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Training deep neural networks for solving machine learning problems is onegreat challenge in the field, mainly due to its associated optimisation problembeing highly non-convex. Recent developments have suggested that many trainingalgorithms do not suffer from undesired local minima under certain scenario,and consequently led to great efforts in pursuing mathematical explanations forsuch observations. This work provides an alternative mathematical understandingof the challenge from a smooth optimisation perspective. By assuming exactlearning of finite samples, sufficient conditions are identified via a criticalpoint analysis to ensure any local minimum to be globally minimal as well.Furthermore, a state of the art algorithm, known as the GeneralisedGauss-Newton (GGN) algorithm, is rigorously revisited as an approximateNewton's algorithm, which shares the property of being locally quadraticallyconvergent to a global minimum under the condition of exact learning.

Towards a Mathematical Understanding of the Difficulty in Learning with Feedforward Neural Networks