This paper focuses on online learning procedures for locally recurrent neural nets with emphasis on multilayer perceptron (MLP) with infinite impulse response (IIR) synapses and its variations which include generalized output and activation feedback multilayer networks (MLN). We propose a new gradient-based procedure called recursive backpropagation (RBP) whose online version, causal recursive backpropagation (CRBP), has some advantages over other online methods. CRBP includes as particular cases backpropagation (BP), temporal BP, Back-Tsoi algorithm (1991) among others, thereby providing a unifying view on gradient calculation for recurrent nets with local feedback. The only learning method known for locally recurrent nets with no architectural restriction is the one by Back and Tsoi. The proposed algorithm has better stability and faster convergence with respect to the Back-Tsoi algorithm. The computational complexity of the CRBP is comparable with that of the Back-Tsoi algorithm, e.g., less that a factor of 1.5 for usual architectures and parameter settings. The superior performance of the new algorithm, however, easily justifies this small increase in computational burden. In addition, the general paradigms of truncated BPTT and RTRL are applied to networks with local feedback and compared with CRBP. CRBP exhibits similar performances and the detailed analysis of complexity reveals that CRBP is much simpler and easier to implement, e.g., CRBP is local in space and in time while RTRL is not local in space.
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