While on some natural distributions, neural-networks are trained efficiently using gradient-based algorithms, it is known that learning them is computationally hard in the worst-case. To separate hard from easy to learn distributions, we observe the property of local correlation: correlation between local patterns of the input and the target label. We focus on learning deep neural-networks using a gradient-based algorithm, when the target function is a tree-structured Boolean circuit. We show that in this case, the existence of correlation between the gates of the circuit and the target label determines whether the optimization succeeds or fails. Using this result, we show that neural-networks can learn the (log n)-parity problem for most product distributions. These results hint that local correlation may play an important role in separating easy/hard to learn distributions. We also obtain a novel depth separation result, in which we show that a shallow network cannot express some functions, while there exists an efficient gradient-based algorithm that can learn the very same functions using a deep network. The negative expressivity result for shallow networks is obtained by a reduction from results in communication complexity, that may be of independent interest.
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