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Documents authored by Canonne, Clément L.

Document
DOI: 10.4230/LIPIcs.ITCS.2024.23
Private Distribution Testing with Heterogeneous Constraints: Your Epsilon Might Not Be Mine

Authors: Clément L. Canonne and Yucheng Sun

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

Abstract
Private closeness testing asks to decide whether the underlying probability distributions of two sensitive datasets are identical or differ significantly in statistical distance, while guaranteeing (differential) privacy of the data. As in most (if not all) distribution testing questions studied under privacy constraints, however, previous work assumes that the two datasets are equally sensitive, i.e., must be provided the same privacy guarantees. This is often an unrealistic assumption, as different sources of data come with different privacy requirements; as a result, known closeness testing algorithms might be unnecessarily conservative, "paying" too high a privacy budget for half of the data. In this work, we initiate the study of the closeness testing problem under heterogeneous privacy constraints, where the two datasets come with distinct privacy requirements. We formalize the question and provide algorithms under the three most widely used differential privacy settings, with a particular focus on the local and shuffle models of privacy; and show that one can indeed achieve better sample efficiency when taking into account the two different "epsilon" requirements.
Cite as

Clément L. Canonne and Yucheng Sun. Private Distribution Testing with Heterogeneous Constraints: Your Epsilon Might Not Be Mine. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{canonne_et_al:LIPIcs.ITCS.2024.23,
  author =	{Canonne, Cl\'{e}ment L. and Sun, Yucheng},
  title =	{{Private Distribution Testing with Heterogeneous Constraints: Your Epsilon Might Not Be Mine}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{23:1--23:24},
  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.2024.23},
  URN =		{urn:nbn:de:0030-drops-195518},
  doi =		{10.4230/LIPIcs.ITCS.2024.23},
  annote =	{Keywords: differential privacy, distribution testing, local privacy, shuffle privacy}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2021.55
Identity Testing Under Label Mismatch

Authors: Clément L. Canonne and Karl Wimmer

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract
Testing whether the observed data conforms to a purported model (probability distribution) is a basic and fundamental statistical task, and one that is by now well understood. However, the standard formulation, identity testing, fails to capture many settings of interest; in this work, we focus on one such natural setting, identity testing under promise of permutation. In this setting, the unknown distribution is assumed to be equal to the purported one, up to a relabeling (permutation) of the model: however, due to a systematic error in the reporting of the data, this relabeling may not be the identity. The goal is then to test identity under this assumption: equivalently, whether this systematic labeling error led to a data distribution statistically far from the reference model.
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Clément L. Canonne and Karl Wimmer. Identity Testing Under Label Mismatch. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{canonne_et_al:LIPIcs.ISAAC.2021.55,
  author =	{Canonne, Cl\'{e}ment L. and Wimmer, Karl},
  title =	{{Identity Testing Under Label Mismatch}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{55:1--55:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.55},
  URN =		{urn:nbn:de:0030-drops-154880},
  doi =		{10.4230/LIPIcs.ISAAC.2021.55},
  annote =	{Keywords: distribution testing, property testing, permutations, lower bounds}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.24
Testing Data Binnings

Authors: Clément L. Canonne and Karl Wimmer

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
Motivated by the question of data quantization and "binning," we revisit the problem of identity testing of discrete probability distributions. Identity testing (a.k.a. one-sample testing), a fundamental and by now well-understood problem in distribution testing, asks, given a reference distribution (model) 𝐪 and samples from an unknown distribution 𝐩, both over [n] = {1,2,… ,n}, whether 𝐩 equals 𝐪, or is significantly different from it. In this paper, we introduce the related question of identity up to binning, where the reference distribution 𝐪 is over k ≪ n elements: the question is then whether there exists a suitable binning of the domain [n] into k intervals such that, once "binned," 𝐩 is equal to 𝐪. We provide nearly tight upper and lower bounds on the sample complexity of this new question, showing both a quantitative and qualitative difference with the vanilla identity testing one, and answering an open question of Canonne [Clément L. Canonne, 2019]. Finally, we discuss several extensions and related research directions.
Cite as

Clément L. Canonne and Karl Wimmer. Testing Data Binnings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{canonne_et_al:LIPIcs.APPROX/RANDOM.2020.24,
  author =	{Canonne, Cl\'{e}ment L. and Wimmer, Karl},
  title =	{{Testing Data Binnings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{24:1--24:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.24},
  URN =		{urn:nbn:de:0030-drops-126277},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.24},
  annote =	{Keywords: property testing, distribution testing, identity testing, hypothesis testing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.29
Testing k-Monotonicity

