Order in the Black Box: Consistency and Robustness of Hidden Neuron Activation of Feed Forward Neural Networks and Its Use in Efficient Optimization of Network Structure

Abstract

Neural networks are widely used for nonlinear pattern recognition and regression. However, they are considered as black boxes due to lack of transparency of internal workings and lack of direct relevance of its structure to the problem being addressed making it difficult to gain insights. Furthermore, structure of a neural network requires optimization which is still a challenge. Many existing structure optimization approaches require either extensive multi-stage pruning or setting subjective thresholds for pruning parameters. The knowledge of any internal consistency in the behavior of neurons could help develop simpler, systematic and more efficient approaches to optimise network structure. This chapter addresses in detail the issue of internal consistency in relation to redundancy and robustness of network structure of feed forward networks (3-layer) that are widely used for nonlinear regression. It first investigates if there is a recognizable consistency in neuron activation patterns under all conditions of network operation such as noise and initial weights. If such consistency exists, it points to a recognizable optimum network structure for given data. The results show that such pattern does exist and it is most clearly evident not at the level of hidden neuron activation but hidden neuron input to the output neuron (i.e., weighted hidden neuron activation). It is shown that when a network has more than the optimum number of hidden neurons, the redundant neurons form clearly distinguishable correlated patterns of their weighted outputs. This correlation structure is exploited to extract the required number of neurons using correlation distance based self organising maps that are clustered using Ward clustering that optimally cluster correlated weighted hidden neuron activity patterns without any user defined criteria or thresholds, thus automatically optimizing network structure in one step. The number of Ward clusters on the SOM is the required optimum number of neurons. The SOM/Ward based optimum network is compared with that obtained using two documented pruning methods: optimal brain damage and variance nullity measure to show the efficacy of the correlation approach in providing equivalent results. Also, the robustness of the network with optimum structure is tested against perturbation of weights and confidence intervals for weights are illustrated. Finally, the approach is tested on two practical problems involving a breast cancer diagnostic system and river flow forecasting.