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We show that the standard stochastic gradient decent (SGD) algorithm isguaranteed to learn, in polynomial time, a function that is competitive withthe best function in the conjugate kernel space of the network, as defined inDaniely, Frostig and Singer. The result holds for log-depth networks from arich family of architectures. To the best of our knowledge, it is the firstpolynomial-time guarantee for the standard neural network learning algorithmfor networks of depth more that two.  As corollaries, it follows that for neural networks of any depth between $2$and $\log(n)$, SGD is guaranteed to learn, in polynomial time, constant degreepolynomials with polynomially bounded coefficients. Likewise, it follows thatSGD on large enough networks can learn any continuous function (not inpolynomial time), complementing classical expressivity results.

SGD Learns the Conjugate Kernel Class of the Network