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A Random Matrix Perspective of Echo State Networks: From Precise Bias--Variance Characterization to Optimal Regularization
Yessin Moakher, Malik Tiomoko, Cosme Louart, Zhenyu Liao
https://arxiv.org/abs/2509.22011

A Random Matrix Perspective of Echo State Networks: From Precise Bias--Variance Characterization to Optimal Regularization
We present a rigorous asymptotic analysis of Echo State Networks (ESNs) in a teacher student setting with a linear teacher with oracle weights. Leveraging random matrix theory, we derive closed form expressions for the asymptotic bias, variance, and mean-squared error (MSE) as functions of the input statistics, the oracle vector, and the ridge regularization parameter. The analysis reveals two key departures from classical ridge regression: (i) ESNs do not exhibit double descent, and (ii) ESNs …