3 Search Results for "Paradise, Orr"


Document
Junta Distance Approximation with Sub-Exponential Queries

Authors: Vishnu Iyer, Avishay Tal, and Michael Whitmeyer

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)


Abstract
Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f:{±1}ⁿ → {±1}: 1) We give a poly(k, 1/(ε)) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k'-juntas, where k' = O(k/(ε²)). 2) In the non-relaxed setting, we extend our ideas to give a 2^{Õ(√{k/ε})} (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k-juntas. To the best of our knowledge, this is the first subexponential-in-k query algorithm for approximating the distance of f to being a k-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in k). Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [Michel Talagrand, 1994].

Cite as

Vishnu Iyer, Avishay Tal, and Michael Whitmeyer. Junta Distance Approximation with Sub-Exponential Queries. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 24:1-24:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{iyer_et_al:LIPIcs.CCC.2021.24,
  author =	{Iyer, Vishnu and Tal, Avishay and Whitmeyer, Michael},
  title =	{{Junta Distance Approximation with Sub-Exponential Queries}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{24:1--24:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.24},
  URN =		{urn:nbn:de:0030-drops-142988},
  doi =		{10.4230/LIPIcs.CCC.2021.24},
  annote =	{Keywords: Algorithms, Complexity Theory, Fourier Analysis, Juntas, Normalized Influence, Property Testing, Tolerant Property Testing}
}
Document
Interactive Proofs for Verifying Machine Learning

Authors: Shafi Goldwasser, Guy N. Rothblum, Jonathan Shafer, and Amir Yehudayoff

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)


Abstract
We consider the following question: using a source of labeled data and interaction with an untrusted prover, what is the complexity of verifying that a given hypothesis is "approximately correct"? We study interactive proof systems for PAC verification, where a verifier that interacts with a prover is required to accept good hypotheses, and reject bad hypotheses. Both the verifier and the prover are efficient and have access to labeled data samples from an unknown distribution. We are interested in cases where the verifier can use significantly less data than is required for (agnostic) PAC learning, or use a substantially cheaper data source (e.g., using only random samples for verification, even though learning requires membership queries). We believe that today, when data and data-driven algorithms are quickly gaining prominence, the question of verifying purported outcomes of data analyses is very well-motivated. We show three main results. First, we prove that for a specific hypothesis class, verification is significantly cheaper than learning in terms of sample complexity, even if the verifier engages with the prover only in a single-round (NP-like) protocol. Moreover, for this class we prove that single-round verification is also significantly cheaper than testing closeness to the class. Second, for the broad class of Fourier-sparse boolean functions, we show a multi-round (IP-like) verification protocol, where the prover uses membership queries, and the verifier is able to assess the result while only using random samples. Third, we show that verification is not always more efficient. Namely, we show a class of functions where verification requires as many samples as learning does, up to a logarithmic factor.

Cite as

Shafi Goldwasser, Guy N. Rothblum, Jonathan Shafer, and Amir Yehudayoff. Interactive Proofs for Verifying Machine Learning. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{goldwasser_et_al:LIPIcs.ITCS.2021.41,
  author =	{Goldwasser, Shafi and Rothblum, Guy N. and Shafer, Jonathan and Yehudayoff, Amir},
  title =	{{Interactive Proofs for Verifying Machine Learning}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{41:1--41:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.41},
  URN =		{urn:nbn:de:0030-drops-135806},
  doi =		{10.4230/LIPIcs.ITCS.2021.41},
  annote =	{Keywords: PAC learning, Fourier analysis of boolean functions, Complexity gaps, Complexity lower bounds, Goldreich-Levin algorithm, Kushilevitz-Mansour algorithm, Distribution testing}
}
Document
Smooth and Strong PCPs

Authors: Orr Paradise

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: - A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim. - A PCP is smooth if each location in a proof is queried with equal probability. We prove that all sets in NP have PCPs that are both smooth and strong, are of polynomial length, and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora, Lund, Motwani, Sudan and Szegedy (JACM, 1998), providing a stronger analysis of the Hadamard and Reed - Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in NP has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of NP witnesses to correct proofs. This improves on the recent construction of Dinur, Gur and Goldreich (ITCS, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (SODA, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.

Cite as

Orr Paradise. Smooth and Strong PCPs. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 2:1-2:41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{paradise:LIPIcs.ITCS.2020.2,
  author =	{Paradise, Orr},
  title =	{{Smooth and Strong PCPs}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{2:1--2:41},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.2},
  URN =		{urn:nbn:de:0030-drops-116875},
  doi =		{10.4230/LIPIcs.ITCS.2020.2},
  annote =	{Keywords: Interactive and probabilistic proof systems, Probabilistically checkable proofs, Hardness of approximation}
}
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