12 Search Results for "Bun, Mark"

Document
DOI: 10.4230/LIPIcs.ITCS.2024.68
Distributional PAC-Learning from Nisan’s Natural Proofs

Authors: Ari Karchmer

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Do natural proofs imply efficient learning algorithms? Carmosino et al. (2016) demonstrated that natural proofs of circuit lower bounds for Λ imply efficient algorithms for learning Λ-circuits, but only over the uniform distribution, with membership queries, and provided AC⁰[p] ⊆ Λ. We consider whether this implication can be generalized to Λ ⊉ AC⁰[p], and to learning algorithms which use only random examples and learn over arbitrary example distributions (Valiant’s PAC-learning model). We first observe that, if, for any circuit class Λ, there is an implication from natural proofs for Λ to PAC-learning for Λ, then standard assumptions from lattice-based cryptography do not hold. In particular, we observe that depth-2 majority circuits are a (conditional) counter example to this fully general implication, since Nisan (1993) gave a natural proof, but Klivans and Sherstov (2009) showed hardness of PAC-Learning under lattice-based assumptions. We thus ask: what learning algorithms can we reasonably expect to follow from Nisan’s natural proofs? Our main result is that all natural proofs arising from a type of communication complexity argument, including Nisan’s, imply PAC-learning algorithms in a new distributional variant (i.e., an "average-case" relaxation) of Valiant’s PAC model. Our distributional PAC model is stronger than the average-case prediction model of Blum et al. (1993) and the heuristic PAC model of Nanashima (2021), and has several important properties which make it of independent interest, such as being boosting-friendly. The main applications of our result are new distributional PAC-learning algorithms for depth-2 majority circuits, polytopes and DNFs over natural target distributions, as well as the nonexistence of encoded-input weak PRFs that can be evaluated by depth-2 majority circuits.
Cite as

Ari Karchmer. Distributional PAC-Learning from Nisan’s Natural Proofs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 68:1-68:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{karchmer:LIPIcs.ITCS.2024.68,
  author =	{Karchmer, Ari},
  title =	{{Distributional PAC-Learning from Nisan’s Natural Proofs}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{68:1--68:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.68},
  URN =		{urn:nbn:de:0030-drops-195964},
  doi =		{10.4230/LIPIcs.ITCS.2024.68},
  annote =	{Keywords: PAC-learning, average-case complexity, communication complexity, natural proofs}
}
Document
DOI: 10.4230/LIPIcs.TQC.2023.1
Approximate Degree Lower Bounds for Oracle Identification Problems

Authors: Mark Bun and Nadezhda Voronova

Published in: LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

Abstract
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string x ∈ {0, 1}ⁿ given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of x. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of Ω(n/log² n) for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.
Cite as

Mark Bun and Nadezhda Voronova. Approximate Degree Lower Bounds for Oracle Identification Problems. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bun_et_al:LIPIcs.TQC.2023.1,
  author =	{Bun, Mark and Voronova, Nadezhda},
  title =	{{Approximate Degree Lower Bounds for Oracle Identification Problems}},
  booktitle =	{18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)},
  pages =	{1:1--1:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-283-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{266},
  editor =	{Fawzi, Omar and Walter, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.1},
  URN =		{urn:nbn:de:0030-drops-183113},
  doi =		{10.4230/LIPIcs.TQC.2023.1},
  annote =	{Keywords: Approximate degree, quantum query complexity, communication complexity, ordered search, polynomial approximations, polynomial method}
}
Document
DOI: 10.4230/LIPIcs.ITP.2022.30
Formalizing Algorithmic Bounds in the Query Model in EasyCrypt

Authors: Alley Stoughton, Carol Chen, Marco Gaboardi, and Weihao Qu

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract
We use the EasyCrypt proof assistant to formalize the adversarial approach to proving lower bounds for computational problems in the query model. This is done using a lower bound game between an algorithm and adversary, in which the adversary answers the algorithm’s queries in a way that makes the algorithm issue at least the desired number of queries. A complementary upper bound game is used for proving upper bounds of algorithms; here the adversary incrementally and adaptively realizes an algorithm’s input. We prove a natural connection between the lower and upper bound games, and apply our framework to three computational problems, including searching in an ordered list and comparison-based sorting, giving evidence for the generality of our notion of algorithm and the usefulness of our framework.
Cite as

