DROPS

Document

DOI: 10.4230/LIPIcs.CCC.2022.21

Random Restrictions and PRGs for PTFs in Gaussian Space

Authors: Zander Kelley and Raghu Meka

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

A polynomial threshold function (PTF) f: ℝⁿ → ℝ is a function of the form f(x) = sign(p(x)) where p is a polynomial of degree at most d. PTFs are a classical and well-studied complexity class with applications across complexity theory, learning theory, approximation theory, quantum complexity and more. We address the question of designing pseudorandom generators (PRGs) for polynomial threshold functions (PTFs) in the gaussian space: design a PRG that takes a seed of few bits of randomness and outputs a n-dimensional vector whose distribution is indistinguishable from a standard multivariate gaussian by a degree d PTF. Our main result is a PRG that takes a seed of d^O(1) log(n/ε) log(1/ε)/ε² random bits with output that cannot be distinguished from an n-dimensional gaussian distribution with advantage better than ε by degree d PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seed length of d^O(log d)) in the degree d. Along the way we prove a few nearly-tight structural properties of restrictions of PTFs that may be of independent interest. Similar results were obtained in [Ryan O'Donnell et al., 2021] (independently and concurrently).

Cite as

Zander Kelley and Raghu Meka. Random Restrictions and PRGs for PTFs in Gaussian Space. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{kelley_et_al:LIPIcs.CCC.2022.21,
  author =	{Kelley, Zander and Meka, Raghu},
  title =	{{Random Restrictions and PRGs for PTFs in Gaussian Space}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{21:1--21:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.21},
  URN =		{urn:nbn:de:0030-drops-165836},
  doi =		{10.4230/LIPIcs.CCC.2022.21},
  annote =	{Keywords: polynomial threshold function, pseudorandom generator, multivariate gaussian}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.34

Improved Pseudorandom Generators for AC⁰ Circuits

Authors: Xin Lyu

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

We give PRG for depth-d, size-m AC⁰ circuits with seed length O(log^{d-1}(m)log(m/ε)log log(m)). Our PRG improves on previous work [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] from various aspects. It has optimal dependence on 1/ε and is only one "log log(m)" away from the lower bound barrier. For the case of d = 2, the seed length tightly matches the best-known PRG for CNFs [Anindya De et al., 2010; Avishay Tal, 2017]. There are two technical ingredients behind our new result; both of them might be of independent interest. First, we use a partitioning-based approach to construct PRGs based on restriction lemmas for AC⁰. Previous works [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] usually built PRGs on the Ajtai-Wigderson framework [Miklós Ajtai and Avi Wigderson, 1989]. Compared with them, the partitioning approach avoids the extra "log(n)" factor that usually arises from the Ajtai-Wigderson framework, allowing us to get the almost-tight seed length. The partitioning approach is quite general, and we believe it can help design PRGs for classes beyond constant-depth circuits. Second, improving and extending [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021], we prove a full derandomization of the powerful multi-switching lemma [Johan Håstad, 2014]. We show that one can use a short random seed to sample a restriction, such that a family of DNFs simultaneously simplifies under the restriction with high probability. This answers an open question in [Zander Kelley, 2021]. Previous derandomizations were either partial (that is, they pseudorandomly choose variables to restrict, and then fix those variables to truly-random bits) or had sub-optimal seed length. In our application, having a fully-derandomized switching lemma is crucial, and the randomness-efficiency of our derandomization allows us to get an almost-tight seed length.

Cite as

Xin Lyu. Improved Pseudorandom Generators for AC⁰ Circuits. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 34:1-34:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{lyu:LIPIcs.CCC.2022.34,
  author =	{Lyu, Xin},
  title =	{{Improved Pseudorandom Generators for AC⁰ Circuits}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{34:1--34:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.34},
  URN =		{urn:nbn:de:0030-drops-165963},
  doi =		{10.4230/LIPIcs.CCC.2022.34},
  annote =	{Keywords: pseudorandom generators, derandomization, switching Lemmas, AC⁰}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.58

