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• W3032056551 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 [SVW15] 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 ∈ {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 n 2 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 n 2 25 memory bits and the scheme is used at most an exponential number of times. Previous works on boundedstorage cryptography assumed that the memory size used by the attacker is at most linear in the time needed for encryption/decryption. ∗Weizmann Institute of Science, Israel, and the Institute for Advanced Study, Princeton, NJ. Research supported by the Israel Science Foundation grant No. 1402/14, by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation, by the Simons Collaboration on Algorithms and Geometry, by the Fund for Math at IAS, and by the National Science Foundation grant No. CCF1412958. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Email: ran.raz.mail@gmail.com" @default.
• W3032056551 created "2020-06-05" @default.
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• W3032056551 date "2016-01-01" @default.
• W3032056551 modified "2023-09-26" @default.
• W3032056551 title "Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning." @default.
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