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In this paper, we theoretically prove that adding one special neuron peroutput unit eliminates all suboptimal local minima of any deep neural network,for multi-class classification, binary classification, and regression with anarbitrary loss function, under practical assumptions. At every local minimum ofany deep neural network with these added neurons, the set of parameters of theoriginal neural network (without added neurons) is guaranteed to be a globalminimum of the original neural network. The effects of the added neurons areproven to automatically vanish at every local minimum. Moreover, we provide anovel theoretical characterization of a failure mode of eliminating suboptimallocal minima via an additional theorem and several examples. This paper alsointroduces a novel proof technique based on the perturbable gradient basis(PGB) necessary condition of local minima, which provides new insight into theelimination of local minima and is applicable to analyze various models andtransformations of objective functions beyond the elimination of local minima.

Elimination of All Bad Local Minima in Deep Learning