Abstract

In this paper, we revisit the recurrent backpropagation (RBP) algorithm (Almeida, 1987; Pineda, 1987), discuss the conditions under which it applies as well as how to satisfy them in deep neural networks. We show that RBP can be unstable and propose two variants based on conjugate gradient on the normal equations (CG-RBP) and Neumann series (Neumann-RBP). We further investigate the relationship between Neumann-RBP and back propagation through time (BPTT) and its truncated version (TBPTT). Our NeumannRBP has the same time complexity as TBPTT but only requires constant memory, whereas TBPTT’s memory cost scales linearly with the number of truncation steps. We examine all RBP variants, along with BPTT and TBPTT, in three different application domains: associative memory with continuous Hopfield networks, document classification in citation networks using graph neural networks, and hyperparameter optimization for fully connected networks. All experiments demonstrate that RBPs, especially the Neumann-RBP variant, are efficient and effective for optimizing convergent recurrent neural networks.