Alley Stoughton, Carol Chen, Marco Gaboardi, and Weihao Qu. Formalizing Algorithmic Bounds in the Query Model in EasyCrypt. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{stoughton_et_al:LIPIcs.ITP.2022.30,
  author =	{Stoughton, Alley and Chen, Carol and Gaboardi, Marco and Qu, Weihao},
  title =	{{Formalizing Algorithmic Bounds in the Query Model in EasyCrypt}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{30:1--30:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.30},
  URN =		{urn:nbn:de:0030-drops-167399},
  doi =		{10.4230/LIPIcs.ITP.2022.30},
  annote =	{Keywords: query model, lower bound, upper bound, adversary argument, EasyCrypt}
}
Document
DOI: 10.4230/LIPIcs.FORC.2022.1
Controlling Privacy Loss in Sampling Schemes: An Analysis of Stratified and Cluster Sampling

Authors: Mark Bun, Jörg Drechsler, Marco Gaboardi, Audra McMillan, and Jayshree Sarathy

Published in: LIPIcs, Volume 218, 3rd Symposium on Foundations of Responsible Computing (FORC 2022)

Abstract
Sampling schemes are fundamental tools in statistics, survey design, and algorithm design. A fundamental result in differential privacy is that a differentially private mechanism run on a simple random sample of a population provides stronger privacy guarantees than the same algorithm run on the entire population. However, in practice, sampling designs are often more complex than the simple, data-independent sampling schemes that are addressed in prior work. In this work, we extend the study of privacy amplification results to more complex, data-dependent sampling schemes. We find that not only do these sampling schemes often fail to amplify privacy, they can actually result in privacy degradation. We analyze the privacy implications of the pervasive cluster sampling and stratified sampling paradigms, as well as provide some insight into the study of more general sampling designs.
Cite as

Mark Bun, Jörg Drechsler, Marco Gaboardi, Audra McMillan, and Jayshree Sarathy. Controlling Privacy Loss in Sampling Schemes: An Analysis of Stratified and Cluster Sampling. In 3rd Symposium on Foundations of Responsible Computing (FORC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 218, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bun_et_al:LIPIcs.FORC.2022.1,
  author =	{Bun, Mark and Drechsler, J\"{o}rg and Gaboardi, Marco and McMillan, Audra and Sarathy, Jayshree},
  title =	{{Controlling Privacy Loss in Sampling Schemes: An Analysis of Stratified and Cluster Sampling}},
  booktitle =	{3rd Symposium on Foundations of Responsible Computing (FORC 2022)},
  pages =	{1:1--1:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-226-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{218},
  editor =	{Celis, L. Elisa},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2022.1},
  URN =		{urn:nbn:de:0030-drops-165243},
  doi =		{10.4230/LIPIcs.FORC.2022.1},
  annote =	{Keywords: privacy, differential privacy, survey design, survey sampling}
}
Document
DOI: 10.4230/LIPIcs.TQC.2020.2
Improved Approximate Degree Bounds for k-Distinctness

Authors: Nikhil S. Mande, Justin Thaler, and Shuchen Zhu

Published in: LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Abstract
An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element Distinctness) is well-understood, there is a polynomial gap between the known upper and lower bounds for all constants k>2. Specifically, the best known upper bound is O (N^{(3/4)-1/(2^{k+2}-4)}) (Belovs, FOCS 2012), while the best known lower bound for k≥ 2 is Ω̃(N^{2/3} + N^{(3/4)-1/(2k)}) (Aaronson and Shi, J. ACM 2004; Bun, Kothari, and Thaler, STOC 2018). For any constant k ≥ 4, we improve the lower bound to Ω̃(N^{(3/4)-1/(4k)}). This yields, for example, the first proof that 4-distinctness is strictly harder than Element Distinctness. Our lower bound applies more generally to approximate degree. As a secondary result, we give a simple construction of an approximating polynomial of degree Õ(N^{3/4}) that applies whenever k ≤ polylog(N).
Cite as

