Noise-Tolerant Learning, the Parity Problem, and the Statistical Query Model
We describe a slightly subexponential time algorithm for learning parity functions in the presence of random classification noise, a problem closely related to several cryptographic and coding problems. Our algorithm runs in polynomial time for the case of parity functions that depend on only the first O(log n log log n) bits of input, which provides the first known instance of an efficient noise-tolerant algorithm for a concept class that is not learnable in the Statistical Query model of Kearns . Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model.In coding-theory terms, what we give is a poly(n)-time algorithm for decoding linear k × n codes in the presence of random noise for the case of k = c log n log log n for some c > 0. (The case of k = O(log n) is trivial since one can just individually check each of the 2k possible messages and choose the one that yields the closest codeword.)A natural extension of the statistical query model is to allow queries about statistical properties that involve t-tuples of examples, as opposed to just single examples. The second result of this article is to show that any class of functions learnable (strongly or weakly) with t-wise queries for t = O(log n) is also weakly learnable with standard unary queries. Hence, this natural extension to the statistical query model does not increase the set of weakly learnable functions.