Task-Driven Selection and Sizing of Simple Cycle Reservoirs

Sparse deterministic reservoirs can reduce deployment cost relative to random echo-state networks (ESNs), but it is unclear when a simple cycle is sufficient for a new forecasting task. We separate this architecture-selection problem from the subsequent choice of cycle size. For any fixed reservoir, optimal squared prediction risk decomposes into irreducible noise and projection error onto the reservoir feature span. For a linear simple cycle represented over a declared finite history, delays congruent modulo the cycle size share state coordinates, yielding an exact covariance-weighted representation loss and a model-conditional minimum-cost certificate. A conservative training-only screen based on this analysis recommends cycles on 23 of 72 benchmark configurations, with no observed recommendation exceeding a validation-selected nonlinear random ESN by more than the prespecified 2% held-out NRMSE margin. The certified regime is narrow, however: a reservoir-free diagnostic developed on controlled tasks fails to transfer reliably to 83 configurations from five real datasets. Importantly, this failure is primarily one of predicting cycle suitability rather than of the architecture itself: a cheaper cycle remains within 2% of the random ESN on 50/83 configurations. Separately, once the cycle family is chosen, a near-best validation rule reduces deployed cycle dimension by 42.0% on 24 held-out configurations while satisfying the 2% practical-equivalence criterion throughout. These results distinguish what can be certified from what must still be validated directly.