Fast Learning Requires Good Memory
Ran Raz · Journal of the ACM · 2018
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 et al. (2016) 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 ( a 1 , b 1 ), ( a 2 , b 2 ) …, 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 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 provably 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 bounded-storage cryptography assumed that the memory size used by the attacker is at most linear in the time needed for encryption/decryption.