Cognitive Computation

, Volume 9, Issue 3, pp 337–350 | Cite as

Echo State Property of Deep Reservoir Computing Networks

  • Claudio GallicchioEmail author
  • Alessio Micheli


In the last years, the Reservoir Computing (RC) framework has emerged as a state of-the-art approach for efficient learning in temporal domains. Recently, within the RC context, deep Echo State Network (ESN) models have been proposed. Being composed of a stack of multiple non-linear reservoir layers, deep ESNs potentially allow to exploit the advantages of a hierarchical temporal feature representation at different levels of abstraction, at the same time preserving the training efficiency typical of the RC methodology. In this paper, we generalize to the case of deep architectures the fundamental RC conditions related to the Echo State Property (ESP), based on the study of stability and contractivity of the resulting dynamical system. Besides providing a necessary condition and a sufficient condition for the ESP of layered RC networks, the results of our analysis provide also insights on the nature of the state dynamics in hierarchically organized recurrent models. In particular, we find out that by adding layers to a deep reservoir architecture, the regime of network’s dynamics can only be driven towards (equally or) less stable behaviors. Moreover, our investigation shows the intrinsic ability of temporal dynamics differentiation at the different levels in a deep recurrent architecture, with higher layers in the stack characterized by less contractive dynamics. Such theoretical insights are further supported by experimental results that show the effect of layering in terms of a progressively increased short-term memory capacity of the recurrent models.


Reservoir computing Deep learning Echo state property Stability analysis Contractivity 


Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of PisaPisaItaly

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