Computational Efficiency of Multi-Step Learning Echo State Networks for Nonlinear Time Series Prediction

The echo state network (ESN) is a representative model for reservoir computing, which has been mainly used for temporal pattern recognition. Recent studies have shown that multi-reservoir ESN models constructed with multiple reservoirs can enhance the potential of the ESN-based approach. In the present study, we investigate computational performance and efficiency of the multi-step learning ESN which is one of the multi-reservoir ESN models and characterized by step-by-step learning processes. We show that the time complexity of the training algorithm of the multi-step learning ESN is equal to or smaller than that of the standard ESN. Our numerical experiments demonstrate that the multi-step learning ESN can achieve better or comparable performance with much less computational time compared to the standard ESN in nonlinear time series prediction tasks. Moreover, we reveal how the model architecture of the multi-step learning ESN is effective in comparison with other possible variant models. The step-by-step learning is applicable to general multi-reservoir systems and hardware for enhancement of their computational ability and efficiency.