  We study the problem of PAC learning one-hidden-layer ReLU networks with $k$
hidden units on $\mathbb{R}^d$ under Gaussian marginals in the presence of
additive label noise. For the case of positive coefficients, we give the first
polynomial-time algorithm for this learning problem for $k$ up to
$\tilde{O}(\sqrt{\log d})$. Previously, no polynomial time algorithm was known,
even for $k=3$. This answers an open question posed by~\cite{Kliv17}.
Importantly, our algorithm does not require any assumptions about the rank of
the weight matrix and its complexity is independent of its condition number. On
the negative side, for the more general task of PAC learning one-hidden-layer
ReLU networks with arbitrary real coefficients, we prove a Statistical Query
lower bound of $d^{\Omega(k)}$. Thus, we provide a separation between the two
classes in terms of efficient learnability. Our upper and lower bounds are
general, extending to broader families of activation functions.
