Surrogate-guided topology search for nonlinear reservoirs

(1) Integration point: implement this as an outer architecture-search loop for an Echo State Network or another nonlinear recurrent reservoir. Fix the number of nodes \(K\), input weights, node nonlinearity, spectral-radius normalization, and ridge-trained linear readout. Let the binary adjacency matrix \(A\) be the only evolved object. Use the paper's 45 binary 3x3 patterns, or extract binary patches from MNIST, with the all-one patch as \(p_{full}\). The reservoir update is \(h_{t+1}=\tanh(Ah_t+W_{in}x_t)\). (2) Pseudocode: initialize 100 connected sparse graphs; for every graph, run the reservoir on each pattern, record time-averaged nonnegative responses \(r_j(p)\), compute \(\hat m_1\), train the linear readout by ridge regression, and measure validation accuracy. Compute \(z(G)\), store the graph and its measured targets, fit an ensemble of five regressors, mutate graphs using edge insertion, deletion, rewiring, or degree-preserving swaps, and exactly evaluate the top candidates by \(s(G)=\mu(G)+\beta u(G)\). Repeat for 20 generations. (3) Compute graph descriptors directly from \(A\): degrees, clustering, shortest paths, eigenvalues, spectral radius, and density. Estimate \(\mu\) and \(u\) from ensemble predictions; these are empirical quantities, not theoretical guarantees. Normalize descriptors and maintain a Pareto archive over predicted accuracy, edge count, and \(\hat m_1\). (4) First experiment: use a 50-node reservoir on MNIST 8x8 images or the 45 binary patterns. Compare random search, degree-matched random search, and surrogate-guided evolution under the same budget of 500 exact reservoir evaluations. The success signal is higher validation accuracy at equal evaluation count, plus fewer edges at matched accuracy and faster attainment of the random-search best score. Run five seeds and ablate both the graph descriptor vector and the \(\hat m_1\) objective.