Detailed comments 1. Line 36 introduces h(x) \phi(\theta x) where \phi is a transfer function, which I guess is intended to model the nonlinear activations in the output layer of the (single-layer) NN. Incorporating \phi breaks Theorems 1 and 2 since the loss \ell(\phi(\theta x), y) will not be a convex function of \theta for most choices of function \phi. I guess the authors meant to say "One linear layer learning games"? 2. Generality of results Regarding Theorems 4 and 5, the ancillary lemmas in the appendix assume that the activation functions f_v are convex and differentiable. This means that the most commonly used functions in practice (ReLU, maxout, max-pooling, sigmoid, and tanh units) are all ruled out along with many others. The results in the paper therefore apply in much less generality than a cursory reading would suggest.