This research investigates online learning for feedforward neural networks utilizing the sign activation function. The paper identifies a margin condition in the first hidden layer as crucial for learnability, demonstrating that the optimal error bound is closely tied to the totally-separable packing number of the input space, showing an exponential dependence on dimension in some cases. To address this dimensionality issue, the authors examine two scenarios: a multi-index model where the function depends on a lower-dimensional projection, achieving better bounds, and a setting with a large margin throughout all layers, yielding bounds dependent on network depth and the number of output labels. The study also provides a method for adaptive learning when these margin parameters are unknown and extends its analysis to the agnostic learning setting.