ECCC-Report TR16-019https://eccc.weizmann.ac.il/report/2016/019Comments and Revisions published for TR16-019en-usFri, 05 Feb 2016 17:30:30 +0200
Paper TR16-019
| Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning |
Ran Raz
https://eccc.weizmann.ac.il/report/2016/019We 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 and shows that for some learning problems a large storage space is crucial.
More formally, in the problem of parity learning, an unknown string $x \in \{0,1\}^n$ was chosen uniformly at random. A learner tries to learn $x$ from a stream of samples $(a_1, b_1), (a_2, b_2) \ldots$, where each~$a_t$ is uniformly distributed over $\{0,1\}^n$ and $b_t$ is the inner product of $a_t$ and $x$, modulo~2. We show that any algorithm for parity learning, that uses less than $\frac{n^2}{25}$ 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/decription of each bit, and is provenly and unconditionally secure as long as the attacker uses less than $\frac{n^2}{25}$ 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/decription.Fri, 05 Feb 2016 17:30:30 +0200https://eccc.weizmann.ac.il/report/2016/019