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This paper introduces the notion of ``Initial Alignment'' (INAL) between aneural network at initialization and a target function. It is proved that if anetwork and a Boolean target function do not have a noticeable INAL, then noisygradient descent on a fully connected network with normalized i.i.d.initialization will not learn in polynomial time. Thus a certain amount ofknowledge about the target (measured by the INAL) is needed in the architecturedesign. This also provides an answer to an open problem posed in [AS20]. Theresults are based on deriving lower-bounds for descent algorithms on symmetricneural networks without explicit knowledge of the target function beyond itsINAL.

An initial alignment between neural network and target is needed for gradient descent to learn