  We prove several hardness results for training depth-2 neural networks with
the ReLU activation function; these networks are simply weighted sums (that may
include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural
network that minimizes the square loss with respect to a given training set. We
prove that this problem is NP-hard already for a network with a single ReLU. We
also prove NP-hardness for outputting a weighted sum of $k$ ReLUs minimizing
the squared error (for $k>1$) even in the realizable setting (i.e., when the
labels are consistent with an unknown depth-2 ReLU network). We are also able
to obtain lower bounds on the running time in terms of the desired additive
error $\epsilon$. To obtain our lower bounds, we use the Gap Exponential Time
Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of
approximating the well known Densest $\kappa$-Subgraph problem in
subexponential time (these hypotheses are used separately in proving different
lower bounds). For example, we prove that under reasonable hardness
assumptions, any proper learning algorithm for finding the best fitting ReLU
must run in time exponential in $1/\epsilon^2$. Together with a previous work
regarding improperly learning a ReLU (Goel et al., COLT'17), this implies the
first separation between proper and improper algorithms for learning a ReLU. We
also study the problem of properly learning a depth-2 network of ReLUs with
bounded weights giving new (worst-case) upper bounds on the running time needed
to learn such networks both in the realizable and agnostic settings. Our upper
bounds on the running time essentially matches our lower bounds in terms of the
dependency on $\epsilon$.