Nikhil S. Mande, Justin Thaler, and Shuchen Zhu. Improved Approximate Degree Bounds for k-Distinctness. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mande_et_al:LIPIcs.TQC.2020.2,
  author =	{Mande, Nikhil S. and Thaler, Justin and Zhu, Shuchen},
  title =	{{Improved Approximate Degree Bounds for k-Distinctness}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{2:1--2:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Flammia, Steven T.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.2},
  URN =		{urn:nbn:de:0030-drops-120613},
  doi =		{10.4230/LIPIcs.TQC.2020.2},
  annote =	{Keywords: Quantum Query Complexity, Approximate Degree, Dual Polynomials, k-distinctness}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.52
How to Find a Point in the Convex Hull Privately

Authors: Haim Kaplan, Micha Sharir, and Uri Stemmer

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
We study the question of how to compute a point in the convex hull of an input set S of n points in ℝ^d in a differentially private manner. This question, which is trivial without privacy requirements, turns out to be quite deep when imposing differential privacy. In particular, it is known that the input points must reside on a fixed finite subset G ⊆ ℝ^d, and furthermore, the size of S must grow with the size of G. Previous works [Amos Beimel et al., 2010; Amos Beimel et al., 2019; Amos Beimel et al., 2013; Mark Bun et al., 2018; Mark Bun et al., 2015; Haim Kaplan et al., 2019] focused on understanding how n needs to grow with |G|, and showed that n=O(d^2.5 ⋅ 8^(log^*|G|)) suffices (so n does not have to grow significantly with |G|). However, the available constructions exhibit running time at least |G|^d², where typically |G|=X^d for some (large) discretization parameter X, so the running time is in fact Ω(X^d³). In this paper we give a differentially private algorithm that runs in O(n^d) time, assuming that n=Ω(d⁴ log X). To get this result we study and exploit some structural properties of the Tukey levels (the regions D_{≥ k} consisting of points whose Tukey depth is at least k, for k=0,1,…). In particular, we derive lower bounds on their volumes for point sets S in general position, and develop a rather subtle mechanism for handling point sets S in degenerate position (where the deep Tukey regions have zero volume). A naive approach to the construction of the Tukey regions requires n^O(d²) time. To reduce the cost to O(n^d), we use an approximation scheme for estimating the volumes of the Tukey regions (within their affine spans in case of degeneracy), and for sampling a point from such a region, a scheme that is based on the volume estimation framework of Lovász and Vempala [László Lovász and Santosh S. Vempala, 2006] and of Cousins and Vempala [Ben Cousins and Santosh S. Vempala, 2018]. Making this framework differentially private raises a set of technical challenges that we address.
Cite as

Haim Kaplan, Micha Sharir, and Uri Stemmer. How to Find a Point in the Convex Hull Privately. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaplan_et_al:LIPIcs.SoCG.2020.52,
  author =	{Kaplan, Haim and Sharir, Micha and Stemmer, Uri},
  title =	{{How to Find a Point in the Convex Hull Privately}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{52:1--52:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.52},
  URN =		{urn:nbn:de:0030-drops-122107},
  doi =		{10.4230/LIPIcs.SoCG.2020.52},
  annote =	{Keywords: Differential privacy, Tukey depth, Convex hull}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.55
The Large-Error Approximate Degree of AC^0

Authors: Mark Bun and Justin Thaler

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial error. First, we prove a tight Omega~(n^{1/2}) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC^0 function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS 2015). Our result also extends to a 2^{Omega~(n^{1/2})} lower bound on the sign-rank of an AC^0 function, improving on the previous best bound of 2^{Omega(n^{2/5})} (Bun and Thaler, ICALP 2016). Second, for any delta>0, we exhibit a function f : {-1,1}^n -> {-1,1} that is computed by a circuit of depth O(1/delta) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error epsilon=1-2^{-Omega(n^{1-delta})}, even by polynomials of degree n^{1-delta}. Our recent prior work (Bun and Thaler, FOCS 2017) proved a similar lower bound, but which held only for error epsilon=1/3. Our result implies 2^{Omega(n^{1-delta})} lower bounds on the complexity of AC^0 under a variety of basic measures such as discrepancy, margin complexity, and threshold weight. This nearly matches the trivial upper bound of 2^{O(n)} that holds for every function. The previous best lower bound on AC^0 for these measures was 2^{Omega(n^{1/2})} (Sherstov, FOCS 2015). Additional applications in learning theory, communication complexity, and cryptography are described.
Cite as

