Second-order backpropagation algorithms for a stagewise-partitioned separable Hessian matrix

Abstract

Recent advances in computer technology allows the implementation of some important methods that were assigned lower priority in the past due to their computational burdens. Second-order backpropagation (BP) is such a method that computes the exact Hessian matrix of a given objective function. We describe two algorithms for feed-forward neural-network (NN) learning with emphasis on how to organize Hessian elements into a so-called stagewise-partitioned block-arrow matrix form: (1) stagewise BP, an extension of the discrete-time optimal-control stagewise Newton of Dreyfus 1966; and (2) nodewise BP, based on direct implementation of the chain rule for differentiation attributable to Bishop 1992. The former, a more systematic and cost-efficient implementation in both memory and operation, progresses in the same layer-by-layer (i.e., stagewise) fashion as the widely-employed first-order BP computes the gradient vector. We also show intriguing separable structures of each block in the partitioned Hessian, disclosing the rank of blocks.