Feedforward and Recurrent Neural Networks Backward Propagation and Hessian in Matrix Form

09/16/2017

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by   Maxim Naumov, et al.

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In this paper we focus on the linear algebra theory behind feedforward (FNN) and recurrent (RNN) neural networks. We review backward propagation, including backward propagation through time (BPTT). Also, we obtain a new exact expression for Hessian, which represents second order effects. We show that for t time steps the weight gradient can be expressed as a rank-t matrix, while the weight Hessian is as a sum of t^2 Kronecker products of rank-1 and W^TAW matrices, for some matrix A and weight matrix W. Also, we show that for a mini-batch of size r, the weight update can be expressed as a rank-rt matrix. Finally, we briefly comment on the eigenvalues of the Hessian matrix.