Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning

Research output: Chapter in Book/Report/Conference proceedingConference contribution

13 Scopus citations

Abstract

We prove that any algorithm for learning parities requires either a memory of quadratic size or an exponential number of samples. This proves a recent conjecture of Steinhardt, Valiant and Wager [15] and shows that for some learning problems a large storage space is crucial. More formally, in the problem of parity learning, an unknown string x ϵ {0,1}n was chosen uniformly at random. A learner tries to learn x from a stream of samples (a1, b1), (a2, b2)..., where each at is uniformly distributed over {0,1}n and bt is the inner product of at and x, modulo 2. We show that any algorithm for parity learning, that uses less than n2/25 bits of memory, requires an exponential number of samples. Previously, there was no non-trivial lower bound on the number of samples needed, for any learning problem, even if the allowed memory size is O(n) (where n is the space needed to store one sample). We also give an application of our result in the field of bounded-storage cryptography. We show an encryption scheme that requires a private key of length n, as well as time complexity of n per encryption/decryption of each bit, and is provenly and unconditionally secure as long as the attacker uses less than n2/25 memory bits and the scheme is used at most an exponential number of times. Previous works on bounded-storage cryptography assumed that the memory size used by the attacker is at most linear in the time needed for encryption/decryption.

Original languageEnglish (US)
Title of host publicationProceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
PublisherIEEE Computer Society
Pages266-275
Number of pages10
ISBN (Electronic)9781509039333
DOIs
StatePublished - Dec 14 2016
Externally publishedYes
Event57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 - New Brunswick, United States
Duration: Oct 9 2016Oct 11 2016

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2016-December
ISSN (Print)0272-5428

Other

Other57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
CountryUnited States
CityNew Brunswick
Period10/9/1610/11/16

All Science Journal Classification (ASJC) codes

  • Computer Science(all)

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  • Cite this

    Raz, R. (2016). Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning. In Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 (pp. 266-275). [7782939] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2016-December). IEEE Computer Society. https://doi.org/10.1109/FOCS.2016.36