Abstract Training deep neural networks for solving machine learning problems is one great challenge in the field, mainly due to its associated optimisation problem being highly non-convex. Recent developments have suggested that many training algorithms do not suffer from undesired local minima under certain scenario, and consequently led to great efforts in pursuing mathematical explanations for such observations. This work provides an alternative mathematical understanding of the challenge from a smooth optimisation perspective. By assuming exact learning of finite samples, sufficient conditions are identified via a critical point analysis to ensure any local minimum to be globally minimal as well. Furthermore, a state of the art algorithm, known as the Generalised Gauss-Newton (GGN) algorithm, is rigorously revisited as an approximate Newton’s algorithm, which shares the property of being locally quadratically convergent to a global minimum under the condition of exact learning