Spectral-Pole-Tuned Decentralized Optimizer

Integrate this into the communication and optimizer step of decentralized neural-network training. Each worker stores parameters \(\theta_i\), a gradient tracker \(s_i\), and the current minibatch gradient \(g_i\). Build the symmetric communication Laplacian \(\mathcal L=D-W\), where \(W_{ij}\) is the edge weight and \(D_{ii}=\sum_jW_{ij}\). Estimate \(\lambda_N\) with 10-20 power iterations on \(\mathcal L\). Estimate \(\lambda_2\) by power iteration after projecting each vector orthogonally to the all-ones vector, or use a small Lanczos eigensolver on the graph matrix. Set \(\kappa=\lambda_N/\max(\lambda_2,10^{-6})\), compute \(\gamma^*\), \(\alpha^*\), and \(\varepsilon^*\) from the displayed equations, and optionally clip \(\varepsilon\) to the displayed feasible interval using a target \(\gamma\geq\gamma^*\). Initialize \(s_i^0=g_i^0\). At each step execute: theta_new = theta - eps * sum_j L[i,j] * theta_j - alpha * s; compute the new minibatch gradient g_new; then execute s_new = s - eps * sum_j L[i,j] * s_j + g_new - g. Use the plain tracker first; only afterward test Adam-style diagonal preconditioning of \(s_i\). The formulas and eigenvalue estimates are computed analytically/numerically from the graph; actual neural-loss stability is measured empirically. Test eight workers training a two-layer MLP on MNIST or CIFAR-10 under ring, grid, and random geometric graphs. Compare standard decentralized gradient tracking, the proposed spectral gains, and a grid-searched best gain pair at equal communication rounds. Record loss versus rounds and transmitted bytes, parameter disagreement \(N^{-1}\sum_i\|\theta_i-\bar\theta\|^2\), tracker disagreement, gradient norms, and divergence frequency. Success is faster loss descent or higher accuracy at equal communication cost without increased instability, especially on poorly connected graphs with large \(\lambda_N/\lambda_2\).