In this work, we show, for the well-studied problem of learning parity under noise, where a learner tries to learn $x=(x_1,ldots,x_n) in {0,1}^n$ from a stream of random linear equations over $mathrm{F}_2$ that are correct with probability $frac{1}{2}+varepsilon$ and flipped with probability $frac{1}{2}-varepsilon$, that any learning algorithm requires either a memory of size $Omega(n^2/varepsilon)$ or an exponential number of samples. In fact, we study memory-sample lower bounds for a large class of learning problems, as characterized by [GRT18], when the samples are noisy. A matrix $M: A times X rightarrow {-1,1}$ corresponds to the following learning problem with error parameter $varepsilon$: an unknown element $x in X$ is 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 for every $i$, $a_i in A$ is chosen uniformly at random and $b_i = M(a_i,x)$ with probability $1/2+varepsilon$ and $b_i = -M(a_i,x)$ with probability $1/2-varepsilon$ ($0<varepsilon< frac{1}{2}$). Assume that $k,ell, r$ are such that any submatrix of $M$ of at least $2^{-k} cdot |A|$ rows and at least $2^{-ell} cdot |X|$ columns, has a bias of at most $2^{-r}$. We show that any learning algorithm for the learning problem corresponding to $M$, with error, requires either a memory of size at least $Omegaleft(frac{k cdot ell}{varepsilon} right)$, or at least $2^{Omega(r)}$ samples. In particular, this shows that for a large class of learning problems, same as those in [GRT18], any learning algorithm requires either a memory of size at least $Omegaleft(frac{(log |X|) cdot (log |A|)}{varepsilon}right)$ or an exponential number of noisy samples. Our proof is based on adapting the arguments in [Raz17,GRT18] to the noisy case.
تحميل البحث