Mark Bun and Justin Thaler. The Large-Error Approximate Degree of AC^0. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 55:1-55:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bun_et_al:LIPIcs.APPROX-RANDOM.2019.55,
  author =	{Bun, Mark and Thaler, Justin},
  title =	{{The Large-Error Approximate Degree of AC^0}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{55:1--55:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.55},
  URN =		{urn:nbn:de:0030-drops-112709},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.55},
  annote =	{Keywords: approximate degree, discrepancy, margin complexity, polynomial approximations, secret sharing, threshold circuits}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.71
Approximate Degree, Secret Sharing, and Concentration Phenomena

Authors: Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, and Christopher Williamson

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
The epsilon-approximate degree deg~_epsilon(f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are: - We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution. - We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg~_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena. As a corollary of this result, we show that for any d <= n/64, any degree d polynomial approximating a symmetric function f to error 1/3 must have coefficients of l_1-norm at least K^{-3/2} * exp ({Omega (deg~_{1/3}(f)^2/d)}). We also show this bound is essentially tight for any d > deg~_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND.
Cite as

Andrej Bogdanov, Nikhil S. Mande, Justin Thaler, and Christopher Williamson. Approximate Degree, Secret Sharing, and Concentration Phenomena. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 71:1-71:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bogdanov_et_al:LIPIcs.APPROX-RANDOM.2019.71,
  author =	{Bogdanov, Andrej and Mande, Nikhil S. and Thaler, Justin and Williamson, Christopher},
  title =	{{Approximate Degree, Secret Sharing, and Concentration Phenomena}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{71:1--71:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.71},
  URN =		{urn:nbn:de:0030-drops-112869},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.71},
  annote =	{Keywords: approximate degree, dual polynomial, pseudorandomness, polynomial approximation, secret sharing}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.30
Sign-Rank Can Increase Under Intersection

Authors: Mark Bun, Nikhil S. Mande, and Justin Thaler

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
The communication class UPP^{cc} is a communication analog of the Turing Machine complexity class PP. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem f, let f wedge f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP^{cc}(f)= O(log n), and UPP^{cc}(f wedge f) = Theta(log^2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP^{cc}, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n^{Omega(log n)}. This matches an upper bound of (Klivans, O'Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.
Cite as

Mark Bun, Nikhil S. Mande, and Justin Thaler. Sign-Rank Can Increase Under Intersection. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bun_et_al:LIPIcs.ICALP.2019.30,
  author =	{Bun, Mark and Mande, Nikhil S. and Thaler, Justin},
  title =	{{Sign-Rank Can Increase Under Intersection}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{30:1--30:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.30},
  URN =		{urn:nbn:de:0030-drops-106067},
  doi =		{10.4230/LIPIcs.ICALP.2019.30},
  annote =	{Keywords: Sign rank, dimension complexity, communication complexity, learning theory}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.35
Approximate Degree and the Complexity of Depth Three Circuits

Authors: Mark Bun and Justin Thaler

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
Threshold weight, margin complexity, and Majority-of-Threshold circuit size are basic complexity measures of Boolean functions that arise in learning theory, communication complexity, and circuit complexity. Each of these measures might exhibit a chasm at depth three: namely, all polynomial size Boolean circuits of depth two have polynomial complexity under the measure, but there may exist Boolean circuits of depth three that have essentially maximal complexity exp(Theta(n)). However, existing techniques are far from showing this: for all three measures, the best lower bound for depth three circuits is exp(Omega(n^{2/5})). Moreover, prior methods exclusively study block-composed functions. Such methods appear intrinsically unable to prove lower bounds better than exp(Omega(sqrt{n})) even for depth four circuits, and have yet to prove lower bounds better than exp(Omega(sqrt{n})) for circuits of any constant depth. We take a step toward showing that all of these complexity measures indeed exhibit a chasm at depth three. Specifically, for any arbitrarily small constant delta > 0, we exhibit a depth three circuit of polynomial size (in fact, an O(log n)-decision list) of complexity exp(Omega(n^{1/2-delta})) under each of these measures. Our methods go beyond the block-composed functions studied in prior work, and hence may not be subject to the same barriers. Accordingly, we suggest natural candidate functions that may exhibit stronger bounds.
Cite as

