  This paper presents a new optimisation approach to train Deep Neural Networks
(DNNs) with discriminative sequence criteria. At each iteration, the method
combines information from the Natural Gradient (NG) direction with local
curvature information of the error surface that enables better paths on the
parameter manifold to be traversed. The method is derived using an alternative
derivation of Taylor's theorem using the concepts of manifolds, tangent vectors
and directional derivatives from the perspective of Information Geometry. The
efficacy of the method is shown within a Hessian Free (HF) style optimisation
framework to sequence train both standard fully-connected DNNs and Time Delay
Neural Networks as speech recognition acoustic models. It is shown that for the
same number of updates the proposed approach achieves larger reductions in the
word error rate (WER) than both NG and HF, and also leads to a lower WER than
standard stochastic gradient descent. The paper also addresses the issue of
over-fitting due to mismatch between training criterion and Word Error Rate
(WER) that primarily arises during sequence training of ReLU-DNN models.
