  We study the parameterized complexity of training two-layer neural networks
with respect to the dimension of the input data and the number of hidden
neurons, considering ReLU and linear threshold activation functions. Albeit the
computational complexity of these problems has been studied numerous times in
recent years, several questions are still open. We answer questions by Arora et
al. [ICLR '18] and Khalife and Basu [IPCO '22] showing that both problems are
NP-hard for two dimensions, which excludes any polynomial-time algorithm for
constant dimension. We also answer a question by Froese et al. [JAIR '22]
proving W[1]-hardness for four ReLUs (or two linear threshold neurons) with
zero training error. Finally, in the ReLU case, we show fixed-parameter
tractability for the combined parameter number of dimensions and number of
ReLUs if the network is assumed to compute a convex map. Our results settle the
complexity status regarding these parameters almost completely.
