Nonreciprocal Two-Replica Optimizer

1. Integration point: replace the SGD update for a selected parameter block \(\theta\) by center state \(c\) and auxiliary state \(r\) with the same shape. Use \(c\) in the forward pass, initialize \(c=\theta\) and \(r=0\), and retain \(r\) in optimizer state. Compute the minibatch gradient \(g_k=\nabla_c L(c_k)\). 2. Pseudocode: set an unconstrained auxiliary command \(\widehat R_k=-\beta g_k-\gamma r_k\); apply either \(R_k=\operatorname{clip}(\widehat R_k,-R_{\max},R_{\max})\) or \(R_k=R_{\max}\tanh(\widehat R_k/R_{\max})\); update \(r_k\leftarrow(1-\eta\rho)r_k+\eta R_k\); update \(c_k\leftarrow c_k-\eta g_k+\eta\alpha r_k\); copy \(c_k\) into the network parameters. Use \(\rho\geq1\), and optionally add minibatch Gaussian noise to \(R_k\) to mimic thermal forcing. 3. Quantities computed directly are gradients, \(r_k\), \(R_k\), and the correction norm \(\|\alpha r_k\|\). The paper supplies the modal decomposition and relaxation condition; \(\alpha,\beta,\gamma\), and \(R_{\max}\) are hyperparameters estimated by a small sweep. For explicit Euler, the isolated auxiliary mode is linearly stable only when \(0<\eta\rho<2\). 4. First experiment: train a two-layer MLP on MNIST using SGD, the same optimizer with \(\alpha=0\), and the nonreciprocal optimizer over \(\alpha\in\{0,0.25,0.5,1\}\), \(\rho\in\{1,2,4\}\), and hard versus smooth saturation. Record loss, gradient norm, auxiliary norm, correction norm, and the power spectrum of parameter motion. The quantitative prediction is a divergence boundary near \(\eta_c=2/\rho\), with auxiliary perturbations growing geometrically by factor \(|1-\eta\rho|\) beyond the boundary. As \(R_{\max}\) increases, hard clipping should show an abrupt jump in mean control magnitude, while smooth tanh saturation should show continuous growth; test whether the measured stability boundary agrees with \(2/\rho\) within 20%.