  This paper aims to overcome a fundamental problem in the theory and
application of deep neural networks (DNNs). We propose a method to solve the
local minimum problem in training DNNs directly. Our method is based on the
cross-entropy loss criterion's convexification by transforming the
cross-entropy loss into a risk averting error (RAE) criterion. To alleviate
numerical difficulties, a normalized RAE (NRAE) is employed. The convexity
region of the cross-entropy loss expands as its risk sensitivity index (RSI)
increases. Making the best use of the convexity region, our method starts
training with an extensive RSI, gradually reduces it, and switches to the RAE
as soon as the RAE is numerically feasible. After training converges, the
resultant deep learning machine is expected to be inside the attraction basin
of a global minimum of the cross-entropy loss. Numerical results are provided
to show the effectiveness of the proposed method.