Mark Bun and Justin Thaler. Approximate Degree and the Complexity of Depth Three Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bun_et_al:LIPIcs.APPROX-RANDOM.2018.35,
  author =	{Bun, Mark and Thaler, Justin},
  title =	{{Approximate Degree and the Complexity of Depth Three Circuits}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{35:1--35:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.35},
  URN =		{urn:nbn:de:0030-drops-94390},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.35},
  annote =	{Keywords: approximate degree, communication complexity, learning theory, polynomial approximation, threshold circuits}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.37
Improved Bounds on the Sign-Rank of AC^0

Authors: Mark Bun and Justin Thaler

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
The sign-rank of a matrix A with entries in {-1, +1} is the least rank of a real matrix B with A_{ij}*B_{ij} > 0 for all i, j. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC^0, answering an old question of Babai, Frankl, and Simon (1986). Specifically, they exhibited a matrix A = [F(x,y)]_{x,y} for a specific function F:{-1,1}^n*{-1,1}^n -> {-1,1} in AC^0, such that A has sign-rank exp(Omega(n^{1/3}). We prove a generalization of Razborov and Sherstov’s result, yielding exponential sign-rank lower bounds for a non-trivial class of functions (that includes the function used by Razborov and Sherstov). As a corollary of our general result, we improve Razborov and Sherstov's lower bound on the sign-rank of AC^0 from exp(Omega(n^{1/3})) to exp(~Omega(n^{2/5})). We also describe several applications to communication complexity, learning theory, and circuit complexity.
Cite as

Mark Bun and Justin Thaler. Improved Bounds on the Sign-Rank of AC^0. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bun_et_al:LIPIcs.ICALP.2016.37,
  author =	{Bun, Mark and Thaler, Justin},
  title =	{{Improved Bounds on the Sign-Rank of AC^0}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{37:1--37:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.37},
  URN =		{urn:nbn:de:0030-drops-63173},
  doi =		{10.4230/LIPIcs.ICALP.2016.37},
  annote =	{Keywords: Sign-rank, circuit complexity, communication complexity, constant-depth circuits}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.625
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness

Authors: Mark Bun and Thomas Steinke

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Abstract
Low-degree polynomial approximations to the sign function underlie pseudorandom generators for halfspaces, as well as algorithms for agnostically learning halfspaces. We study the limits of these constructions by proving inapproximability results for the sign function. First, we investigate the derandomization of Chernoff-type concentration inequalities. Schmidt et al. (SIAM J. Discrete Math. 1995) showed that a tail bound of delta can be established for sums of Bernoulli random variables with only O(log(1/delta))-wise independence. We show that their results are tight up to constant factors. Secondly, the “polynomial regression” algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be efficiently learned with respect to log-concave distributions on R^n in the challenging agnostic learning model. The power of this algorithm relies on the fact that under log-concave distributions, halfspaces can be approximated arbitrarily well by low-degree polynomials. In contrast, we exhibit a large class of non-log-concave distributions under which polynomials of any degree cannot approximate the sign function to within arbitrarily low error.
Cite as

Mark Bun and Thomas Steinke. Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 625-644, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bun_et_al:LIPIcs.APPROX-RANDOM.2015.625,
  author =	{Bun, Mark and Steinke, Thomas},
  title =	{{Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{625--644},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.625},
  URN =		{urn:nbn:de:0030-drops-53274},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.625},
  annote =	{Keywords: Polynomial Approximations, Pseudorandomness, Concentration, Learning Theory, Halfspaces}
}
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