18 Search Results for "Bun, Mark"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.40
Private Counting of Distinct Elements in the Turnstile Model and Extensions

Authors: Monika Henzinger, A. R. Sricharan, and Teresa Anna Steiner

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Privately counting distinct elements in a stream is a fundamental data analysis problem with many applications in machine learning. In the turnstile model, Jain et al. [NeurIPS2023] initiated the study of this problem parameterized by the maximum flippancy of any element, i.e., the number of times that the count of an element changes from 0 to above 0 or vice versa. They give an item-level (ε,δ)-differentially private algorithm whose additive error is tight with respect to that parameterization. In this work, we show that a very simple algorithm based on the sparse vector technique achieves a tight additive error for item-level (ε,δ)-differential privacy and item-level ε-differential privacy with regards to a different parameterization, namely the sum of all flippancies. Our second result is a bound which shows that for a large class of algorithms, including all existing differentially private algorithms for this problem, the lower bound from item-level differential privacy extends to event-level differential privacy. This partially answers an open question by Jain et al. [NeurIPS2023].
Cite as

Monika Henzinger, A. R. Sricharan, and Teresa Anna Steiner. Private Counting of Distinct Elements in the Turnstile Model and Extensions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 40:1-40:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{henzinger_et_al:LIPIcs.APPROX/RANDOM.2024.40,
  author =	{Henzinger, Monika and Sricharan, A. R. and Steiner, Teresa Anna},
  title =	{{Private Counting of Distinct Elements in the Turnstile Model and Extensions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{40:1--40:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.40},
  URN =		{urn:nbn:de:0030-drops-210335},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.40},
  annote =	{Keywords: differential privacy, turnstile model, counting distinct elements}
}
Document
DOI: 10.4230/LIPIcs.TQC.2024.11
A Direct Reduction from the Polynomial to the Adversary Method

Authors: Aleksandrs Belovs

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other. It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known. In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.
Cite as

Aleksandrs Belovs. A Direct Reduction from the Polynomial to the Adversary Method. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{belovs:LIPIcs.TQC.2024.11,
  author =	{Belovs, Aleksandrs},
  title =	{{A Direct Reduction from the Polynomial to the Adversary Method}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.11},
  URN =		{urn:nbn:de:0030-drops-206814},
  doi =		{10.4230/LIPIcs.TQC.2024.11},
  annote =	{Keywords: Polynomials, Quantum Adversary Bound}
}
Document
Invited Paper
DOI: 10.4230/LIPIcs.MFCS.2024.4
From TCS to Learning Theory (Invited Paper)

Authors: Kasper Green Larsen

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
While machine learning theory and theoretical computer science are both based on a solid mathematical foundation, the two research communities have a smaller overlap than what the proximity of the fields warrant. In this invited abstract, I will argue that traditional theoretical computer scientists have much to offer the learning theory community and vice versa. I will make this argument by telling a personal story of how I broadened my research focus to encompass learning theory, and how my TCS background has been extremely useful in doing so. It is my hope that this personal account may inspire more TCS researchers to tackle the many elegant and important theoretical questions that learning theory has to offer.
Cite as

Kasper Green Larsen. From TCS to Learning Theory (Invited Paper). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 4:1-4:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{larsen:LIPIcs.MFCS.2024.4,
  author =	{Larsen, Kasper Green},
  title =	{{From TCS to Learning Theory}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{4:1--4:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.4},
  URN =		{urn:nbn:de:0030-drops-205603},
  doi =		{10.4230/LIPIcs.MFCS.2024.4},
  annote =	{Keywords: Theoretical Computer Science, Learning Theory}
}
Document
DOI: 10.4230/LIPIcs.ITC.2024.4
Pure-DP Aggregation in the Shuffle Model: Error-Optimal and Communication-Efficient

