## Abstract

Reservoir computing devices are promising as energy-efficient machine learning hardware for real-time information processing. However, some online algorithms for reservoir computing are not simple enough for hardware implementation. In this study, we focus on the first order reduced and controlled error (FORCE) algorithm for online learning with reservoir computing models. We propose a two-step FORCE algorithm by simplifying the operations in the FORCE algorithm, which can reduce necessary memories. We analytically and numerically show that the proposed algorithm can converge faster than the original FORCE algorithm.

### Keywords

- Reservoir computing
- FORCE learning
- Edge computing
- Nonlinear time series generation

This work was partially supported by JSPS KAKENHI Grant Numbers JP20J13556 (HT), JP20K11882 (GT), and JST CREST Grant Number JPMJCR19K2, Japan.

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## Acknowledgements

The authors thank Dr. K. Fujiwara for stimulating discussions.

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## Appendix

### Appendix

We used the parameters setting shown in Table 1 in all the simulations. \(\varDelta t\) denotes the time step width used when we convert continuous-time systems into discrete-time systems. (i.e., the discrete time *n* corresponds to the continuous time \(n\varDelta t\).)

In Fig. 2, we used the following equation for the periodic teacher signal *d*(*t*):

In Fig. 3, we used the following equation for the periodic teacher signal *d*(*t*):

In Fig. 4, we used the following equation for the periodic teacher signal *d*(*t*) with the variable period *T*:

and we searched the minimum appropriate \(T_0\) with the interval of 0.1 s.

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Tamura, H., Tanaka, G. (2020). Two-Step FORCE Learning Algorithm for Fast Convergence in Reservoir Computing. In: Farkaš, I., Masulli, P., Wermter, S. (eds) Artificial Neural Networks and Machine Learning – ICANN 2020. ICANN 2020. Lecture Notes in Computer Science(), vol 12397. Springer, Cham. https://doi.org/10.1007/978-3-030-61616-8_37

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