On Noise-Tolerant Learning of Sparse Parities and Related Problems

13 years 12 days ago

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We consider the problem of learning sparse parities in the presence of noise. For learning parities on r out of n variables, we give an algorithm that runs in time poly log 1 δ , 1 1−2η n(1+(2η)2 +o(1))r/2 and uses only r log(n/δ)ω(1) (1−2η)2 samples in the random noise setting under the uniform distribution, where η is the noise rate and δ is the conﬁdence parameter. From previously known results this algorithm also works for adversarial noise and generalizes to arbitrary distributions. Even though eﬃcient algorithms for learning sparse parities in the presence of noise would have major implications to learning other hypothesis classes, our work is the ﬁrst to give a bound better than the brute-force O(nr ). As a consequence, we obtain the ﬁrst nontrivial bound for learning r-juntas in the presence of noise, and also a small improvement in the complexity of learning DNF, under the uniform distribution.

Elena Grigorescu, Lev Reyzin, Santosh Vempala

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