On PAC Learning Algorithms for Rich Boolean Function Classes

Servedio, Rocco A.

doi:10.1007/11750321_42

Cited by 5 publications

References 35 publications

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Private Data Release via Learning Thresholds

Hardt

Rothblum

Servedio

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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This work considers computationally efficient privacypreserving data release. We study the task of analyzing a database containing sensitive information about individual participants. Given a set of statistical queries on the data, we want to release approximate answers to the queries while also guaranteeing differential privacyprotecting each participant's sensitive data.Our focus is on computationally efficient data release algorithms; we seek algorithms whose running time is polynomial, or at least sub-exponential, in the data dimensionality. Our primary contribution is a computationally efficient reduction from differentially private data release for a class of counting queries, to learning thresholded sums of predicates from a related class.We instantiate this general reduction with algorithms for learning thresholds, obtaining new results for differentially private data release. As two examples, taking {0, 1}d to be the data domain (of dimension d), we obtain differentially private algorithms for:1. Releasing all k-way conjunction counting queries (or k-way contingency tables). For any given k, the resulting data release algorithm has bounded error as long as the database is of size at) (ignoring the dependence on other parameters). The running time is polynomial in the database size. The best sub-exponential time algorithms known prior to our work required a database of sizeÕ(d k/2 ) [Dwork McSherry Nissim and Smith 2006].

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Private Data Release via Learning Thresholds

Hardt

Rothblum

Servedio

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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On the Hardness of PAC-learning Stabilizer States with Noise

Gollakota

Liang

2022

Quantum

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We consider the problem of learning stabilizer states with noise in the Probably Approximately Correct (PAC) framework of Aaronson (2007) for learning quantum states. In the noiseless setting, an algorithm for this problem was recently given by Rocchetto (2018), but the noisy case was left open. Motivated by approaches to noise tolerance from classical learning theory, we introduce the Statistical Query (SQ) model for PAC-learning quantum states, and prove that algorithms in this model are indeed resilient to common forms of noise, including classification and depolarizing noise. We prove an exponential lower bound on learning stabilizer states in the SQ model. Even outside the SQ model, we prove that learning stabilizer states with noise is in general as hard as Learning Parity with Noise (LPN) using classical examples. Our results position the problem of learning stabilizer states as a natural quantum analogue of the classical problem of learning parities: easy in the noiseless setting, but seemingly intractable even with simple forms of noise.

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