Authors: Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, k-monotone functions are natural generalizations of the classical monotone functions, which are the 1-monotone functions. Motivated by the recent interest in k-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of k-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone. Our results include the following: 1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0,1}^d, for k >= 3; 2. We demonstrate a separation between testing and learning on {0,1}^d, for k=\omega(\log d): testing k-monotonicity can be performed with 2^{O(\sqrt d . \log d . \log{1/\eps})} queries, while learning k-monotone functions requires 2^{\Omega(k . \sqrt d .{1/\eps})} queries (Blais et al. (RANDOM 2015)). 3. We present a tolerant test for functions f\colon[n]^d\to \{0,1\}$with complexity independent of n, which makes progress on a problem left open by Berman et al. (STOC 2014). Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques. Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.
Cite as

Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer. Testing k-Monotonicity. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 29:1-29:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{canonne_et_al:LIPIcs.ITCS.2017.29,
  author =	{Canonne, Cl\'{e}ment L. and Grigorescu, Elena and Guo, Siyao and Kumar, Akash and Wimmer, Karl},
  title =	{{Testing k-Monotonicity}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{29:1--29:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.29},
  URN =		{urn:nbn:de:0030-drops-81583},
  doi =		{10.4230/LIPIcs.ITCS.2017.29},
  annote =	{Keywords: Boolean Functions, Learning, Monotonicity, Property Testing}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.27
An Adaptivity Hierarchy Theorem for Property Testing

Authors: Clément L. Canonne and Tom Gur

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries. In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of "rounds of adaptivity" it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in k+1 rounds, where the queries in the i'th round may depend on the answers obtained in the previous i-1 rounds. Then, we ask the following question: Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity? We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every n in N and 0 <= k <= n^{0.99} there exists a property Pi_{n,k} of functions for which (1) there exists a k-adaptive tester for Pi_{n,k} with query complexity tilde O(k), yet (2) any (k-1)-adaptive tester for Pi_{n,k} must make Omega(n) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.
Cite as

Clément L. Canonne and Tom Gur. An Adaptivity Hierarchy Theorem for Property Testing. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 27:1-27:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{canonne_et_al:LIPIcs.CCC.2017.27,
  author =	{Canonne, Cl\'{e}ment L. and Gur, Tom},
  title =	{{An Adaptivity Hierarchy Theorem for Property Testing}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{27:1--27:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.27},
  URN =		{urn:nbn:de:0030-drops-75164},
  doi =		{10.4230/LIPIcs.CCC.2017.27},
  annote =	{Keywords: Property Testing, Coding Theory, Hierarchy Theorems}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.28
Distribution Testing Lower Bounds via Reductions from Communication Complexity

Authors: Eric Blais, Clément L. Canonne, and Tom Gur

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef (Computational Complexity, 2012), we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds. Our main result is concerned with testing identity to a specific distribution p, given as a parameter. In a recent and influential work, Valiant and Valiant (FOCS, 2014) showed that the sample complexity of the aforementioned problem is closely related to the 2/3-quasinorm of p. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's K-functional. We show that this quantity is closely related to the size of the effective support of p (loosely speaking, the number of supported elements that constitute the vast majority of the mass of p). This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.
Cite as

Eric Blais, Clément L. Canonne, and Tom Gur. Distribution Testing Lower Bounds via Reductions from Communication Complexity. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 28:1-28:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blais_et_al:LIPIcs.CCC.2017.28,
  author =	{Blais, Eric and Canonne, Cl\'{e}ment L. and Gur, Tom},
  title =	{{Distribution Testing Lower Bounds via Reductions from Communication Complexity}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{28:1--28:40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.28},
  URN =		{urn:nbn:de:0030-drops-75366},
  doi =		{10.4230/LIPIcs.CCC.2017.28},
  annote =	{Keywords: Distribution testing, communication complexity, lower bounds, simultaneous message passing, functional analysis}
}
Document
DOI: 10.4230/LIPIcs.STACS.2016.25
Testing Shape Restrictions of Discrete Distributions

Authors: Clément L. Canonne, Ilias Diakonikolas, Themis Gouleakis, and Ronitt Rubinfeld

