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A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the followinglearning problem: An unknown element $x \in X$ is chosen uniformly at random. Alearner 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$b_i = M(a_i,x)$.  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 biasof at most $2^{-r}$. We show that any learning algorithm for the learningproblem corresponding to $M$ requires either a memory of size at least$\Omega\left(k \cdot \ell \right)$, or at least $2^{\Omega(r)}$ samples. Theresult holds even if the learner has an exponentially small success probability(of $2^{-\Omega(r)}$).  In particular, this shows that for a large class of learning problems, anylearning algorithm requires either a memory of size at least $\Omega\left((\log|X|) \cdot (\log |A|)\right)$ or an exponential number of samples, achieving atight $\Omega\left((\log |X|) \cdot (\log |A|)\right)$ lower bound on the sizeof the memory, rather than a bound of $\Omega\left(\min\left\{(\log|X|)^2,(\log |A|)^2\right\}\right)$ obtained in previous works [R17,MM17b].  Moreover, our result implies all previous memory-samples lower bounds, aswell as a number of new applications.  Our proof builds on [R17] that gave a general technique for provingmemory-samples lower bounds.

Extractor-Based Time-Space Lower Bounds for Learning