)] and the related problem MKtP of computing time-bounded Kolmogorov complexity. In [Oliveira and Santhanam, FOCS 2018], [Oliveira, Pich, and Santhanam, CCC 2019], and Williams, STOC 2019], it was shown that minor (n 1+ε -style) lower bounds for MCSP [2 o(m) ] or MKtP [2 o(m) ] would imply breakthrough circuit lower bounds such as NP ⊂ P /poly , NP ⊂ NC 1 , or EXP ⊂ P /poly .We consider the question: What is so special about MCSP and MKtP? Why do they admit this striking phenomenon? One simple property is that all variants of MCSP (and MKtP) considered in prior work are sparse languages. For example,-sparse. We show that there is a hardness magnification phenomenon for all equally-sparse NP languages. Formally, suppose there is an ε > 0 and a language L ∈ NP which is-sparse, and L / ∈ Circuit[n 1+ε ]. Then NP does not have n k -size circuits for all k. We prove analogous theorems for De Morgan formulas, B 2 -formulas, branching programs, AC 0 [6] and TC 0 circuits, and more: improving the state of the art in NP lower bounds against any of these models by an ε factor in the exponent would already imply NP lower bounds for all fixed polynomials. In fact, in our proofs it is not necessary to prove a (say) n 1+ε circuit size lower bound for L: one only has to prove a lower bound against n 1+ε -time n ε -space deterministic algorithms with n ε advice bits. Such lower bounds are well-known for non-sparse problems.Building on our techniques, we also show interesting new hardness magnifications for search-MCSP and search-MKtP (where one must output small circuits or short representations of strings), showing consequences such as ⊕P (or PP, PSPACE, and EXP) is not contained in P /poly (or NC 1 , AC 0 [6], or branching programs of polynomial size). For instance, if there is an ε > 0 such that search-MCSP[2 βm ] does not have De Morgan formulas of size n 3+ε for all constants β > 0, then ⊕P ⊂ NC 1 .

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“…Tell [Tel17] constructed a quantified derandomization algorithm for TC circuits with depth d and n 1+exp(−d) wires, and showed that a modest improvement of his algorithm would imply standard derandomization of TC 0 , and consequently NEXP ⊆ TC 0 .…”

Section: Related Workmentioning

confidence: 99%

Toward Super-Polynomial Size Lower Bounds for Depth-Two Threshold Circuits

Chen

2018

Preprint

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Proving super-polynomial size lower bounds for TC 0 , the class of constant-depth, polynomial-size circuits of Majority gates, is a notorious open problem in complexity theory. A major frontier is to prove that NEXP does not have poly-size THR • THR circuit (depth-two circuits with linear threshold gates).In recent years, R. Williams proposed a program to prove circuit lower bounds via improved algorithms. In this paper, following Williams' framework, we show that the above frontier question can be resolved by devising slightly faster algorithms for several fundamental problems:Shaving Logs for ℓ 2 -Furthest-Pair. An n 2 poly(d)/ log ω(1) n time algorithm for ℓ 2 -Furthest-Pair in R d for polylogarithmic d implies NEXP has no polynomial size THR • THR circuits. The same holds for Hopcroft's problem, Bichrom.-ℓ 2 -Closest-Pair and Integer Max-IP. • Shaving Logs for Approximate Bichrom.-ℓ 2 -Closest-Pair. An n 2 poly(d)/ log ω(1) n time algorithm for (1 + 1/ log ω(1) n)-approximation to Bichrom.-ℓ 2 -Closest-Pair or Bichrom.-ℓ 1 -Closest-Pair for polylogarithmic d implies NEXP has no polynomial size SYM • THR circuits.• Shaving Logs for Modest Dimension Boolean Max-IP. An n 2 / log ω(1) n time algorithm for Bichromatic Maximum Inner Product with vector dimension d = n ε for any small constant ε would imply NEXP has no polynomial size THR • THR circuits. Note there is an n 2 polylog(n) time algorithm via fast rectangle matrix multiplication.

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Quantified derandomization of linear threshold circuits

Cited by 15 publications

References 72 publications

Sharp threshold results for computational complexity

Sharp threshold results for computational complexity

Hardness Magnification for all Sparse NP Languages

Toward Super-Polynomial Size Lower Bounds for Depth-Two Threshold Circuits

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