Memory-sample lower bounds for learning parity with noise

In this work, we show, for the well-studied problem of learning parity under noise, where a learner tries to learn x = (x1,..., xn) ∈ {0, 1}ⁿ from a stream of random linear equations over F2 that are correct with probability ¹₂ + ε and flipped with probability ¹₂ − ε (0 < ε < ¹₂ ), that any learning algorithm requires either a memory of size Ω(n²/ε) or an exponential number of samples. In fact, we study memory-sample lower bounds for a large class of learning problems, as characterized by [8], when the samples are noisy. A matrix M : A × X → {−1, 1} corresponds to the following learning problem with error parameter ε: an unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a1, b1), (a2, b2)..., where for every i, ai ∈ A is chosen uniformly at random and bi = M(ai, x) with probability 1/2 + ε and bi = −M(ai, x) with probability 1/2 − ε (0 < ε < ¹₂ ). Assume that k, ℓ, r are such that any submatrix of M of at least 2^−k · |A| rows and at least 2^−ℓ · |X| columns, has a bias of at most 2^−r. We show that any learning algorithm for the learning problem corresponding to M, with error parameter ε, requires either a memory of size at least Ω (^k_ε^·ℓ), or at least 2^Ω(r⁾ samples. The result holds even if the learner has an exponentially small success probability (of 2⁻Ω(r⁾). In particular, this shows that for a large class of learning problems, same as those in [8], any learning algorithm requires either a memory of size at least Ω ( ^(log |X|)·(log |A|⁾ ) or an exponential number of noisy samples. ε Our proof is based on adapting the arguments in [21, 8] to the noisy case.