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Abstract
We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution D over [n] and a property P, the goal is to distinguish between D in P and l_{1}(D,P)>epsilon. We develop a general algorithm for this question, which applies to a large range of "shape-constrained" properties, including monotone, log-concave, t-modal, piecewise-polynomial, and Poisson Binomial distributions. Moreover, for all cases considered, our algorithm has near-optimal sample complexity with regard to the domain size and is computationally efficient. For most of these classes, we provide the first non-trivial tester in the literature. In addition, we also describe a generic method to prove lower bounds for this problem, and use it to show our upper bounds are nearly tight. Finally, we extend some of our techniques to tolerant testing, deriving nearly-tight upper and lower bounds for the corresponding questions.
Cite as

Clément L. Canonne, Ilias Diakonikolas, Themis Gouleakis, and Ronitt Rubinfeld. Testing Shape Restrictions of Discrete Distributions. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{canonne_et_al:LIPIcs.STACS.2016.25,
  author =	{Canonne, Cl\'{e}ment L. and Diakonikolas, Ilias and Gouleakis, Themis and Rubinfeld, Ronitt},
  title =	{{Testing Shape Restrictions of Discrete Distributions}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.25},
  URN =		{urn:nbn:de:0030-drops-57260},
  doi =		{10.4230/LIPIcs.STACS.2016.25},
  annote =	{Keywords: property testing, probability distributions, statistics, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.449
A Chasm Between Identity and Equivalence Testing with Conditional Queries

Authors: Jayadev Acharya, Clément L. Canonne, and Gautam Kamath

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

Abstract
A recent model for property testing of probability distributions enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain. In particular, Canonne, Ron, and Servedio showed that, in this setting, testing identity of an unknown distribution D (i.e., whether D = D* for an explicitly known D*) can be done with a constant number of samples, independent of the support size n - in contrast to the required sqrt(n) in the standard sampling model. However, it was unclear whether the same held for the case of testing equivalence, where both distributions are unknown. Indeed, while Canonne, Ron, and Servedio established a polylog(n)-query upper bound for equivalence testing, very recently brought down to ~O(log(log(n))) by Falahatgar et al., whether a dependence on the domain size n is necessary was still open, and explicitly posed by Fischer at the Bertinoro Workshop on Sublinear Algorithms. In this work, we answer the question in the positive, showing that any testing algorithm for equivalence must make Omega(sqrt(log(log(n)))) queries in the conditional sampling model. Interestingly, this demonstrates an intrinsic qualitative gap between identity and equivalence testing, absent in the standard sampling model (where both problems have sampling complexity n^(Theta(1))). Turning to another question, we investigate the complexity of support size estimation. We provide a doubly-logarithmic upper bound for the adaptive version of this problem, generalizing work of Ron and Tsur to our weaker model. We also establish a logarithmic lower bound for the non-adaptive version of this problem. This latter result carries on to the related problem of non-adaptive uniformity testing, an exponential improvement over previous results that resolves an open question of Chakraborty, Fischer, Goldhirsh, and Matsliah.
Cite as

Jayadev Acharya, Clément L. Canonne, and Gautam Kamath. A Chasm Between Identity and Equivalence Testing with Conditional Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 449-466, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{acharya_et_al:LIPIcs.APPROX-RANDOM.2015.449,
  author =	{Acharya, Jayadev and Canonne, Cl\'{e}ment L. and Kamath, Gautam},
  title =	{{A Chasm Between Identity and Equivalence Testing with Conditional Queries}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{449--466},
  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.2015.449},
  URN =		{urn:nbn:de:0030-drops-53178},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.449},
  annote =	{Keywords: property testing, probability distributions, conditional samples}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.512
Learning Circuits with few Negations

Authors: Eric Blais, Clément L. Canonne, Igor C. Oliveira, Rocco A. Servedio, and Li-Yang Tan

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

Abstract
Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).
Cite as

Eric Blais, Clément L. Canonne, Igor C. Oliveira, Rocco A. Servedio, and Li-Yang Tan. Learning Circuits with few Negations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 512-527, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2015.512,
  author =	{Blais, Eric and Canonne, Cl\'{e}ment L. and Oliveira, Igor C. and Servedio, Rocco A. and Tan, Li-Yang},
  title =	{{Learning Circuits with few Negations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{512--527},
  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.2015.512},
  URN =		{urn:nbn:de:0030-drops-53214},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.512},
  annote =	{Keywords: Boolean functions, monotonicity, negations, PAC learning}
}
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