Authors: Badih Ghazi, Ravi Kumar, and Pasin Manurangsi

Published in: LIPIcs, Volume 304, 5th Conference on Information-Theoretic Cryptography (ITC 2024)

Abstract
We obtain a new protocol for binary counting in the ε-DP_shuffle model with error O(1/ε) and expected communication Õ((log n)/ε) messages per user. Previous protocols incur either an error of O(1/ε^1.5) with O_ε(log n) messages per user (Ghazi et al., ITC 2020) or an error of O(1/ε) with O_ε(n²) messages per user (Cheu and Yan, TPDP 2022). Using the new protocol, we obtained improved ε-DP_shuffle protocols for real summation and histograms.
Cite as

Badih Ghazi, Ravi Kumar, and Pasin Manurangsi. Pure-DP Aggregation in the Shuffle Model: Error-Optimal and Communication-Efficient. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ghazi_et_al:LIPIcs.ITC.2024.4,
  author =	{Ghazi, Badih and Kumar, Ravi and Manurangsi, Pasin},
  title =	{{Pure-DP Aggregation in the Shuffle Model: Error-Optimal and Communication-Efficient}},
  booktitle =	{5th Conference on Information-Theoretic Cryptography (ITC 2024)},
  pages =	{4:1--4:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-333-1},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{304},
  editor =	{Aggarwal, Divesh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2024.4},
  URN =		{urn:nbn:de:0030-drops-205127},
  doi =		{10.4230/LIPIcs.ITC.2024.4},
  annote =	{Keywords: Differential Privacy, Shuffle Model, Aggregation, Pure Differential Privacy}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.18
Pseudorandomness, Symmetry, Smoothing: I

Authors: Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that - achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. - have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. - rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.
Cite as

Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola. Pseudorandomness, Symmetry, Smoothing: I. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 18:1-18:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{derksen_et_al:LIPIcs.CCC.2024.18,
  author =	{Derksen, Harm and Ivanov, Peter and Lee, Chin Ho and Viola, Emanuele},
  title =	{{Pseudorandomness, Symmetry, Smoothing: I}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{18:1--18:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.18},
  URN =		{urn:nbn:de:0030-drops-204144},
  doi =		{10.4230/LIPIcs.CCC.2024.18},
  annote =	{Keywords: pseudorandomness, k-wise uniform distributions, small-bias distributions, noise, symmetric tests, thresholds, Krawtchouk polynomials}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.13
Learning Low-Degree Quantum Objects

Authors: Srinivasan Arunachalam, Arkopal Dutt, Francisco Escudero Gutiérrez, and Carlos Palazuelos

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We consider the problem of learning low-degree quantum objects up to ε-error in 𝓁₂-distance. We show the following results: (i) unknown n-qubit degree-d (in the Pauli basis) quantum channels and unitaries can be learned using O(1/ε^d) queries (which is independent of n), (ii) polynomials p:{-1,1}ⁿ → [-1,1] arising from d-query quantum algorithms can be learned from O((1/ε)^d ⋅ log n) many random examples (x,p(x)) (which implies learnability even for d = O(log n)), and (iii) degree-d polynomials p:{-1,1}ⁿ → [-1,1] can be learned through O(1/ε^d) queries to a quantum unitary U_p that block-encodes p. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded polynomials.
Cite as

Srinivasan Arunachalam, Arkopal Dutt, Francisco Escudero Gutiérrez, and Carlos Palazuelos. Learning Low-Degree Quantum Objects. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arunachalam_et_al:LIPIcs.ICALP.2024.13,
  author =	{Arunachalam, Srinivasan and Dutt, Arkopal and Escudero Guti\'{e}rrez, Francisco and Palazuelos, Carlos},
  title =	{{Learning Low-Degree Quantum Objects}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.13},
  URN =		{urn:nbn:de:0030-drops-201563},
  doi =		{10.4230/LIPIcs.ICALP.2024.13},
  annote =	{Keywords: Tomography}
}
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}
}
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