Pseudorandom Generators for Read-Once Monotone Branching Programs

Authors: Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract

Motivated by the derandomization of space-bounded computation, there has been a long line of work on constructing pseudorandom generators (PRGs) against various forms of read-once branching programs (ROBPs), with a goal of improving the O(log² n) seed length of Nisan’s classic construction [Noam Nisan, 1992] to the optimal O(log n). In this work, we construct an explicit PRG with seed length Õ(log n) for constant-width ROBPs that are monotone, meaning that the states at each time step can be ordered so that edges with the same labels never cross each other. Equivalently, for each fixed input, the transition functions are a monotone function of the state. This result is complementary to a line of work that gave PRGs with seed length O(log n) for (ordered) permutation ROBPs of constant width [Braverman et al., 2014; Koucký et al., 2011; De, 2011; Thomas Steinke, 2012], since the monotonicity constraint can be seen as the "opposite" of the permutation constraint. Our PRG also works for monotone ROBPs that can read the input bits in any order, which are strictly more powerful than read-once AC⁰. Our PRG achieves better parameters (in terms of the dependence on the depth of the circuit) than the best previous pseudorandom generator for read-once AC⁰, due to Doron, Hatami, and Hoza [Doron et al., 2019]. Our pseudorandom generator construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989; Gopalan et al., 2012]. We give a randomness-efficient width-reduction process which proves that the branching program simplifies to an O(log n)-junta after only O(log log n) independent applications of the Forbes-Kelley pseudorandom restrictions [Michael A. Forbes and Zander Kelley, 2018].

Cite as

Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan. Pseudorandom Generators for Read-Once Monotone Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 58:1-58:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2021.58,
  author =	{Doron, Dean and Meka, Raghu and Reingold, Omer and Tal, Avishay and Vadhan, Salil},
  title =	{{Pseudorandom Generators for Read-Once Monotone Branching Programs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{58:1--58:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.58},
  URN =		{urn:nbn:de:0030-drops-147513},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.58},
  annote =	{Keywords: Branching programs, pseudorandom generators, constant depth circuits}
}

Document

DOI: 10.4230/LIPIcs.CCC.2019.16

Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas

Authors: Dean Doron, Pooya Hatami, and William M. Hoza

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract

We give an explicit pseudorandom generator (PRG) for read-once AC^0, i.e., constant-depth read-once formulas over the basis {wedge, vee, neg} with unbounded fan-in. The seed length of our PRG is O~(log(n/epsilon)). Previously, PRGs with near-optimal seed length were known only for the depth-2 case [Gopalan et al., 2012]. For a constant depth d > 2, the best prior PRG is a recent construction by Forbes and Kelley with seed length O~(log^2 n + log n log(1/epsilon)) for the more general model of constant-width read-once branching programs with arbitrary variable order [Michael A. Forbes and Zander Kelley, 2018]. Looking beyond read-once AC^0, we also show that our PRG fools read-once AC^0[oplus] with seed length O~(t + log(n/epsilon)), where t is the number of parity gates in the formula. Our construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989]. We assume by recursion that we already have a PRG for depth-d AC^0 formulas. To fool depth-(d + 1) AC^0 formulas, we use the given PRG, combined with a small-bias distribution and almost k-wise independence, to sample a pseudorandom restriction. The analysis of Forbes and Kelley [Michael A. Forbes and Zander Kelley, 2018] shows that our restriction approximately preserves the expectation of the formula. The crux of our work is showing that after poly(log log n) independent applications of our pseudorandom restriction, the formula simplifies in the sense that every gate other than the output has only polylog n remaining children. Finally, as the last step, we use a recent PRG by Meka, Reingold, and Tal [Meka et al., 2019] to fool this simpler formula.

Cite as

Dean Doron, Pooya Hatami, and William M. Hoza. Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 16:1-16:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{doron_et_al:LIPIcs.CCC.2019.16,
  author =	{Doron, Dean and Hatami, Pooya and Hoza, William M.},
  title =	{{Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{16:1--16:34},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.16},
  URN =		{urn:nbn:de:0030-drops-108382},
  doi =		{10.4230/LIPIcs.CCC.2019.16},
  annote =	{Keywords: Pseudorandom generators, Constant-depth formulas, Explicit constructions}
}

4 Search Results for "Kelley, Zander"

Random Restrictions and PRGs for PTFs in Gaussian Space

Abstract

Cite as

Improved Pseudorandom Generators for AC⁰ Circuits

Abstract

Cite as

Pseudorandom Generators for Read-Once Monotone Branching Programs

Abstract

Cite as

Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas

Abstract

Cite as

Thanks for your feedback!

Could not send message